Analytic solutions to compartmental models of the HIV/AIDS epidemic

For many years compartmental models have provided useful insights into the spread of epidemics. Such models are usually fairly easy to set up, but even the simpler models have the disadvantage that they are intractable to analytic solution. In this paper we examine models of the HIV/AIDS epidemic, and show that the equations may be linearized in a piecewise manner over time, thus allowing analytic solutions to be obtained. Indications of the usefulness of this approach are provided. In particular, an analytic solution gives insight into the mechanism of the epidemic, together with a clearer picture of the sensitivity of results to changes in parameter values. Further, the processes of parameter estimation and the methodology of back-calculation also benefit from the provision of functional forms for the state variables.