Infected Equilibrium Research Articles

Mathematical analysis of an HIV infection model including quiescent cells and periodic antiviral therapy

In this paper, we revisit the model by Guedj et al. [J. Guedj, R. Thibaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, Biometrics 63 (2007) 198–206; J. Guedj, R. Thibaut and D. Commenges, Practical identifiability of HIV dynamics models, Bull. Math. Biol. 69 (2007) 2493–2513] which describes the effect of treatment with reverse transcriptase (RT) inhibitors and incorporates the class of quiescent cells. We prove that there is a threshold value [Formula: see text] of drug efficiency [Formula: see text] such that if [Formula: see text], the basic reproduction number [Formula: see text] and the infection is cleared and if [Formula: see text], the infectious equilibrium is globally asymptotically stable. When the drug efficiency function [Formula: see text] is periodic and of the bang–bang type we establish a threshold, in terms of spectral radius of some matrix, between the clearance and the persistence of the disease. As stated in related works [L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol. 69 (2007) 2027–2060; P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bull. Math. Biol. 71 (2009) 189–210.], we prove that the increase of the drug efficiency or the active duration of drug must clear the infection more quickly. We illustrate our results by some numerical simulations.

A bacteriophage model based on CRISPR/Cas immune system in a chemostat.

Clustered regularly interspaced short palindromic repeats(CRISPRs) along with Cas proteins are a widespread immune system across bacteria and archaea. In this paper, a mathematical model in a chemostat is proposed to investigate the effect of CRISPR/Cas on the bacteriophage dynamics. It is shown that the introduction of CRISPR/Cas can induce a backward bifurcation and transcritical bifurcation. Numerical simulations reveal the coexistence of a stable infection-free equilibrium with an infection equilibrium, or a stable infection-free equilibrium with a stable periodic solution.

Mathematical analysis of an influenza A epidemic model with discrete delay

Recently, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. In this paper, we present a delay differential model to describe influenza A (H1N1) dynamics. We begin by presenting the model with a brief discussion, followed by proving the positivity and boundedness of the model solution. We establish sufficient conditions for the global stability of the equilibria (the infection free equilibrium and the infected equilibrium), these conditions are obtained by means of the Lyapunov LaSalle invariance principle of the system. Also we have carried out bifurcation analysis along with an estimated length of delay to preserve the stability behavior. In particular, we show the threshold dynamics in the sense that if (reproduction number) R 0 < 1 the infectious population disappear so the disease dies out, while if R 0 > 1 the infectious population persist. Sensitivity analysis of the influenza A (H1N1) model reveals which parameter values have a major impact on the model dynamics. Numerical simulations with application to H1N1 infection are given to verify the analytical results.

Effects of HIV infection on CD4+ T-cell population based on a fractional-order model

In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4+ cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4+ T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results.

Modeling and bifurcation analysis of a viral infection with time delay and immune impairment

The rich dynamics of a viral infection model is studied under the assumption that the immune response is proposed based on some important biological meanings. In this paper, we study the dynamical behaviour of delayed viral infection with immune impairment model, viewing the infection rate of HIV-1 infection as a saturated function of the viral load, to formulate HIV-1 model dynamics. We explicitly link the individual and the viral population scale, and derive the basic reproduction number \(R_0\) for the coupled system. The effect of time delay on stability and Hopf bifurcation for the infected equilibrium have been studied. By fixing the immune delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. Using center manifold argument and normal form theory, we derive explicit formulae to determine the stability and direction of the limit cycles of the model. To analyze the model and perform a detailed global dynamics analysis, two Lyapunov functionals are constructed to prove the global asymptotical stability of the disease-free and infected equilibria. Theoretical results indicate that \(R_0\) provides a threshold value determining whether or not the disease dies out. Numerical simulations are carried out to explain the mathematical conclusions.

Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

STABILITY AND BIFURCATION ANALYSIS OF A VIRAL INFECTION MODEL WITH DELAYED IMMUNE RESPONSE

In this paper, we study a viral infection model with an immunity time delay accounting for the time between the immune system touching antigenic stimulation and generating CTLs. By calculation, we derive two thresholds to determine the global dynamics of the model, i.e., the reproduction number for viral infection R0 and for CTL immune response R1. By analyzing the characteristic equation, the local stability of each feasible equilibrium is discussed. Furthermore, the existence of Hopf bifurcation at the CTL-activated infection equilibrium is also studied. By constructing suitable Lyapunov functionals, we prove that when R0 ≤ 1, the infection-free equilibrium is globally asymptotically stable,when R0>1 and R1 ≤ 1, the CTLinactivated infection equilibrium is globally asymptotically stable,Numerical simulation is carried out to illustrate the main results in the end.

