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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. 1 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. 2 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. 1 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) 4 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. 6