Overview lecture 2
Basic
mathematical
epidemiology
I
Introduction to the problem of virulence evolution
I
Predicting evolutionary outcomes with mathematical models
I
Trade-offs
I
transmission mode, intra-host competition, local adaptation
Examples:
I
Myxomatosis in rabbits
I
Fig-wasps and nematodes
I
Bacteria and Phage
I
Water fleas and their parasites
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Evolution of parasite virulence
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Parasites are defined here broadly to include viruses,
bacteria, etc as well as macroparasites
I
Why are parasites harmful?
I
Trade-offs between infectivity and virulence
I
Factors influencing the evolution of virulence
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Definition of virulence
Basic
mathematical
epidemiology
The virulence of a parasite is defined as the fitness costs to
the hosts that are induced by the parasite.
Possible costs:
I
Mortality
I
Morbidity
I
Reduction of host fecundity (i.e. castration)
I
Parasite induced change of host behaviour (i.e.
increased predation by definitive host)
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Conventional view
Parasites should evolve to be harmless, since killing of their hosts
also ends their opportunities for transmission. Hence, longstanding
host-parasite association should be avirulent.
Empirical support:
I
HIV, avian flu, West Nile virus, etc are recent, virulent host
parasite associations
Empirical contradiction:
I
Measles: In human population for approx. 10,000 years, but
remains highly virulent
I
Nematodes of fig-wasps: Fig-wasps and their nematode
parasites have been found preserved in amber. This
host-parasite association is highly virulent today.
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Virulence of myxoma virus
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence of myxoma virus
1975
Virulence evolution
1965
1960
Heterogeneities
1955
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
1950
Year
1970
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
1
2
3
Virulence grade
4
5
Maximisation of R0
Model of two competing strains:
dS/dt = λ − δS − (b1 I1 + b2 I2 )S + q1 R1 + q2 R2
dI1 /dt = b1 SI1 − (δ + v1 + r1 )I1
dI2 /dt = b2 SI2 − (δ + v2 + r2 )I2
dR1 /dt = r1 I − δR1 − q1 R1
dR2 /dt = r2 I − δR2 − q2 R2
where, v1 and v2 are the parasite induced host mortalities
(i.e. virulence).
Note: Constant birth rate λ, ⇒ total host population is not constant
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Invasion
Basic
mathematical
epidemiology
Basic reproduction numbers:
(1)
R0
λb1
=
δ(δ + v1 + r1 )
and
(2)
R0
λb2
=
δ(δ + v2 + r2 )
At infected equilibrium, when only strain 1 is present (i.e.
I2 = 0)
δ + v1 + r1
?
S =
b1
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Invasion of strain 2, if
dI2 /dt > 0
⇒
b2
1
> ?
δ + v2 + r2
S
⇒ Natural selection maximises R0 .
⇒
(2)
R0
(1)
> R0
No trade-off between virulence and other
parameters
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Basic reproduction number:
λb
R0 =
δ(δ + v + r )
R0 is maximal if v → 0
⇒ Evolution towards avirulence.
Assumption: There is no constraint between v and other
parameters determining R0 .
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Trade-offs
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Three trade-offs:
(i) b = αv
(ii) b = αv
2
√
(iii) b = α v
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Basic reproduction numbers:
(i) R0 =
λαv
δ(δ + r + v )
√
2
(ii) R0 =
λαv
δ(δ + r + v )
(iii) R0 =
λα v
δ(δ + r + v )
Trade-offs
0.4
0.6
Basic SIR model
Estimating R0
Avian flu
0.2
Virulence evolution
0.0
infectivity, b
Basic
mathematical
epidemiology
0
2
4
6
8
10
virulence, v
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
6
4
2
0
R0
8
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
0
2
4
6
virulence, v
8
10
Optimal virulence for diminishing returns
trade-off
Trade-off:
√
b=α v
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Basic reproduction number:
√
λα v
R0 =
δ(δ + r + v )
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Optimal virulence:
dR0 /dv = 0
⇒
αλ
δ+r −v
√
=0
2
2δ v (δ + r + v )
Hence dR0 /dv is zero, if the enumerator equals zero.
vopt = δ + r
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Key assumptions of the virulence model
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
1. Infection ∝ SI
⇒
population is well-mixed.
2. Transmission of the pathogen is contact dependent.
3. Trade-offs only between the rate of infectivity, b, and
virulence, v .
4. Immunity to superinfection.
5. No intra-host competition.
6. Equilibrium analysis
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Host density and evolution of virulence
Basic
mathematical
epidemiology
Hypothesis: Virulence should increase with increasing host
density, since increased host density reduces cost of virulence.
Model: Host density ↑ if either λ ↑ or δ ↓
Assumption: Diminishing returns trade-off
Result:
vopt = δ + r
Conclusion: Optimal virulence is not affected by changes in
λ, and decreases with decreasing δ. Contradiction to
hypothesis.
