Optimal control for therapeutic drug treatment on a delayed model incorporating immune response

Abstract

Millions of people get infected every year by viral pathogens. Newly emergent diseases such as Ebola, Swine-flu, HIV/AIDS, etc. are spreading worldwide at an alarming rate. We introduced a delayed mathematical model with immune response and therapeutic drug treatment to understand the dynamics of pathogenimmune interaction. Here, we are considering the innate immune response and the two major component of the acquired immune response, namely, cytotoxic T lymphocytes (CTLs) and humoral immunity. This model also incorporates the absorption of pathogens i.e. loss of pathogens and its related mechanisms. Further, an optimal control model is formulated with two optimal controls i.e. maximization of uninfected cells count and minimization of cost of treatments. This is done by using the Pontryagins' Maximum Principle. Existence of non-negative equilibria is established and their stability behavior is studied using theory of ordinary differential equations. Further, numerical simulations are carried out to exemplify the qualitative results.