On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China

Heterogeneity in sexual behavior is known to play an important role in the spread of HIV. In 1986, a mathematical model based on ordinary differential equations was introduced to take into account the distribution of sexual activity. Assuming proportionate mixing, it was shown that the basic reproduction number $R_0$ determining the epidemic threshold was proportional to $M+V/M$, where $M$ is the mean and $V$ the variance of the distribution. In the present paper, we notice that this theoretical distribution is different from the one obtained in behavioral surveys for the number of sexual partnerships over a period of length $\tau$. The latter is a ''mixed Poisson distribution'' whose mean $m$ and variance $v$ are such that $M=m/\tau$ and $V=(v-m)/\tau^2$. So $M+V/M=(m+v/m-1)/\tau$. This way, we improve the link between theory and data for sexual activity models of HIV/AIDS epidemics. As an example, we consider data concerning sex workers and their clients in Yunnan, China, and find an upper bound for the geometric mean of the transmission probabilities per partnership in this context.