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Basic reproduction number for varying λ or δ
30
Basic
mathematical
epidemiology
R0
20
λ=3 v 5
λ=2 v 5
λ= v 5
Basic SIR model
Estimating R0
Avian flu
0
10
Virulence evolution
0
2
4
6
8
10
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
6
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
4
δ=1
δ=2
δ=3
0
2
R0
8
Heterogeneities
0
2
4
6
8
10
virulence, v
Note: For larger λ more virulent parasites can spread in the population, but vopt does not change.
Experiment with increased host mortality in
Daphnia
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
number of lines
% replaced (per week)
host background mortality
time from infection to death
within host growth rate
transmission
Replacement
6
70-80%
high
51.3 ± 2.3 (d)
0.27 ± 0.02(1/d)
0.6 ± 0.1
Virulence evolution
Non-Replacement
Intro virulence
Predicting virulence
6
evolution
Trade-offs
0%
Transmission mode
low
Intra-host competition
43.8 ± 2.6 (d) Local adaptation
Heterogeneities
0.34 ± 0.02 (1/d)Risk structure
in koalas
0.3 ± 0.1 Chlamydia
Spatial structure
Rabies in foxes
Alternative interpretation: High multiplicity of infection causes
high virulence
Source: D. Ebert & K.L. Mangin, Evolution, 1997
Treatment and evolution of virulence
Basic
mathematical
epidemiology
Note:
I
Parasite transmission ends when host dies or recovers
I
Treatment increases recovery rate
For diminishing-returns trade-off we have
vopt = δ + r
⇒ Increasing r should result in higher optimal virulence.
Caution:
Treatment induces incidence and prevalence of infection, but
may have the unwanted side-effect of selecting for higher
virulence.
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Horizontal versus vertical transmission
Basic
mathematical
epidemiology
Horizontal transmission: Transmission between unrelated
individuals
Vertical transmission: Transmission from parent to offspring
Hypothesis:
Parasites that transmit exclusively vertically should evolve to
be avirulent.
Intuition:
Vertically transmitted, virulent parasite ⇔ dominant,
deleterious gene
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Fig-wasps and nematodes
Species specific interaction: Fig species
m
fig wasp species
m
nematode species
Fig wasp life cycle: Mother enters fig syconium
⇒ lay eggs + dies
⇒ offsprings hatch and
mate
⇒ offsprings leave fig.
Nematode life cycle: Nematode enters in body
cavity of fig wasp and
consumes host’s tissue
⇒ 6-7 adult nematodes
emerge from dead host mate
and lay eggs
⇒ offspring crawl onto
newly hatched fig wasps.
Natural history: Different species of fig wasps differ in the proportion of broods that are founded
by a single mother.
Vertical/horizontal transmission: The number of fig wasps that enter a syconium determines the
opportunities for horizontal transmission.
Source: E.A. Herre, Science 1993
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Fig-wasps and nematodes
Basic
mathematical
epidemiology
1.00
●
●
●
●
0.95
●
●
●
0.90
●
0.85
Relative reproductive sucess (inf/uninf)
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
●
●
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
●
0.4
0.6
0.8
1.0
Proportion of single foundress broods
Reproductive success of 11 species of fig wasps increases with increasing degree of vertical transmission.
Bacteria and phage
Experimental system: E. coli and filamentous phage f1
engineered to carry antibiotic resistance gene
Natural history of phage:
I
Filamentous phages establish permanent infection without
killing the host
I
Horizontal transmission requires cells carrying F-pili
I
Within 1 hours of infection phage produces 30-50 replicative
forms (RF)
I
Phage infection results in disassembly of F-pili. Hence there
is no superinfection.
I
Vertical transmission occurs when a cell divides (and RFs are
distributed over both cells)
I
Efficiency of vertical transmission 99-99.9%
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Experimental control of transmission mode
Horizontal transmission:
Phage are separated from cells and used to infect a fresh
population of E. coli.
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Vertical transmission:
Phage and bacteria are cultured in presence of antibiotic.
⇒ Only cells that are infected by phage can survive
Since infected cells cannot be superinfected
⇒ Exclusive vertical transmission
Experimental protocol:
24 day-experiment
L8 lines: Horizontal transmission following 8 days vertical
transmission
L1 lines: Horizontal transmission following 1 day vertical
transmission
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Bacteria and phage
2
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
● ●
●
●
●
Virulence evolution
●
●
●
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
●
0
●●
●
●
−1
log phage titer
1
●●
●
●
−2
●
Heterogeneities
●
●
● ●
●●●
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
●
−2
−1
0
1
2
log infected cell density
L1 lines (filled circles) show higher virulence (i.e. lower infected cell densities) than L8 lines (open
circles). Note: Negative correlation between fecundity (i.e. horizontal transmissibility) and virulence.
(Source: Messenger et al, Proc Roy Soc B, 1999)
Intra-host competition
Note: Parasites compete not only for transmission between
hosts, but also for survival within a host. Selection for traits
increasing the survival within the host occurs irrespective of their
effect on transmission.
