Epidemic Models

Last update: 13 Dec 2024 20:00
First version: 31 May 2018

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...)

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]

Nino Boccara, Modeling Complex Systems [ Review]
Stephen P. Ellner and John Guckenheimer, Dynamic Models in Biology

A. J. Lotka, Elements of Mathematical Biology, ch. VIII [It's become conventional to attribute epidemic models of this sort to Ross's work on malaria, and while that was certainly very important, Lotka's contemporary review of the literature, originally published in 1924, makes it clear that this was much more of a collective effort (as one might expect)]

Roy M. Anderson and Robert M. May, "Population biology of infectious diseases"
- "Part I", Natur 280 (1979): 361--367
- "Part II", Nature 280 (1979): 455--461
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
Jonas L. Juul, Kaare Græsbøll, Lasse Engbo Christiansen, Sune Lehmann, "Fixed-time descriptive statistics underestimate extremes of epidemic curve ensembles", Nature Physics 17 (2021): 5--8, arxiv:2007.05035
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.]
Robert M. May, Sunetra Gupta, Angela R. McLean, "Infectious Disease Dynamics: What Characterizes a Successful Invader?", Philosophical Transactions: Biological Sciences 356 (2001): 899+901--910
Omar Melikechi, Alexander L. Young, Tao Tang, Trevor Bowman, David Dunson, James Johndrow, "Limits of epidemic prediction using SIR models", arxiv:2112.07039 [My comments]
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.]

CRS, Lecture slides on epidemic models, 16 April 2020

N. T. Bailey, The Mathematical Theory of Epidemics
Lamia Belhadji, "Ergodicity and hydrodynamic limits for an epidemic model", arxiv:0710.5185
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
Stefan Hohenegger, Francesco Sannino, "Renormalisation Group Methods for Effective Epidemiological Models", arxiv:2402.16409
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
Ning Ning and Edward Ionides, "Systemic infinitesimal over-dispersion on graphical dynamic models", Statistics and Computing 34 (2024): 147
Victor M. Panaretos, "Partially observed branching processes for stochastic epidemics", Journal of Mathematical Biology 54 (2007): 645--668
Ana Pastore y Piontti, Nicola Perra, Luca Rossi, Nicole Samay and Alessandro Vespignani, Charting the Next Pandemic: Modeling Infectious Disease Spreading in the Data Science Age
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)
Emilia Vynnycky and Richard G. White, An Introduction to Infectious Disease Modelling

CRS, "Fisher Information and (Near) Partial Identification for SIR Models and Logistic Growth Curves"