基于分数阶微分方程的手机病毒传播模型研究

随着手机普及率的提高，手机病毒的传播也愈发严重。因此，对于病毒传播规律的研究必不可少。本文建立并研究了一类基于分数阶微分方程的手机病毒传播模型，利用分数阶微分方程的相关理论，详细分析了平衡点的存在性及其局部稳定性，并通过数值试验验证了理论结果的正确性。通过研究，我们得到在基本再生数小于1的情况下，未感染平衡点是局部渐进稳定的，病毒会消亡；在基本再生数大于1时，感染平衡点局部渐近稳定，病毒将扩散。根据所得到的理论结果给出了控制手机病毒传播的有效措施，为手机病毒的预测、控制和防治提供了重要的参考依据。 With the increasing popularity of mobile phones, the spread of mobile virus has become increas-ingly serious. Therefore, the study of the spread of the virus is necessary. In this paper, we first establish an epidemic model of mobile phone virus based on the fractional differential equation. Then by means of the theory of fractional differential equations, we analyse the existence and sta-bility condition of the equilibrium of the model. It is showed that if the basic reproduction number is less than 1, the infection free equilibrium is locally asymptotically stable and virus will die out, and if the basic reproductive number is greater than 1, the infection equilibrium is stable and the virus will spread. Some numerical experiments are carried out to confirm the obtained results. In addition, some effective measures are given to control the spread of the mobile virus, which pro-vides referential support for virus prediction, control and prevention.

Dynamics of a Stochastic Viral Infection Model with Immune Response

The aim of this work is to study the dynamical behavior of a stochastic viral infection model which is formulated by four nonlinear stochastic differential equations to describe the interactions between virus, host cells, and immune response represented by cytotoxic T lymphocytes (CTL) cells. The infection transmission process is modeled by Hattaf-Yousfi incidence function which includes many special cases existing in the literature. The positivity of solutions is investigated. In addition, the extinction of the infection is established in terms of the basic reproduction number R 0 . Moreover, sufficient conditions for the fluctuation of solutions around the two infection equilibria are obtained. Numerical simulations are presented to illustrate our theoretical results.

Analysis of a New Delayed HBV Model with Exposed State and Immune Response to Infected Cells and Viruses.

We propose a comprehensive delayed HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and then study the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model by numerical simulation.

Mathematical Model for Cyber Attack in Computer Network

This investigation focuses to develop an e-SEIRS (susceptible, exposed, infectious, recovered) epidemic computer network model to study the transmission of malicious code in a computer network and derive the approximate threshold condition (basic reproduction number) to examine the equilibrium and stability of the model. The authors have simulated the results for various parameters used in the model and Runge-Kutta Fehlberg fourth-fifth order method is employed to solve system of equations developed. They have studied the stability of crime level to equilibrium and found the critical value of threshold value determining whether or not the infectious free equilibrium is globally asymptotically stable and endemic equilibrium is locally asymptotically stable. The simulation results using MATLAB agree with the real life situations.

Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses

A virus dynamics model with intracellular state-dependent delay and nonlinear infection rate of Beddington-DeAngelis functional response is studied. The technique of Lyapunov functionals is used to analyze stability of an interior infection equilibrium which describes the case of both CTL and antibody immune responses activated. We consider first a particular biologically motivated class of discrete state-dependent delays. Next, the general case is investigated.

Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays

In this paper, the dynamical behaviors of a viral infection model with cytotoxic T-lymphocyte (CTL) immune response, immune response delay and production delay are investigated. The threshold values for virus infection and immune response are established. By means of Lyapunov functionals methods and LaSalle's invariance principle, sufficient conditions for the global stability of the infection-free and CTL-absent equilibria are established. Global stability of the CTL-present infection equilibrium is also studied when there is no immune delay in the model. Furthermore, to deal with the local stability of the CTL-present infection equilibrium in a general case with two delays being positive, we extend an existing geometric method to treat the associated characteristic equation. When the two delays are positive, we show some conditions for Hopf bifurcation at the CTL-present infection equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations are performed in order to illustrate the dynamical behaviors of the model.

