A Generating Function Approach to HIV Transmission with Dynamic Contact Rates

¹ Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM
² Department of Physics, University of Michigan, Ann Arbor, MI
³ Department of Epidemiology, University of Michigan, Ann Arbor, MI

^⋆ Corresponding author. E-mail: eoromero@lanl.gov

Abstract

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.