Disease ecology – SIR models
ECOL 8310, 11/17/2015
Why do ecologists study host-parasite
interactions?
• Major regulator of wildlife population dynamics
Control
Trichostrongylus tenuis
Lagopus lagopus scoticus
1 Treatment
2 Treatments
Why do ecologists study host-parasite
interactions?
• Cause of decline for threatened and endangered
species
Mustela nigripes
Omnivorous marsupials
Even-toed ungulates
Why do ecologists study host-parasite
interactions?
• Zoonoses are the main source of emerging
infectious diseases in humans
Jones et al. 2008
Epidemics
Poliomyelitis in the USA
Canine parvovirus in wolves
in Minnesota
Sequential epidemics
Measles in England
Rabies in red foxes in France
45 nm
Culex pipiens
Culex quinquefasciatus
West Nile virus (extreme complications) in
humans
Key Concepts
Models of disease dynamics
•
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The logic behind a compartment model
Processes and rates
Density-dependent transmission
Disease invasion and R0
Threshold population sizes
Frequency-dependent transmission
Versions of SIR models
Vaccination (Did Not Get To This)
Compartmental model
• Individuals in the population are classified
– This matches epidemiological data
• What are the possible classes?
• What are the rates at which individuals move
between them?
Susceptible - Infected
βSI
S
I
The transmission term, β, determines how quickly individuals in the S
compartment move into the I compartment
Unstable equilibrium: (S*,I*) = (N,0)
Stable equilibrium: (S*,I*) = (0,N)
Transmission between
hosts
Contact rate
Population size, N
Athens
Atlanta
NYC
We often assume a linear relationship between contact rate and
population size
e.g. the chances of bumping into someone on the sidewalk in
different populations
Contact rate = cN
Transmission between hosts
Transmission requires:
i) Contact between individuals
✔ (cN)
ii) The ‘right’ sort of contact (between an S and an I)
iii) The parasite establishes in the new host
For one infectious individual that makes contact with a random individual in the
population, the chance that it is with a susceptible individual is S/N
So for I infectious individuals, this scales up to (S/N)*I
And we assume there is some (biologically determined) chance that this sort of contact
allows the parasite to establish in the new host (call this chance ‘a’)
Transmission requires:
i) Contact between individuals
✔ (cN)
✔ (S/N)*I
ii) The ‘right’ sort of contact (between an S and an I)
iii) The parasite establishes in the new host
✔ (a)
Expression for transmission = (cN)*(S/N)*I*a
Research papers usually make the notation simpler by replacing the constants
c*a by β, which is called the transmission rate
Expression for transmission = βSI
Infectious Period
Infected
Recovered
γI
I
R
This is the length of time that an individual is capable of transmitting infection to
susceptible individuals
Commonly, we use 1/γ for the infectious period, and rate of movement from I
compartment to R compartment is γI
After this time, the host’s immune system clears the virus, and this simple model
assumes that hosts remain recovered
SIR: The basic compartmental
model
S
βSI
I
γI
R
• Examples: measles, chickenpox
• Anything that only needs one course of vaccine
• Equilibria depend on R0, can be endemic or epidemic
SIR model with births and deaths
S
dS
= d N - b SI - mS
dt
dI
= b SI - g I - m I
dt
dR
= g I - mR
dt
I
R
Here, birth rate and death rate are different.
What happens to the total host population size?
S
dN
= (d - m )N
dt
I
R
Either grows or declines (both exponentially)
S
dS
= d N - b SI - mS
dt
dI
= b SI - g I - m I
dt
dR
= g I - mR
dt
I
R
Suppose the parasite can cause additional host mortality
How would we model that?
S
I
R
dS
= d N - b SI - mS
dt
dI
= b SI - g I - m I - a I
dt
Suppose the parasite can cause additional host mortality
dR
= g I - mR
How would we model that?
dt
S
dN
= (d - m )N - a I
dt
I
R
Now the parasite might stop exponential growth
(an example of regulation)
dN
= (d - m )N - a I
dt
Now the parasite might stop exponential growth
(an example of regulation)
S
I
R
The number of infected individuals will increase if more is flowing into the “I”
compartment than is flowing out (the same as saying dI/dt>0)
dS
= d N - b SI - mS
dt
dI
= b SI - g I - m I
dt
dR
= g I - mR
dt
 SI   I   I  0
S
 SI   I   I  0
 SI   I   I
 SI
1
 I  I
S
1
 
I
R
R0=Basic reproduction number
Early on, infected
N
 1 individuals are rare and
so S~N (the
 
population size)
Basic reproductive number (R0): the expected number of secondary cases
caused by the first infectious individual in a wholly susceptible population.
This acts as a threshold criterion because disease invasion can succeed only
if R0>1.
Example:
R0=2
Supposing this pathogen continued causing infections, where each infected
individual gives rise to 2 new infections (each week, say). How long until all of us
(>6 billion people) would have been infected?
Why don’t we all get infected in reality?
Disease invasion & threshold
populations
R0 
N
1
 
Suppose transmission rate and infectious period are fixed, the disease
might invade some populations but not others
NT 
Why?
 

