Download Incidence functions and population thresholds

Outline
Incidence functions
Incidence functions and
population thresholds
Density dependence
Population thresholds
Jamie Lloyd-Smith
-
Invasion
-
Persistence
-
Thresholds and host extinction
Center for Infectious Disease Dynamics
Pennsylvania State University
The incidence rate
S
Density-dependent transmission
Incidence rate = f (S,I)
S
I
Incidence rate = f (S,I)
f ( S, I ) = c( N ) p
Incidence term in models describes the rate that new infections arise.
f(S,I) = Force of infection × S
I
SI
N
If contact rate is linearly density-dependent:
Force of infection, λ = c(N) p I/N
c (N) = contact rate (possibly density-dependent)
p = probability of transmission given contact
I/N = prob. that randomly-chosen partner is infectious
f ( S, I ) = c( N ) p
So
Then
c(N)
f(S,I)
= kN
= kN p SI/N
= βMA SI
where βMA = kp
Æ “Mass action” transmission. Also known as density-dependent or,
confusingly, “pseudo-mass action” (see McCallum et al, 2001)
SI
N
Frequency-dependent transmission
S
Incidence rate = f (S,I)
f ( S, I ) = c( N ) p
I
SI
N
If contact rate is constant with respect to density:
Then
c(N)
f(S,I)
= c0
= c0 p SI/N
= βFD SI/N
where βFD = c0 p
Æ “Frequency-dependent” transmission. Also known as the
standard incidence or, confusingly, “true mass action” (see
McCallum et al, 2001)
McCallum et al (2001) Trends Ecol Evol 16: 295-300.
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Many choices – what to do?
Saturating transmission
Classically it was assumed that transmission rate increases with
population size, because contacts increase with crowding.
Æ mass action (βSI) was dominant transmission term
Hethcote and others argued that rates of sexual contact are
determined more by behaviour and social norms than by density, and
favoured frequency-dependent transmission for STDs.
Since the 1990s, this has been a topic of active research using
experimental epidemics, field systems, and epidemiological data.
Deredec et al (2003) Ann Zool Fenn 40: 115-130.
Detecting density dependence
Evidence for FD vs MA transmission
How can we test for density dependence in transmission?
• Fit models with different transmission functions to epidemic time series.
• Look at indicators for transmission ∝ N in epidemiological data:
With increased transmission rate, we expect:
↑ estimates of R0
↑ exponential growth rate of epidemic, r
↓ proportion susceptible following epidemic, or at steady state
↓ mean age of infection in endemic setting
Fitting models to data from cowpox in bank voles and wood mice
Æ FD model is better fit than MA (though neither is perfect)
Begon et al (1999) Proc Roy Soc B 266: 1939-1945.
Evidence for FD vs MA transmission
Evidence for FD vs MA transmission
Model results:
density-dependent transmission
Measles in England and Wales
(recall that MA predicts that R0 ∝ N)
9
8
8
7
7
6
5
4
3
p<0.001
2
1
0
p<0.001
0.6
0.8
1
1.2
Population size, N
1.4
1.6
Mean age of infection (y)
Æ roughly FD transmission
Mean age of infection (y)
• R0 is ~ constant vs population size
Lepto data
9
6
5
p=0.54
4
p=0.35
3
2
1
0
0.6
5
x 10
0.8
1
1.2
Population size, N
1.4
1.6
5
x 10
Leptospirosis in California sea lions
Mean age of infection does not decrease with N
Bjornstad et al (2002) Ecol Monog. 72: 169-184
Æ transmission not density-dependent.
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Evidence for FD vs MA transmission Æ neither?
Evidence for FD vs MA transmission
Model results:
density-dependent transmission
Lepto data
0.7
0.35
p=0.19, slope CI = (-4.4e-006, 1e-006)
p=5.9e-006, slope CI = (4.7e-007, 8.6e-007)
Exponential growth rate
0.6
0.25
0.2
0.15
0.1
0
0.5
0.5
0.4
0.3
0.2
0.1
0.05
1
1.5
2
Size of mixing population
2.5
0
0.4
3
0.6
5
x 10
0.8
1
1.2
Size of mixing population
1.4
1.6
5
x 10
Leptospirosis in California sea lions
PiGV in Plodia (Indian meal moth)
Epidemic growth rate does not increase with N
Transmission rate is not FD or MA – need complex functional forms.