Mathematical Modeling Applied to Understand the Dynamical Behavior of HIV Infection

The study of viral dynamics of HIV/AIDS has resulted in a deep understanding of host-pathogenesis of HIV infection from which numerous mathematical modeling have been derived. Most of these models are based on nonlinear ordinary differential equations. In Bangladesh, the rate of increase of HIV infection comparing with the other countries of the world is not so high. Bangladesh is still considered to be a low prevalent country in the region with prevalence and shown the local and global stability at disease free and chronic infected equilibrium points. Also we have shown that if the basic reproduction number , then HIV infection is cleared from T cell population and it converges to disease free equilibrium point. Whereas if , then HIV infection persists.

Dynamics of a Delayed HIV-1 Infection Model with Saturation Incidence Rate and CTL Immune Response

In this paper, we investigate the dynamics of a five-dimensional virus model incorporating saturation incidence rate, CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model by proving the positivity and boundedness of the solutions. Our model admits three possible equilibrium solutions, namely the infection-free equilibrium [Formula: see text], the infectious equilibrium without immune response [Formula: see text] and the infectious equilibrium with immune response [Formula: see text]. Moreover, by analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcation at the equilibrium point [Formula: see text] are established, respectively. Further, by using fluctuation lemma and suitable Lyapunov functionals, it is shown that [Formula: see text] is globally asymptotically stable when the basic reproductive numbers for viral infection [Formula: see text] is less than unity. When the basic reproductive numbers for immune response [Formula: see text] is less than unity and [Formula: see text] is greater than unity, the equilibrium point [Formula: see text] is globally asymptotically stable. Finally, some numerical simulations are carried out for illustrating the theoretical results.

A mathematical model of HIV-1 infection within host cell to cell viral transmissions with RTI and discrete delays

A within host viral infection model with both virus to cell and cell to cell transmissions and three discrete delays using reverse transcriptase inhibitors is investigated. The effect of time delay on stability of the equilibria of the infection model has been studied. By choosing the immune delay as the bifurcation parameter, we establish the sufficient condition for local stability of chronic infection equilibrium, Hopf bifurcation occurs when the immune delay passes a critical value. The global stability analysis of the model is carried out in terms of the basic reproduction number $$R_0$$ . If $$R_0 < 1$$ , the infection-free (semi-trivial) equilibrium is globally stable for both ODE and DDE model; if $$R_0>1$$ , the chronic infection (positive) equilibrium exists and is globally stable for both ODE and DDE model. Numerical simulations are obtained to validate our analytical findings by varying the system parameter.

Stability and optimal control of a delayed HIV model

We propose and investigate a delayed model that studies the relationship between HIV and the immune system during the natural course of infection and in the context of antiviral treatment regimes. Sufficient criteria for local asymptotic stability of the infected and viral free equilibria are given. An optimal control problem with time delays both in state variables (incubation delay) and control (pharmacological delay) is then formulated and analyzed, where the objective consists to find the optimal treatment strategy that maximizes the number of uninfected CD4 + T cells as well as cytotoxic T lymphocyte immune response cells, keeping the drug therapy as low as possible. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy

In this paper, we construct an HIV infection model which includes latent infection, logistic growth for healthy CD4+ T-cells, and antiretroviral therapy. We obtain the global asymptotic stability of the uninfected equilibrium by constructing a Lyapunov function, and we give a sufficient condition for the local asymptotic stability of the infected equilibrium. We also use the latin hypercube sampling technique to identify the key parameters in determining the stability of the infected equilibrium. By numerical simulations, we observe that the model without logistic growth would underestimate the number of infectious virions, while the model without latent infection would overestimate the number of infectious virions.

Analysis of time delay in viral infection model with immune impairment

In this paper, a class of virus infection models with CTLs response is considered. We incorporate three delays with immune impairment into the virus infection model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By analyzing corresponding characteristic equation, the local stability of each feasible equilibria and the existence of Hopf bifurcation at the infected equilibrium are addressed, respectively. By fixing the immune delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. Using center manifold argument and normal form theory, we derive explicit formulae to determine the stability and direction of the limit cycles of the model. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of steady states at $$\tau _3<\tau _3^*$$ . By theoretical analysis and numerical simulations, we show that the immune impairment rate has a dramatic effect on the infected equilibrium of the DDE model.

Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions

In this paper, we first propose a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions. Then, we consider the discretization of the model by using nonstandard finite difference scheme. It is then followed by the investigation of the global stability of the equilibria of the discrete model. Our study shows that if the basic reproduction number R0≤1, then the infection-free equilibrium is globally asymptotically stable; however if R0>1, then the infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate our theoretical results.