Mathematical Analysis of Epidemiological Models III

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1 Intro Computing R Complex models Mathematical Analysis of Epidemiological Models III Jan Medlock Clemson University Department of Mathematical Sciences 27 July 29

2 Intro Computing R Complex models What is R? Basic Reproduction Number Net Reproductive Rate the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible (Anderson & May, 1991)

3 Intro Computing R Complex models Why is R important? For a wholly susceptible host population, R > 1 pathogen can invade. R < 1 pathogen cannot invade. When a pathogen is present in the population, often, but not always, R < 1 pathogen will die out of the population.

4 Intro Computing R Complex models The effective reproduction number, R If the population is not wholly susceptible, then we have R, the effective reproduction number. Pathogen already present Vaccinated population

5 Intro Computing R Complex models How to compute R? Heuristic methods Systematic method P. van den Driessche & James Watmough, 22, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 18:

6 Intro Computing R Complex models Example model for STI M S M E M I M R F S F E F I F R dm S dm E dm I dm R = ω M M R β M F I F M S = β M F I F M S τ M M E df S df E = ω F F R β F M I M F S = β F M I M F S τ F F E = τ M M E γ M M I df I = τ FF E γ F F I = γ M M I ω M M R df R = γ F F I ω M F R

7 Intro Computing R Complex models Procedure Decide which states are infected We need to decide which states are infected and which are uninfected. In the STI model, Infected: M E, F E, M I, F I Uninfected: M S, F S, M R, F R

8 Intro Computing R Complex models Procedure Find disease-free equilibrium (or other equilibrium) Set dx = for all model state variables to find equilibrium. Also, for disease-free equilibrium, there are no infected people.

9 Intro Computing R Complex models Procedure Find disease-free equilibrium (or other equilibrium) = ω M M R β M F M S = ω F F R β F M F S = β M F M S τ M = β F M F S τ F = τ M γ M = τ F γ F = γ M ω M M R = γ F ω F F R M S = F S = P 2 M E = F E = M I = F I = M R = F R = M = F = P 2

10 Intro Computing R Complex models Procedure Decide which terms are new infections From the right-hand sides of the equations for the infected states, decide which terms represent new infections, F. The remainder are V. dx = F V F is the rate of production of new infections. V is the transition rates between states.

11 Intro Computing R Complex models Procedure Decide which terms are new infections dm E F I = β M F M S τ M M E df E M I = β F M F S τ F F E dm I = τ M M E γ M M I df I = τ FF E γ F F I F β I M F M S τ M M E M β I F = F M F S, V = τ F F E τ M M E + γ M M I τ F F E + γ F F I

12 Intro Computing R Complex models Procedure Take derivatives at equilibrium F = df dx = df 1 dx 1. df n df dx 1 n dx n df 1 dx n. V = dv dx = dv 1 dx 1. dv n dv dx 1 n dx n dv 1 dx n. These are the rates for new infections and transitions near the equilibrium.

13 Intro Computing R Complex models Procedure Take derivatives at equilibrium At the disease-free equilibrium, M S = F S = M = F = P 2, M E = F E = M I = F I = M R = F R = F β I M F M M S β S M M β I F = F M F F β M S, F = F β S F M = β F τ M M E τ M τ V = F F E τ M M E + γ M M I, V = τ F τ M γ M τ F F E + γ F F I τ F γ F

14 Intro Computing R Complex models Procedure Find V 1 V 1 gives the times spent in each state. In general, finding the inverse is difficult by hand, but computer algebra (Sage, Maple, Mathematica) takes care of that. 1 τ M V 1 1 = τ F 1 1 γ M γ M 1 1 γ F γ F

15 Intro Computing R Complex models Procedure Find FV 1 FV 1 gives the total production of new infections over the course of an infection. β M β F F = FV 1 β F =, V 1 = β M γf γ M 1 τ M 1 τ F 1 1 γ M γ M 1 1 γ F γ F β M γf β F γ M

16 Intro Computing R Complex models Procedure Find ρ(fv 1 ) The largest eigenvalue λ gives the fastest growth of the infected population. ( FV 1) N λ N v for large N. So R = λ. β M β γf M γf FV 1 β F β = γ M F γ M { } σ(fv 1 βf β M βf β M βf β M ) =,, = R = γ M γ F γ M γ F γ M γ F

17 Intro Computing R Complex models Alternative interpretation If we had chosen only F E & F I to be infected states, then R = β Fβ M γ M γ F

18 Intro Computing R Complex models More complex models Flu S I R ds a = λ a S a di a = λ as a (γ a + ν a )I a, dr a λ a = σ a N 17 α=1 φ aα β α I α, = γ a I a, for a = 1,..., 17

19 Intro Computing R Complex models I a are infected states More complex models Flu Equilibrium is everyone susceptible, with given age structure New-infection term is λ a S a, so F = λ S, V = (γ + ν) I Then And F = {[ σ S ] } β T φ, V = diag (γ + ν) N FV 1 = {[ σ (γ + ν) S ] } β T φ N

20 Intro Computing R Complex models More complex models Flu Putting in parameter values from the pandemics, we get 1918 R = R = 1.3 Proportion infected Time (days)