Global stability of a delayed virus model with latent infection and Beddington–DeAngelis infection function

https://doi.org/10.1016/j.aml.2020.106463Get rights and content

Highlights

  • A delayed model with latent infection and Beddington–DeAngelis is formulated.

  • Local asymptotic stability of the equilibrium of the model is studied.

  • Global stability of the equilibrium is derived by constructing Lyapunov function.

Abstract

In this paper, a virus model with latent infection, Beddington–DeAngelis infection function and three time delays is proposed, and the basic reproduction number is obtained. The global dynamics of the model is analyzed by estimating the modulus of the characteristic equation and constructing an appropriate Lyapunov function. When the basic reproduction number is less than one, the virus-free steady state is globally asymptotically stable. When the basic reproduction number is greater than one, the infected steady state is globally asymptotically stable and the virus-free steady state is unstable.

Introduction

People’s life and health have been threatened by virus infection (such as HIV, hepatitis B/C virus) for a long time. Many researchers use mathematical models to describe the process of the virus invasion in human body, and predict the further development trend of the virus. Early classical virus infection models include three populations [1]: the target cells, the infected cells and the virus particles. However, in the medical literature [2], it was pointed out that the latent reservoir (that is latent infection) was the main obstacle to eradicate the virus. Therefore, the four-dimensional mathematical model including the latent infection seems more reasonable [3].

In recent years, the latent infection models have been extended to include intracellular delay and cell-to-cell infection mode, and the theoretical analysis results have also been gradually improved [4], [5]. However, the infection function in these models is bilinear. Beddington [6] and DeAngelis et al. [7] introduced a generalized infection function, say Beddingon–DeAngelis infection function βT(t)V(t)1+aT(t)+bV(t) (β is the infection rate, and a,b0 are the inhibition constants), which has been widely used in the ecological model and the infectious disease model. Huang et al. [8], [9] firstly considered the Beddington–DeAngelis infection function into the three-dimensional HIV infection model, and theoretically analyzed the global stability. Recently, Miao et al. have extended this functional response to a five-dimensional virus infection model with two kinds of immune responses, and have investigated the existence of Hopf bifurcation [10]. However, all these models did not involve latent infection [8], [9], [10].

In this paper, we propose a virus model with the latent infection and the Beddington–DeAngelis infection function, which has not been studied in the previous mathematical models. Here, the Beddington–DeAngelis infection function is more general, which contains the bilinear and Holling type II infection rates in the existing references [3], [4], [11]. Both the latent infection and the Beddington–DeAngelis infection function, are introduced into the virus model at the same time, which brings great challenge to the theoretical analysis. The purpose of this paper is to construct an appropriate Lyapunov function to analyze the global dynamical properties of this kind of models.

The rest of this paper is organized as follows. In Section 2, a delayed virus model with both the latent infection and the Beddington–DeAngelis infection function is formulated. In Sections 3 Global asymptotic stability of the virus-free steady state, 4 Global asymptotic stability of the infected steady state, the global asymptotic stability of the virus-free and the infected steady states is derived, respectively. Finally, we summarize our work in Section 5.

Section snippets

Model formulation

With both the latent infection and the Beddington–DeAngelis infection function, a delayed virus model can be formulated as Ṫ(t)=λd1T(t)+rT(t)1T(t)TmaxβT(t)V(t)1+aT(t)+bV(t),L̇(t)=ηemτ1βT(tτ1)V(tτ1)1+aT(tτ1)+bV(tτ1)αL(t)d2L(t),İ(t)=(1η)emτ2βT(tτ2)V(tτ2)1+aT(tτ2)+bV(tτ2)+αL(t)d3I(t),V̇(t)=kI(tτ3)d4V(t),with initial values T(θ)=ψ1(θ),L(0)=ψ2,I(θ)=ψ3(θ),V(θ)=ψ4(θ)forθ[τ,0], τ=max{τ1,τ2,τ3}. Here, ψ2 is a given non-negative constant, ψ1(θ), ψ3(θ), ψ4(θ)C([τ,0], R+) with R+=[

Global asymptotic stability of the virus-free steady state

To study the stability at the steady state Ē (T̄,L̄,Ī,V̄), we let Y1(t)=T(t)T̄,Y2(t)=L(t)L̄,Y3(t)=I(t)Ī,Y4(t)=V(t)V̄, and Y(t)=(Y1(t),Y2(t),Y3(t),Y4(t)). The linearization of system (1) at Ē(T̄,L̄,Ī,V̄) is Ẏ1(t)=θY1(t)JY4(t),Ẏ2(t)=Hηemτ1Y1(tτ1)+Jηemτ1Y4(tτ1)(α+d2)Y2(t),Ẏ3(t)=H(1η)emτ2Y1(tτ2)+J(1η)emτ2Y4(tτ2)+αY2(t)d3Y3(t),Ẏ4(t)=kY3(tτ3)d4Y4(t),where, θ=d1r12T̄Tmax+H,H=βV̄(1+bV̄)(1+aT̄+bV̄)2,J=βT̄(1+aT̄)(1+aT̄+bV̄)2.

Theorem 3.1

If R0<1, the virus-free steady state E0 is

Global asymptotic stability of the infected steady state

Theorem 4.1

If R0>1 and d1r+rTTmax>0, then the infected steady state E is locally asymptotically stable for τ10, τ20, τ30.

Proof

The characteristic equation of the linearized system (6) at the infected steady state E is ξ+d1r12TTmax+H(ξ+α+d2)(ξ+d3)(ξ+d4)=kJeξτ3ξ+d1r12TTmaxαηe(m+ξ)τ1+(ξ+α+d2)(1η)e(m+ξ)τ2,where, J=βT(1+aT)(1+aT+bV)2,H=βV(1+bV)(1+aT+bV)2. Eq. (10) is equivalent to the following equation ξ+d1r12TTmax+Hξ+d1r12TTmax=kJeξτ3αηe(m+ξ)τ1+(ξ+α+d2)(1η)e(m+ξ)τ2(ξ+α+d2)(

Conclusion

In this paper, we have formulated a virus model incorporating the target cell logistic growth, the latently infected cell and the Beddington–DeAngelis infection function. The latent delay, the activation delay and the maturation delay during the process of the virus infection are also included in the model. Theoretically, by constructing the appropriate Lyapunov function, we obtain that the virus-free steady state is globally asymptotically stable for the three time delays when the basic

CRediT authorship contribution statement

Yan Wang: Investigation, Writing - original draft, Supervision. Minmin Lu: Investigation, Methodology. Jun Liu: Writing - review & editing.

Acknowledgments

The authors thank the editor and referees for their useful suggestions, which have greatly helped them to improve their study. This work is supported by National Natural Science Foundation of China (Nos. 11401589, 11801566, 11871473), the Fundamental Research Funds for the Central Universities (No. 18CX02049A), and Shandong Provincial Natural Science Foundation (No. ZR2019MA010).

References (16)

There are more references available in the full text version of this article.

Cited by (13)

View all citing articles on Scopus
View full text