Population Biology for UWIS students
(701-0307-01)
Applications to and lessons from infectious diseases
Roland R. Regoes
HS 2010
Contents
Foreword
viii
1 Basic mathematical epidemiology
1.1 SIR models . . . . . . . . . . . . . . . . . . . .
1.2 Basic reproduction number . . . . . . . . . . . .
1.3 R0 and age of infection . . . . . . . . . . . . . .
1.4 Vaccination . . . . . . . . . . . . . . . . . . . .
1.5 Parametrizing epidemiological models . . . . . .
1.5.1 Estimating R0 from reported cases . . .
1.5.2 Estimating R0 from seroprevalence data
1.5.3 Example: avian influenza in Thai chicken
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2 Advanced mathematical epidemiology
2.1 Heterogeneity in Risk structure . . . . . . . . . . . .
2.1.1 Basic reproduction number . . . . . . . . . . .
2.1.2 Impact of heterogeneity on disease dynamics .
2.1.3 Chlamydia infections in koala’s . . . . . . . .
2.2 Spatial heterogeneity . . . . . . . . . . . . . . . . . .
2.2.1 A simple model in one-dimension . . . . . . .
2.2.2 Example: the spatial spread of rabies in the
population . . . . . . . . . . . . . . . . . . . .
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European
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3 Stochasticity in epidemics
3.1 Galton-Watson branching process . . . . . . . . . . . . . . . . .
3.1.1 A specific example . . . . . . . . . . . . . . . . . . . . .
3.1.2 Analytics . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 The distribution of secondary cases for various infections
3.2 Stochastic SIR model . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Critical community size . . . . . . . . . . . . . . . . . . . . . . .
3.4 Example: Porcine reproductive and repiratory syndrome . . . .
4 Evolution of virulence
4.1 Why should parasites be harmful? . . . . . . . .
4.2 Maximization of the basic reproduction number
4.3 Trade-offs between infectivity and virulence . .
4.4 Key assumptions of the virulence model . . . .
4.5 Host density and the evolution of virulence . . .
ii
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1
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Contents
4.6
4.7
4.8
4.9
Treatment and the evolution of virulence . . . . . . . . . . . . . .
Horizontal versus vertical transmission . . . . . . . . . . . . . . .
Intra-host competition . . . . . . . . . . . . . . . . . . . . . . . .
Local adaption, host heterogeneity, and cross-species transmission
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60
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65
67
List of Figures
1.1
1.2
1.3
1.4
1.5
Graphical scheme of an SIR model . . . . . . . . . . . . .
Timecourses of the SIR model . . . . . . . . . . . . . . . .
Graphical illustration of the basic reproduction number R0
Type I and II mortality . . . . . . . . . . . . . . . . . . . .
Likelihood for age-stratified example seroprevalence data .
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1
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4
7
14
2.1
2.2
2.3
2.4
Diagram of an SIR model with many classes . . . . . . . . . . . . . . . .
Timecourses of the SI model with high and low risk groups . . . . . . . .
Time course of the risk-structure model by May and Anderson (1988) . .
Impact of sterility and infection-induced mortality on the population size
of koalas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples for the spatial spread of infections . . . . . . . . . . . . . . . .
Travelling wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical solution of the one-dimensional spatial SI model . . . . . . . .
Numerical solution of the rabies model . . . . . . . . . . . . . . . . . . .
Spatial data on the rabies epizootic . . . . . . . . . . . . . . . . . . . . .
18
19
24
Epidemics and stochasticity . . . . . . . . . . . . . . . . . . . . . .
Galton-Watson branching process . . . . . . . . . . . . . . . . . . .
Geometric distribution of secondary infections . . . . . . . . . . . .
Simulation of the Galton-Watson branching process . . . . . . . . .
Emergence versus R0 . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of the relation between probability generating function
extinction probability . . . . . . . . . . . . . . . . . . . . . . . . . .
36
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40
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
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and
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43
List of Figures
3.7
Figure 1 from Lloyd-Smith et al Nature 2005. Their legend reads: “Evidence for variation in individual reproductive number ν. a, Transmission
data from the SARS outbreak in Singapore in 2003 (ref. 5). Bars show
observed frequency of Z, the number of individuals infected by each case.
Lines show maximum-likelihood fits for Z Poisson (squares), Z geometric (triangles), and Z negative binomial (circles). Inset, probability
density function (solid) and cumulative distribution function (dashed) for
gamma-distributed ν (corresponding to Z negative binomial) estimated
from Singapore SARS data. b, Expected proportion of all transmission
due to a given proportion of infectious cases, where cases are ranked by
infectiousness. For a homogeneous population (all ν = R0 ), this relation is linear. For five directly transmitted infections (based on k values
in Supplementary Table 1), the line is concave owing to variation in ν.
c, Proportion of transmission expected from the most infectious 20% of
cases, for 10 outbreak or surveillance data sets (triangles). Dashed lines
show proportions expected under the 20/80 rule (top) and homogeneity
(bottom). Superscript v indicates a partially vaccinated population. d,
Reported superspreading events (SSEs; diamonds) relative to estimated
reproductive number R (squares) for twelve directly transmitted infections. Lines show 5–95 percentile range of Z, Poisson(R), and crosses
show the 99th-percentile proposed as threshold for SSEs. Stars represent SSEs caused by more than one source case. ’Other’ diseases are:
1, Streptococcus group A; 2, Lassa fever; 3, Mycoplasma pneumonia; 4,
pneumonic plague; 5, tuberculosis. R is not shown for ‘other’ diseases,
and is off-scale for monkeypox. See Supplementary Notes for details.” .
3.8 Figure 2 a and b from Lloyd-Smith et al Nature 2005. Their legend reads:
“Outbreak dynamics with different degrees of individual variation in infectiousness. a, The individual reproductive number ν is drawn from a
gamma distribution with mean R0 and dispersion parameter k. Probability density functions are shown for six gamma distributions with R0 = 1.5
(’k = Inf’ indicates k −→ ∞). b, Probability of stochastic extinction of an
outbreak, q, versus population-average reproductive number, R0 , following introduction of a single infected individual. The value of k increases
from top to bottom (values and colours as in a). “ . . . . . . . . . . . .
3.9 Timecourses in the stochastic SIR model . . . . . . . . . . . . . . . . . .
3.10 Statistics on the stochastic SIR model . . . . . . . . . . . . . . . . . . .
3.11 Critical community size of measles . . . . . . . . . . . . . . . . . . . . .
3.12 Fig 6.10 from Keeling and Rohani, 2008 . . . . . . . . . . . . . . . . . .
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51
4.1
4.2
4.3
4.4
4.5
53
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59
62
64
Evolution of virulence in myxoma virus . . . . . . . . . . . . . . .
Hypothetical trade-offs between virulence and infectivity . . . . .
basic reproduction number as a function of host birth or mortality
Horizontal versus vertical transmission in fig wasps . . . . . . . .
Horizontal versus vertical transmission in bacteriophages . . . . .
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rate
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44
List of Figures
4.6
4.7
4.8
Virulence and serial transfer of Salmonella in mice . . . . . . . . . . . . .
Local adaptation and the evolution of virulence . . . . . . . . . . . . . .
Virulence and attenuation by serial passage of a fungus . . . . . . . . . .
vi
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69
70
List of Tables
0.1
Infectious diseases of wildlife . . . . . . . . . . . . . . . . . . . . . . . . . viii
1.1
1.2
1.3
1.4
Average age of first infection for various childhood
Critical proportions for herd immunity . . . . . .
Bird flu cases in chicken flocks in Thailand 2004 .
R0 estimated in chicken flocks in Thailand 2004 .
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3.1
Examples of stochastic processes . . . . . . . . . . . . . . . . . . . . . . .
37
vii
diseases
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Foreword
Interactions between populations and interactions of the populations with their environment typically are highly nonlinear and thus their dynamical behaviour frequently defies
intuition. Therefore mathematical models are important tools to establish the factors
underlying the temporal changes in the abundance of natural populations. Traditionally
the models and methods developed in population biology have been used to describe
populations of animals or plants. In this context models play a crucial role, for example,
in helping to establish how natural populations respond to habitat fragmentation and
are thus relevant for issues regarding conservation biology. More recently the methods
and models have been adopted to describe the population biology of infectious diseases
both within a host population and within an individual host.
The application of population biological concepts and formalisms to the study of
infectious diseases is interesting to environmental scientists for two reasons. First, it
provides excellent case studies for population biology in non-standard scenarios. In these
non-standard scenarios the essentials of the concepts become more apparent. We can
think about population biology by comparing and contrasting populations of species in
ecosystems with populations of infections in susceptible populations. Second, infectious
diseases are an important factor that have been shaping host populations throughout
their evolution, and are also the cause of recent species extinction (see Table 0.1).
Pathogen
Host
Location
Impact
Phocine distemper virus
seals
North Sea
Marburg & Ebola virus
primates
Canine parvovirus
dogs
Sub-Saharan Africa, Indonesia, Phillipines
Europe, USA
Varroa jacobsoni
bees
killed half of the seals in
North Sea
high mortality in nonhuman primates
suspected cause of grey
wolfe population decline
catastrophic
massmortality
Batrachotryium
dendrobatidis
Viral chorioretinitis
Aphanomyces astaci
amphibians
kangoroos
crayfish
panglobal except Australasia and Central
Africa
Australia, Central and
North America
Australia
Europe
amphibian extinction
potential extinctions
extinction threat
Table 0.1: Infectious diseases of wildlife and their impact. From Daszak et al, Emerging
infectious diseases of wildlife — threats to biodiversity and human health. Science 2000.
viii
CHAPTER 0. FOREWORD
In contrast to most other areas of biology mathematical models play a central role
in ecology and evolution. Therefore it is important to be familiar with the underlying
concepts and methods. The degree of mathematical sophistication in the population
genetic and population biological literature can be high. However, in a course such as
this one, it will not be possible to go into great mathematical detail, nor is it possible
to give complete derivations of all mathematical concepts introduced. The emphasis of
this course will therefore be on developing an intuitive understanding and familiarity
with the concepts used in population biology.
ix
1 Basic mathematical epidemiology
1.1 SIR models
In general the population dynamics of infectious diseases can often be usefully described
in terms of so called SIR models. Here the population is subdivided into susceptible (i.e
non-infected, non-immune) hosts (S), infected hosts (I), and recovered (immune) hosts
(R). A graphical scheme of an SIR model is given in figure 1.1. Mathematically this
scheme translates into the following equations
dS/dt = B(S, I, R) − δS − βSI + qR
dI/dt = βSI − aI − rI
dR/dt = rI − δR − qR
qR
Birth
Loss of
immunity
B(S,I,R)
Recovereds
R
Infecteds
I
Susceptibles
S
Natural
δS
death
(1.1)
(1.2)
(1.3)
βSI
aI
Infection
rI
δR
Natural
death
Recovery
Natural
and disease
induced
death
Figure 1.1: Graphical scheme of the SIR model.
The model is based on the following assumptions. (i) Susceptible individuals are
born at a rate B(S, I, R) which is assumed to be a function of the densities of the
susceptible, infected and recovered hosts. (ii) Susceptibles are infected at a rate given
1
100
40
60
80
S
I
R
0
20
Population size
80
60
20
40
S
I
R
0
Population size
100
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
0
5
10
15
20
25
0
Days
100
200
300
400
500
Days
Figure 1.2: Timecourses of the SIR model. For this graph eqs 1.1–1.3 were numerically solved.
B(S, I, R) was chosen such that the total population size remained constant. Parameters
were: δ = 10−4 /d, β = 0.01/d, a = 0.1/d, r = 0.3/d, q = 0.01/d. The left graph shows the
timecourse during the first 25 days, while the graph on the right shows the timecourse until
day 500. Are the equilibrium levels in agreement with the theoretical expectation in eq. 1.7?
by the product of the densities of susceptible and infected hosts, times a proportionality
constant β describing the infectivity rate per contact between the two types of host.
The assumption to model the infection rate proportional to SI is justified if both hosts
types are well mixed and the encounter between the two host types is random. This
assumption is often called mass-action kinetics and derives from chemical kinetics. (iii)
Susceptible and recovered hosts die at a rate δ which describes the natural death rate due
to causes unrelated to infection. (iv) Infected hosts die at a rate a which includes both
the natural death rate plus the disease induced death rate. (iv) Infected hosts recover
at a rate r. (v) Recovered hosts lose immunity at a rate q and become susceptible again.
The above model may describe qualitatively the dynamics of infectious diseases with
contact dependent transmission (such as the common cold). However, it is important
to note, that the above model still ignores many possibly relevant biological aspects.
For example, age structure of the population is ignored. The infectivity, death rate,
and rate of recovery may all depend on the age of the infected individual. Moreover,
spatial structure is ignored. Often transmission to cohabiting individuals (i.e. members
of the family) occurs with increased probability. All of these issues have been addressed
in detail using more complex models. We ignore these higher levels of complexity and
focus on the above simple model for didactic reasons.
Figure 1.2 shows timecourses of the SIR model defined in eqs. 1.1–1.3. There are two
dynamical phases. First, there is a transient phase during which the populations change.
For the initial conditions S(0) = 99, I(0) = 1, R(0) = 0 that were used in Figure 1.2
the number of susceptibles decreases, and the number of infected and recovered hosts
increases. After the transient phase the dynamics equilibrates, ie population sizes are
attained the do not change in time. Because the SIR model (eqs. 1.1–1.3) contains a nonlinear infection term, βSI, it is not possible to derive explicite mathematical expressions
for the transient dynamics. The equilibrium levels can, however, be calculated (see
2
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
below).
Often the total population size remains roughly constant over the period of interest
(such as the time for an epidemic to occur). In terms of the above model this implies
that N = S + I + R is constant and therefore dN/dt = dS/dt + dI/dt + dR/dt = 0.
Summing up equations 1.1-1.3 we thus obtain for the birth rate:
B(S, I, R) = δ(S + R) + aI
(1.4)
If the total population size is assumed to be constant, we can drop one of the three
population variables, say R, since R is given by N − S − I. We thus obtain
dS/dt = (δ + q)(N − S − I) + aI − βSI
dI/dt = βSI − aI − rI
(1.5)
(1.6)
The model has two equilibrium solutions
(i) S ? = N, I ? = 0 and (ii) S ? =
a + r ? (βN − a − r)(δ + q)
,I =
β
β(δ + q + r)
(1.7)
The first equilibrium represents the case where none of the individuals are infected
(uninfected equilibrium). The second equilibrium represents the case where a fraction
of the individuals are infected (infected equilibrium). Note, that there is no equilibrium
with all of the individuals in the population being infected1 .
1.2 Basic reproduction number
We now address under which conditions the SIR model (eqs. 1.5 and 1.6) converges to
the uninfected or the infected equilibrium. To this end we need to determine whether
the growth rate of the infecteds, dI/dt, is larger or smaller than 0. From equation 1.6
we derive that the population of infecteds grows if (and only if)
βS
>1
a+r
We want to determine under which conditions an epidemic can spread in the population. Thus we consider a scenario in which a (infinitesimally) small number of infected
individuals is placed into a population in which all other individuals are susceptible.
Mathematically this implies substituting S = N for the population density of susceptibles (i.e. the equilibrium population density when all individuals are uninfected) in the
above equation. We obtain
βN
R0 =
>1
(1.8)
a+r
1
SIR models can be explored as a module in the ”Modelling course in population and evolutionary
biology” that takes place every summer term
3
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
Figure 1.3: Graphical illustration of the basic reproduction number R0 . Here a sick frog (red) is
placed into an otherwise susceptible population of 9 frogs (green). Over the entire duration
that the frog is infected, this frog infects 3 other individuals. Thus, in this hypothetical
scenario, the basic reproduction number is 3.
R0 is the so-called basic reproduction number 2 . Although most people in the field use
the term basic reproduction rate, some prefer the term basic reproductive number, since
strictly speaking R0 is not a rate. (The quantities βN and a+r both have the dimension
time−1 and therefore the unit of time cancels out in the expression of R0 . R0 is thus a
number and not a rate).
Above we have determined the basic reproduction number for the given SIR model
(eqs. 1.5 and 1.6). More generally, the definition is:
The basic reproduction number is the average number of
secondary infected hosts when one primary infected host
is placed into a wholly susceptible population.
A graphical illustration is given in figure 1.3. An infection can spread in the population
if R0 > 1. Conversely an infection will die out if R0 < 1. Hence if R0 > 1 the population
will converge to the infected equilibrium and if R0 < 1 the population will converge to
the uninfected equilibrium.
An intuitive explanation for the basic reproduction number as given in eq. 1.8 can
be obtained as follows: N is the population size of susceptibles (when all individuals
in the population are susceptible). The rate at which individuals leave the infected
compartment is given by a + r, describing the combined loss of infecteds through death
and recovery. Hence the inverse of a + r is the average duration of an infection. Thus
over the entire duration of the infection an infected individual will infect βN/(a + r)
individuals.
