Chronic-infection Equilibrium Research Articles

Stability analysis of a fractional model for the transmission of the cochineal

Scale insects are parasitic insects that attack many indoor and outdoor plants, including cacti and succulents. These insects are among the frequent causes of diseases in cacti: for the reason that they are tough, multiply in record time and could be destructive to these plants, although they are considered resistant. Mealybugs feed on the sap of plants, drying them out and discoloring them. In this research, we propose and investigate a fractional model for the transmission of the Cochineal. In the first place, we prove the positivity and boundedness of solutions in order to ensure the well-posedness of the proposed model. The local stability of the disease-free equilibrium and the chronic infection equilibrium is established. Numerical simulations are presented in order to validate our theoretical results.

Global stability of a delayed HIV-1 dynamics model with saturation response with cure rate, absorption effect and two time delays

In this paper, we proposed a new HIV-1 infection model with saturation infection rate by incorporating, cure rate, absorption effect, and two-time delays namely intracellular time delay that representing the time viral passage into a target cell and creation of new infectious particles and maturation time delay representing the required time for newly produce virus to mature and the infect the susceptible cell. The mathematical investigation shows that the basic reproduction number R0 of the model totally decides the steadiness properties. Utilizing the characteristic equation of the model and Routh-Hurwitz steadiness basis, we affirmed that the infection-free equilibrium and the chronic disease equilibrium are locally asymptotically steady in case R0 ≤ 1 and R0 > 1, individually. If R0 ≤ 1, using the appropriate Lyapunov functions and LaSalle’s invariance principle, it has been shown that the infection-free equilibrium of the model is globally asymptotically stable. If R0 > 1 the model is persistent. Conversely, if R0 > 1, the infection-free equilibrium is unstable and a unique chronic infection equilibrium exists. We show that, if R0 > 1, the chronic infection is globally asymptotically stable. Moreover, numerical simulations are performed to verify the theoretical results.

Dynamics analysis of an HIV infection model with latent reservoir, delayed CTL immune response and immune impairment

In this paper, we propose an HIV model with latent reservoir, delayed CTL immune response and immune impairment in which both virus-to-cell infection and cell-to-cell viral transmission are considered. By using Lyapunov functionals and LaSalle's invariance principle, it is verified that when time delay is equal to zero, the global threshold dynamics of the model is determined by the basic reproduction ratio. With the help of uniform persistence theory for infinite dimensional systems, we obtain the uniform persistence when the basic reproduction ratio is greater than unity. By choosing time delay τ as a bifurcation parameter and analyzing the corresponding characteristic equation of the system, we establish the existence of Hopf bifurcation at the chronic-infection equilibrium. Numerical simulations are carried out to illustrate the corresponding theoretical results.

Backward bifurcation and stability analysis in a within-host HIV model with both virus-to-cell infection and cell-to-cell transmission, and anti-retroviral therapy

Researches have shown that in addition to direct virus-to-cell infection, viral particles can also be transferred from a productively-infected cell to an uninfected cell through the formation of virological synapses. In order to reduce the viral load in infected individuals, different classes of antiretroviral drugs have been developed, including reverse transcriptase inhibitor (RTI), integrase inhibitor (II), protease inhibitor (PI) and so on. In this paper, we incorporate the mitotic proliferation of target cells which is described by the logistic term, both virus-to-cell infection and cell-to-cell transmission, the intracellular delay and RTI-based therapy into an in-host HIV infection model. Through mathematical analysis, we find that the model undergoes a backward bifurcation when the turn-over rate coefficient of productively-infected cells is smaller than its mitotic proliferation rate coefficient. When the turn-over rate coefficient of productively-infected cells is greater than its mitotic proliferation rate coefficient, the existence of Hopf bifurcation at the chronic-infection equilibrium with and without the intracellular delay is established, respectively. Numerical simulations suggest that the dynamics of the model be sensitive to parameter values and initial conditions, which may be of great significance to control HIV infection. We also show numerical evidence to support the fact that the smaller the therapy efficacy, the higher the viral load in infected individuals.

