Population dynamics of infectious diseases Arjan Stegeman

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

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

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

SIR Models Population consists of: SusceptibleInfectiousRecovered individuals

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

Greenwood assumption Constant probability of infection (Force of infection)

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

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

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

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

Results of Markov chain model

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

Results of Markov chain model

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

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

SIR model with Reed Frost assumption Probability of infection upon contact (p) Contacts are with rate  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

SIR Model with Reed Frost assumption Rate of infection of susceptibles depends on the number of infectious individuals  SI/N I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) S I R

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

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

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

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

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

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

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.

Transmission between individuals  /  = Basic Reproduction ratio, R 0 Average number of secondary cases caused by 1 infectious individual during its entire infectious period in a fully susceptible population

Reproduction ratio, R 0 R 0 = 3 R 0 = 0.5

Stochastic threshold theorem The probability of a major outbreak Prob major = 1 - 1/R 0

Final size distribution for R 0 = 0.5 infection fades out after infection of 1 or a few individuals (minor outbreaks only)

Final size distribution for R 0 = 3 R 0 > 1 : infection may spread extensively (major outbreaks and minor outbreaks)

Deterministic threshold theorem: Final size as function of R 0

Transmission in an open population S IR  *S*I/N  *I  = infection rate parameter  = recovery rate parameter  = replacement rate parameter  *S  *I  *R  *N

Courses of infection in an open population S I 1: Minor outbreak (R 0 1) 2: Major outbreak (R 0 > 1) 3: Endemic infection (R 0 > 1)

Infection can become endemic when the number of animals in a herd is at least: M. paratuberculosis:  = 0.003;  = ; R 0 = 10 N min = 5 BHV1 :  = 0.07;  = , R 0 = 3.5 N min = 110

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

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

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

Population dynamics of infectious diseases Interaction between agent - host & contact structure between hosts determine the transmission Quantitative approach: R 0 plays the central role