2
The concept of R0 goes back to Klaus Dietz (University of Tübingen), Roy Anderson (Imperial
College, London) and Robert May (University of Oxford)
4
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
1.3 R0 and age of infection
We will now derive that for childhood infections with life-long immunity, the basic
reproduction number is approximately given by
R0 =
L
A
(1.9)
where L is the life expectancy of individuals in the population and A is the average
age at which a host contracts the infection. We will refer to A as the age of infection.
(Sometimes age of infection denotes the time that has elapsed since the infection started
within a host, but we use this term exclusively for the age of the host at which it contracts
the infection.) Life-long immunity implies that q = 0 in the SIR model (eqs. 1.5 and
1.6). We now do the following elementary calculations
R0 =
=
=
=
=
βN
a+r
β(S + I + R)
a+r
a + r + βI + βR
a+r
βI(1 + r/δ)
1+
a+r
βI δ + r
1+
δ a+r
(1.10)
Note, that we are using here the population sizes at the infected equilibrium, since we
want to express R0 as a function of these values. In particular we have used in this
calculation that at the infected equilibrium S = (a + r)/β and R = (r/δ)I if q = 0.
Let us first discuss the expression (δ + r)/(a + r). Clearly, we have δ < a since
the parameter a contains both the natural and the disease induced mortality. Hence
(δ + r)/(a + r) < 1. Moreover, for childhood infections it is reasonable to assume that
the rate of recovery r is much larger than the natural death rate δ, but also than the
mortality rate of infecteds a. Therefore (δ + r)/(a + r) ≈ 1. Thus we obtain
R0 ≈ 1 +
βI
δ
(1.11)
Note, that δ is the (average) natural death rate of uninfecteds. Hence the reciprocal
1/δ is the average life expectancy L of uninfecteds. On the other hand βI is the rate
at which susceptibles become infected. Hence its reciprocal 1/(βI) is the average age at
which susceptibles first get infected. Altogether we obtain
R0 ≈ 1 +
L
L
≈
A
A
if L >> A
(1.12)
It is important to spell out the assumptions underlying this calculation. As mentioned
already above this result was derived assuming life long immunity (i.e q = 0) which is
5
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
a reasonable assumption for childhood diseases, but not for other types of infections3 .
More importantly, the above calculation assumes that all parameters such as infectivity,
recovery, natural death rate and disease associated death rate do not depend on the
age of the individual, which is clearly a biologically unjustified assumption. The SIR
model (eqs. 1.5 and 1.6) upon which our calculation is based ignores age structure
in the population. Basically, it assumes what ecologists refer to as Type II mortality,
namely that the death rate is constant over age. In the developed world the death rate
of humans is fairly low until age 60 or higher and then begins to increase markedly with
age. In the developing world child mortality is high, followed by a moderate mortality
in intermediate ages (which is actually currently changing because of AIDS), and a high
mortality at old age. Thus a more realistic mortality (at least for the developed world)
is the so called Type I mortality, which assumes a zero mortality rate until age L and an
infinitely high mortality afterwards. Figure 1.4 illustrates these two types of mortality
functions.
Age structure can be incorporated into population dynamical models by various techniques. The Leslie matrix approach subdivides the population into discrete classes with
regard to age. Partial differential equations can be used to describe the population
dynamics with continuous age distributions. Importantly, it can be shown with these
approaches that the basic reproduction number R0 is also given by L/A for the more
realistic Type I mortality.
For some infections it is also necessary to incorporate that the probability of infection
depends on age. For example, the risk of infection by sexual transmitted diseases is high
at intermediate ages, but vanishes at young age. Moreover, the model (eqs. 1.5 and
1.6) assumes that uninfected and infected individuals always give birth to uninfected
offspring, which is only justified for horizontally transmitted infections4 .
Table 1.1 lists the average age of infection for various childhood diseases in different
countries. The relation R0 = L/A makes clear that childhood diseases must have high
basic reproduction numbers, because of the early average age of infection. The data
in table 1.1 also makes clear that the basic reproduction number may be different in
different localities because of differences in the age of first infection. (Note that the
expected life-span will also differ between countries.)
1.4 Vaccination
Next we want to determine the effect of vaccination. In particular, we want to ask what
fraction of the population needs to be vaccinated in order to eradicate the disease. In
the previous section we derived that R0 = βN/(a + r). Vaccination has the effect of
transferring individuals directly from the susceptible to the immune class. So we ask to
what level S ? do we have to reduce the susceptible population such that the disease can
3
It is straightforward to repeat the above calculation assuming that q > 0. If q >> δ then essentially
R0 ≈ L/(Aq).
4
Horizontal transmission refers to transmission between individuals independent of whether they are
related or not. In contrast, vertical transmission refers to transmission from mother to child.
6
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
1.0
Typical survivorship curves
0.6
Type II
0.4
% surviving
0.8
Type I
0
20
40
60
80
100
age
Figure 1.4: Type I (dashed line) and Type II (solid line) mortality. Type II mortality assumes a constant death rate independent of age, while Type I mortality assume a vanishing
death rate until age L and infinitely high death rate afterwards. Type II mortality is a
better approximation for the survivorship of human populations in the developed world.
Type III mortality (not shown here) assumes a high mortality at young ages that decreases
markedly with increasing age. Type III mortality may for example describe survivorship in
fish populations where most individuals never make it to adulthood.
7
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
Infectious disease
Measles
Rubella
Chicken Pox
Polio
Pertussis
Mumps
A
Location and time period
5–6
4–5
2–3
2–3
1–2
9–10
9–10
6–7
2–3
6–8
12–17
4–5
6–7
USA, 1955–1958
England/Wales, 1948–1968
Morocco, 1963
Ghana, 1960–1968
Senegal, 1964
Sweden, 1965
USA 1966–1968
Poland, 1970–1977
Gambia, 1976
USA, 1921–1928
USA 1955
England/Wales, 1948–1968
England/Wales, 1975–1977
Table 1.1: Average age at infection, A, for various childhood diseases in different geographical
localities and time periods (Source, Anderson & May, Infectious Diseases of Humans, Oxford
University Press)
no longer spread through the population. The threshold condition is given by
R0vaccinated
βS ?
=1=
a+r
Solving for S ? we obtain
S? =
a+r
β
(1.13)
(1.14)
which is exactly the population of susceptible individuals at the infected equilibrium.
Thus the critical proportion of individuals that need to be vaccinated is given by
pc =
N − S?
a+r
1
=1−
=1−
N
βN
R0
(1.15)
Hence, the higher the basic reproduction number, the higher is the fraction of individuals
that need to be vaccinated in order to eradicate the disease. Note, however, that in no
case it is necessary to vaccinate the entire population. Once the population of susceptibles falls below a critical level (given by pc N ) the entire population is protected against
infection. This concept is called herd immunity. Importantly, this type of immunity is
not a property of the individual, but a property of the population only. Table 1.2 gives
the critical proportion for several important infectious diseases.
These considerations have a number of important implications:
(i) First of all the data in table 1.2 shows why it is so hard to eradicate diseases such
as malaria or certain childhood diseases. The vaccination needs to cover a very large
8
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
Infectious disease
pc
Malaria (P. falciparum in hyperendemic regions)
99%
Measles
90–95 %
Pertussis
90–95 %
Human parvovirus infections
90–95 %
Chicken pox
85–90 %
Mumps
85–90 %
Rubella
82–87 %
Polio
82–87 %
Diptheria
82–87 %
Scarlet fever
82–87 %
Small pox
70–80 %
Table 1.2: Critical proportion pc of individuals that need to be vaccinated in order to eradicate
a disease through herd immunity. (Source: Anderson & May, Infectious Diseases of Humans,
Oxford University Press)
fraction of the population. Moreover, the above calculation assumes that a vaccine offers
100% protection, which is not the case for many vaccines.
(ii) There is a conflict between the interests of the individual and the interests of the
community. Because vaccines can have side-effects5 parents frequently decide not to have
their children vaccinated, essentially because they estimate that the risk of infection by
a childhood disease (and its complications) may be lower than the risk of side-effects
through vaccination. However, the risk of infection is only low because so many children
are vaccinated. Hence, from the perspective of the individual it may be rational not to
vaccinate one’s children. However, on the basis of the population it is clear that it is
necessary to maintain a high coverage of vaccination, since childhood infection can often
lead to very severe or even fatal outcomes. For example, measles caused 600,000 deaths
in 2002).
(iii) Vaccination essentially has the effect of lowering the basic reproduction number
by decreasing the number of susceptibles in the population. However, since the basic
reproduction number is given by the ratio of the life expectancy, L, and the average
age of infection, A, lowering the basic reproduction number has the effect of increasing
the average age of infection. Thus while the protection though vaccines leads to a lower
number of infections, an unwanted side effect of vaccination can be that some childhood
infections have more severe effects when they occur in older individuals. Rubella, for
example, can lead to severe complications during pregnancy. Measles can lead to male
sterility in adults. It is often suggested that the history of poliomyelitis in developed
countries represents such a case of unwanted side-effects of vaccination. When contracted
at an early age polio is less likely to have serious consequences. However, increased
5
Most side-effects of vaccines against childhood infections are harmless (such as a transient fever).
Severe complications are extremely rare.
9
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
standards of hygiene led to a drop of poliovirus infections (which are transmitted through
the oral-fecal route of infection), which in turn increased the average age of infection.
However, in older individuals the infection is more likely to lead to paralysis and other
complications and therefore the epidemic became more troublesome as the incidence
declined. These considerations caution that population biological effects of vaccination
policies need to be carefully investigated for new vaccination policies.
1.5 Parametrizing epidemiological models
How does one determine the parameters of an epidemiological model, such as the SIR
model in eq. 1.1–1.3? To make a mathematical model consistent with the data one needs
data. What data are available for epidemiological models?
In this section, we are going to introduce some of the basic concepts needed for parameter estimation. This is related to the field of statistical inference that you may
have encountered in the context of hypothesis testing. Statistical inference, however, is
a broader concept that also conatins parameters estimation.
Let’s look at our SIR model in eqs. 1.1–1.3. It has three variables (S, I, R) and five
parameters (δ, β, a, r, q) if we assume constant population size. To estimate five parameters from empirical data is challenging and would require an extensive dataset. For
this reason, the strategy that is most commonly adopted is that, if possible, parameters
are determined indepedently. For example, the death rate of hosts, δ, is related to the
average life time, and can therefore be estimated without any epidemics in a species.
The death rate constant a and recovery rate constant r can be determined by the observation of infected individual hosts. The immunity loss rate constant q can be infered
by looking at the long term history of recovered individual hosts.
The parameter that cannot be determined by observation of a single host is the transmission rate constant β because it measures the spread of the infection through a population. Therefore, population-level observations are required to determine β. Because
there is a simple relationship between β and the basic reproduction number R0 , one
often estimates R0 instead of β.
R0 is determined either from early stages of the epidemic, or from the endemic equilibrium. Both have advantages and disadvantages as we discuss in this section. The data
that are most commonly used to estimate R0 are resported cases and sero-prevalence
data (ie data on hosts who have antibodies against the pathogen).
1.5.1 Estimating R0 from reported cases
Imagine, we have a population of size N in which there is an infection in its endemic
equilibrium. If all cases of the disease are reported we essentially know the endemic
equilibrium level I ? (eq. 1.7 (ii)).
In eq. 1.7 we calculated the equilibrium level of infected hosts from our SIR model
10
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
(eqs. 1.1–1.3) with constant population size as:
I? =
Using R0 =
βN
a+r
(βN − a − r)(δ + q)
β(δ + q + r)
we get:
I
?
δ+q
=
N
δ+q+r
1
1−
R0
(1.16)
Thus, measuring I ? and knowing δ, r, and q, we can determine R0 .
This method estimates R0 via the term 1 − 1/R0 . This term is very close to one
for large R0 . For that reason, we cannot estimate R0 with large confidence when it is
large. Furthermore, most infections are under-reported, ie fewer cases are reported than
actually occur. Under-reporting leads to a bias in the estimate of R0 .
Alternatively, we can estimate R0 from the intial dynamics of the epidemics. From
the equation for infected hosts of our SIR model (eq. 1.2), we can derive:
I(t) ≈ I(0)e(R0 −1)(a+r)t
(1.17)
Using this equation, we can estimate R0 from the initial increase of infected cases. For
this we need measurements on at least two different time-points.
The problem with this method is that we need an epidemic. The estimation of the basic
reproduction number of, for example, pandemic influenza — should a new pandemic
influenza strain arise at all — will be possible only after an epidemic has taken off
somewhere. However, the assessment of man public health interventions depends on R0 .
A vicious circle!
A second problem of this method to estimate R0 is that the number of infected hosts is
very low in the beginning of an epidemic, and low numbers are often associated with large
stochastic effect. This will make the estimation difficult. A way around this obstacle is
to use data from multiple outbreaks.
Lastly, it is important to catch really only the initial stages of the epidemic. If one
uses data from too late into the epidemic, non-linear effects of the dynamics may come
into play, such as the exhaustion of susceptibles that slows down the increase of infected
cases.
A last way of estimating the basic reproduction number from reported cases exploits
the relationship between R0 and the average age of infection A (see eq. 1.9):
R0 ≈ 1 +
1
L
≈1+
A
δA
To use this relation, it is necessary that the age of each reported case is reported as well.
Knowing the average life-span of an individual host L, we can then estimate R0 .
However, this method is only valid if all of the assumptions that went into the derivation of relation 1.9. These were discussed at length after the derivation. In short, the
relation is only valid for diseases with life-long immunity and age-independent transmission, recovery, and virulence parameters.
11
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
1.5.2 Estimating R0 from seroprevalence data
Pathogens induce immune responses in their hosts. Although there is a plethora of immune effectors that can be measured, an easy and common one are pathogen-specific
antibodies. Many clinical tests for diseases test for the presence or absence of antibodies
specific to the pathogen in question for example, amebiasis, anthrax, brucellosis, Human immunodeficiency virus (HIV), fungal infection, measles, rubella, RSV, syphilis,
tularemia, and various types of viral hepatitis.
If a host has antibodies against a pathogen it is called sero-positive. 6 The number of
people with pathogen-specific antibodies in a population is called the sero-prevalence.
Assume we have a population with N people. If we divide the population into susceptibles, infecteds and recovereds, then only the susceptible individuals do not have
antibodies against the pathogen in question. Hence, in the endemic equilibrium the
seroprevalence is:
N − S ? = N (1 −
1
)
R0
(1.18)
Hereby, S ? is the equilibrium level of susceptibles as given by eq. 1.7(ii). This relation can
then be used to estimate the basic reproduction number from the sero-prevalence. Unlike
for reported cases (eq. 1.16), we do not need to determine any additional parameters.
But, we again have a term 1 − 1/R0 which makes the estimation of large basic reporduction numbers difficult. Furthermore, we have tacitly assumed that antibodies are
protective and losing immunity means losing antibodies. If some recovered individuals
became susceptible again, but would still have some low, yet detectable, antibody level,
eq. 1.18 would no longer hold. A last problem with estimating R0 with eq. 1.18 is that,
for diseases with lifelong immunity, older people are more likely to have been exposed an
therefore more likely to be sero-prevalent. Therefore, to avoid biases in determining the
total sero-prevalence in your population, all age classes need to be sampled uniformously.
A last method to estimate R0 involves age-stratified sero-prevalence data, ie data on
the fraction of hosts sero-prevalent in each age class. The probability that host is still
susceptible (sero-negative) at age α is:
Using A =
1
βI
≈
1
,
δ(R0 −1)
P (α) = e−α/A
(1.19)
P (α) ≈ e−αδ(R0 −1)
(1.20)
we get:
If we have n seronegative hosts with ages α1 , . . . , αn , and m seropositive hosts with ages
then one can combine all the probabilities of sero-positivity and -negativity in
0
α10 , . . . , αm
6
“sero” refers to the blood serum — the part of the blood between the cells which contains the antibody
molecules.
12
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
one function, called the likelihood.
(R0 ) and the data:
L(R0 ) =
n
Y
7
e
The likelihood is a function of the model parameters
−αi δ(R0 −1)
×
i=1
m Y
0
1 − e−αi δ(R0 −1)
(1.21)
i=1
Qn
Hereby i=1 means the product of the expression on its right for i = 1, . . . , n. The best
estimate of R0 is obtained by maximising L(R0 ).
Assume, for example, that we observe two sero-positive hosts that are 2 and 4 years
old, and two sero-negative hosts that are 3 and 5 years old. Assume that the average
life-span of this host is 10 years, ie δ = 0.1/year. Then we have:
L(R0 ) = e−2year×0.1/year×(R0 −1) × e−4year×0.1/year×(R0 −1)
×(1 − e−3year×0.1/year×(R0 −1) ) × (1 − e−5year×0.1/year×(R0 −1) )
(1.22)
This function is plotted in Figure 1.5. The optimum for this specific example would be
R0 = 3.13
According to Keeling and Rohani (2008), this is the best way to estimate the basic
reproduction number from endemic data. It is most efficient to sample around the
The likelihood is a statistical construct used for fitting of models to data, and the estimation of
model parameters. It has been introduced by R. A. Fisher in 1922 in the paper entitled “On
the mathematical foundations of theoretical statistics”. It is calculated as the probability of the
data assuming certain model parameters. Technically, the likelihood is not a probability, though.
