Time-Dependent Casual Encounters Games and HIV Spread

In Tully et al. (Math Biosci Eng AIMS, 2015, to submitted) the authors model and investigate casual sexual encounters between two members of a population with two possible HIV states: positive and negative, using a Nash game framework in which players try to maximize their expected payoff resulting out of a possible encounter. Each player knows their own HIV status, but do not know the HIV status of a potential partner. They do however have a personal assessment of the risk that the potential partner may be HIV positive. Last but not least, each player has a ranked list of preferences of potential types of sexual outcomes: unprotected, protected, or no sexual outcome. In Tully et al. (Math Biosci Eng AIMS, 2015, to submitted), the game model is studied via 1- and 2-dimensional sensitivity analyses on parameters such as the utility values of unprotected sex of an HIV negative individual with an HIV positive, and values of personal risk (of encountering an HIV positive partner) perception. In this work, we introduce time as a variable which affects players’ risk perceptions, and thus their strategies. Given that HIV transmission happens when an HIV positive player has a non-zero probability (strategy) of having unprotected sex with a HIV negative player, we are also able to keep track of the time evolution of the overall fraction of HIV positive individuals in the population, as reflected as an outcome of repeated casual encounters. We model a continuous time dynamic game (as in Cojocaru et al. (J Optim Theory Appl 127(3):549–563, 2005)) where we compute the stable strategies of each player based on a dynamical system defined on a set of functions. We observe that with change in choices the HIV prevalence in the population increases.