Within-host model of an HTLV-1 infection of CD4+ T-cells incorporating an adult T-cell leukemia development: stability analysis and optimal control by prevention and two treatments

Human T-cell leukemia/lymphotropic virus type 1 (HTLV-1) is a pathogenic retrovirus that infects CD4+ T-cells. HTLV-1 associates with various diseases including the development of adult T-cell leukemia (ATL) and myelopathy/tropical spastic paraparesis (HAM/TSP). Currently, there is no cure or neutralizing vaccine for HTLV-1 infection. To better understanding of HTLV-1 infection dynamics for CD4+ T-cells, in this study we propose a within-host model with six classes of cells includes uninfected, latently infected, actively infected CD4+ T-cells, leukemia cells, HTLV-1-specific CD8+ T-cells or CTLs, and free virus. Mathematical analysis is performed in detail starting with non-negativity and the boundedness of the solutions are verified, three equilibrium points (infection-free, immune-free and chronic infection) are computed, the model basic reproduction number is calculated, and its sensitivity is performed. When the basic reproduction number is less than one, the infection can be eliminated, otherwise the infection persists. Both local and global stability of all equilibrium points are analyzed, where we obtain that each equilibrium point is stable under some specific conditions. Next, an optimal control problem is applied into the model to investigate the optimal strategy of three considered controls which are preventive control, treatment effort for leukemia cells and antivirus drugs for HTLV-1. Our numerical results demonstrates that preventive control has shown to reduce the HTLV-1 infection of CD4+ T-cells, the treatment control for leukemia cells can reduce the leukemia cells, and the antivirus drugs for HTLV-1 can reduce the actively infected CD4+ T-cells and free virus. However, a combination of all three controls gives the most effective results for reducing the overall HTLV-1 infection of CD4+ T-cells.