Æ transmission not density-dependent.
Interpret in terms of host heterogeneity and effects of density on behaviour.
So what should we do?
Population thresholds in epidemic dynamics
Despite its fundamental importance, the issue of how to formulate
the transmission term in simple models is unresolved.
Some pointers:
R0 has been the central concept in epidemic dynamics since ~1980,
thanks largely to the work of Anderson & May.
(see the history of R0 by Heesterbeek 2002, Acta Biotheoretica)
• FD transmission is generally thought to be more appropriate
than MA in large well-mixed populations.
• In quite small populations, transmission is generally thought to
exhibit some density dependence and MA is acceptable.
• Think about population structure and mechanisms of mixing at
the scales of space and time you’re thinking about.
Is a very simple model appropriate?
(more on this in the next lecture)
Population threshold for disease invasion
Long before this, people studying epidemic dynamics have focused
on population thresholds.
• Population threshold for invasion (Kermack & McKendrick 1927):
host population size below which parasite cannot invade.
• Population threshold for persistence, or the critical community size
(Bartlett 1957, Black 1966): host population size below which
parasite cannot persist long-term.
Population threshold for disease invasion
Under density-dependent transmission, R0 = βND
or in fact R0 any increasing function of N.
Æ R0>1 corresponds to a population threshold N>NT.
Under frequency-dependent transmission, R0 = βD.
Æ No threshold N for R0>1.
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Probability
Probabilityof
of extinction
extinction
Exponential growth rate
0.3
0.8
0.6
R0 < 1
Disease
dies out
0.4
R0 > 1
R0 > 1
Disease can
invade/persist
Disease can
invade/persist
0.2
0
0
1
2
3
4
Basic reproductive number,R0
NT R
0
5
Host population size, N
Host population size, N
3
Susceptibility threshold for disease invasion
Population thresholds for invasion: evidence
Recall: under any form of transmission, Reffective = R0 × S/N.
Despite its conceptual simplicity , real-world evidence for invasion
thresholds is hard to find, for several reasons
Æ For Reffective > 1, must have S/N > 1/R0.
• failed invasions are difficult to observe
• demographic stochasticity leads to variation in outbreak sizes
Reff > 1
Disease
dies out
Disease can
invade/persist
1/R0
Proportion susceptible, S/N
• when R0<1, limited chains of transmission can still occur
• when R0>1, epidemic can still die out by chance.
N=100
R0=0.9
0
This phenomenon is the basis for herd immunity.
Stochastic variation in outbreak size
R0=1.5
Frequency
Reff < 1
100
Outbreak size
0
100
Outbreak size
Stochastic variation in outbreak size
Measles outbreaks in
vaccinated populations, UK
Outbreak size
Posterior distribution
on Reff under two
models
Branching process models allow analysis of outbreak size to make
inference about the effective reproductive number.
Farrington et al (2003) Biostatistics 4: 279-295.
Lloyd-Smith et al (2005) Trends Ecol Evol 20: 511-519.
Population thresholds for persistence
Even if parasite is able to invade (R0>1), this does not guarantee its
persistence in the long term.
Persistence thresholds – another view
Broken chains of transmission can arise in two ways:
1.
Birth
There are two broad mechanisms whereby a disease can fail to
persist, or fade out:
Endemic fadeout: random fluctuations around the endemic
equilibrium can cause extinction of the parasite.
Epidemic fadeout: following a major epidemic, the susceptible pool
is depleted and the parasite runs out of individuals to infect.
Critical Community Size is population size above which a disease
can persist long-term (yes, this definition is vague).
Courtesy of
Ottar Bjornstad
1. Susceptible bottleneck
2. Transmission bottleneck
Lost immunity
2.
