Existence of order-1 periodic solutions for a viral infection model with state-dependent impulsive control

Huilan Wang,Qizhen Xiao,Binxiang Dai

doi:10.1186/s13662-019-1967-x

Abstract

It is well known that the drug treatment is always combined with the injection of immune factors. In this paper, a virus infection model with state-dependent impulsive control is considered. Firstly, by deriving three categories of Bendixson domain and using the methods of geometry and successor function, we establish some criteria for the existence of positive order-1 periodic solution for a general model, which extends the existing results in the literature. Further, the criteria are used to obtain the existence of positive order-1 periodic solutions in the two cases that the positive equilibrium point is on the left or right side of the pulse line, respectively. Finally, an example is presented to illustrate our results.

Highlights

With respect to the mathematical analysis of virus copies in vivo, differential equations are important tools modeling the evolution mechanism of normal cells and virus [1,2,3,4]
The positive constant a is the natural death rate of free virus. f (x) is the growth rate at which new target cells are generated, which incorporating the natural death rate of the cells; g(x) represents the rate at which an uninfected cell infected by virus
We are aiming to establish some criteria for the existence of order-1 periodic solution based on the Bendixson domain types

Summary

Introduction

With respect to the mathematical analysis of virus copies in vivo, differential equations are important tools modeling the evolution mechanism of normal cells and virus [1,2,3,4]. Proof Provided that there exists an x0 ∈ (h, h ) such that v0(x0) = φ(x0), the trajectory P0Q0 will intersect with the trajectory initiated from (x0, φ(x0)) which is tangent to the line x = x0 It will contradict the uniqueness of the solution to system (1.1). Proof If ω0 > th, according to Lemma 3.3, O–(W0) will intersect with x = hat unique point W0–, which implies that all the trajectories, initiated from the points under W0– in N , will not hit the line x = h. W0– > P0, which implies that all the points that is, ω0– > v0 It follows in W0Th will be mapped from onto the segment below W0– by impulsive map I, and the trajectories initiated from segment under W0– will not hit M any more. G(h )h g(h)h < 1, which implies the order-1 periodic solution (X(t), V (t)) is orbitally asymptotically stable

Example

Conclusion and discussion