Stochastic near-optimal control for drug therapy in a random viral model with cellular immune response

Viruses are responsible of illness. The purpose of drug therapy is to fight bacterial infection and to achieve definite outcomes that improve the patient's quality of life. Mathematical models combined with clinical studies can be useful in forecasting infection and understanding viral infections mechanism among host cells. In this work, incorporating stochastic fluctuations, we consider a viral infection model to describe the role of lytic and nonlytic immune responses. Lytic immunity is defined as the destruction of infected cells. Nonlytic immunity is defined as the inhibition of viral replication by soluble mediators secreted by immune cells. Mainly, we investigate near optimal control for drug therapy. We show sufficient and necessary conditions for the near optimality. Then, by means of adjoint equations, we estimate the error bound for the near optimality. Numerical illustrations indicate that the antiviral drug therapy may induce a significant decrease of the peaks of infected cells and free virions.