The values of the model parameters which maximize the likelihood are called maximum likelihood
estimators.
As a simple example, consider a simple coin toss experiment in which a coin is toss 10 times and
6 times the outcome is “head”. If we denote the probability of the coin falling on “head” as pH the
likelihood is:
10
L(p) =
p6H (1 − pH )10−6
6
0.00
0.10
0.20
This likelihood is maximized at p̂H = 0.6 as can be seen when we plot L versus p:
Likelihood
7
0.0
0.2
0.4
0.6
pH
13
0.8
1.0
0.00 0.02 0.04 0.06 0.08
Likelihood, L
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
1
2
3
4
5
6
7
8
9
10
R0
Figure 1.5: The likelihood in eq. 1.22 as a function of R0 . The optimum is at R0 = 3.13 and
marked by the dashed line.
average age of infection since these age classes contain mist information on how well the
disease spreads. In too young and old hosts the seroprevalence will be close to 0% or
100%, respectively.
1.5.3 Example: avian influenza in Thai chicken
As an illustration, we are going to look at avian influenza. You may have heard a lot
about avian influenza in recent years. There is the concern that avian influenza strains
may cause a new pandemic in the human population. Wild birds are the natural host for
influenza A viruses and usually do not get sick from infection. Some avian influenza A
strains, called highly pathogenic avian influenza A (HPIA-A), are highly virulent to poultry, and some of these strains are even pathogenic in wild birds and humans. The main
subtypes of HPAI-A are H5N1, H7N7, and H7N3, named after the types of their viral proteins hemaglutinin and neuraminidase. Since 1997 there are regular outbreaks of HPAI-A
in Asia, Africa and Europe (see http://www.cdc.gov/flu/avian/gen-info/flu-viruses.htm).
In 2007, Tiensin et al published an article in the J Inf Dis, in which they estimated
the basic reproduction number of the H5N1 subtype of avian influenza on Thai chicken
farms from the year 2004. They had data on the number of dead chicken per day, R.
In Table 1.3 you see the data for one flock with initially N = 4100 chicken. On one day
they had 55 dead chicken, the next day 115, then 203, 415 and finally 837. After that
the flock was culled. These data were then used to “back-calculate” the values for C,
14
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
Table 1.3: Bird flu cases in chicken flocks in Thailand 2004. This table shows how one can
infer S, I and R using only the data on sick chicken C. The inferred values for S, I, and
R depend on what infectious period one assumes. (This is Table 1 from Tiensinet al, J Inf
Dis 2007).
15
CHAPTER 1. BASIC MATHEMATICAL EPIDEMIOLOGY
Table 1.4: Transmission rates β and basic reproduction numbers R0 estimated in chicken flocks
in Thailand 2004. (This is Table 2 from Tiensin et al, J Inf Dis 2007).
S, I, R, and N assuming different durations of infectiousness (1,2,3,or 4 days). Note,
that unlike above the total population size in the model by Tiensin et al is not assumed
constant. C denotes the population of newly infected, but not yet infectious chicken
and is just a helper variable. Also note that R does not denote the recovered, but the
removed or dead class.
The expected value for newly infected cases is given by a term familiar from the SIR
model:
E(C) =
βSI
∆t
N
(1.23)
Hereby the time step ∆t = 1 day. This allows us to estimate β and R0 by hand. For
day 1 with infectious period of 1 day we have:
115 =
β × 4045 × 55
× 1 day
4100
(1.24)
For β we therefore get:
β=
115 × 4100
≈ 2/day
4045 × 55 × day
and
R0 = β × infectious period ≈ 2
(1.25)
To incorporate all the available data the authors use generalized linear models, which
is a sophisticated statistical method beyong the scope of this course. But the simple
calculation above shows you how the estimation works in principle. Table 1.4 shows the
results of the anaylsis of Tiensin et al. The estimate for R0 ranges from around 2 to 5.
Therefore, at least 80% of chicken should be vaccinated to prevent H5N1 spreading in
this setting. What you also see is that the estimate of R0 is almost independent of the
choice of the infectious period, while the estimate the transmission rate constant β is
not.
16
2 Advanced mathematical
epidemiology
In this chapter, we are going to look at more sophisticated mathematical models in
mathematical epidemiology, which incorporate some of the complexities of real life. In
most of our discussions so far, we have assumed that the populations are homogeneous
and well-mixed. In reality, however, individuals in a population vary in many respects.
The two aspect of inter-individual variation we consider in this chapter are differences
of disease risk and spatial structure. The point of a more realistic description of the
population dynamics of host-pathogen systems is not just an increased realism by itself.
More refined models allow us to also consider more elaborate strategies of disease control.
For example, in addition to asking what fraction of the population one should vaccinate,
one can ask how vaccination should be targeted to make optimal use of vaccines.
2.1 Heterogeneity in Risk structure
Let us consider heterogeneity in risk structure first. Risk structure can be described by
introducing multiple classes of hosts. Hereby the classification can be anything, such
as age, sex, spatial location etc. Figure 2.1 shows a flow chart of an SIR model with
three classes. There are many more arrows than in an SIR model with just one class.
In particular, 9 infection rates have to be specified.
Let’s start with the simplest possible epidemiological model which ignores host demography, virulence and immunity:
dS/dt = −βSI + rI
dI/dt = +βSI − rI
(2.1)
(2.2)
Let’s now split up the population into hosts at high risk (indexed by H), and hosts at
low risk (indexed by L):
dSH /dt
dIH /dt
dSL /dt
dIL /dt
=
=
=
=
−βHH SH IH − βHL SH IL + rIH
+βHH SH IH + βHL SH IL − rIH
−βLH SL IH − βLL SL IL + rIL
+βLH SL IH + βLL SL IL − rIL
(2.3)
(2.4)
(2.5)
(2.6)
For simplicity, we assume that the high and low risk populations do not differ in their
recovery rates. Figure 2.2 show the time-courses of S and I in eq. 2.3–2.6.
17
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
S
I
R
S
I
R
S
I
R
Figure 2.1: Introducing multiple classes of hosts into an SIR model. The classes could be by
age, sex etc.
In the homogeneous SIR model, we had two dynamical phases of the time-course: one,
during which the number of infecteds increased, and a second, during which the level of
infecteds was in equilibrium. In a model with a heterogeneous host population, we have
three phases. During the first phase the proportions of infected hosts in each risk class
change and converge to a quasi-steady state. Quasi because in the state, which they
eventually attain, the absolute numbers of infected hosts are not stable, but increase
exponentially. Their proportion in the population is stable, though. The quasi-steady
state that is attained during the first phase is analogous to the stable age distribution
which is attained in age-structured demographic models (see Keyfitz and Caswell 2005,
chapter 10). The particular shape of this first transient phase depends on the initial
conditions of the dynamical systems. During the second phase the level of the infected
hosts increase in parallel. Keeling and Rohani call this the slaved phase. During the
last phase, the dynamics saturates and asymptotically converges towards its final state
in which all populations are constant. In models with host demography this final state
is the endemic equilibrium in which there is a stable level of infected hosts. In our
model the final state of the infected population is 0 because the susceptible population
is exhausted and not replenished by births. The last two phases correspond to the two
phases in homogeneous models. The first transient phase exists only in heterogeneous
models.
18
transient
10−3
10−2
IH
IL
asymptote
slaved
10−4
Infected hosts
10−1
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
0
5
10
15
20
25
30
Weeks
Figure 2.2: Time-courses of the SI model with high and low risk groups as defined in eqs. 2.3–
2.6. In contrast to the homogeneous model, there are three dynamic phases: first, a transient
phase at which the infection in the different risk groups goes into a quasi-steady state.
second, a phase during which the infecteds in each risk groups increase in parallel. last,
a phase in which the number of infecteds asymtotically converges towards its equilibrium.
The parameters for this plot are: βHH = 2.5/year, βHL = 0.2/year, βLH = 0.3/year, βLL =
0.1/year, and r = 0.3/year. We used the inital consitions: SH (0) = 0.3, SL (0) = 0.699,
IH (0) = 0, IL (0) = 0.001
19
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
2.1.1 Basic reproduction number
How can we define the basic reproduction number in models with heterogeneous risk
structure?
In homogeneous model R0 was defined as “the average number of secondary infected
hosts arising from one primary infected host in an entirely susceptible population” In
our SI model with high and low risk groups, we could put either a high or a low risk
individual into the population. And, obviously, a high risk individual will cause more
infections than a low risk individual. To define R0 , we therefore need to average over high
and low risk primary infected hosts. But the average must not be taken according to the
proportion of risk classes in the susceptible population, but rather their proportions in
the infected population. As we have discussed above, the proportions of infected hosts
in the different classes change in the first transient phase of the dynamics. During this
phase, the proportions also depend on the intial conditions. Therefore, we can define the
average primary infected only in the second “slaved phase” during which the infecteds
in different risk classes have attained their quasi-steady state.
These insights culminate in the following definition:
In an epidemiological model with heterogeneous host population the basic reproduction number is the average number of secondary infected hosts arising from one average
primary infected host in an entirely susceptible population
once initial transients have decayed.
How is the basic reproduction number calculated in a epidemiological model with a
heterogeneous host population?
To this end, let us assume that, initially, the number of susceptible hosts are SH = nH
and SL = nL in the high and low risk classes, respectively. We can rewrite eqs. 2.4 and
2.6 as:
dIH /dt = (βHH nH − r)IH + βHL nH IL
(2.7)
dIL /dt = βLH nL IH + (βLL nL − r)IL
(2.8)
IH
βHH nH − r
βHL nH
If we define I =
and the Jacobian matrix J =
we
IL
βLH nL
βLL nL − r
can write eqs. 2.7 and 2.8 in matrix notation:
dI/dt = J I
(2.9)
Decoupling the system of differential equations 2.9 can be accomplished by diagonalizing the Jacobian matrix J (see Imboden & Koch, 2003: Systemanalyse, pp82). 1 The
1
Remember: If you have a matrix M =
a
c
b
d
it can be diagonalized by solving the eigenvector
equation:
Mu
20
=
λu
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
dominant (ie largest) eigenvalue, λ, of J is
q
1
2
2
λ =
n2L βLL
− 2 nH βHH nL βLL + 4 nH βHL nL βLH + n2H βHH
2
+nL βLL + nH βHH ) − r
(2.10)
and is related to the basic reproduction number according to the following relation:
R0 = 1 + λ/r
(2.11)
The dominant eigenvector gives you the ratio of high and low risk infecteds in the slaved
s
phase IH
/ILs . We can calculate the basic reproduction number R0 as the weighted sum
of the number of secondary infection caused by a high risk primary infected host, RH ,
and the number of secondary infection caused by a low risk primary infected host, RL ,
s
R0 = RH IH
+ RL ILs
(2.12)
During the slaved phase the total number of infecteds increases as eλt = e(R0 −1)rt
As a numerical example, consider the simulation shown in Figure 2.2. With the
parameters βHH = 2.5/year, βHL = 0.2/year, βLH = 0.3/year, βLL = 0.1/year, and
r = 0.3/year, and for nH = 0.3, nL ≈ 0.7 we get as the dominant eigenvalue:
λ = 0.468/year
(2.13)
And for the basic reproduction number we get:
R0 = 1 + 0.468/0.3 = 2.56
(2.14)
2.1.2 Impact of heterogeneity on disease dynamics
There are two famous results in models with risk structure: heterogeneity in risk accelerates early spread of infections, but leads to lower total number of infecteds. We derive
which is equivalent to solving
det(M − λI)
=
0
This gives rise to the so-called characteristic equation for λ.
For our 2 × 2 matrix M the characteristic equation reads:
a−λ
b
det
= (a − λ)(d − λ) − bc
c
d−λ
=
λ2 − (a + d)λ + (ad − bc)
and has the solutions:
λ=
p
1
(a + d ± (a + d)2 − 4(ad − bc))
2
21
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
these two results now. To that end we use a model which May and Anderson used in a
paper in Parasitology 1988 to model the spread of HIV infection.
The model they used was simple. It neglected host demography, and recovery. The
hosts population was subdivided according to the number of contacts, i of the hosts in
each class per unit time. The model equations are:
X
dSi /dt = −
βij Si Ij
(2.15)
j
dIi /dt =
X
βij Si Ij − aIi
(2.16)
j
Hereby, Si and Ii denote the proportion of susceptible and infected hosts with i contacts
per unit time, respectively.
May and Anderson used a common trick to reduce the number of parameters which
have to be specified to define this model. They defined the transmission rate constants
as:
ij
k knk
βij = b P
(2.17)
P
where nk is the proportion of individuals in class k.
k knk is the mean number of
contacts. b is the average transmission rate per time-unit. The transmission rate scales
with ij, ie a person with five contacts transmits to a person with 2 contacts 10 times as
much as a person with one contact to a person with one contact.
Why is this clever? Because one has to define only one transmission rate constant b,
and the rest is given by generic assumptions about the contact network.
The basic reproduction number in this model can be calculated without recurrence
to the complex diagonalization described above. Assume initially Si = ni . After the
introduction of the infection the dynamics soon enters the slaved phase in which the
infecteds in different classes increase in parallel. In this phase we have:
ini
Ii
≈P
I
k knk
22
(2.18)
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
Basic reproduction number is:
P
βij Si Ij
aI
1X
jnj
=
βij ni P
a ij
k knk
ij
jnj
1X
bP
ni P
=
a ij
k knk
k knk
P
P 2
b
i ini
j j nj
=
×
P
a
( k knk )2
R0 =
ij
b M (M 2 + V )
×
a
M2
2
b M +V
=
×
a
M V
b
M+
=
a
M
=
(2.19)
P
P
Hereby M = i ini denotes mean number of contacts, and V = i (i − M )2 ni denotes
the variance in the number of contacts. 2
Eq. 2.19 is a famous result derived by May and Anderson in 1988, and can also be
found in Keeling and Rohani 2008, p 66. I means that the basic reproduction number in
a host population with heterogeneous risk is larger than you would expect based on the
mean transmission. The basic reproduction number increases linearly with the variance
even if the mean number of contact remains the same.
To illustrate this important result let’s assume two risk classes i = 1, 10. Let’s also
assume 90% of the population are in risk class i = 1 and 10% are in risk class i = 10
The mean number of contacts for this case is:
X
knk = 1 × 0.9 + 10 × 0.1 = 1.9
(2.20)
M=
k
And the mean square number of contacts is:
X
k 2 nk = 1 × 0.9 + 100 × 0.1 = 10.9
M2 + V =
(2.21)
k
The transmission matrix is:
b
β=
1.9
1 10
10 100
(2.22)
The proportion of infecteds in each risk class during the slaved phase is:
I1 /I = 0.9/1.9 = 9/19 and
I10 /I = 1/1.9 = 10/19
2
From the definition of V one can calculate easily that
eq. 2.19
23
P
i
(2.23)
(2.24)
i2 ni = M 2 + V which we used to derive
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
B
1e−02
S
I
1e−04
Population fraction
1e+00
A
0
20
40
60
80
100
1e−02
SH
SL
IH
IL
1e−04
Population fraction
1e+00
Years
0
20
40
60
80
100
Years
Figure 2.3: Comparison of timecourses of a homogeneous mixing model and the heterogeneous
mixing model by May and Anderson (1988). (A) homogeneous mixing model (parameters:
b = 0.1 × 5.73/year, a = 0.1/year). (B) heterogeneous mixing model by May and Anderson
(1988) b = 0.1/year, a = 0.1/year, other parameters as in the numerical example.
Finally, the basic reproduction number is:
b M2 + V
b 10.9
b
×
= ×
= × 5.73
(2.25)
a
M
a
1.9
a
In contrast, the basic reproduction number in a homogeneous population with the same
mean number of contacts would be:
b
b
R0homo = × M = × 1.9
(2.26)
a
a
Figure 2.3B shows the timecourse of the SI model eqs. 2.15–2.16 for the parameters
of our example above. The level of susceptible hosts in the low risk class saturates at
approximately 0.4 and that in the high risk class at 0. This asymptotic behaviour of
the heterogeneous mixing differs models with homogeneous risks. In a homogeneous
model the levels of susceptibles would asymptotically coverge towards 0.003, which is
much lower than 0.4. Thus, in models with heterogeneous risk, the asymtotic level of
susceptibles remains higher than in a homogeneous model. The reason for that is that
the infection “burns out” in the class at high risk.
R0 =
24
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
2.1.3 Chlamydia infections in koala’s
As an example of a model with risk structure, we consider chlamydia infections in koala’s
(Phascolarctos cinereus). Koala’s are marsupial herbivores which inhabit the coastal
region of eastern and southern Australia. Chlamydia is an intracellular bacterium which
is mostly sexually transmitted. In koala’s as well as in humans the infection can sterilize
females. The particular strain infecting koala’s is Chlamydophila psittaci (Augustine, J
Appl Ecol 1998).
