A simple SI model with two age groups and its application to us HIV epidemics: To treat or not to treat?

The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years or more, then the number of infected adults may actually increase in the next 20 years or so. This creates a dilemma: to treat or not to treat?