Analysis, simulation and applications of a delayed target limited cell model under the cumulative effects of vaccine and treatment

In this paper, we investigate mathematically and numerically a modified diffusive target cell limited model in one spatial dimension. The model considers the presence of delay transmission and virus production. Moreover, the model includes the effects from treatment either by mucosal vaccination or antiviral treatment. The cumulative effects are coupled by assuming a Beddington–DeAngelis-type incidence rate on the target cells. The mathematical model is studied analytically through a linear stability analysis. The disease-free and endemic equilibrium solutions are calculated, and the next generation matrix technique is employed to determine the basic reproduction number. Moreover, the stability properties of the equilibria are also derived. A traveling-wave analysis is performed next, and conditions for the existence of positive solutions are rigorously derived. From the numerical point of view, we perform also a traveling-wave analysis via a reliable implicit finite-difference scheme. Various physically relevant implications are derived from our simulations. Some of these consequences could have practical value in order to establish policies on the treatment of epidemic diseases. The proposed approach seems to be a good tool for preliminary test before experimentation in general and specifically may be an alternative to evaluate the within host dynamics in a low-income country where there is a crucial lack of logistics.