The heterogeneity in risk structure in this example has two aspects. First, we will
assume that the transmission occurs from one sex to the other, ie from male to female,
and female to male, but never from male to male, or female to female. This is a reasonable
assumption for a sexually transmitted disease in a wildlife species. Second, we assume
that female koalas are slightly more susceptible to infection than male koalas. This is
often the case for sexually transmitted diseases and due to the differences in anatomy
of primary sexual organs of males and females in mammalian species.
The differential equations describing chlamydia infections are:
dXF /dt
dXM /dt
dYF /dt
dYM /dt
=
=
=
=
r(XF + αYF ) − rXF N − βF M XF YM /N
r(XF + αYF ) − rXM N − βM F XM YF /N
+βF M XF YM /N − rYF N − mYF
+βM F XM YF /N − rYM N − mYM
(2.27)
(2.28)
(2.29)
(2.30)
Hereby, r is the logistic growth rate, β the transmission rate constant, α the effect of
infection on fecundity, m the disease-induced mortality, and N the total population size:
N = X F + XM + YF + YM .
The infection terms are divided by the total population size N . This is due to the
assumption that koalas have a constant number of sexual mates per unit time irrespective
of the population size. So, for example, females are infected at a rate βF M XF YM /N .
The term YN /N (which is the frequency of infected males in the population) assures
that in a bigger population the infection rate is not bigger.
The basic reproduction number can be derived as follows. For the sake of simplicity,
let us assume that intially N = 1. Let us set the sex ratio of our koala population to 1:1,
ie XF (0) = XM (0) = 1/2. As always, we derive the basic reproduction number from the
equations describing the infected populations at the very beginning of the epidemic:
dYF /dt = βF M YM /2 − rYF − mYF
dYM /dt = βM F YF /2 − rYM − mYM
(2.31)
(2.32)
In matrix form this equation is:
dY /dt = J Y
where Y =
YF
YM
(2.33)
and the Jacobian matrix, J , is:
J =
−(r + m) βF M /2
βM F /2 −(r + m)
25
(2.34)
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
Figure 2.4: Impact of sterility and infection-induced mortality on the equilibrium population
size of reproducing females XF + αYF . From Keeling & Rohani, 2008, Fig. 3.8.
The dominant eigenvalue of J is:
λ = −(r + m) +
1p
βF M βM F
2
(2.35)
(2.36)
which lead to:
√
R0 =
βF M βM F
2(r + m)
(2.37)
There are two insights we can gain from this basic reproduction number. First, R0
independent of α, the parameter which describes how sterilizing chlamydia infections
are for females. Second, the infection-induced mortality virulence m must not be too
high for chlamydia persistence. This result is very relevant for the evolution of the
host-pathogen system as we discuss in the next chapter.
Lastly, let us look at the impact of sterility and infection-induced mortality on the
equilibrium populaton size of koalas. The equilibria of model 2.27–2.30 are very difficult
to solve analytically. To calculate equilibria, one has to resort to numerical simulations
of eqs. 2.27–2.30 for diffent values of α and m. Figure 2.4 shows the results of such
simulations. For low fertility of infected females the koala population is predicted to
go extinct. The lower the fertility, the lower the population size. The dependence on
the infection-induced mortality m is not as simple. There is a level of m for which the
negative impact is greatest. If m is too large or too small, the population of reproducing
females is less severely affected. This is due to the effect of m on the basic reproduction
number: for large m the infection does not spread properly, and for low m it spreads
but there is not much harm done.
26
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
A
B
C
Figure 2.5: Examples for the spatial spread of infections. (A) plague in Europe 1347-1350,
(B) phocine distemper in North Sea seals in 2002, and (C) Devil Facial Tumor Disease in
Tazmania.
2.2 Spatial heterogeneity
In this section, we turn to another important aspect of epidemics which we have neglected
so far: spatial heterogeneity. Many infectious diseases have a strong spatial component.
Examples are the plague or black death in humans, rabies in red foxes and racoons,
foot-and-mouth disease in cattle, tuberculosis in badgers, phocine distemper virus in
North Sea seals, and facial tumor disease in Tasmanian devils. Also, many diseases of
plants have a strong spatial component Classical example are Dutch elm disease, and
sudden oak death.
The plague in the middle ages is a classical example of an epidemic which spread
through Europe in a wave-like fashion. One plague epidemic started in Italy in December
1347 and spread North-East with a speed of approximately 600 kilometers per year. It
reached the Baltic sea in the end of 1350 (see Figure 2.5A). After the major epidemic
had waned, there were secondary outbreaks some years later, for example, in 1356 in
Germany.
Another classical example is the spread of rabies (“Tollwut”). In the 20th century
there was a epidemic wave going through Europe, starting in Poland 1939 and moving
South-West at a speed of approximately 30–60 kilometers per year. In Europe the most
important host of rabies are red foxes. Although rabies is extinct, or at least under
control in many European countries it remains a problem in America, Asia and Africa.
In the USA, for example, rabies is spread mostly by racoons (“Waschbären”) and causes
human deaths until today. For that reason the mathematical modelling of rabies in
raccons is pursued heavily in the United States.
There are also more recent, less classical examples for epizootics (ie wildlife epidemics)
with a strong spatial component. One is phocine distemper virus which is related to
27
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
measles virus and infects seals (Phoca vitulina). There were two outbreaks in recent
history, one in 1988 and another in 2002. Both of these epidemics started on the Danish
island of Anholt end spread first West and the North with a speed of 2000-3000 kilometers per year (see Fig 2.5B). Another example is Devil Facial Tumor Disease. This
is a curious Disease of Tazmanian Devils, the last marsupial carnivor. The infectious
agents are cancerous Tazmanian Devil cells which are transmitted from devil to devil
by fighting, mostly during the mating season. Usually the transmission of cells from
another individual (called an “allograft” in immunology) is rejected by the immune system. However, due to low population sizes of Tazmanian Devils, their genetic diversity
has shrunken, and their immune systems cannot recognize the tumor cells as foreign
anymore. Due to the high mortality it induces this disease is threatening the already
decimated population of Tazmanian Devils with extinctions. Devil Facial Tumor Disease
spreads with a speed of 5–10 kilometers per year through Tazmania (see Fig 2.5C). This
disease exemplifies how the decimation of a species set off a evolutionary and ecological
feedback that exacerbates the species’ extinction.
In this section, we introduce basic concepts of the spatial modelling of epidemics. The
main focus will be to understand the determinants of the epidemic wave speed.
2.2.1 A simple model in one-dimension
How can we deal with space mathematically?
Formally, there are different ways to deal with spatial aspects. The easiest to understand way are individual-based simulations, in which the spatial information of every
individual is tracked. Such simulations can give a very detailed picture of the spread
of a disease. However, they are not suitable to derive quantities of interest analytically,
such as the wave spead or the basic reproduction number. For that reason we do not
discuss any individual-based models here.
Alternatively, there are so-called metapopulation models. Such models assume that
there are patches in which the epidemics spreads according to a simple model, such as
an SIR model, and that the patches are connected by migration of individuals between
them. Such models are often used in evolutionary ecology and epidemiology. There
are also analytical results which can be obtained from metapopulation fomulations of a
problem.
Lastly, there are diffusion models, which involve partial differential equations. In
such models host are assumed to interact locally and to diffuse in space. The classical
examples of spatially spreading epidemics have been formulated as diffusion models,
that’s why we discuss them here. In addition to classical examples of spatially spreading
epidemics, diffusion models are used often in population genetics. Therefore, it may be
generally useful to get acquainted to such models.
Let us discuss a simple model for the spatial spread of epidemics in one dimension.
For the sake of simplicity, let us consider a disease with no recovery, ie we have only susceptible and infected populations. In diffusion models the variables for the susceptible
and infected populations are not just function of the time, t, but also of their spatial location, x: S(x, t) and I(x, t). We assume a one-dimensional space, and as a consequence
28
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
x is a scalar (ie a real number). For two-dimensional spatial system, the spatial location
would be given by a two-dimensional vector.
A simple model with diffusion is given by the following equations:
∂S
∂ 2S
= −βSI + D 2
∂t
∂x
∂ 2I
∂I
= +βSI − aI + D 2
∂t
∂x
(2.38)
(2.39)
These are so-called partial differential equations, or PDEs for short. The reason for this
name is that we are dealing with function of t and x. We can therefore differentiate these
functions with respect to either t or x. These derivatives are called partial derivatives.
∂
∂
and ∂x
denote partial derivatives with respect to time t and space x,
In eqn 2.38–2.39, ∂t
∂2S
respectively. The term D ∂x2 is a diffusion term (see the Fick’s second law in Imboden
& Koch, 2003: Systemanalyse, p207 eq 8.26). In short, this term assumes assumes
conservation of matter and that the flux is proportional to the difference (gradient)
of the density (Fick’s first law). D is given in units of distance square per time, eg
km2 /year.
Even though eqs. 2.38–2.39 may seem very cryptic to you now, there is a large literature on what kind of behavior such diffusion models can display and how one can obtain
an analytical understanding. Common to diffusion models are so-called travelling wave
solutions. In our system this means that an infection starts somewhere and then travels
away from its point of origin at constant speed.
Knowing that diffusion systems can display travelling wave behavior we make the
following ansatz:
S(x, t) = S(x − ct) = S(z)
I(x, t) = I(x − ct) = I(z)
(2.40)
(2.41)
This sais that S and I depend on x and t in a very special way. The dependence is in
fact such that we can sum it up in one single variable z = x − ct.
How does this describe a travelling wave? Look at Figure 2.6. To hypothetical densities of infected hosts across the spatial dimension x are plotted. The black curve
corresponds to a time t1 and the red curve to a time t2 . If this curve moves at constant
speed c in the direction indicated then the speed is:
c=
x2 − x1
t2 − t1
(2.42)
It follows that:
x1 − ct1 = x2 − ct2 =: z
(2.43)
The variable z sums up space and time into one. Plotting a travelling wave solution
as a function of this composite variable z takes the movement out of the wave, and it
becomes static.
29
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
I t 1 t2
wave direction
x
x1 x 2
Figure 2.6: Travelling wave solution.
Mathematically the ansatz in eqs. 2.40–2.41 is also useful. It convert the partial
differential equations into ordinary differential equations, which are easier to handle.
Substituting the ansatz into our model equations 2.38 and 2.39 we obtain:
− c2 S 0 = −βSI − DS 00
−c2 I 0 = βSI − aI − DI 00
(2.44)
(2.45)
Hereby, the dash denotes the derivative with respect to the variable z. The ordinary
differential equations above are still not trivial but allow to gain analytical insights into,
for example the wave speed.
This brief outline about how to handle PDEs is not meant to be a comprehensive
introduction into the topic of PDEs, or diffusion models. Rather, we introduce as much
as is needed to understand the solutions, which we simply state below. The take home
messages are that diffusion models often have travelling wave solutions, that these solutions, through an ansatz, lead to a simplification of the PDEs. An brief introduction
into the ansatz is also necessary since the solutions are often expressed in terms of the
composite variable z. If you are interested in more of the details of solving PDEs, you
can find them in the book by Murray on Mathematical Biology, 2nd edition, chapter 20,
which I copied for you.
Figure 2.7 shows a numerical solution of the simple spatial SI model in eqs. 2.38 and
2.39. It shows S and I as functions of the composite variable z. It can be read in
two ways. First, we could fix the time t. Then z is basically the spatial dimension x.
In this interpretation the curves in Figure 2.7 show you a snapshot of the travelling
wave solution. You can envisage the wave traveling right in time. Second, we could
fix the spatial dimension x. Then z is negatively proportional to t. According to this
interpretation Figure 2.7 gives you the change of the population densities at a given
location x through time, albeit reverse time, ie the time-axis points left.
It can be derived that the wave speed of the travelling wave solution is:
r
a
)
(2.46)
c ≥ 2 βS0 D(1 −
βS0
30
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
0.8
t=7.5yrs
t=10yrs
0.6
t=5yrs
0.2
0.4
S
I
0.0
Host densities
1.0
A
−600
−400
−200
0
200
400
600
Spatial dimension in km
B
Figure 2.7: Numerical solution of the one-dimensional spatial SI model defined in eqs. 2.38
and 2.39. (A) shows the spatial densities of susceptible and infected hosts for different time
points in the simulation. (B) shows the denisty of infected hosts over space and time. From
this plot you can read of the wave speed of 100km/year. (Parameters in this simulation
were: β = 40/year, D = 100km2 /year, and a = 20/year)
31
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
The wave speed is positive if:
βS0
>1
a
(2.47)
In this spatial SI model the basic reproduction number is defined as:
R0 :=
βS0
a
(2.48)
In its form this is very similar to what we had in the models without space. Interestingly,
in spatial models R0 > 1 is equivalent to the existence of a traveling wave solution.
The condition R0 > 1 can be translated into three critical entities for the existence
of an epidemic wave. First, there is a critical density of susceptible hosts, Sc = a/β,
below which the epidemic will not spread as a travelling wave. Second, there is a critical
transmission rate constant, βc = a/S0 , below which there is no travelling wave solution.
Wildlife epidemics can be stopped by either reducing the density of the susceptible
population, or by treating them with drugs or vaccines. Thereforem the critical density
of susceptible hosts, Sc , and the critical transmission rate constant, βc , is important
to know if one intends to stop an epidemic wave. Lastly, there is a critical mortality,
ac = βS0 , above which there will not be an epidemic wave. Hence, also in spatial models
a pathogen which is too virulent will not spread.
2.2.2 Example: the spatial spread of rabies in the European red fox
population
The classical example of a wildlife disease that spreads geographically is rabies. Rabies
is caused by a virus and infects the central nervous system of its hosts. It is transmitted
by direct contact. Rabies virus can infect many species, among them red foxes, racoons,
skunks, bats, dogs, and rarely humans. In humans it is treatable only during the incubation stage by therapeutic vaccination. There was an epidemic in red foxes in Europe
in the middle of the 20th century that started in Poland 1939 and moved South-West
with a speed of 30–60 kilometers per year.
As the simplest model to describe the rabies epidemic in red foxes we define:
∂S
= −βSI
∂t
∂I
∂ 2I
= +βSI − aI + D 2
∂t
∂x
(2.49)
(2.50)
The only difference of this model and the simple one-dimensional spatial SI model in
eqs. 2.38 and 2.39 is that there is no diffusion term for susceptible hosts. This is due to
the fact that red foxes remain in their territory normally. Only rabid foxes that lost their
sense of orientation will walk around randomly. This random walk is well described by a
∂2I
diffusion term D ∂x
2 . This model ignores the demography of the red foxes. It also ignores
the long incubation period of rabies of 12–150 days between infection and symptoms of
rabidity. Figure 2.8 show a numerical solution of the rabies model in eqs. 2.49 and 2.50.
32
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
Figure 2.8: Numerical solution of the rabies model in eqs. 2.49 and 2.50. You can see the wave
speed is 50km/year. Parameters in this simulation were: β = 40/year, D = 100km2 /year,
and a = 73/year)
33
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
Without a derivation we state the minimal wave speed of an epidemic, which can be
derived from eqs. 2.49 and 2.50:
p
c = 2 D(βS0 − a)
(2.51)
From this we now calculate the basic reproduction number R0 .
Red foxes have a density of approximately 2 individuals per square kilometer. From the
average duration of infection of 5 days the virulence parameter can be estimated as a =
0.2/day = 73/year (see Murray, chapter 20, p668, Table 20.1). The diffusion parameter
has been estimated as D ≈ 100km2 /year 3 Lastly, as we have already stated, the speed
of the epidemic was 30–60 kilometers per year. We will work with c = 50km/year.
From these parameters we can calculate:
β=
c2
4D
+a
= 40km2 /year
S0
(2.53)
(2.54)
and the basic reproduction number:
R0 =
βS0
≈ 1.1
a
(2.55)
4
.
Let us consider the limitations of the simple rabies model in eqs. 2.49 and 2.50. Figure 2.9 shows actual data of the rabies epidemics. We see that in addition to the big
first wave there are subsequent waves of smaller amplitude. These recurring waves are
a very common feature of spatial epidemics. However, they are not predicted by our
simple model. To explain these phenomena the demography of the red foxes has to be
incorporated into the model. Due to the high mortality the infection in induces, it runs
out of susceptible hosts after the first wave. The disease can then not maintain itself on
such a low host population and decreases in prevalence. The fox population can then
recover. The new births serve as fuel for the recurrent flares of the epidemic.
3
Murray (chapter 20, p680) gives a range of 50–330 km2 / year. There are several ways to estimate
diffusion rate constants. The most common is to observe how far rabid foxes get after a certain time.
The distance from the origin is predicted to increase less than linearly with time. More precisely, if
you plot the square of the distance from origin versus the time that the rabid fox has been diffusing,
2
then the relationship should be linear. The diffusion coefficient can be estimated as D = distance
4×time .
As an example, consider you observe one rabid fox for 5 days (= 1/73 year), and after this time
she is 2 kilometers from where she started. Then D can be estimated as:
D=
4
4km2
√
= 73km2 /year
4 × 5 days
(2.52)
The clever reader may have noticed a slight discrepancy between our one-dimensional model rabies
model and the definition of S(0) in the units of individuals per km2 . In a one-dimensional model
the density should be given in individuals per kilometer. But the biological problem is, of course,
two-dimensional rather than one-dimensional, as foxes cannot be confined to a line. It is therefore
useful to think of the problem as foxes in a stripe of 1 kilometer width, diffusing only in one direction.