Susceptible
S
Infected
I
Recovered Susceptible
R
S
Transmission
Epidemic fadeout: Parasite extinction occurring because susceptible numbers are so
low immediately following an epidemic that small stochastic fluctuations can remove all
parasites. (Susceptible bottleneck)
Endemic fadeout: Parasite extinction occurring because endemic numbers of infected
individuals are so low that small stochastic fluctuations can remove all parasites.
(Transmission bottleneck)
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Endemic fadeout
Endemic fadeout
Stochastic SIR model with FD transmission and R0=4.
10 simulations are shown. + signs show times when disease fades out.
Mean time to endemic fadeout for stochastic SIR model with R0=4 and
different rates of demographic turnover.
1. No sharp threshold in N; there’s a gradual trend of longer persistence.
4000
1
2. Demographic rates are as important as N, if not more so.
0.6
2000
0.4
1000
Infectious, I
0
0
2000
0.2
Mean time to disease fadeout
0.8
3000
Fraction fadeout
Population size
Susceptible, S
0
6000
4000
20000
(i) Fast
turnover
(ii)
15000
10000
5000
Slow turnover (iii)
0
0
5000
10000
15000
20000
Population size (N )
Time
Lloyd-Smith et al (2005) Trends Ecol Evol 20: 511-519.
Endemic fadeout
Epidemic fadeout
Quasi-stationary distribution: distirbution of I conditioned on non-extinction.
Fast births and deaths
Stochastic SIR model with FD transmission and R0=4
(exact same model as for endemic fadeout, but now started from I=1
instead of I=I*.)
Slow births and deaths
6000
2500
5000
2000
4000
1500
3000
1000
2000
S
500
1000
I
0
See work by I. Nasell for mathematical development.
0
Quasi-stationary
distribution of I
Epidemic fadeout
200
1. No sharp threshold in N; there’s a gradual trend of longer persistence.
400
600
0
800
Time
The classic example
Probability that disease persists through the first post-epidemic trough
Infected hosts (I )
Quasi-stationary
distribution of I
Susceptible hosts (S )
10 simulations are shown. + signs show times when disease fades out.
UK ~ 60M people
of epidemic fadeout:
measles
2. Demographic rates are as important as N, if not more so.
Probability of persistence
Denmark ~ 5M people
1
(i) Fast
turnover
0.8
(ii)
0.6
Note how measles is not
endemic in Iceland, but
instead has periodic
outbreaks dependent on
re-introduction of the virus.
0.4
(iii)
Slow turnover
0.2
0
0
5000
10000
15000
20000
Iceland ~ 0.3M people
25000
Population size (N )
5
The classic example of epidemic fadeout: measles
Proportion of time extinct
0.7
0.6
Recurrent
Epidemics
Endemic
0.5
0.4
Extinction risk: can a disease drive its host extinct?
Density-dependent
transmission
Frequency-dependent
transmission
Population threshold NT
protects host from
disease-induced
extinction.
No threshold Æ diseaseinduced extinction is possible
(Getz & Pickering 1984).
0.3
Critical Community Size
~300-500k
0.2
R0 < 1
R0 > 1
R0 > 1
0.1
NT
0
10 000
100 000
1 000 000
10 000 000
population size
Æ Direct extinction due to disease in single host is unlikely, but
diseases can cause bottlenecks such that genetic diversity
and Allee effects become important.
Measles in England and Wales
Extinction risk: can a disease drive its host extinct?
Extinction risk: multiple host species and spillover
Spillover from reservoir can threaten endangered populations
What if the host population
itself has a threshold
density below which it
cannot persist?
Then the outcome
depends on the relative
values of the threshold
population size for disease
extinction vs host
extinction.
Deredec & Courchamp (2003)
Ann. Zool. Fenn. 40:115-130.
Woodroffe, 1999
1997 Mediterranean monk seal die-off
in Mauritania.
>100 monk seals died (~1/3 of global
population), probably due to dolphin
morbillivirus (a relative of measles)
that spilled over from another species.
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