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.