The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression (original) (raw)

A Delay Virus Dynamics Model with General Incidence Rate

Differential Equations and Dynamical Systems, 2013

In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and intracellular delay is studied. Lyapunov functionals are constructed and LaSalle invariance principle for delay differential equation is used to establish the global asymptotic stability of the disease-free equilibrium and the chronic infection equilibrium. The results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number R 0 and under some assumptions on the general incidence function. Our results extend the known results on delay virus dynamics considered in other papers and suggest useful methods to control virus infection. These results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models.

Global Stability of an HIV-1 Infection Model with General Incidence Rate and Distributed Delays

International Scholarly Research Notices, 2014

In this work an HIV-1 infection model with nonlinear incidence rate and distributed intracellular delays and with humoral immunity is investigated. The disease transmission function is assumed to be governed by general incidence rate (,). The intracellular delays describe the time between viral entry into a target cell and the production of new virus particles and the time between infection of a cell and the emission of viral particle. Lyapunov functionals are constructed and LaSalle invariant principle for delay differential equation is used to establish the global asymptotic stability of the infection-free equilibrium, infected equilibrium without cells response, and infected equilibrium with cells response. The results obtained show that the global dynamics of the system depend on both the properties of the general incidence function and the value of certain threshold parameters 0 and 1 which depends on the delays.

Global stability analysis of a delayed HIV model with saturated infection rate

MATEC Web of Conferences

In this paper, the global stability of a delayed HIV model with saturated infection rate infection is investigated. We incorporate two discrete delays into the model; the first describes the intracellular delay in the production of the infected cells, while the second describes the needed time for virions production. We also derive the global properties of this two-delay model as function of the basic reproduction number R0. By using some suitable Lyapunov functions, it is proved that the free-equilibrium point is globally asymptotically stable when R0 ≤ 1, and the endemic equilibrium point is globally asymptotically stable when R0 ≥ 1. Finally, in order to support our theoretical findings we have illustrate some numerical simulations.

A class of delayed viral models with saturation infection rate and immune response

In this paper, a class of three delayed viral dynamics models with immune response and saturation infection rate are proposed and studied. By constructing suitable Lyapunov functionals, we derive the basic reproduction number R 0 and the corresponding immune response reproduction numbers for the viral infection models, and establish that the global dynamics are completely determined by the values of the related basic reproduction number and immune response reproduction numbers.

A class of diffusive delayed viral infection models with general incidence function and cellular proliferation

Arabian Journal of Mathematics

We propose and analyze a new class of three dimensional space models that describes infectious diseases caused by viruses such as hepatitis B virus (HBV) and hepatitis C virus (HCV). This work constructs a Reaction–Diffusion-Ordinary Differential Equation model of virus dynamics, including absorption effect, cell proliferation, time delay, and a generalized incidence rate function. By constructing suitable Lyapunov functionals, we show that the model has threshold dynamics: if the basic reproduction number \mathcal {R}_{0}(\tau ) \le 1 R0(τ)≤1,thentheuninfectedequilibriumisgloballyasymptoticallystable,whereasifR 0 ( τ ) ≤ 1 , then the uninfected equilibrium is globally asymptotically stable, whereas ifR0(τ)1,thentheuninfectedequilibriumisgloballyasymptoticallystable,whereasif\mathcal {R}_{0}(\tau ) > 1$$ R 0 ( τ ) > 1 , and under certain conditions, the infected equilibrium is globally asymptotically stable. This precedes a careful study of local asymptotic stability. We pay particular attention to prove boundedness, positivity, existence and uniqueness of the solution to the obtained initial and boundary value probl...

Global stability for a delayed HIV-1 infection model with nonlinear incidence of infection

Applied Mathematics and Computation, 2012

In this paper, a delayed HIV-1 infection model with nonlinear incidence of infection is reinvestigated. It is shown that if the reproduction number R > 1, then the system is permanent, and the infective equilibrium of the system is globally asymptotically stable. Thus, the global dynamics of the system is completely determined by the reproduction number R. The results obtained enrich and improve the corresponding results given by Wang et al. [X. Wang, Y. Tao, X. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dynamics 62 (2010) 67-72]. The conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper [D. Li, W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007) 683-691].

A discrete viral infection model with both modes of transmission and distributed delays

International Journal of Dynamical Systems and Differential Equations, 2021

The aim of this work is to propose and analyse a discrete virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the first distributed delays describes the time needed for infected cells to produce new virions, and the second portrays the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modelled by general incidence functions for both modes. Furthermore, we prove that the proposed discrete model has the same dynamics as the corresponding continuous model, such as positivity, boundedness and global behaviours of solutions with no restriction on the time step size. Moreover, numerical simulations are given to illustrate and confirm our main analytical results.

Qualitative Analysis of a Generalized Virus Dynamics Model with Both Modes of Transmission and Distributed Delays

International Journal of Differential Equations

We propose a generalized virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the distributed delays describe (i) the time needed for infected cells to produce new virions and (ii) the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, the qualitative analysis of the model is rigorously established and many known viral infection models with discrete and distributed delays are extended and improved.

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R 0 for the viral infection, and establish that the global dynamics are completely determined by the values of R 0 . If R 0 ≤ 1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R 0 > 1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R 0 > 1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.

Mathematical analysis of delayed HIV-1 infection model for the competition of two viruses

Cogent Mathematics

In this research article, a new mathematical delayed human immunodeficiency virus (HIV-1) infection model with two constant intracellular delays, is investigated. The analysis of the model is thoroughly discussed by the basic reproduction numbers R 0 and R s. For R 0 < 1, the infection-free equilibrium (E 0) is shown to be locally as well as globally stable. Similarly, the single-infection equilibrium (E s) is proved to be locally as well as globally asymptotically stable if 1 < R 0 < R s. Our derived results show that the incorporation of even small intracellular time delay can control the spread of HIV-1 infection and can better the quality of the life of the patient. Finally, numerical simulations are used to illustrate the derived theoretical results.