1. Disease ecology – SIR models ECOL 8310, 11/17/2015

2. Why do ecologists study host-parasite interactions? Major regulator of wildlife population dynamics Control 1 Treatment 2 Treatments Trichostrongylus tenuis Lagopus lagopus scoticus

3. Why do ecologists study host-parasite interactions? Cause of decline for threatened and endangered species Even-toed ungulates Omnivorous marsupials Mustela nigripes

4. Why do ecologists study host-parasite interactions? Zoonoses are the main source of emerging infectious diseases in humans Jones et al. 2008

5. Epidemics Poliomyelitis in the USA Canine parvovirus in wolves in Minnesota

6. Sequential epidemics Measles in EnglandRabies in red foxes in France

7. 45 nm Culex quinquefasciatus Culex pipiens West Nile virus (extreme complications) in humans

8. Key Concepts Models of disease dynamics The logic behind a compartment model Processes and rates Density-dependent transmission Disease invasion and R 0 Threshold population sizes Frequency-dependent transmission Versions of SIR models Vaccination (Did Not Get To This)

9. 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?

10. S S I I Susceptible - Infected βSI 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)

11. Contact rate Population size, N 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 AthensAtlantaNYC Contact rate = cN Transmission between hosts

12. Transmission requires: i) Contact between individuals ii) The ‘right’ sort of contact (between an S and an I) iii) The parasite establishes in the new host ✔ (cN) Transmission between hosts

13. 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’)

14. Transmission requires: i) Contact between individuals ii) The ‘right’ sort of contact (between an S and an I) iii) The parasite establishes in the new host ✔ ✔ ✔ Expression for transmission = (cN)*(S/N)*I*a (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

15. γI 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 I I R R Infectious Period Infected Recovered

16. βSI γI S S I I R R SIR: The basic compartmental model Examples: measles, chickenpox Anything that only needs one course of vaccine Equilibria depend on R 0, can be endemic or epidemic

17. S S I I R R Here, birth rate and death rate are different. What happens to the total host population size? SIR model with births and deaths

18. S S I I R R Either grows or declines (both exponentially)

19. S S I I R R Suppose the parasite can cause additional host mortality How would we model that?

20. S S I I R R Suppose the parasite can cause additional host mortality How would we model that?

21. S S I I R R Now the parasite might stop exponential growth (an example of regulation)

22. Now the parasite might stop exponential growth (an example of regulation)

23. S S I I R 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)

24. S S I I R R Early on, infected individuals are rare and so S~N (the population size) R 0 =Basic reproduction number

25. Basic reproductive number (R 0 ): 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 R 0 >1. Example: R 0 =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?

26. Suppose transmission rate and infectious period are fixed, the disease might invade some populations but not others Why? Disease invasion & threshold populations

27. Epidemic burnout vs. Endemic Causes of parasite decline: Immunity Density below R 0 = 1 Stochasticity Causes of endemic persistence: Births Immigration loss of immunity pathogen evolution reintroduction

28. Polio HIV/AIDS Measles Pandemic flu SARS 20 18 16 14 12 10 8 6 4 2 0 Whooping cough R0R0

29. From Wikipedia

30. Contact rate Population size Constant Frequency-dependent transmission: Alternative to density-dependent transmission

31. Transmission requires: i)Contact between individuals ii)The ‘right’ sort of contact (between an S and an I) iii)The parasite establishes in the new host ✔ ✔ ✔ Expression for transmission = (c)*(S/N)*I*a (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

32. Transmission Density- and frequency-dependent transmission are (useful) idealizations N Contact rate Reality is probably more complicated. Difficult to measure in practice…

33. Predictions based on transmission Density- and frequency-dependent transmission N R0R0

34. βSI/N γI S S I I R 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) Early on, infected individuals are rare and so S~N (the population size) R0R0 R0 with Frequency Dependence

35. With frequency-dependent transmission there is no threshold population size Models predict that the existence of threshold populations depends on the type of transmission

36. R 0 & age at infection It is possible to derive that if R 0 >1 and there is a supply of susceptibles, then the equilibrium proportion of susceptibles is S/N=1/R 0 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 R 0 =L/A R 0 is inversely proportional to mean age at infection This is also true when using more complicated models (e.g. not assuming rectangular age distribution)

37. Attack rate (better: attack ratio) = proportion of population that get infected during an outbreak. Intuitively, attack rate increases as R 0 increases. Here we see evidence that this also corresponds to a lowering of the mean age at infection. R 0 increasing

38. Predictions based on transmission Density- and frequency-dependent transmission N Mean age at infection

39. Leptospirosis in sea lions DD model of lepto predicts mean age at infection decreases with N Not detected in data Lepto in sealions not DD (DD)

40. Measles Bjørnstad et al. 2002, Ecol. Monographs Measles in 60 cities of England & Wales Recall that density-dependent transmission implies R 0 is proportional to N R 0 appears independent of host population size  supporting frequency- dependent transmission

41. Critical community size (CCS) Smallest population of susceptible hosts below which disease goes extinct, and above which disease persists Includes stochasticity S S I I R R

42. Epidemic fade-outs e.g. Measles Denmark ~ 5M people UK ~ 60M people Iceland ~ 0.3M people Note how measles is non-endemic in Iceland Cliff et al. 1981

43. Persistence vs population size Measles in England and Wales Proportion of time extinct population size Critical Community Size ~300-500k Recurrent Epidemics Endemic

44. 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 N T >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)

45. 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 R 0, which gives a lot of useful information and predictions (e.g. will the disease invade?)

46. time of infection time since infection incubation diseased susceptible exposed/latent infectious recovered pathogen medical status infection status Compartmental models

47. time of infection time since infection susceptible exposed/latent infectious recovered immune response pathogen infection status S S E E I I R R Including realism