Global stability of a delayed virus model with latent infection and Beddington–DeAngelis infection function
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 ( is the infection rate, and 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 with initial values . Here, is a given non-negative constant, , , , with
Global asymptotic stability of the virus-free steady state
To study the stability at the steady state , we let , and . The linearization of system (1) at is where,
Theorem 3.1 If , the virus-free steady state is
Global asymptotic stability of the infected steady state
Theorem 4.1 If and , then the infected steady state is locally asymptotically stable for , , .
Proof The characteristic equation of the linearized system (6) at the infected steady state is where,
Eq. (10) is equivalent to the following equation
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)
- et al.
Global properties for virus dynamics model with Beddington–DeAngelis functional response
Appl. Math. Lett.
(2009) - et al.
Global analysis for delay virus dynamics model with Beddington–DeAngelis functional response
Appl. Math. Lett.
(2011) - et al.
Dynamics of two time delays differential equation model to HIV latent infection
Physica A
(2019) - et al.
A delay-dependent model with HIV drug resistance during therapy
J. Math. Anal. Appl.
(2014) - et al.
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
Math. Biosci.
(2002) Complete global stability for an SIR epidemic model with delay-distributed or discrete
Nonlinear Anal. R. World. Appl.
(2010)- et al.
Mathematical analysis of HIV-1 dynamics in vivo
SIAM Rev.
(1999) - et al.
Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy
Proc. Natl. Acad. Sci.
(1997)
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