Qualitative Analysis of a Fractional-Order for a Within-Host Infection Dynamics with Adaptive Immunity Using Caputo Derivative

The objective of this paper is to introduce and study a within-host viral infection model via both modes of viral transmission mechanism; virus-to-cell and cell-to-cell. The proposed model incorporates the role of adaptive immunity in limiting viral pathogen spread, including humoral and cellular immune responses. This paper focuses on describing the long memory effect using the Caputo derivative. This study begins with the investigation of the well-posedness of our mathematical model concerning demonstrating the existence, positivity and boundedness of solutions. Then, it moves to introduce all the problem’s steady states, which are determined by specific reproduction numbers. Subsequently, the work proceeds to demonstrate the global stability of the five equilibria points. To evaluate the theoretical findings of global stability, we employ a numerical technique based on the fundamental theorem of fractional calculus, in addition to utilizing a three-step Lagrange polynomial interpolation method. Numerical simulations have shown the effect of drug therapies on the dynamical behavior of the fractional-order system.