We develop a Susceptibles, Infected and Recovered (SIR) type mathematical model of HIV epidemiology to explore a possible mechanism by which mass incarceration can lead to increased HIV incidence. The results are particularly relevant for the African American community in the United States that represents only 12% of the total population but accounts for 45% of HIV diagnoses and 40% of the incarcerated population. While most explanations of the link between mass incarceration or anything else that leads to a population with a low ratio of males to females and higher HIV burden are based on the complicated idea of sexual concurrency, we propose a much simpler mechanism based on the idea of sexual activity compensation. The pool of men will increase their sexual activity to meet the demands of the female population. Through mathematical analysis and numerical simulation, we demonstrate that these assumptions produce a situation in which mass incarceration lead to higher HIV incidence.
We also develop an optimal control model of SIR type. In this model, the control is education or information given to the public to manage a disease outbreak when effective treatments or vaccines are not readily available or too costly to be widely used. We study stability analysis and use optimal control theory on the system of differential equations to achieve the goal of minimizing the infected population. We illustrate our results with some numerical simulations.