Global Existence of a Virus Infection Model with Saturated Chemotaxis

Ruijing Li

doi:10.11648/j.ijssam.20210602.12

Abstract

In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.

Highlights

As the virological, immunological and mathematical plates become interlocked, mathematics is playing an increasingly important role in biology
The vast majority of existing research on evolution of a virus infection model was almost described by ordinary differential equations (ODEs) [7, 18], this leads to the ignorance of spatial variations, which means the ODEs are not suitable for obtaining spatial information about the distribution of infected cells
Kirschner et al studied optimal chemotherapy strategy in an early treatment background which depicted the interaction of the immune system with the human immunodeficiency virus (HIV) by the optimal control theories and methods, where the immune system is governed by ODEs [13]

Summary

Introduction

Immunological and mathematical plates become interlocked, mathematics is playing an increasingly important role in biology. To the best of our knowledge, the optimal control problem of virus infection models with saturated chemotaxis (1) has not been studied. With the addition of the arguments in previous studies [11, 16, 17, 22, 24, 30, 33], the aim of this paper is to consider a virus infection model with saturated chemotaxis. Under appropriate regularity assumptions on the initial data, via Lp-estimate techniques, we show that the epidemic system (1) exists at least one global weak solution This result generalizes and improves Theorem 1.1 [11]. Ruijing Li: Global Existence of a Virus Infection Model with Saturated Chemotaxis. 4, the optimal control problem of the system (70) is considered and the existence of the optimal pair is obtained

Preliminaries

A Priori Estimate

Passing to the Limit

Application to Optimal Control

Conclusions