Fractional HBV infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity

In this work, a fractional order hepatitis B virus infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity will be examined. The adaptive immunity that we will consider will be represented by the cellular and the humoral immune responses. In our model, the two modes of infection will be symbolized by two saturated incidence functions. We started our study by proving the existence, uniqueness and boundedness of the positive solutions. Next, we have formulated the free-equilibrium and the endemic equilibria of our model. By using Lyapunov’s method and LaSalle’s invariance principle, we have shown the global stability of each equilibrium. Numerical simulation is also given to support our theoretical results and to show the effect of the fractional derivative order on the convergence toward the equilibrium points. More precisely, numerical results have confirmed our theoretical findings about the equilibria stability. We have noticed that for smaller fractional derivative order values, the variables of our model converge more quickly to their corresponding steady states. However, for a higher values of the fractional derivative order, the convergence becomes very slowly, this means a long memory effect. In other words, the fractional derivative order has no effect on the equilibria stability but only on the convergence speed toward the equilibria. In order to show the importance of incorporating to saturated infection rates, a numerical comparison between the dynamical behavior of the model with two saturated incidence and two bilinear incidence rates is curried out.