Maximum principle for age and duration structured systems: A tool for optimal prevention and treatment of HIV

Age and duration since infection are considered in a model of optimal control of the spread of Human Immunodeficiency Virus (HIV) in countries with high prevalence. Prevention and medical treatment are selected so as to maximize an economic objective function. The model extends the classical McKendrick equation. Necessary optimality conditions in the form of Pontryagin's global maximum principle and numerical solution based on them are presented. ‘Critical’ initial prevalence is established numerically for which there are two optimal medical treatments: one intense and another less demanding. It is shown that treatment alone can be counterproductive: increase in treatment must be accompanied by increase in prevention.