This manuscript presents didactic formulations of continuous and discrete time versions of the SIR compartmental modeling paradigm (S=susceptible, I=infectious, R=recovered) used to capture the dynamics of epidemiological processes. These SIR formulations are typically deterministic (i.e., systems of ordinary differential or difference equations), but they can also be stochastic. SIR models assume epidemiological homogeneity of individuals (i.e., all individuals are equally susceptible, contagious when infected, and are subject to the same rate constants). They also assume the population is well-mixed (i.e., not subject to spatial structure). This latter assumption can be relaxed by imposing a metapopulation structure on the population: each subpopulation can modeled by an SIR model, with individuals moving between subpopulations at rates controlled by a propensity for individuals to move (as a function of their disease class), the distance the populations are separated in space, and the relative attractiveness of the different subpopulations to which individuals can move.
In their Epidemics paper, Getz and coauthors review the continuous and discrete deterministic and discrete stochastic formulations of the SIR dynamical systems models. They also demonstrate how to extend these models to a metapopulation setting using NMB network and mapping tools. Their exposition of using Numerus Model Builder to easily and rapidly construct these models is supported by videos (accessible at YouTube: see manuscript for links) of the construction process. The models presented in this paper are also downloadable for NMB users (see link above).
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