In this chapter, a series solution of a nonlinear fractional-order mathematical model of human T-cells lymphotropic virus-1 (HTLV-1) infection of CD4+ T-cells is obtained by using a strong and capable technique so-called Homotopy Analysis Method (HAM). The proposed model is a system of nonlinear ordinary differential equations that divides CD4+ T-cells into four components: uninfected cells, latently infected cells, actively infected cells and leukemia cells. The fractional model is more general than the classical one, as in the fractional model, the next state depends not only upon its current state but also upon all of its historical states. The homotopy analysis method (HAM) is applied for a strongly nonlinear fractional-order system as it utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter and allows to obtain a oneparametric family of explicit series solutions. By using the homotopy series solutions, firstly, several β-curves are plotted to demonstrate the regions of convergence, then the square residual errors are obtained for different values of these regions. Secondly, the numerical solutions are presented to show the accuracy of the applied homotopy analysis method. In this chapter, a detailed proof of the convergence of this method for nonlinear fractional-order model of HTLV-1 infection of CD4+ T-cells is also given. The results indicate that the HAM is accurate and capable to obtain an accurate approximate analytical solution for HTLV-1 infection of CD4+ T-cells.
Keywords: Human T-cells lymphotropic virus-1 (HTLV-1) infection of CD4+ T-cells, Homotopy analysis method, ħ-curve, β-curves, Convergence-control parameter, Least square.
