A generating function approach to HIV transmission with dynamic contact rates

The basic reproduction number, R₀, is often defined as the average number of infections generated by a newly infected individual in a fully susceptible population. The interpretation, meaning, and derivation of R₀ are controversial. However, in the context of mean field models, R₀ demarcates the epidemic threshold below which the infected population approaches zero in the limit of time. In this manner, R₀ has been proposed as a method for understanding the relative impact of public health interventions with respect to disease eliminations from a theoretical perspective. The use of R₀ is made more complex by both the strong dependency of R₀ on the model form and the stochastic nature of transmission. A common assumption in models of HIV transmission that have closed form expressions for R₀ is that a single individual's behavior is constant over time. In this paper we derive expressions for both R₀ and probability of an epidemic in a finite population under the assumption that people periodically change their sexual behavior over time. We illustrate the use of generating functions as a general framework to model the effects of potentially complex assumptions on the number of transmissions generated by a newly infected person in a susceptible population. We find that the relationship between the probability of an epidemic and R₀ is not straightforward, but, that as the rate of change in sexual behavior increases both R₀ and the probability of an epidemic also decrease.