Notebooks

## Epidemic Models

19 Feb 2020 16:12

These are a specialized class of stochastic process, originally inspired by epidemiology, but widely applied in the social sciences, e.g., to model the spread of information through social contagion ("going viral", as we say). The most basic form divides the members of the population into two classes, the "susceptible" and the "infected"; contact between a susceptible person and an infected one can, with some probability, make the susceptible person infected. This is called an "SI" model. A natural refinement is to make the period of infectiousness finite, with a distinction between a formerly infectious person becoming susceptible again ("SIS"), or recovered or otherwise removed from the population ("SIR"); a delayed period between becoming infected and becoming infectious; whether the probability of transmission depends on the total number of infected individuals or depends on details of geography and social networks; etc., etc.

(In fact, epidemic models on networks get their own notebook...)

Recommended, big picture:
• M. S. Bartlett
• Nino Boccara, Modeling Complex Systems [ Review]
• Stephen P. Ellner and John Guckenheimer, Dynamic Models in Biology
Recommended, close-ups:
• D. J. Daley, D. G. Kendall, "Stochastic Rumours", IMA Journal of Applied Mathematics 1 (1965): 42--55 [The interesting twist here on a standard SIR model is that they assume you stop spreading the rumor on encountering someone who's already heard it, so $I + I \rightarrow 2R$ and $I+R \rightarrow 2R$. This, naturally, makes it very hard for the rumor to reach everyone...]
• R. M. May and R. M. Anderson, "The Transmission Dynamics of Human Immunodeficiency Virus (HIV)", Philosophical Transactions of the Royal Society of London B 321 (1988): 565--607 [This remarkable paper is the oldest one I can find which works out the consequences for an SIR model of randomly-varying but uncorrelated node degrees (section 4.1). Further commentary under epidemics on networks. JSTOR.]