This way also the definition of our population densities per km2 make sense.
34
CHAPTER 2. ADVANCED MATHEMATICAL EPIDEMIOLOGY
Figure 2.9: Spatial data on the rabies epizootic. This is Figure 20.5 from Murray, Mathematcail
Biology, 2nd ed.
35
3 Stochasticity in epidemics:
emergence, persistence, and
extinction
In this chapter, we are introducing basic concepts of stochastic processes. Stochastic
processes describe the interaction of populations allowing for randomness in the events
that occur. The disturbance stochasticity causes to a system is usually largest when
the population sizes are small. Interestingly, allowing for such randomness does not
destroy the predictability of the system completely. Thus, stochasticity is distinctly
different from deterministic chaos, which is found in the growth dynamics described by
the logistic equation in discrete time.
infecteds
persistence
extinction
time
emergence
Figure 3.1: Epidemics and stochaticity. The figure show the different phases during an epidemic/endemic when stochastic extinction may occur.
In mathematical epidemiology, stochasticity plays an important role during the initial
stages of an epidemic. During that phase, population sizes are small and an infection
may die out by chance. Once an epidemic has occured and the endemic equilibrium has
been attained, extinction of the infection may also occur. This is often referred to as
stochastic fade-out. (Look at Figure 3.1).
What is a stochastic process? How can we formalize it mathematically?
All of the models we have been dealing with can be formulated as stochastic processes.
In the SIR model, for example, we had a model that described how the variables S, I,
and R changed over time. Time was a continuous (real) number, and the populations
were also measure by a continous, positive number. In stochastic process theory, the
36
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
Table 3.1: Examples for stochastic processes with real or integer index and state spaces.
index space
integer
state space
integer Galton-Watson
branching process &
Markov chains
real
Fisher’s
geometric
model
real
continuous-time
Markov
process;
stochastic SIR model
Brownian motion
range of the independent variable (time) is referred to as the index space. The range
of the dependent variables (population sizes) are called the state space. Table 3 show
examples of stochastic processes with different choices of index and state space.
3.1 Galton-Watson branching process
As an example of a stochastic process, let us consider the Galton-Watson branching
process. This is one of the simplest stochastic processes and has been used widely in
population biology, population genetics, and to model thermo-nuclear reactions. 1 It
is fairly straight-forward to calculate the extinction probability in this process, which
makes it very useful to study the emergence of infections in population (emergence being
the opposite of extinction).
t=0
t=1
t=2
t=3
t=4
t=5
t=6
X=1
X=1
X=2
X=4
X=5
X=3
X=4
Figure 3.2: Galton-Watson branching process. Integer index space (time/generations), integer
state space (population size). Individuals are not interacting.
1
Interestingly, the problem for which this process has been invented does not originate from any of
these fields. Galton and Watson were interested in the extinction of family names. At there time,
there was the hypothesis that families of famous men went extinct due to a loss in fertility. Galton
and Watson tried to calculate the average rate at which family names would go extinct (irrespective
of whether they were headed by famous men or not).
37
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
0.2
0.0
0.1
Probability
0.3
0.4
Distribution of secondary cases: R0 = 1.5
0
1
2
3
4
5
6
7
8
9
10
Number of secondary cases
Figure 3.3: An example for a distribution of secondary cases. This is a geometric distribution
with mean R0 = 1.5. For this distribution the probability of causing i secondary infections
R0i
is given by pi = (R0 +1)
i+1 .
The Galton-Watson branching process is the stochastic version of geometric growth. 2
Figure 3.2 shows a sketch of this process. It assumes that there is no interaction between
individuals, and that each individual is randomly assigned offspring, according to some
offspring distribution (see Figure 3.3 for an example). In the context of epidemics, we
are going to speak about secondary infections, rather than offspring. We are going to
use this process to describe the intial phase of an epidemic when susceptibles are still
abundant.
3.1.1 A specific example
As an example process, let us consider the process with a geometrically distributed
number of secondary infections (see Figure 3.3). 3 In this distribution, the probability
2
Geometric growth is a discrete-time growth model. If you, for example, assume every individual has
2 offspring and dies at birth then, starting with one individual, you get the following sequence of
population sizes: 1, 2, 4, 8, 16, 32, ...
3
A geometric distribution has nothing to do with the geometric growth model. A geometric distribution
is the distribution you get when you toss a coin until you get “head”. If pH is the probability of
a coin toss resulting in “head” then the probability that you need i tosses before you toss head
H
is pi = (1 − pH )i pH , i = 0, 1, 2, . . . . The mean of this distribution is 1−p
pH . If, in the context of
38
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
Ri
0
of causing i secondary infections is given by pi = (R0 +1)
i+1 . Hereby, R0 is the average
number of secondary infections an individual causes. Calling this R0 is justified since
we are modelling the initial phases of the infection.
If we simulate this process, we observe that a high fraction of processes die out even
if the average number of secondary infections is larger than one. For example, of the
100 simulations in Figure 3.4, which were started with one infected individual and an
average of 1.5 secondary infections, 66 went extinct.
15
10
5
0
Infected population size
20
66 out of 100 epidemics go extinct
0
5
10
15
Generations
Figure 3.4: Simulation of the Galton-Watson branching process. For these simulations we
assumed that initially there is one infected individual. The number of secondary infections
each individual causes is drawn from a geometric distribution with mean R0 = 1.5. The
thick solid lines show simulations that went extinct, the thin grey lines show simulations
that did not go extinct.
The finding that epidemics die out even if the average number of secondary infections,
R0 , is larger than one is one of the most important differences between deterministic and
stochastic approaches. While in a deterministic model we always predicted an epidemic
of R0 > 1, in a stochastic model this is not true (see Figure 3.5). One can calculate
the probability of emergence for the geometric distribution of secondary infections as
1 − 1/R0 . If R0 < 1, both deterministic and stochastic models both predict extinction.
4
Ri
0
infections, you set this mean to R0 you get: pi = (R0 +1)
i+1 .
4
Note, however, that in the dertministic models we discussed in previous chapters, extinction never
39
1.0
0.4
0.6
0.8
deterministic
0.2
GW branching process
0.0
Probability of emergence
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
0
1
2
3
4
5
Basic reproduction number R0
Figure 3.5: Probability of emergence versus the number of secondary infections, R0 . While
in a determinisic model an infectious disease emerges with certainty, in the Galton-Watson
branching process with geometric number of secondary cases the probability of emergence
is 1 − 1/R0 .
3.1.2 Analytics
The Galton-Watson branching process is useful because the extinction probability can
be derived analytically. Essential for the solution of this problem is the construct of the
probability generating function. The probability generating function is a way to describe
a probability distribution.
Assume we have a probability distribution given by the probabilities pj , where j =
0, 1, 2, 3, . . . . The probability generating function for this distribution is defined as:
f (s) =
∞
X
p j sj ,
0≤s≤1
(3.1)
j=0
The variable s has no meaning at all. It is called a dummy variable.
Why is this construct useful?
• You can extract every probability pi from f (s). For example, f (0) = p0 , f 0 (0) = p1
and f 00 (0) = p2 .
really occured. The number/fraction of infected hosts in these models simply decreased exponentially
to zero. The population size 0 was never reached in finite time.
40
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
• You can extract P
every moment of the probability
distribution from f (s). For
P∞
∞
0
example, f (1) = j=0 pj = 1 and f (1) = j=0 jpj = mean.
• The sum of two independent random variables, one described by f (s), the other
described by g(s) is distributed according to a probability distribution defined by
the product f (s) × g(s).
Beyond the fact that the probability generating function is a concise way to mathematically express a probability distribution, it allows us to calculate the extinction probability for a Galton-Watson branching process. One can derive that, for the Galton-Watson
branching process, the probability generating function for the distribution of cases in
the mth generation is:
f m (s)
(3.2)
Read this as “function f applied m times to s” 5
Let us illustrate this result by considering a specific distribution: the geometric distribution of secondary cases. For this distribution, the probability generating function
is:
f (s) =
∞
X
j=0
=
5
R0j
sj
j+1
(R0 + 1)
(3.8)
1
1 + R0 (1 − s)
(3.9)
This can be derived as follows. Define the single-generation transition probabilities P (i, j) as the
probability that i infected individuals give rise to j secondary infections. Obviously, P (1, j) = pj .
Let us denote the transition probabilities across two generations as P2 (i, j) (this is the probability
that i infections give rise to j tertiary infections). The probability generating function of the number
of cases in the second disease generation in the Galton-Watson branching process is then:
∞
X
P2 (1, j)sj
=
j=0
=
=
∞ X
∞
X
j=0 k=0
∞
X
P (1, k)P (k, j)sj
P (1, k)
k=0
∞
X
∞
X
P (k, j)sj
(3.3)
(3.4)
j=0
k
P (1, k) [f (s)]
(3.5)
k=0
= f (f (s))
(3.6)
(3.7)
To go from the second to the third line in this derivation, we need to use the result that the probability
generating function of a Galton-Watson branching process started with k infected individuals is
k
[f (s)] . This follows from the general result that the probability generating function of the sum
of two independent random variables is the product of the probability generating function of each
individual variable.
The general result for the mth generation follows by induction.
41
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
1
The extinction probability in the first generation is simply f (0) = 1+R
. The probabil0
ity generating function of the distribution of tertiary cases, i.e. in the second disease
generation, is:
f 2 (s) =
1
1 + R0 (1 − 1+R01(1−s) )
(3.10)
The extinction probability in the second disease generation is then simply:
f 2 (0) =
1
1 + R0 (1 −
1
)
1+R0
=
R02
R0 + 1
+ R0 + 1
(3.11)
In this simple case, we can check the result by calculating the probability of extinction
in the second disease generation from the probabilities pi . Extinction in the second
generation can arise because the disease becomes extinct already in the first generation.
1
The probability of this event is p0 = 1+R
. Alternatively, the number of infected hosts
0
could be i in the second generation, and none of these individuals passed on the disease
to anybody. The probability of this event is pi pi0 . Summing these two probabilities, we
get:
p0 +
∞
X
i=1
pi pi0 =
∞
X
pi pi0 = f (p0 ) = f (f (0))
(3.12)
i=0
The probability of ultimate extinction, q can be defined by considering infinitely many
generations:
lim f m (0) =: q
m→∞
(3.13)
It immediately follows that this extinction probability q is the smallest fixed point of
the probability generating function: 6
f (q) = q
(3.14)
Figure 3.6 illustrates this result geometrically. To calculate the probability of extinction, you can simply iterate the probability generating function f . You start with the
probability that the process goes extinct in the first generation p0 = f (0). The ultimate
probability of extinction must be larger than that. Then we apply f to p0 . This brings
us a little higher. This is the probability that the process goes extinct in the second
disease generation. And so on. This way you converge to q, the probability of ultimate
extinction.
There is one more insight we can derive from the Figure. You see that there is a fix
point lower than 1 only if f is steeper at 1 than the diagonal. Mathematically, we can
express this as the condition:
f 0 (1) > 1
6
This is because f (q) = f (limm→∞ f m (0)) = limm→∞ f m+1 (0) = limm→∞ f m (0) = q.
42
(3.15)
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
1
f
●
f32(0)
f (0)
f1(0)
0
f1(0) f2(0)
q
0
1
s
Figure 3.6: Geometry of the relation between probability generating function and extinction
probability
Interestingly, f 0 (1) is the mean number of secondary cases, i.e. the basic reproduction
number R0 . This means that the process does not go extinct with certainty only if
R0 > 1.
As a specific example, let us again consider the geometric distribution of secondary
cases. In this case, we can calculate the probability of extinction as:
q=
1
1
⇐⇒ q =
1 + R0 (1 − q)
R0
(3.16)
Thus, for a disease with R0 = 2 there is a 50% chance that the process goes extinct
(compare Figure 3.5).
3.1.3 The distribution of secondary cases for various infections
To illustrate the use of the Galton-Watson branching process in the context of infectious
diseases, let us discuss a paper by Lloyd-Smith et al, Nature 2005. In this paper, the
authors assembled data on the number of secondary cases individuals, who were infected
with various pathogens, had caused.
In Figure 1a in the paper by Lloyd-Smith et al (see fig 3.7a), the authors show the
distribution for SARS. As you can see the distribution is very wide. Most infected
people did not infect anybody else, but a few individuals spread the virus to more than
20 others. These individuals represent the superspreaders that were talked about a lot
when the SARS epidemic was discussed in the media.
43
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
Figure 3.7: Figure 1 from Lloyd-Smith et al Nature 2005. Their legend reads: “Evidence
for variation in individual reproductive number ν. a, Transmission data from the SARS
outbreak in Singapore in 2003 (ref. 5). Bars show observed frequency of Z, the number of individuals infected by each case. Lines show maximum-likelihood fits for Z Poisson
(squares), Z geometric (triangles), and Z negative binomial (circles). Inset, probability density function (solid) and cumulative distribution function (dashed) for gamma-distributed ν
(corresponding to Z negative binomial) estimated from Singapore SARS data. b, Expected
proportion of all transmission due to a given proportion of infectious cases, where cases are
ranked by infectiousness. For a homogeneous population (all ν = R0 ), this relation is linear.
For five directly transmitted infections (based on k values in Supplementary Table 1), the
line is concave owing to variation in ν. c, Proportion of transmission expected from the
most infectious 20% of cases, for 10 outbreak or surveillance data sets (triangles). Dashed
lines show proportions expected under the 20/80 rule (top) and homogeneity (bottom). Superscript v indicates a partially vaccinated population. d, Reported superspreading events
(SSEs; diamonds) relative to estimated reproductive number R (squares) for twelve directly
transmitted infections. Lines show 5–95 percentile range of Z, Poisson(R), and crosses show
the 99th-percentile proposed as threshold for SSEs. Stars represent SSEs caused by more
than one source case. ’Other’ diseases are: 1, Streptococcus group A; 2, Lassa fever; 3,
Mycoplasma pneumonia; 4, pneumonic plague; 5, tuberculosis. R is not shown for ‘other’
diseases, and is off-scale for monkeypox. See Supplementary Notes for details.”
44
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
Figure 3.8: Figure 2 a and b from Lloyd-Smith et al Nature 2005. Their legend reads: “Outbreak dynamics with different degrees of individual variation in infectiousness. a, The
individual reproductive number ν is drawn from a gamma distribution with mean R0 and
dispersion parameter k. Probability density functions are shown for six gamma distributions
with R0 = 1.5 (’k = Inf’ indicates k −→ ∞). b, Probability of stochastic extinction of an
outbreak, q, versus population-average reproductive number, R0 , following introduction of a
single infected individual. The value of k increases from top to bottom (values and colours
as in a). “
Using similar data for many other diseases, the authors estimated the dispersion parameter k which described how homogeneously the virus was spread by different people.
The case k = 0 described a scenario in which every infected individual spread the disease
at the same rate. In this case, the distribution of secondary cases would be a Poisson
distribution with mean ν (the “individual R0 ”). Values of k > 0 described a heterogeneous host population, in which each infected individual i transmitted on average to νi
individuals.
In Figure 2 in the paper by Lloyd-Smith et al (see fig 3.8), the authors show how
different distributions of secondary cases influence the probability of emergence. Fig 3.8a
shows different assumptions about the dispersion parameter k. Fig 3.8b shows the
probability of extinction for the different values for k in Fig 3.8a. The take-home message
here is that variation in spread between individuals increases the probability of extinction
(and therefore reduces the probability of emergence).
You could put this result provocatively as “superspreaders are good because they
make the probability of emergence of a disease less likely”. However, this is only true
if you compare the disease with superspreaders to a one with homogeneous spread and
with the same R0 . If you could simply remove superspreaders from a host population it
would always be good because the resulting reduction of the mean transmission usually
outweighs the negative effects of homogeneous spread on emergence.
45
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
3.2 Stochastic SIR model
The Galton-Watson process is a very simple stochastic model which ignores densitydependent effects, such as slowing growth rates as the population size increases, and
interactions between populations, such as predator-prey interactions or the mass-action
infection term we encountered in the SIR model.
