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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 nonlinear 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: Susceptible Infectious individuals Recovered 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 *I IR*S S dS IR * S dt I dI IR * S * I dt R dR *I dt IR = Incidence rate = recovery rate parameter (1/infectious period) Transition matrix Markov chain To S I R S PSS PSI PSR From I PIS PII PIR R PRS PRI PRR P = probability to go from a state at time t to a state at time t+1 Markov chain modeling Starting vector* S I R To S I R S PSS PSI PSR From I PIS PII PIR R PRS PRI PRR * number of S, I and R at the start of the modeling Example Markov chain modeling Starting vector* 99 1 0 To S I R S 0.90 0.10 0.00 From I 0.00 0.80 0.20 R 0.00 0.00 1.00 * number of S, I and R at the start of the modeling Results of Markov chain model Time step S I R 0 99 1 0 1 =99*0.9+1*0+0*0= =99*0.1+1*0.8+0*0= 99*0+1*0.2+0*1= 89.1 10.7 0.2 Example Markov chain modeling Starting vector * 89.1 10.7 0 .2 To S I R S 0.90 0.10 0.00 From I 0.00 0.80 0.20 R 0.00 0.00 1.00 * number of S, I and R at the end of time step 1 Results of Markov chain model Time step S I R 0 99 1 0 1 99*0.9+1*0+0*0= 99*0.1+1*0.8+0*0= 99*0+1*0.2+0*1= 89.1 10.7 0.2 2 89.1*0.9+10.7*0+0.2 89.1*0.1+10.7*0.9+ *0=80.2 0.2*0=17.5 89.1*0+10.7*0.2+0.2 *1=2.3 Course of number of S, I and R animals in a closed population (Greenwood assumption) Number of animals 100 80 60 40 20 0 0 5 10 15 20 25 30 Number of time steps S I R 35 40 45 50 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) 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 SIR Model with Reed Frost assumption Rate of infection of susceptibles depends on the number of infectious individuals S SI/N I I R = 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) SIR Model with Reed Frost assumption I t 1 St (1 q ) It It+1= number of new infectious individuals at t+1 q = probability to escape from infection • (formulation in text books, pseudomass action) I t 1 St (1 e I t / N ) • (formulation according to mass action) Example: Classical Swine Fever virus transmission among sows housed in crates • = 0.29; Susceptible has a probability of: I (1 e N ) to become infected in one time step • = 0.10; Infectious has a probability of (1 e ) 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) 80 60 40 20 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 10 0 Number of animals 100 Number of time steps S I R 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) 80 60 40 20 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 10 0 Number of animals 100 Number of time steps S I R Course of number of S, I and R animals in a closed population (1 run stochastic SIR model) Number of animals 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Number of time steps S I R Course of number of S, I and R animals in a closed population (1 run stochastic SIR model) Number of animals 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Number of time steps S I R 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, R0 Average number of secondary cases caused by 1 infectious individual during its entire infectious period in a fully susceptible population Reproduction ratio, R0 R0 = 3 R0 = 0.5 Stochastic threshold theorem prob major The probability of a major outbreak 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 Prob major = 1 - 1/R0 0 1 2 3 4 5 R0 6 7 8 9 10 Final size distribution for R0 = 0.5 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 probability infection fades out after infection of 1 or a few individuals (minor outbreaks only) final size Final size distribution for R0 = 3 R0 > 1 : infection may spread extensively (major outbreaks and minor outbreaks) 0,3 0,2 0,15 0,1 0,05 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 probability 0,25 final size Final size Deterministic threshold theorem: Final size as function of R0 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 ln( 1 p) R p 0 0,5 1 1,5 2 2,5 3 3,5 Reproduction ratio 4 4,5 5 5,5 6 Transmission in an open population m*N I S *S*I/N m*S R *I m*I = infection rate parameter = recovery rate parameter m = replacement rate parameter m*R 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 Infection can become endemic when the number of animals in a herd is at least: m R0 N *( ) m R0 1 M. paratuberculosis: = 0.003; m = 0.0009; R0 = 10 Nmin = 5 BHV1 : = 0.07; m = 0.0009, R0 = 3.5 Nmin = 110 Transmission in an open population R0 m At endemic equilibrium (large population) N R0 S 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: R0 plays the central role