## Epidemic Models

*28 Sep 2020 10:03*

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 being exposed and becoming infectious (SEIR); 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...)

See also: Agent-based Modeling; Complex Networks; Dynamics; Ecology; Evolution; Interacting Particle Systems; Memes and the "epidemiology of beliefs"; Statistics

- Recommended, big picture:
- M. S. Bartlett
- Stochastic Population Models in Ecology and Epidemiology
- "The Relevance of Stochastic Models for Large-Scale Epidemiological Phenomena", Journal of the Royal Statistical Society C
**13**(1964): 2--8

- D. J. Daley and J. Gani, Epidemic Modelling: An Introduction
- Lisa Sattenspiel (with Alun Lloyd), The Geographic Spread of Infectious Diseases: Models and Applications [While obviously focused on the spatial aspects, this actually presumes no particular background in epidemic models, and so can serve as a good introduction. JSTOR]

- Recommended, good treatments in broader textbooks:
- 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...] - Peter Sheridan Dodds, Duncan J. Watts, "A generalized model of social and biological contagion", Journal of Theoretical
Biology
**232**(2005): 587--604, arxiv:1705.10783 - 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.] - Joel C. Miller, "A note on the derivation of epidemic final sizes",
Bulletin of Mathematical Biology
**74**(2012): 2125--2141 [PubMed. Thanks to Carl Bergstrom, somewhere or other, for the pointer.]

- Modesty forbids me to recommend:
~~Lecture slides on epidemic models, 16 April 2020~~

- To read:
- N. T. Bailey, The Mathematical Theory of Epidemics
- Lamia Belhadji, "Ergodicity and hydrodynamic limits for an epidemic model", arxiv:0710.5185
- Romulus Breban, Raffaele Vardavas and Sally Blower,
"Linking population-level models with growing networks: A class of epidemic models", Physical Review E
**72**(2005): 046110 - Tom Britton and Etienne Pardoux (eds.), Stochastic Epidemic Models with Inference
- Odo Diekmann, Hans Heesterbeek and Tom Britton, Mathematical Tools for Understanding Infectious Disease Dynamics
- Peter Sheridan Dodds, "Slightly generalized Generalized Contagion: Unifying simple models of biological and social spreading", arxiv:1708.09697
- Romain Guy, Catherine Larédo, Elisabeta Vergu, "Approximation of epidemic models by diffusion processes and their statistical inference", arxiv:1305.3492
- Valerie Isham and Graham Medley (eds.), Models for Infectious Human Diseases: Their Structure and Relation to Data
- Matt J. Keeling and Pejman Rohani, Modeling Infectious Diseases in Humans and Animals
- Amanda A. Koepke, Ira M. Longini, Jr., M. Elizabeth Halloran, Jon Wakefield, Vladimir N. Minin, "Predictive Modeling of Cholera Outbreaks in Bangladesh", arxiv:1402.0536
- Maia Martcheva, An Introduction to Mathematical Epidemiology
- Victor M. Panaretos, "Partially observed branching processes for
stochastic
epidemics", Journal
of Mathematical Biology
**54**(2007): 645--668 - Steffen Unkel, C. Paddy Farrington, Paul H. Garthwaite, Chris Robertson, Nick Andrews, "Statistical methods for the prospective detection of infectious disease outbreaks: a review", Journal of the Royal Statistical Society A forthcoming (2011)