Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response

This paper investigates a fractional-order viral infection model with saturated infection rate and cellular immune response. The cellular immunity will be represented by cytotoxic T-lymphocytes (CTL) cells. In order to study mathematically the infection model, we will suggest five fractional differential equations describing the interaction between the uninfected cells, the latently infected cells, the infected cells, the CTL cells and the free viruses. A saturated infection rate will be taken into consideration to represent the viral infection. First, the positivity and boundedness of solutions for non-negative initial data are proved. Next, by constructing suitable Lyapunov functions, the global stability of the disease free equilibrium and the endemic equilibria are established depending on the basic reproduction number $R_0$ and the CTL immune response reproduction number $R_{CTL}$. Finally, numerical simulations are performed in order to show the dynamics behavior of the viral infection and to support the theoretical results.