As an example of a more complex stochastic model, we now discuss the stochastic
version of the SIR model. In the first chapter, we introduced the deterministic SIR
model, which was given by the following set of ordinary differental equations (1.1-1.3):
dS/dt = B(S, I, R) − δS − βSI + qR
dI/dt = βSI − aI − rI
dR/dt = rI − δR − qR
In the stochastic version of this model, we assume that the state space is integer
valued, i.e. that S, I, and R can only take values 0, 1, 2, 3, . . . . A certain state (S, I, R)
may then change if an individual is born, dies, becomes infected, recovers, or looses
her/his immunity. All these events occur at the rates that we have in the deterministic
SIR model equations (1.1-1.3). The following table list all possible events and the rates
at which they occur:
B(S,I,R)
S
−−−−−→
S
−−−→
S+1
δS
S
I
I
I
R
R
S
R
βSI
S−1
−−−→
aI
−−−→
rI
−−−→
δR
−−−→
qR
−−−→
(birth)
(death S)
S−1
I +1
I −1
I −1
R+1
R−1
S+1
R−1
(infection)
(death I)
(3.17)
(recovery)
(death R)
(immunity loss)
This stochastic process is too complicated to be understood analytically. It can only
be simulated. But how do we simulate this process? Let us assume that the system is in
the state (S = 99, I = 1, R = 0). Then, with the parameters given in Figure 3.9 we have,
B(S, I, R) = δ(S + R) + aI = 99/(1000d) + 1/(10d) ≈ 0.2/d, δS = 99/(1000d) ≈ 0.1/d,
βSI = 99/(100d) ≈ 1/d, aI = 0.1/d, rI = 0.3/d, δR = 0, qR = 0. This means, the
most likely process to occur is infection. But also birth, death of a susceptible host, or
death or recovery of the infected hosts may occur. Since there are no recovered hosts,
the events related to recovered hosts cannot occur.
To simulate one step of this process on the computer, we need to determine two things:
the time at which the next event happens, and which process (birth, death, etc) it is.
Let us call the process rates a1 , a2 , . . . , a7 . To calculate the time of the next event, we
46
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
B
80
60
40
20
0
Population size
100
A
0
5
10
15
20
25
Days
Figure 3.9: A few runs of the stochastic SIR model with N = 100 (A) and N = 1000 (B). The
simulation was performed using the Gillespie algorithm. Parameters for this simulations
were. δ = 10−4 /d, β = 0.01/d, a = 0.1/d, r = 0.3/d, q = 0.01/d
P
draw an exponentially distributed number with mean 1/ 7i=1 ai (in our example, this
mean would be 1d/1.7). 7 The second step is to calculate which event occurs. To that
end, we draw a random number
P7 from 8a uniformous distribution over the interval 0 to
the sum of all process rates, i=1 ai . If this random number is between 0 to a1 then
we choose the first process. If the random number is between a1 to a2 then we choose
the second process. An so on. This algorithm to simulate the stochastic process is called
Gillespie algorithm.
Figure 3.9 shows such simulations of the stochastic SIR model with a population size
of N = 100 and N = 1000. What we see is that there is quite a lot of variation between
individual runs of these models. This is despite the fact that the parameters and initial
values of the variables are identical across runs. What we also see is that the variation
is much larger if the population size is smaller. This is a general result for stochastic
processes.
Since every run is different one can do a lot of statistics with the output of this
stochastic simulation. I have plotted the distribution of outbreak sizes and durations
in Figure 3.10 for the simulation with N = 100. You can see that there are a lot of
instances in which the outbreak goes extinct almost immediately. Once the population
of infected host is large enough such that the infection evaded early extinction, an
outbreak occurs which infects between 100 and 150 hosts in total 9 and lasts between
7
Computers can draw random numbers. It is actually quite difficult to programm a computer to draw
a really random number, but since this is an important task, many implementations of random
number generators exist. The language R for statistical computing, for example, has a large variety
of reliable random number generators. To draw a exponentially distributed random number with
mean 1/1.7 in R you can issue the command: rexp(1,rate=1.7).
8
To draw a uniformously distributed random number between 0 and 1.7 in R you can issue the
command: runif(1, min=0, max=1.7).
9
More than 100 hosts can be infected in total because hosts are born during the time of the epidemic.
47
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
B
100
50
Frequency
150
0
0
50
Frequency
250
150
A
0
50
100
150
0
Outbreak size
10
20
30
40
50
60
70
Duration of outbreak
Figure 3.10: Statistics on the stochastic SIR model with N = 100. (A) shows the distribution
of outbreak sizes. (B) shows the distribution of the outbreak durations. For this Figure I
performed 1000 simulations. Parameters for this simulations were the same as in Fig 3.9.
15 to 40 days. Interestingly, in these simulations with N = 100 the epidemics comes to
a halt eventually. This behaviour is contrary to that of the deterministic SIR model, in
which the infection did not go extinct of R0 > 1 but attained the endemic equilibrium.
This phenomenon is observed empirically and is termed stochastic fade-out.
3.3 Critical community size
The phenomenon of fade-outs in stochastic epidemiological models leads to the concept
of the critical community size. The critical community size is defined as the population
size necessary to persistently maintain an infectious disease. This concept was introduced
into mathematical epidemiology by Bartlett in 1957. For measles this size is between
300’000 to 500’000 inhabitants (see Figure 3.11(a)).
Bartlett divided the dynamics into three types. Type I dynamics display no fade-outs
because the population size is large, such as in London with its many million inhabitants
(see Figure 3.11(b)). Type II dynamics is characterized by occasional fade-outs. But
these fade-outs are rare, and the epidemics still occur regularly (see Figure 3.11(c)).
Lastly, Type III dynamics display very frequent fade-outs — so frequent that the resulting dynamics looses any regularity (see Figure 3.11(d)).
A corrollary of the fairly high critical community size of measles is that a disease like
measles could not have been maintained in the human population before the agricultural
revolution about 5000 to 6000 years ago, because population sizes before that were too
low. Interesting in this context is that monkeys, while susceptible to measles, do not
contract this disease in their natural habitat. The tribal structure of monkey populations is thought to protect them from measles (Black, 1966 J Theor Biol ). Francis Black
48
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
REVIEWS
3
cases of the disease - three weeks
was chosen to allow for under-reporting of cases. Figure I shows the
average number of fade-outs of
measles per year for 60 communities in England and Wales before
vaccination (1946-1968). The
CCS is clearly -300 000-500 000:
data from American cities and
isolated islands demonstrate a
similar value for the CCS. However, despite the accuracy of deterministic models at predicting the
large-scale dynamics of epidemics s
these models cannot directly address the stochastic question of
persistence 6,v. The CCS provides
a natural measure of stochastic
persistence for any community.
•
•
"b'"
0
1 O0
200
300
400
500
600
700
800
I
I
900
1000
Population size (thousands)
(b)
1945
1947
1949
The basic models
Year
One of the simplest models commonly used for simulating measles
dynamics is the SEIR model. This
is a compartmental model and
splits the population according to
whether individuals are susceptible, exposed, infectious or have
recovered from the infection2,~,8,
(Fig. 2a). In practice, a more com1945
1947
1949
plex formulation incorporating
age structure and allowing for the
Year
seasonal aggregation of children
(d) ~ . . . . . . . . . .
.
.
.
.
.
.
in school is used to model the detailed patterns s,~°. This RAS (realistic age-structured) model captures the overall biennial pattern
of epidemics and the input of vaccination extremely well. However
when this model is made stochastic
- so that each event (e.g. birth
1945
1947
1949
death or transmission of infection
Year
occurs randomly - frequent fade
Fig. 1. (a) The average number of fade-outs (local extinctions) of measles per year against population
outs in the inter-epidemic troughs
size for 60 communities in England and Wales. Parts (b)-(d} demonstrate the three levels of persistare seen and the CCS for the mode
ence and dynamics as defined by Bartlett4. (b) Type I dynamics (e.g. London, population 3.4 million)
is found to be significantly large
are regular: measles is endemic and does not fade-out. (¢) With type II dynamics (e.g. Nottingham,
Figure 3.11: Critical
community
size
of
measles
epidemics.
Fig
1
from
Keeling,
population 300 000) there are fade-outs in the troughs - represented by black dots - but the epithan Trends
that observed 7. Thus, despite
demics
are
regular.
(d)
Type
III
dynamics
(e.g.
Teignmouth,
population
11000)
display
irregular
epiMicrobiol 1997. His legend reads: “(a) The average number of fade-outs (local extinctions)
the overall realism of the model
demics with long fade-outs between them.
it still Parts
overestimates the CCS by
of measles per year against population size for 60 communities in England and Wales.
an order of magnitude 7'1°.
(b)-(d) demonstratewell
theas three
levels
of persistence
and the CCS
for the
model
and
Three main
processes
have
recently been proposed
the contact
and infection
rates. For seen
small poputo deal
with this problem.
Each acts :as an extension to
there is4 .a high
probability
that the disease
dynamics as defined lations,
by Bartlett
(b) Type
I dynamics
(e.g. will
London,
population
3.4 million)
the standard models, with the ultimate goal of predict
be eradicated
by chance,
as the population
is found to be significantly
larger
are and
regular:
measlesincreases
is endemic
and does not fade-out.
ing the remarkably high persistence of measles. These
so does the persistence. Both the persistence and inva(c) With type II dynamics
(e.g.areNottingham,
300 000)
fade-outs in
ideas there
all rely are
on incorporating
greater biological real
sion thresholds
important if we population
are to fully underterms of heterogeneity
at different scales. Thi
stand vaccination.
the troughs — represented
by black dots — but the epidemics ism,
are inregular.
(d) Type III
need to examine processes at a variety of scales paral
The persistence of measles is usually addressed in
dynamics (e.g. Teignmouth,
population 11000) display irregular epidemics with long fadeterms of the critical community size (CCS), which is lels recent work in theoretical ecology T M (Fig. 2).
outs between them.”the smallest population without any extinctions (a temLarge-scale heterogeneities and metapopulations
porary absence of the disease). In his seminal work on
In large communities, there is never complete mixin
the CCS, Bartlett 4 defined a fade-out (local extinction)
between all individuals. This leads to variations in th
as three or more consecutive weeks with no reported
(c)
TRENDS
IN MICROBI()LOGY
49
514
VOL.
5
NO.
12
DECEMBER
195)7
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
also argued that infectious diseases that have not evolved in the Americas have played
an important role during the invasion of that continent by Europeans. The invaders imported various very transmissible infectious agents that wiped out entire communities of
native Americans because they had no pre-existing immunity to these infectious agents.
3.4 Example: Porcine reproductive and repiratory
syndrome
As an illustration of these general principles let us consider porcine reproductive and
repiratory syndrome. 10 This is a disease of pigs caused by positive single-stranded RNA
virus. There were outbreaks on pig farms in the US 1986, and the Netherlands & UK
1991 The disease is endemic since then. The virus causes reproductive problems in sows
and respiratory problems in piglets, hence the name of this syndrome. 11 Unlike footand-mouth disease, PRRS spreads slowly within farms, and reaches only low prevalence
The basic reproduction number of PRRS has been estimated as R0 ≈ 3.
Nodelijk et al (2000) described the disease within a farm by the following SIR model
with waning immunity (this is equation 6.5 in Keeling and Rohani, 2008):
dX/dt = µN + wZ − µX − βXY
dY /dt = βXY − γY − µY
dZ/dt = γY − wZ − µZ
(3.18)
(3.19)
(3.20)
Here, X, Y , and Z refer to the number of susceptible, infected and recovered pigs
respectively. The parameter µ denotes replacement rate constant of lifestock, w the
immunity loss rate constant, β the transmission rate constant, and γ the recovery rate
constant.
Figure 3.12 shows statistics from simulations. You can see that PRRS outbreak size
and duration are predicted to be very variable. However, when sampling across many
farms one should see more stochastic fade-outs in small farms than in big farms. This
prediction is consistent with empirical observation (Evans et al, BMC Vet Res 2008).
10
11
This material is taken from the book by Keeling & Rohani, 2008, pp214.
It is also refered to mystery swine disease. The German name of this disease is Seuchenhafter
Spätabort.
50
CHAPTER 3. STOCHASTICITY IN EPIDEMICS
Figure 3.12: Fig 6.10 from Keeling and Rohani, 2008. Their legend reads: “The simulated
behaviour of PRRS epidemics with event-driven (demographic) stochasticity. The top lefthand graph shows the distribution of the total number of cases, including re-infections from
an introduction of 3 infected animals in a farm with 115 sows. The top right-hand graph
shows the duration for a similar scenario. Results are from 100,000 simulations. The lower
graphs consider how these quantities change with the number of animals on the farm; both
show an exponential increase with the number of sows. (R0 = 3, γ = 6.517 per year,
µ = 0.6257 per year, and w = 0.2607 per year). Shaded regions give the 90% confidence
intervals associated with a single epidemic.”
51
4 Evolution of virulence
4.1 Why should parasites be harmful?
Infectious diseases vary greatly in the effect on their hosts. While some infectious
pathogens (such as Ebola virus, HIV-1, avian influenza, and many others) induce very
high host mortality rates, many other infectious pathogens cause mild diseases or even
go completely unnoticed. Moreover, some pathogens can cause death a few days after infection, whereas others only cause death after years of infection. The pathogenic effects,
however, do not only differ greatly between different pathogen species, but can also vary
substantially between different strains of a single pathogen species. Since there is likely
a heritable component to variation in parasite virulence, it is a challenge to evolutionary
biologists to establish the factors which determine the evolution of parasite virulence.
In this chapter, we will employ population biological methods as a useful tool to address
how different factors may affect the evolution of parasite virulence. Moreover, we will
discuss some key experimental findings.
Until recently, medical doctors and biologists where brought up on the notion that
in the long term parasites1 should evolve to be harmless to their hosts. This so called
conventional view is based on the notion, that parasites typically require a living host
for their transmission. Since the death of the host usually implies the end of a parasite’s
opportunities for transmission, it was argued that, evolutionarily speaking, it is in the
parasite’s interest to keep its host alive. Harmful parasites were thus presumed to be
recent parasites that have not yet evolved to avirulence.
Evolutionary biologists have questioned the conventional view both on theoretical and
on experimental grounds. First, there are examples of diseases that are known to have a
long association with their hosts, but have remained virulent until today. It is believed,
for example, that measles has become endemic in the human populations around 10,000
years, when the widespread use of agriculture lead to increased population densities and
larger communities2 . Measles is transmitted exclusively from humans to humans and
despite 10,000 years of coevolution with its host, measles has remained highly virulent.
Another striking example is that fig wasps together with their parasitic nematodes have
been found in million year old amber. This fig wasp-nematode association is known to
be highly virulent today. Hence, either virulence has not declined over a million years
1
We use the term parasite here in a broad sense to include viruses, bacteria, protists and macroparasites.
2
Comment: We have learned in the previous chapter that the basic reproduction number increases
with the (susceptible) population density. It is believed that measles was unable to persist in the
human population prior to the population increase associated with the agricultural revolution.
52
CHAPTER 4. EVOLUTION OF VIRULENCE
1965
1950
1955
1960
Year
1970
1975
Virulence of myxoma virus
1
2
3
4
5
Virulence grade
Figure 4.1: Evolution of virulence of myxoma virus after its introduction into the population
of wild rabbits in Australia. Virulence grade 1 is the highest and grade 5 is the lowest. The
proportion of individuals in a given virulence grade are indicated by increasing grey levels.
Thus virulence decreased rapidly over the first few years, but then remained stably at an
intermediate level after 1964. Thus contrary to the conventional view, the virus did not
evolve to avirulence. (Source: Fenner, Proc. Roy . Soc., 1983)
of evolution, or alternatively virulence must have increased over time. Both types of
explanation are in contrast to the conventional view.
One of the most widely known experimental studies of virulence was done in the
1950’s by the introduction of a highly virulent strain of myxoma virus (a pox virus)
into the Australian rabbit population as a means to control the population explosion
of rabbits (see figure 4.1). In the first years after the introduction the virulence of
myxoma virus declined rapidly and eventually settled at an intermediate level rather
than evolving towards zero. Thus this study provides strong experimental evidence
against the conventional view3 .
Before progressing further it is necessary to provide a concise definition of the term
virulence. In evolutionary biology the virulence of a parasite is defined as the fitness costs
to the hosts that are induced by the parasite4 . Parasites can affect host fitness in various
ways such as increasing mortality, increasing morbidity, reducing host reproductivity5 or
3
The decrease in virulence could also be explained by an increase in resistance in the rabbit population.
But also this interpretation of the data argues against the conventional view.
4
Note that in other fields such as microbiology and plant pathology the term virulence is used to refer
to the ability of a parasite to infect a certain tissue or a certain host.
5
Jukka Jokela, for example, works on trematode parasites that infect the fresh water snail Potamopyrgus antipodarum. These parasites consume the reproductive tissue of their hosts and therefore
sterilize the host.
53
CHAPTER 4. EVOLUTION OF VIRULENCE
by modulating host behavior6 . In what follows we will use the term virulence to denote
the parasite induced host mortality unless stated otherwise.
4.2 Maximization of the basic reproduction number
Consider two strains of a parasite competing for transmission in host population as
described by the following model
dS/dt
dI1 /dt
dI2 /dt
dR1 /dt
dR2 /dt
=
=
=
=
=
λ − δS − (b1 I1 + b2 I2 )S + q1 R1 + q2 R2
b1 SI1 − (δ + v1 + r1 )I1
b2 SI2 − (δ + v2 + r2 )I2
r1 I − δR1 − q1 R1
r2 I − δR2 − q2 R2
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
This is a straightforward extension of the SIR model (eqs. 1.1 - 1.3) discussed in the
previous chapter. Here, I1 and I2 denote the densities of hosts infected with parasite
strain 1 and 2, respectively. We assume that susceptible hosts, S, immigrate at a
constant rate λ and die with a natural death rate δ as this represents the simplest
dynamical equation that leads to a stable density of susceptible hosts (S = λ/δ) in
absence of infection (i.e. I1 = I2 = 0). This assumption is made for the sake of
mathematical simplicity, but has no consequence for the results derived further below.
