Dynamics of an HTLV-I infection model with delayed CTLs immune response

Abstract

This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R0 and R1, basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E0 is globally stable if R0≤1, and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E1 is globally stable if R1≤1<R0, which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E2 is globally asymptotic stable if R1>1, which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R0 and R1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h, removal rate of virus-specific CTLs d4, spontaneous infected CD4+T cell activation d2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored.