Population dynamics of infectious diseases

1. Population dynamics of infectious diseases Arjan Stegeman

2. Introduction to the population dynamics of infectious diseases • Getting familiar with the basic models • Relation between characteristics of the model and the transmission of pathogens

3. Modelling population dynamics of infectious diseases • model : simplified representation of reality • mathematical: using symbols and methods to manipulate these symbols

4. Why mathematical modeling ? • Factors affecting infection have a non-linear dependence • Insight in the importance of factors that affect the spread of infectious agents • Provide testable hypotheses • Extrapolation to other situations/times

5. SIR Models Population consists of: Susceptible Infectious Recovered individuals

6. SIR Models • Dynamic model : • S, I and R are variables (entities that change) that change with time, • parameters (constants) determine how the variables change

7. Greenwood assumption • Constant probability of infection (Force of infection)

8. SIR Model with Greenwood assumption *I IR*S S I R IR = Incidence rate  = recovery rate parameter (1/infectious period)

9. Transition matrix Markov chain P = probability to go from a state at time t to a state at time t+1

10. Markov chain modeling Starting vector* * number of S, I and R at the start of the modeling

11. Example Markov chain modeling Starting vector* * number of S, I and R at the start of the modeling

12. Results of Markov chain model

13. Example Markov chain modeling Starting vector * * number of S, I and R at the end of time step 1

14. Results of Markov chain model

15. Course of number of S, I and R animals in a closed population (Greenwood assumption)

16. Drawback of the Greenwood assumption • Number of infectious individuals has no influence on the rate of transmission

17. SIR model with Reed Frost assumption • Probability of infection upon contact (p) • Contacts are with rate e per unit of time • Contacts are at random with other individuals (mass action assumption), thus probability that an S makes contact with an I equals I/N p

18. SIR Model with Reed Frost assumption Rate of infection of susceptibles depends on the number of infectious individuals I S R SI/N I  = infection rate parameter (Number of new infections per infectious individual per unit of time)  = recovery rate parameter (1/infectious period) N = total number of individuals (mass action)

19. SIR Model with Reed Frost assumption • (formulation in text books, pseudomass action) • (formulation according to mass action) It+1= number of new infectious individuals at t+1 q = probability to escape from infection

20. Example: Classical Swine Fever virus transmission among sows housed in crates •  = 0.29; Susceptible has a probability of: • to become infected in one time step •  = 0.10; Infectious has a probability of • to recover in one time step

21. Course of number of S, I and R animals in a closed population (reed Frost assumption with mass action)

22. Deterministic - Stochastic • Deterministic models: all variables have at each moment in time for a particular set of parameter values only one value • Stochastic models: stochastic variables are used which at each moment in time can have many different values each with its own probability

23. Course of number of S, I and R animals in a closed population (reed Frost assumption with mass action)

24. Course of number of S, I and R animals in a closed population (1 run stochastic SIR model)

25. Course of number of S, I and R animals in a closed population (1 run stochastic SIR model)

26. Stochastic models • Preferred above deterministic models because they show variability in outcomes that is also present in the real world. This is especially important in the veterinary field, because we often work with populations of limited size.

27. Transmission between individuals b/a = Basic Reproduction ratio, R0 Average number of secondary cases caused by 1 infectious individual during its entire infectious period in a fully susceptible population

28. Reproduction ratio, R0 R0 = 3 R0 = 0.5

29. Stochastic threshold theoremThe probability of a major outbreak Prob major = 1 - 1/R0

30. Final size distribution for R0 = 0.5 infection fades out after infection of 1 or a few individuals (minor outbreaks only)

31. Final size distribution for R0 = 3 R0 > 1 : infection may spread extensively (major outbreaks and minor outbreaks)

32. Deterministic threshold theorem: Final size as function of R0

33. Transmission in an open population I R S m*N *I *S*I/N m*S m*I m*R b = infection rate parameter a = recovery rate parameter m = replacement rate parameter

34. Courses of infection in an open population 2: Major outbreak (R0 > 1) I 3: Endemic infection (R0 > 1) 1: Minor outbreak (R0 < 1 of R0 > 1) S

35. Infection can become endemic when the number of animals in a herd is at least: M. paratuberculosis: a = 0.003; m = 0.0009; R0 = 10 Nmin = 5 BHV1 : a = 0.07;m= 0.0009, R0 = 3.5 Nmin = 110

36. Transmission in an open population At endemic equilibrium (large population)

37. Assumptions • Mass action (transmission rate depends on densities) • Random mixing • All S or I individuals are equal (homogeneous)

38. SIR model can be adapted to: • SI model • SIS model • SIRS model • SLIR model • etc.

39. Population dynamics of infectious diseases • Interaction between agent - host & contact structure between hosts determine the transmission • Quantitative approach: R0 plays the central role