The parameters b1 and b2 describe the infectivities of strain 1 and 2, respectively. In the
SIR model (eqs. 1.1 - 1.3) we used the parameter a to describe the combined mortality of
infected hosts arising from natural causes and disease-associated causes of death. Here,
we explicitly account for the virulences v1 and v2 , that describe the mortality rates
associated with infection by parasite strain 1 or 2.
In the previous chapter we derived the basic reproduction number in the context of
the SIR model. Substituting N = S = λ/δ for the density of susceptible individuals in
the absence of infection into eq. ??. We thus obtain for the basic reproduction numbers
of the two parasite strains
(1)
R0 =
λb1
δ(δ + v1 + r1 )
and
(2)
R0 =
λb2
δ(δ + v2 + r2 )
Let us assume first that only parasite strain 1 circulates in the population. Provided
> 1 the equilibrium density of susceptible cells is the given by
R10
S? =
δ + v1 + r1
b1
(see eq. 1.7). Consider now that a small amount of hosts infected with parasite strain
2 are placed into the population (in which parasite strain 1 is already at equilibrium).
6
Helminths often manipulate the behavior of their intermediate hosts to make them more susceptible
to predation and thereby to increase their own transmission to the definitive host.
54
CHAPTER 4. EVOLUTION OF VIRULENCE
The population of hosts infected with strain 2 can increase if
dI2 /dt > 0
⇒
b2
1
> ?
δ + v2 + r2
S
⇒
(2)
(1)
R0 > R0
Hence, strain 2 can invade into the host population infected by strain 1, if the basic
reproduction number of strain 2 is larger than that of strain 1. It is obvious that the
converse is also true, i.e. that strain 1 can only invade strain 2, if its basic reproductive
rate is larger than that of strain 2. Taken together, we can thus say that the parasite with
the larger basic reproduction number can invade and replace the other strain. The model
does not allow for coexistence of both strains in equilibrium. More generally, we have
thus derived that the parasite with maximal basic reproduction number outcompetes all
competitors. Hence, natural selection is expected to maximise the basic reproduction
number.
4.3 Trade-offs between infectivity and virulence
If there are no constraints that couple virulence to infectivity rate or recovery rate,
we expect natural selection to lead to increasing rates of infectivity (b) and decreasing
virulence (v) and decreasing recovery rate (v) since this simultaneously maximises the
duration of the infection and the transmission per contact with a susceptible hosts. Thus
this simple model shows that we expect evolution towards avirulence as postulated by
the conventional view, if there are no constraints between the parameters. In general
terms, however, it is likely that these parameters are intrinsically coupled. For example,
both the infectivity rate per contact and the virulence may increase with increasing
parasite load. Similarly, slow recovery may require escape from the immune responses
and thus may be related to increasing virulence.
To illustrate this situation let us consider different functional forms that may represent
possible trade-offs between the infectivity parameter b and the virulence parameter v.
In particular, we consider three cases:
(i) The infectivity rate increases linearly with virulence (i.e. b = αv, where α is a
proportionality constant).
(ii) The infectivity rate increases faster than linearly with virulence (i.e. b = αv 2 for
example).
√
(iii) The infectivity rate increases slower than linearly with virulence (i.e. b = α v for
example).
These trade-offs between virulence and infectivity are shown in figure 4.2. Substituting
these trade-offs into the basic reproduction number we obtain
√
λαv
λαv 2
λα v
(i) R0 =
(ii) R0 =
(iii) R0 =
δ(δ + r + v)
δ(δ + r + v)
δ(δ + r + v)
55
CHAPTER 4. EVOLUTION OF VIRULENCE
For case (i) and (ii) R0 is maximal if for maximal virulence (i.e. v = ∞). The optimal
level of virulence for case (iii) can be obtained by differentiation:
dR0 /dv = 0
⇒
αλ δ + r − v
√
=0
2δ v(δ + r + v)2
Hence dR0 /dv is zero, if the enumerator of the second fraction equals zero. We thus
obtain for the optimal virulence
vopt = δ + r
In summary, the analysis of the simple two-strain SIR model suggests that natural selection should lead to ever increasing levels of virulence, if the infectivity rate increases
linearly or faster with increasing virulence. If, however, the infectivity rate increases
slower than linearly, then natural selection should lead to an intermediate level of virulence. If there are no constraints between infectivity rate and virulence, then natural
selection should lead to avirulence. Perhaps, the biologically most plausible scenario is
that there are diminishing returns in terms of infectivity rate for higher and higher levels
of virulence (i.e. case (iii)). For many parasites both the harm done to the host and the
probability of transmission increase with increasing parasite load7 . At very high parasite loads, however, the harm to the hosts continues to increase, while the probability
of transmission is expected to saturate. For this case, the above model thus predicts
the evolution towards intermediate levels of virulence (as was for example observed for
myxoma virus, see figure 4.1).
4.4 Key assumptions of the virulence model
The discussion of the SIR model in the previous section provides a formal basis to
investigate how different factors may affect the evolution of virulence. In particular,
the model helped to delineate the conditions under which we expect natural selection
to lead to avirulence. Before moving ahead it is important to consider the assumptions
that underlie the analysis above:
1. Infection is assumed to be proportional to the product of S and I. As discussed
previously, this term assumes that the population is well-mixed (i.e. that the
population has no spatial structure). We will discuss the effects of spatial structure
in section 4.9.
2. The model describes horizontal, contact-dependent transmission of the pathogen,
because the infection proportional to the product of S and I. Many diseases,
however, are not directly transmitted via contact but instead have water-borne
(e.g. Cholera), food-borne (e.g. Hepatitis A), or vector-borne (e.g. malaria)
transmission. We will discuss below how the transmission mode may affect the
7
A correlation between virulence and transmission has been confirmed for a variety of pathogens,
but may not be common to all parasites. An example for a correlation between transmission and
virulence for a bacteriophage is shown in figure 4.5.
56
0.6
0.4
0.2
0.0
infectivity, b
CHAPTER 4. EVOLUTION OF VIRULENCE
0
2
4
6
8
10
8
10
6
4
0
2
R0
8
virulence, v
0
2
4
6
virulence, v
Figure 4.2: Examples for three tradeoffs between virulence and infectivity are shown in the
upper graph. The solid line assumes that the infectivity rate is proportional to the virulence
(i.e. b ∝ v). The dashed line assumes that infectivity increases more than linearly with virulence (i.e. b ∝ v 2 ). The dotted line assumes that the infectivity increases slower than linearly
√
with increasing virulence (i.e. b ∝ v). The basic reproduction numbers corresponding to
these three cases are shown in the bottom graph. Hence, if the infectivity rate increases less
than linearly with increasing virulence, then the basic reproduction number has a maximum
at intermediate values of virulence. Otherwise, the basic reproduction number increases
with increasing virulence.
57
CHAPTER 4. EVOLUTION OF VIRULENCE
evolution of virulence. In particular, we will discuss in section 4.7 how vertical
(i.e. parent to offspring) versus horizontal transmission affects the evolution of
virulence.
3. We have considered only trade-offs between the rate of infectivity, b, and virulence,
v. It is also possible that there are trade-offs between the rate of recovery, r, and
virulence (or all three parameters b, r and v may be depend on each other).
4. The model assumes that all infected hosts are infected by a single parasite strain
only. Hence, infection provides immunity against superinfection by other strains
of the same parasite (or coinfection by other parasites). In the model there is no
competition between distinct strains within an individual hosts. We will discuss
the consequences of intra-host competition in section 4.8 below. In particular we
will show, that under these condition it is no longer the basic reproduction number
that determines the success in competition.
5. To derive the result that natural selection maximizes the basic reproduction number we used that the host population is infected by strain 1 only (and is in the
corresponding infected equilibrium) and then studied the conditions under which
strain 2 can invade. The important assumption underlying the derivation of this
result is that the density of susceptibles is determined by the presence of the parasite. While this is certainly the case in the simple SIR model, this may not
generally be the case. Moreover, the density of susceptible hosts may not be in
equilibrium, but could for example increase. Hence, if there is no feedback of
the parasite on population density, then there is strictly speaking no competition
between different parasite strains. To study the evolution of virulence under nonequilibrium conditions is beyond the scope of this script, but it is important to
add a cautionary note that the evolution of virulence may depend on whether the
total host population is in equilibrium8 .
4.5 Host density and the evolution of virulence
It is frequently stated that virulence should increase with increasing host density, since
increased host density reduces the costs of virulence. The intuition for this claim is that
the increased opportunities of transmission relieve the constraints to keep the host alive
for long times. We will now investigate this claim first on the basis of the results from
the models. In our model (eqs. 4.1 - 4.5) the host density can increase either because
of an increased birth (or immigration) rate λ or because of a decrease in the natural
mortality rate δ. Assuming a diminishing returns trade-off between infectivity rate and
virulence (i.e. case (iii) in section 4.3) we have derived that the optimal virulence is
given by vopt = δ + r (see eq. 4.3). This implies, that the optimal level of virulence
depends on δ but not on λ. Hence, the model suggests that increasing the birth rate has
8
The effect of non-equilibrium conditions is discussed for example in Lenski & May, J. Theor. Biol.,
1994 or Bonhoeffer et al., Proc. Roy. Soc. 1996
58
0
10
R0
20
30
CHAPTER 4. EVOLUTION OF VIRULENCE
0
2
4
6
8
10
8
10
6
4
0
2
R0
8
virulence, v
0
2
4
6
virulence, v
Figure 4.3: The top plot shows the basic reproduction number for a diminishing returns tradeoff for three increasing values of the birth rate b (in order solid line, dashed line and dotted
line). The bottom plot shows the basic reproduction number for three increasing values of
the natural mortality rate δ (again in order solid line, dashed line, dotted line). Hence, the
optimal level of virulence (indicated by the thin dotted line) is not affected by changes in
the birth rate, but increases with increasing death rate.
no effect on the optimal level of virulence, while decreasing the death rate δ is expected
to lead to a decrease in virulence. This is the opposite of what was claimed further above
on intuitive grounds, and highlights the use of formal models to delineate the factors
affecting the evolution of virulence. On second thought, the result of the model does
also make intuitive sense. The shorter the natural life span of a host, the faster the
parasite needs to exploit the host for its own transmission. Thus in short living hosts
parasites should evolve to become more virulent.
The problem with the argument at the outset of the previous paragraph is that while
increasing host density does allow more virulent pathogens to exist, it does not necessarily affect the evolutionary optimal level of virulence. This can be seen as follows. The
basic reproduction number is proportional to the birth rate λ (see eq. 4.2). Thus increasing the birth rate changes the magnitude of R0 , but does not shift its maximum with
regard to virulence (see fig. 4.3). Note, however, that parasites that previously were not
capable of spreading in the population (because of a basic reproduction number smaller
than one) may be capable of spreading after an increase in b, because of their increased
basic reproduction number. In other words, an increased population density may allow
parasites to spread that were previously too virulent. However, the increased population
density (if due to an increase in λ) does not have an effect on the evolutionarily optimal
level of virulence that the parasite should eventually attain.
59
CHAPTER 4. EVOLUTION OF VIRULENCE
The effect of an increasing extrinsic mortality rate on the evolution of virulence has
been tested with water fleas (Daphnia magna) infected by microsporidian gut parasites
(Glugoides intestinalis) by Dieter Ebert and Katrina Mangin9 . The water fleas were
grown together with the parasites in small beakers under conditions of low and high
host mortality. Since from the point of view of the parasite it does not matter whether
hosts are killed or simply removed from the system, host mortality can be easily manipulated by replacing regularly a certain fraction of the hosts in the beakers. Contrary,
to the prediction of the simple SIR model, they found that parasites that were kept
under conditions of high host mortality did not evolve high virulence. The discrepancy
between the model and the experiment turned out to be most likely due to the fact that
multiplicity of infection rather than host mortality determined virulence in this system.
It turned out that the average duration of infection was significantly longer in hosts
under low mortality conditions, presumablyleading to increased probability of multiple
infection. We will discuss the consequences of multiplicity of infection for the evolution
of virulence further below.
4.6 Treatment and the evolution of virulence
Infected hosts can recover from infection naturally or due to treatment. Both cases
are reflected in the recovery rate parameter r in the model 4.1-4.5. Assuming again
a diminishing returns trade-off with between virulence and infectivity rate the optimal
level of virulence (case (iii) in section 4.3) depends on r (see eq. 4.3). Thus the model
suggest that increasing the rate of recovery by medical intervention may lead to the
unwanted side effect of an increased optimal level of virulence. The intuition is the same
as for increased host mortality, since from the perspective of the parasite transmission
opportunities end irrespective of whether the patient dies or recovers. One needs to
emphasise, however, that the main effect of treatment is to reduce the duration of an
infection and thereby decrease the prevalence of the disease. Thus treatment will in
most cases be beneficial in terms of the overall reduction of the burden of disease in
a population, even if it is conceivable that medical interventions could also lead to an
increased level of virulence10 . In this context, it is interesting to note, that intensive
antibiotic therapy as is common for example in intensive care units in hospitals does
indeed select for highly virulent and highly resistant bacteria (although it needs to be
pointed out the the evolution of virulence and the evolution of antibiotic resistance are
not equivalent processes).
In summary, the model suggests that treatment could lead to increased levels of virulence. While this is indeed a conceivable side effect of therapy, it is necessary to point out
that many other factors (some of which we will discuss below) may also affect virulence.
Hence, before you convince your friends to throw away their medications, continue to
9
10
Source: Ebert & Mangin, Evolution, 1997.
This is analogous to the situation discussed in the section on vaccination (section 1.4) where it was
argued that vaccination against certain childhood infections can have the unwanted side-effect of
increasing the average severity of a disease, but its overall effect is to reduce the burden of disease.
60
CHAPTER 4. EVOLUTION OF VIRULENCE
read. Generally, the current scientific understanding of the factors that may influence
the evolution of virulence is too poor to make reliable medical recommendations.
4.7 Horizontal versus vertical transmission
One of the most straightforward predictions regarding the evolution of virulence is that
parasites that transmit only vertically from parent to offspring should evolve to become
avirulent, since in a loose analogy exclusively vertically transmitted, virulent parasites
can be regarded as a dominant deleterious gene and should thus be eliminated by natural
selection. Mathematical models have confirmed this basic prediction, but have also
predicted more counterintuitive outcomes provided there is a trade-off for efficiency of
vertical transmission and virulence11 . However, instead of focussing on the mathematical
intricacies of the effect of vertical versus horizontal transmission we will discuss here two
experiments that addressed the evolution of virulence in the context of varying degrees
of horizontal and vertical transmission.
The first experimental test of the hypothesis that increasing levels of vertical transmission correlate with decreasing levels of virulence was based on a field study of figpollinating wasps and their nematode parasites that was carried out in Panama by Allen
Herre12 . The life cycle of fig-wasps begins when one or more females enter the so-called
syconium, which is the enclosed inflorescence that eventually ripens to become the fig
fruit. Inside the syconium, the pollen bearing fig wasps pollinate the flowers, lay eggs
and die. As the fig fruit ripens the newly hatched fig wasps mature and mate within the
fig, before they leave and the cycle begins anew. The bodies of the dead mother wasps
remain in the fig and can thus be counted.
Fig wasps are generally highly specific to distinct species of fig trees. Each fig wasp
species in turn is associated with a specific species of nematodes. The nematode’s life
cycle is thus intimately connected with its host species. The nematodes are carried with
their host species into the fig. The nematodes reside in the body cavity of the fig wasps,
where they consume their hosts’ tissue. Once the nematodes have matured inside the
fig wasps around 6-7 adult nematodes emerge from the body of a dead wasp, mate and
lay eggs. The nematodes of the next generation then crawl onto to the newly hatching
fig wasps that mature inside the ripening fruit.
The opportunities for horizontal transmission of nematodes depends on the number
of fig wasps that enter a syconium. If only a single fig wasps enters the syconium, then
transmission is purely vertical. With increasing number of fig wasps inside a fig the
opportunities for horizontal transmission rise.
Allen Herre collected field data for 11 different species of fig wasps that are each specific for a single fig tree species and have all a specific nematode parasite. He scored the
proportion of broods founded by a single wasp against the virulence of the nematode
(measured as the life time reproductive success of nematode infected wasps relative to
11
12
See: Lipsitch et al, Evolution, 1996
Source: E.A. Herre, Science 1993
61
CHAPTER 4. EVOLUTION OF VIRULENCE
1.00
●
●
●
●
0.95
●
●
●
0.90
●
0.85
Relative reproductive sucess (inf/uninf)
uninfected wasps). In agreement with the hypothesis he found that the virulence increases with decreasing proportion of single foundress broods (see figure 4.4). However,
it should be kept in mind that there are also alternative explanations for these observations. In particular, the increasing level of virulence with decreasing proportion of single
foundress broods could also be due to higher multiplicity of infection and thus due to
increased intra-host competition (see section 4.8).
