Modeling the intracellular pathogen-immune interaction with cure rate

Abstract

Many common and emergent infectious diseases like Influenza, SARS, Hepatitis, Ebola etc. are caused by viral pathogens. These infections can be controlled or prevented by understanding the dynamics of pathogen-immune interaction in vivo. In this paper, interaction of pathogens with uninfected and infected cells in presence or absence of immune response are considered in four different cases. In the first case, the model considers the saturated nonlinear infection rate and linear cure rate without absorption of pathogens into uninfected cells and without immune response. The next model considers the effect of absorption of pathogens into uninfected cells while all other terms are same as in the first case. The third model incorporates innate immune response, humoral immune response and Cytotoxic T lymphocytes (CTL) mediated immune response with cure rate and without absorption of pathogens into uninfected cells. The last model is an extension of the third model in which the effect of absorption of pathogens into uninfected cells has been considered. Positivity and boundedness of solutions are established to ensure the well-posedness of the problem. It has been found that all the four models have two equilibria, namely, pathogen-free equilibrium point and pathogen-present equilibrium point. In each case, stability analysis of each equilibrium point is investigated. Pathogen-free equilibrium is globally asymptotically stable when basic reproduction number is less or equal to unity. This implies that control or prevention of infection is independent of initial concentration of uninfected cells, infected cells, pathogens and immune responses in the body. The proposed models show that introduction of immune response and cure rate strongly affects the stability behavior of the system. Further, on computing basic reproduction number, it has been found to be minimum for the fourth model vis-a-vis other models. The analytical findings of each model have been exemplified by numerical simulations.