The Deterministic Limit of Infectious Disease Models with Dynamic Partners

This paper analyzes the large population dynamics of an infectious disease model with contacts that occur during partnerships. The model allows for concurrent partnerships following a very broad class of dynamic laws. Previous work, with a stochastic version of the model, computed the reproductive number, the initial growth rate, and the final size. In the present paper, the deterministic system that is the limit for large populations is constructed. The construction is unusual in requiring two different scaling factors. Next, the approximation used by Watts and May for a related model is compared with the exact solution. This approximation is most accurate at the beginning of the epidemic and when partnerships are short. Lastly, the model is generalized to allow dependencies among partnerships. This generalization permits proportional mixing with an arbitrary distribution on the number of partners.