Global dynamics of a virus infection model with repulsive effect

,Hui Li,Manjun Ma

doi:10.3934/dcdsb.2019030

Abstract

This paper is devoted to investigate a virus infection model with a spatially heterogeneous structure and nonlinear diffusion. First we establish the properties of the basic reproduction number \begin{document}$ R_0 $\end{document} for infected cells and free virus particles. Then we prove that the comparison principle can be applied to an auxiliary system with quasilinear diffusion under appropriate conditions. Then the sufficient conditions for the globally asymptotical stability of infection-free steady state are obtained, which indicates that \begin{document}$ R_0 is necessary for infected cells and free virus particles to be extinct. Next we prove the existence of positive non-constant steady states and the persistence of infected cells and free virion where \begin{document}$ R_0>1 $\end{document} is required. Finally, it is shown that, for the spatially homogeneous case when the infected cells rate of change of the repulsive effect is small enough, \begin{document}$ R_0 $\end{document} is the only determinant of the global dynamics of the underlying virus infection system. The obtained results give an insight into the optimal control of the virion.

Highlights

The within-host dynamics of virus infection and spread has been effectively described by mathematical models
= kw − mv, where u is uninfected susceptible host cells, w represents infected host cells, and v stands for free virus particles; a, β, b, k and m are positive constants, and the constant λ ≥ 0 is the production rate of susceptible cells
We prove the existence of positive steady states of system (3)-(7)

Summary

Introduction

The within-host dynamics of virus infection and spread has been effectively described by mathematical models. We will deal with the properties of the basic reproduction number, the comparison principle for a quasilinearly parabolic auxiliary system, the globally asymptotical stability of infection-free steady state, the existence of positive non-constant steady states and the uniform persistence of (3)-(7) with spatially heterogeneous structure. For the component function I(t, x) of a solution (T, I, V ) of (3)-(7), by Lemma 2.1 and condition (5), there exists a positive constant M such that. By the proof of Lemma 2.3, we know that system (11) has a globally attractive solution T ∗(x) in X1+\{0}. From Theorem 1.2.2 in [18] it follows that κ is a steady state solution of S(t), that is, κ = T ∗(x), and ω(T0, I0, V0) = κ × {(0, 0)} = {(T ∗(x), 0, 0)}, which implies that (T ∗(x), 0, 0) is globally attractive for Φ(t) in X3+. (T ∗(x), 0, 0) is locally stable for (3)-(7) in X3+\{0}. complete

The proof is

We now define two functions

Discussion and conclusion