Susceptible-infected-removed epidemic models with dynamic partnerships

Abstract

The author extends the classical, stochastic, Susceptible-Infected-Removed (SIR) epidemic model to allow for disease transmission through a dynamic network of partnerships. A new method of analysis allows for a fairly complete understanding of the dynamics of the system for small and large time. The key insight is to analyze the model by tracking the configurations of all possible dyads, rather than individuals. For large populations, the initial dynamics are approximated by a branching process whose threshold for growth determines the epidemic threshold, R ₀, and whose growth rate, λ, determines the rate at which the number of cases increases. The fraction of the population that is ever infected, Ω, is shown to bear the same relationship to R ₀ as in models without partnerships. Explicit formulas for these three fundamental quantities are obtained for the simplest version of the model, in which the population is treated as homogeneous, and all transitions are Markov. The formulas allow a modeler to determine the error introduced by the usual assumption of instantaneous contacts for any particular set of biological and sociological parameters. The model and the formulas are then generalized to allow for non-Markov partnership dynamics, non-uniform contact rates within partnerships, and variable infectivity. The model and the method of analysis could also be further generalized to allow for demographic effects, recurrent susceptibility and heterogeneous populations, using the same strategies that have been developed for models without partnerships.