Optimal Drug Therapy in a Multi-Pathways HIV-1 Infection Model with Immune Response Delay

A mathematical model is critical to understanding the host-pathogen interactions. In this paper, we study a multi-pathways in-host HIV-1 infection model in the presence of immune response delay and three controls. The controls may be constant or time-dependent. In the case of constant controls, it is shown that the infection-free equilibrium of the system is locally and globally asymptotically stable when the basic reproduction number is less than unity. The stability of the infected equilibrium may be lost through Hopf bifurcation if the length of immune response delay is longer. We define a suitable objective functional in the time-dependent controls to maximize the cell counts of healthy CD4+ T cells and CTLs and derive the optimality of our delay-induced control problem. We examine and compare the effect of mono and multi-drug therapies through numerical simulations. We demonstrate that removing infection is not possible using any mono-drug therapy. It is also shown that the blocker that inhibits the synapse formation during cell-to-cell disease transmission should be used while using multi-drug therapy to clear the infection. However, this control is not an efficient blocker in the mono-drug treatment protocol. Our study reveals that if CTL’s response is quicker, CD4+T cells count may remain stable but becomes unstable if response time increases.