●
●
●
0.4
0.6
0.8
1.0
Proportion of single foundress broods
Figure 4.4: Virulence as a function of the proportion of single foundress broods in 11 species
of fig wasps each infected with a host species specific nematode. If broods are founded by a
single wasp, then nematodes are transmitted purely vertically. With decreasing proportion
of single foundress broods the reproductive success of infected relative to uninfected wasps
decreases. Hence, virulence increases with increasing opportunities for horizontal transmission. The linear regression is highly significant (R2 = 0.81, p = 0.00016).
Another test of the hypothesis that increasing levels of vertical transmission should be
associated with lower virulence has been performed by Sharon Messenger and coworkers13
using E. coli as the host and the filamentous bacteriophage f1 as the parasite. The
bacteriophage was engineered to carry a gene that confers resistance to an antibiotic.
Thus bacteria infected with a phage are resistant to that antibiotic. Filamentous phages
(in contrast to most other known phages) establish a permanent infection, during which
they constantly reproduce without killing the host.
Infection of a bacterium via horizontal transmission requires the presence of F-pili on
their host cell (i.e. requires cells carrying the conjugal F-plasmid). Upon entering a cell
the single-stranded phage DNA is converted by host enzymes into a double stranded
replicative form. The pool of replicative forms increases rapidly to 30-50 copies in
the first hour of infection and settles at an equilibrium of 5-15 copies per cell 2 hours
13
Source: S.L. Messenger, I.J. Molineux, and J.J. Bull, Proc. Roy.Soc, 1999
62
CHAPTER 4. EVOLUTION OF VIRULENCE
after infection. Vertical transmission occurs when an infected bacterium divides. The
replicative forms are presumably distributed randomly over the dividing cells. Due to
the high copy number of replicative forms vertical transmission in this system is close
to perfect (i.e. more than 99% of cells remain infected after a cell division).
In this system the degree of horizontal and vertical transmission can be controlled
experimentally. How this works in detail will become clear further below. In essence,
to ensure exclusively horizontal transmission free phage is separated from the cells and
used to infect a new population of uninfected bacteria. To ensure exclusive vertical
transmission bacteria and phage are cultured in the presence of the antibiotic. Under
these conditions all cells that are not infected are killed by the antibiotic. In the experimental protocol the bacteria where grown for 24 days. Each day the bacteria were
grown for 24 h in the presence of the antibiotic and then diluted 1000-fold and placed
into fresh growth medium (containing antibiotics). The experiment design consists of
two treatment arms that differ in the opportunities for horizontal transmission. The
free phages were either harvested daily (L1 lines) or every eighths day (L8 lines) and
were transfered to a new uninfected population of bacteria. Thus there were 24 steps
of horizontal transmission in the L1 lines and only three in the L8 lines. After 24 days
the virulence of the bacteriophages in both treatment arms was measured using the
logarithm of the bacterial density as a marker for virulence. (Note, that increasing log
cell density corresponds to increasing avirulence.) Figure 4.5 shows that the L8 lines
had consistently lower virulence (i.e. higher log infected cell densities) than the L1 lines.
Moreover the L1 lines had consistently higher fecundity as measured by the log phage
titer produced during one hour of growth in fresh medium. The figure also provides
evidence for a positive correlation between fecundity (i.e. horizontal transmissibility)
and virulence.
A relevant feature of the experimental system is that superinfection cannot occur, since
a consequence of phage gene expression is that the F-pili are permanently disassembled
on the host cell. Hence, shortly after infection the phage actively prevents superinfection.
Moreover, given the relatively low copy number of replicative forms per cell and the
high fidelity of virus replication (which is performed by the replication machinery of E.
coli, there is presumably limited intra-host diversity. Thus in this experimetal setup it
can be ruled out that the increased virulence that is associated with higher horizontal
transmission is caused by increased intra-host competition (see section 4.8).
Both experimental tests lend support to the hypothesis that natural selection should
act to decrease virulence the more a parasite’s transmission is vertical as opposed to
horizontal (although an alternative interpretation is possible for the fig wasp example).
However, purely vertically transmitted parasites can also be virulent if they possess
the ability to increase their representation in the next generation sufficiently to offset
the effects of the selection against infected hosts. Wolbachia, for example, are such an
exception to the general rule. These intracellular bacteria can induce feminization in
their insect hosts. Since they are exclusively transmitted from mothers to their female
offspring, the advantage gained by a distortion of the sex ratio can compensate for the
cost in terms of virulence.
63
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CHAPTER 4. EVOLUTION OF VIRULENCE
● ●
●
●
●
●
●
●
●
0
●●
●
●
−1
log phage titer
1
●●
●
●
−2
●
●
●
● ●
●●●
●
−2
−1
0
1
2
log infected cell density
Figure 4.5: Log phage titer versus log infected cell density in L1 lines (filled circles) and L8
lines (open circles). For a detailed description of the experiment see main text. Log phage
titer is used here as a proxy for fecundity. Virulence increases with decreasing infected cell
density after 24 hours of growth. The figure provides evidence for a positive correlation
between fecundity (i.e. horizontal transmissibility) and virulence. Moreover, the virulence
of the L8 lines is consistently lower than that of the L1 lines, demonstrating that virulence
decreases with decreasing opportunity of horizontal transmission.
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CHAPTER 4. EVOLUTION OF VIRULENCE
4.8 Intra-host competition
So far we considered that competition between distinct parasite strains occurs exclusively
at the level of transmission between hosts. In particular, the simple SIR model (eqs. 4.1 4.5) did not allow for the superinfection of an already infected host by a second parasite.
Many parasites, however, face strong competition within an individual infected hosts.
Intra-host competition arises whenever there is phenotypic variation between parasites
in the same host. Such phenotypic variation can arise due to several processes:
Mutation: Many parasites (in particular viruses) have high mutation rates resulting in
a high phenotypic diversity within an individual infected host
Superinfection: Infected individuals may become re-infected by a related pathogen thus
generating intra-host diversity.
Coinfection: Host may be infected by several unrelated parasites simultaneously, which
can reside in the same or in different tissues. Unrelated parasites residing in
different tissues can nevertheless be in competition, since ultimately both depend
on the survival of the host.
Intrahost competition is expected to be strong if
(i) there is high intra-host diversity (as can be generated by mutation, superinfection,
or coinfection).
(ii) different strains differ markedly in their fitness within the host. This can be for
example the case if some mutants can escape from the control by the host’s immune
responses.
(iii) there are many rounds of replication between the time of infection and the time of
transmission to the next host. This is the case if the turnover rate of the pathogen
is high and there is a long interval between infection and transmission.
Within-host competition can lead to increased virulence. This has been documented
for many pathogens in so called serial transfer experiments, in which pathogens are
transferred experimentally between infected hosts, thus releasing the selection for efficient natural transmission. When the host population is genetically sufficiently homogeneous such serial transfer experiments typically result in the selection of more virulent
pathogens (see fig. 4.6). Hence, there is a correlation between virulence and intra-host
competitive ability.
Intra-host competition is expected to select typically for increased virulence. More
generally, however, intra-host competition will select a competitively superior variant
irrespective of its eventual effect on host survival. Hence, competition for survival within
a host and competition for transmission between hosts, may have opposing effects on the
evolution of virulence. While inter-host competition may (often, but not generally) select
for a low or intermediate level of virulence, intra-host competition may select for high
levels of virulence. Importantly, when taking intra-host competition into consideration
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CHAPTER 4. EVOLUTION OF VIRULENCE
60
●
40
●
●
●
●
●
20
% dead mice
80
●
●
0
●
2
4
6
8
10
number of passages
Figure 4.6: Serial transfer of Salmonella typhimurium in mice. Repeated experimental passage of the pathogen to new mice leads to increasing virulence. Hence, there is a correlation
between virulence and intra-host competitive ability. (Source: I.A. Sutherland, Exp. Parasitol., 1996)
the overall optimal level of virulence is no longer determined by the maximization of the
basic reproduction number, since the basic reproduction number only takes into account
the competition for transmission between hosts.
This can be illustrated by the following simple model of superinfection:
dS/dt = λ − δS − b1 SI1 − b2 SI2
dI1 /dt = b1 SI1 − (δ + v1 )I1 − σI1 I2
dI2 /dt = b2 SI2 − (δ + v2 )I2 + σI1 I2
(4.6)
(4.7)
(4.8)
The parameters are as in the SIR model (eqs. 4.1 - 4.5). For simplicity, we neglect that
infecteds can recover. The model makes the additional assumption that strain 2 can
superinfect hosts that are already infected with strain 1. Hence, we assume that strain
2 has a competitive advantage within a superinfected host resulting in a net rate, σ of
production of host infected by strain 2 per contact between both types of infected hosts.
Provided the basic reproduction number of strain 1 is larger than 1, the equilibrium
values in absence of parasite strain 2 (i.e. I2 = 0) are given by S ? = (δ + v1 )/b1 and
I1? = λ/(δ + v1 ) − δ/b1 . Substituting these values into eq. 4.8, we obtain that strain 2
can invade (i.e. its growth rate is positive for small I2 ) if
(1)
(2)
R0
>
(1)
R0
λσ(R0 − 1)
−
(δ + v2 )(δ + v1 )
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CHAPTER 4. EVOLUTION OF VIRULENCE
(1)
(2)
where R0 = λb1 /(δ(δ + v1 )) and R0 = λb2 /(δ(δ + v2 )) are the basic reproduction
(1)
numbers of the respective strains. Since R0 is by definition larger than one, the second
term in the above equation is always negative. Hence, the basic reproduction number of
strain 2 can be smaller than that of strain 1 provided its competitive advantage within
the host compensates for its smaller basic reproduction number. In summary, intra-host
competition can lead to levels of virulence that are higher than optimal for maximal
transmission between hosts. Further above we derived that selection for maximal transmission (i.e. maximal R0 ) is expected to lead to avirulence if there is no constraint
between the infectivity rate and virulence. Taking into account intra-host competition,
one would expect evolution towards intermediate levels of virulence even in this scenario,
since intra-host competition would select for higher virulence (due to a correlation between intra-host competitive ability and virulence), while inter-host competition would
select for lower virulence (in order to keep the host alive longer).
As pointed out further above intra-host competition leads to the selection of properties
that confer a fitness advantage within the host irrespective of its consequences for the
transmission of the parasites to new hosts. This phenomenon has been termed short
sighted evolution, although it should be pointed out that natural selection never acts with
foresight. Such short-sighted evolution can for example explain the virulence of polio
virus. Polio virus normally replicates in the gut of infected hosts. When the infection
is confined to the gut, the disease is typically harmless. However, occasionally during
an infection the virus also infects nervous tissue, which leads to the severe neurological
consequences that are associated with poliomyelitis. Clearly, infection of the nervous
tissue is not of advantage for transmission to new hosts, since the virus can not be
transmitted directly between the nervous tissue of two hosts. Instead, the infection of
nervous tissue can be better understood selective advantage within a host due to the
ability colonise of a new niche in a host14 .
4.9 Local adaption, host heterogeneity, and
cross-species transmission
It is often believed that parasites that cross a species border are highly virulent. Indeed,
diseases such as HIV, SARS, avian influenza, West Nile virus, and Ebola virus are
characterized by high virulence. However, this belief is likely due to an observational
bias. Clearly, only those infections that cause virulent effects will be noticed, while many
cross-species transmission with mild or no effects may go unnoticed.
A similar argument is frequently made regarding the virulence of parasites that have
moved between different populations of the same species. In this context it is often
stated that parasites are more virulent when entering a novel population that has not
previously encountered the parasite. Many examples can be given that seem to support
this view. Introduction of measles into the population of native American Indians led to
14
The concept of short-sighted evolution was proposed in B. R. Levin & J. J. Bull, Trends in Microbiology, 1994.
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CHAPTER 4. EVOLUTION OF VIRULENCE
devastating epidemics. Similarly the introduction of small pox into the Aztec population
by the Spanish conquistadores may have actually contributed more to the victory of
Hernando Cortez15 , then his much acclaimed skills as a conqueror. A further example
is the introduction of chestnut blight fungus into the population of American chestnuts.
The epidemic was started by the introduction of the fungus to the botanical garden
in Brooklyn, New York around 1900 . By the 1940 the fungus had essentially wiped
out the American Chestnut and has since reduced a formerly magnificent tree on which
a large timber industry relied to the existence as a shrub. Although these examples
are impressive, they should not obscure our view on whether we generally expect new
host-pathogen associations to be more virulent than long-lasting ones.
One study that attempted to address this question16 was based on strains of water
fleas (Daphnia magna) that were isolated from natural ponds which were up to 3000
km apart. In addition three strains of the horizontally transmitted parasite Pleistophora
intestinalis were isolated from three ponds that were in close proximity to each other.
In contrast to the common belief that new host-pathogen associations should be more
virulent, this experiment supports the view that locally co-adapted host parasite system
tend to be characterized by higher virulence than novel host parasite associations (see
figure 4.7).
There are also strong arguments against the view that cross-species transmissions
should generally be associated with high virulence. It has been frequently shown that
increasing adaptation towards one host species lowers virulence in other host species.
Figure 4.8, for example, shows the serial passage of a fungus on barley that was originally
adapted for growth on wheat. Here increases in virulence on barley were associated with
loss of virulence on wheat.
Serial passage in a homogeneous host population typically leads to increasing virulence
(see for example fig. 4.6). However, under natural circumstance host populations will
typically be heterogeneous. The observation that increases in virulence in one host type
may decrease virulence in other host types suggest that host heterogeneity may be an
important factor preventing the evolution of high virulence17 .
The fact, that passage in a new host results in attenuated virulence in the original
host has in fact been employed for the development of life attenuated vaccines18 . The
development of the life-attenuated Sabin19 vaccine against polio virus is a good example.
15
Hernando Cortez (1485-1547) was a Spanish explorer who is famous mainly for his march across
Mexico and his conquering of the Aztec Empire in Mexico
16
Source: D. Ebert, Science, 1994.
17
For detailed investigation of the effects of host heterogeneity see R.R. Regoes, M. A. Nowak & S.
Bonhoeffer, Evolution, 2000.
18
Life attenuated vaccines typically induce the most protective immune responses, but in contrast to
killed vaccines, life attenuated vaccine involve the risk that a replication competent pathogen is
infected into patients, which can potentially revert to restore its virulence
19
Albert Sabin, was born in Poland in 1906. He and his family emigrated to the United States in 1921
to escape anti-Semitism. Sabin’s research included work on pneumonia, encephalitis, toxoplasmosis,
viruses, sandfly fever, dengue and cancer. He began to work on poliomyelitis after World War II.
Sabin scoured the world looking for weak strains of polio virus, found three, and began to develop
his oral, ”live” vaccine, administered at first on a lump of sugar or in a teaspoonful of syrup. In
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CHAPTER 4. EVOLUTION OF VIRULENCE
●
80
●
60
●
40
●
●
●
●
●
●
20
% mortality (after 42 days)
●
●
●
●
0
●
0.0
0.5
●
1.0
1.5
2.0
●●
2.5
3.0
3.5
distance, log(x+1)
Figure 4.7: Local adaptation in of the cytoplasmic parasite of Pleistophora intestinalis to its
host Daphnia magna. Three parasite strains isolated from ponds in close proximity to each
other were tested for their effect on hosts isolated from ponds with increasing distance from
the the pond from which the parasites were sampled. Increasing geographic distance is correlated with decreasing virulence, implying that locally adapted host-parasite combinations
tend to be more virulent. Therefore this data is evidence against the conventional view that
co-adapted host parasite associations should be less virulent.
The Sabin vaccine used to immunize against the disease poliomyelitis is a live poliovirus,
which does not lead to disease (except to about 1-2 in a million people vaccinated)
because it is genetically weakened so that the immune response can control it. The
attenuated vaccine strains came from wild, virulent strains of poliovirus, but they were
evolved by Albert Sabin to become attenuated. Essentially, he grew the viruses outside
of humans, and as the viruses became adapted to those non-human conditions, they lost
their ability to cause disease in people.
1957 the World Health Organization (WHO) decided Sabin’s vaccine deserved world-wide testing.
But at home in the United States, Sabin had a hard time convincing the U.S. Public Health Service
that his method was any better than Jonas Salk’s ”killed” vaccine method. An advantage of Dr.
Sabin’s oral vaccine, especially in less developed countries, is ease of administration: no shots. But
two other pluses are even more important. First, the live vaccine gives both intestinal and bodily
immunity; the killed vaccine gives only bodily immunity and allows the immune person to still serve
as a carrier or transmitter. Second, the Sabin vaccine produces lifelong immunity without the need
for a booster shot or vaccination.
69
30
40
50
●
10
20
●
●
0
# fungal colonies on barley
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CHAPTER 4. EVOLUTION OF VIRULENCE
0
20
40
●
60
# fungal colonies on wheat
Figure 4.8: Change of the number of colonies of a wheat adapted strain of the fungus Septoria
nodorum after serial passage on barley. The data show that increased virulence (as measured
by the number of fungal colonies) on a new host is accompanied by changes in virulence on
the original host. The arrows indicate the direction of adaptation in successive rounds of
serial passage starting from strain that was adapted for growth on wheat.
70