Heterogeneous sexual activity models of HIV transmission in male homosexual populations

A proportionate mixing one-sex model of sexual transmission of HIV is described, in which sexual activity (new partners per unit time) is defined as a continuous variable in a set of integro-partial-differential equations. A discrete activity-class approximation is developed by matching equilibrium state and rate variables as closely as possible with the continuous-variable model, and consists only of ordinary differential equations. Activity-class boundaries are arbitrary, and each class is characterized by a single level of activity. If there are N classes, the level of activity in N - 1 of them is such that the steady-state susceptible class sub-population is equal to the population in the equivalent section of the continuous model. The activity level for the remaining class is chosen such that the condition for endemicity of the infection in the approximation is equal to that for the equivalent continuous-variable model; this minimizes errors in the steady-state population. The relationship between the discrete and continuous-variable models is explored, via numerical and analytical studies, in order to evaluate the accuracy of the approximation.