Factors that generate diversity within a host:
I
High mutation rate (e.g. viruses)
I
Superinfection (i.e. reinfection by a related parasite strain)
I
Coinfection (i.e. infection by a different parasite species)
Intra-host competition is strong if
I
there is high intra-host diversity
I
different strains differ markedly in their fitness within the
host.
I
there are many rounds of replication between the time of
infection and the time of transmission to the next host.
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Serial transfer increases virulence
Basic
mathematical
epidemiology
●
60
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
●
40
●
●
●
●
●
Heterogeneities
20
% dead mice
80
Virulence evolution
●
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
0
●
2
4
6
8
10
number of passages
Note: Serial transfer of Salmonella typhimurium in mice leads to increasing virulence. Hence, there is a
correlation between virulence and intra-host competitive ability. (Source: Sutherland, Exp Parasitol,
1996)
Basic SIR model
Estimating R0
Avian flu
Intra-host competition model
Superinfection model:
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
where σ of production of host infected by strain 2 per contact
between both types of infected hosts.
Assumption: Strain 2 has a intra-host competitive advantage
after superinfection. σ describes net production of host infected by
strain 2 per contact between both types of infected hosts.
Invasion of strain 2: (into equilibrium of strain 1)
(1)
(2)
R0
(1)
λσ(R0 − 1)
(1)
> R0 −
(δ + v2 )(δ + v1 )
(2)
where R0 = λb1 /(δ(δ + v1 )) and R0 = λb2 /(δ(δ + v2 )).
(2)
(1)
Note: Strain 2 can invade even if R0 < R0 . Hence, R0 does
not determine competitive success, when there is intra-host
competition.
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Summary
Conclusion: Intra-host competition can lead to levels of
virulence that are higher than optimal for maximal
transmission between hosts.
Note: Intra-host competition selects for parasite survival
irrespective of its consequences for between host
transmission. Intra-host and interhost competition may have
opposing effect on evolution of virulence:
Intra-host competition: virulence ↑ because virulence may
correlate with intra-host competitive ability
Inter-host competition: virulence ↓ because low virulence
may prolong opportunities for transmission
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Polio as a example for short-sighted evolution
Intra-host competition may lead to adapations that are not
beneficial for transmission (i.e. short-sighted evolution)
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Example:
Polio virus usually infects gut tissue. Occasionally it crosses
tissue borders and infects neuronal tissue. This represents
the colonisation of a new intra-host niche, but is a dead end
for transmission, since the virus is not transmitted from
neuronal tissue. Hence, virulence is a consequence of
“short-sighted evolution”.
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Virulence and novel host-parasite associations
Hypothesis:
Novel host-parasite associations are characterised by high
virulence.
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
Examples: (Mostly just-so stories)
Cross-species transmission: HIV, Ebola, West-Nile virus, Avian
influenza, Chestnut blight fungus.
Naive host population: Measles in American Indians, Small pox in
Aztecs, Yellow fever during the construction of the
Panama canal.
Problem: Bias in reporting. Avirulent new host-parasite
associations may go unnoticed.
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Local adaptation in Daphnia
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
I
Isolation of strains of water fleas (Daphnia magna) from
ponds that were up to 3000 km apart.
I
Isolation of three strains of the horizontally transmitted
parasite Pleistophora intestinalis from three ponds in
close proximity to each other.
I
Experiment: Infection of all Daphnia strains with all
parasite strains.
Virulence evolution
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Experimental results
Basic
mathematical
epidemiology
100
Basic SIR model
Estimating R0
Avian flu
●
Virulence evolution
●
80
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
60
●
40
●
●
●
Heterogeneities
●
●
●
20
% mortality (after 42 days)
●
●
●
●
0
●
0.0
0.5
●
1.0
1.5
2.0
●●
2.5
3.0
distance, log(x+1)
Conclusion: Locally adapted host-parasite combinations are more virulent!
3.5
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Serial passage on novel host
Basic SIR model
Estimating R0
Avian flu
●
50
Virulence evolution
30
40
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
20
●
10
Heterogeneities
●
0
# fungal colonies on barley
60
Basic
mathematical
epidemiology
0
20
40
●
60
# fungal colonies on wheat
Conclusion: Serial passage of a wheat-adapted fungus on barley, increases virulence in barley, but
decreases virulence on wheat. Hence, transmission in a heterogeneous host population may prevent
evolution of high virulence. (Source: B.M. Cunfer, Ann. Appl. Biol., 1984)
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes
Attenuation of life vaccines
Basic
mathematical
epidemiology
Basic SIR model
Estimating R0
Avian flu
Virulence evolution
I
I
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
vaccines
Sabin vaccine against polio was created by passage of
wild polio strains in a new environment.
Intro virulence
Predicting virulence
evolution
Trade-offs
Transmission mode
Intra-host competition
Local adaptation
Heterogeneities
Risk structure
Chlamydia in koalas
Spatial structure
Rabies in foxes