Epidemic burnout vs. Endemic
Causes of parasite decline:
Immunity
Density below R0 = 1
Stochasticity
Causes of endemic persistence:
Births
Immigration
loss of immunity
pathogen evolution
reintroduction
R0
20
18
16
14
12
10
8
6
4
2
0
Measles
Whooping cough
Polio
HIV/AIDS
Pandemic flu
SARS
From Wikipedia
Frequency-dependent transmission:
Alternative to density-dependent transmission
Contact rate
Constant
Population size
Transmission requires:
i)Contact between individuals
✔ (c)
ii)The ‘right’ sort of contact (between an S and an I)
iii)The parasite establishes in the new host
✔ (S/N)*I
✔ (a)
Expression for transmission = (c)*(S/N)*I*a
Research papers usually make the notation simpler by replacing the constants
c*a by β, which is called the transmission rate
Expression for transmission = βSI/N
Transmission
Density- and frequency-dependent transmission are (useful) idealizations
Contact rate
N
Reality is probably more complicated. Difficult to measure in practice…
Predictions based on transmission
Density- and frequency-dependent transmission
R0
N
R0 with Frequency Dependence
dS
= - b SI / N
dt
dI
= b SI / N - g I
dt
dR
=gI
dt
S
βSI/N
I
γI
R
The number of infected individuals will increase if more is flowing into the “I”
compartment than is flowing out (the same as saying dI/dt>0)
b SI / N - g I > 0
b SI / N > g I
b SI / N
>1
gI
bS / N
>1
g
Early on, infected individuals are rare and so S~N (the
population size)
b
>1
g
R0
Models predict that the existence of
threshold populations depends on the
type of transmission
With frequency-dependent transmission there is no threshold
population size
R0 & age at infection
It is possible to derive that if R0>1 and there is a supply of susceptibles, then
the equilibrium proportion of susceptibles is S/N=1/R0
Suppose the average age at infection is A
Individuals < A are susceptible
Individuals > A are immune
For simplicity assume a rectangular age distribution up to life expectancy L
Then the proportion susceptible = A/L
This means R0=L/A
R0 is inversely proportional to mean age at infection
This is also true when using more complicated models (e.g. not assuming
rectangular age distribution)
R0 increasing
Attack rate (better: attack ratio) = proportion of population that get infected during an
outbreak. Intuitively, attack rate increases as R0 increases. Here we see evidence that this
also corresponds to a lowering of the mean age at infection.
Predictions based on transmission
Density- and frequency-dependent transmission
Mean age
at
infection
N
Leptospirosis in
sea lions
(DD)
DD model of lepto predicts mean age at
infection decreases with N
• Not detected in data
• Lepto in sealions not DD
•
Measles
• Measles in 60 cities of England & Wales
• Recall that density-dependent
transmission implies R0 is proportional to
N
• R0 appears independent of host
population size  supporting frequencydependent transmission
Bjørnstad et al. 2002, Ecol. Monographs
Critical community size (CCS)
Smallest population of susceptible hosts
below which disease goes extinct, and
above which disease persists
Includes stochasticity
S
I
R
UK ~ 60M people
Epidemic fade-outs
e.g. Measles
Denmark ~ 5M people
Iceland ~ 0.3M people
Cliff et al. 1981
Persistence vs population size
Measles in England and Wales
Proportion of time extinct
0.7
0.6
Recurrent
Epidemics
Endemic
0.5
0.4
0.3
Critical Community Size
~300-500k
0.2
0.1
0
10 000
100 000
population size
1 000 000
10 000 000
Frequency vs. density dependent:
Frequency-dependent
 No threshold population
size for invasion
 R0 and mean age at
infection (A) do not depend
on N
 Classical example: STD
Density dependent
 NT>γ/β
 Ro should increase with N,
A should decrease with N
 Classical example:
respiratory disease
These are ‘idealizations’. In reality, transmission probably something in between (e.g.
cowpox in voles) or more complex (e.g. network)
Summary
• Compartment models like the SIR model are based on common
features of microparasite infections (peak viremia, immune cell
dynamics)
• We write them as flow charts and / or equations
• Density-dependent transmission predicts a threshold population
size needed for epidemics
• Frequency-dependent transmission has no such threshold
• Although difficult to measure in nature, there is some evidence for
threshold populations in several microparasite infections
• Specifying models helps us to define an R0, which gives a lot of
useful information and predictions (e.g. will the disease invade?)
Compartmental models
medical status
diseased
incubation
infection status
susceptible
exposed/latent
infectious
recovered
pathogen
time of
infection
time since infection
Including realism
S
E
I
R
infection status
susceptible
exposed/latent
infectious
recovered
immune response
pathogen
time of
infection
time since infection