Uniform Persistence and Global Attractivity in a Delayed Virus Dynamic Model with Apoptosis and Both Virus-to-Cell and Cell-to-Cell Infections

In this paper, we study the global dynamics of a delayed virus dynamics model with apoptosis and both virus-to-cell and cell-to-cell infections. When the basic reproduction number R0>1, we obtain the uniform persistence of the model, and give some explicit expressions of the ultimate upper and lower bounds of any positive solution of the model. In addition, by constructing the appropriate Lyapunov functionals, we obtain some sufficient conditions for the global attractivity of the disease-free equilibrium and the chronic infection equilibrium of the model. Our results extend existing related works.

Stability of a delayed HIV-1 dynamics model with Beddington-DeAngelis functional response and absorption effect with two delays

This research examined the stability properties of a set of delay differential equations for HIV-I infection that includes a Beddington-DeAngelis functional response. The model has incorporated with two-time delays namely intracellular time delay and maturation time delay, and the absorption effect in order to make the processes more biologically sensitive. We prove that the infection free equilibrium and the chronic infection equilibrium be locally asymptotically stable if the basic reproduction number R 0 ≤ 1 and R 0 > 1 respectively, by using the characteristic equations of dynamics model and Routh Hurwitz stability criterion. It is shown that, if R 0 ≤ 1, the infection-free equilibrium is globally asymptotically stable by using appropriate Lyapunov function and LaSalle’s invariance principle. Further, we have established the conditions for the permanence of the system. In addition, numerical simulations are performed to illustrate the theoretical results.

Global dynamics of a delayed HTLV-I infection model with Beddington-DeAngelis incidence and immune impairment

In this paper, we investigate a delayed HTLV-I infection model with Beddington-DeAngelis incidence, CTL immune response and immune impairment. In the model, the intracellular delay and immune response delay are considered. By calculation, we obtain the basic reproductive ratio. By analyzing corresponding characteristic equations, the local stabilities of feasible equilibria are established. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that when the basic reproductive ratio is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproductive ratio is greater than unity and immune response delay is equal to zero, the chronic-infection equilibrium is globally asymptotically stable. When the basic reproductive ratio is greater than unity and the intracellular delay is equal to zero, immune response delay will destabilize stability and lead to Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.

Dynamics of a Within-Host Virus Infection Model with Multiple Pathways: Stability Switch and Global Stability

In this paper, we consider an HIV infection model with virus-to-cell infection, cell-to-cell transmission, intracellular delay and mitosis of uninfected cells. The basic reproduction number is calculated by using the method of the next generation matrix. By comparison arguments, it is proved that when the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, the existence of Hopf bifurcation and stability switch at the chronic-infection equilibrium of the model with or without intracellular delay is established. Further, by constructing Lyapunov functionals, sufficient conditions are obtained for the global asymptotic stability of the chronic-infection equilibrium when the cell-to-cell transmission is negligible. Numerical simulations are carried out to illustrate the main theoretical results. The normal form is calculated to determine the bifurcation direction and stability, as well as amplitude and period of bifurcating periodic solutions when the intracellular delay is absent.

Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection

<p style='text-indent:20px;'>This paper studies an <inline-formula><tex-math id="M1">\begin{document}$ (n+2) $\end{document}</tex-math></inline-formula>-dimensional nonlinear HIV dynamics model that characterizes the interactions of HIV particles, susceptible CD4<inline-formula><tex-math id="M2">\begin{document}$ ^{+} $\end{document}</tex-math></inline-formula> T cells and <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula>-stages of infected CD4<inline-formula><tex-math id="M4">\begin{document}$ ^{+} $\end{document}</tex-math></inline-formula> T cells. Both virus-to-cell and cell-to-cell infection modes have been incorporated into the model. The incidence rates of viral and cellular infection as well as the production and death rates of all compartments are modeled by general nonlinear functions. We have revealed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. The basic reproduction number <inline-formula><tex-math id="M5">\begin{document}$ \Re_{0} $\end{document}</tex-math></inline-formula> is determined which insures the existence of the two equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the model's equilibria. The global asymptotic stability of the two equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. We have proven that if <inline-formula><tex-math id="M6">\begin{document}$ \Re_{0}\leq1 $\end{document}</tex-math></inline-formula>, then the infection-free equilibrium is globally asymptotically stable, and if <inline-formula><tex-math id="M7">\begin{document}$ \Re _{0}>1 $\end{document}</tex-math></inline-formula>, then the chronic-infection equilibrium is globally asymptotically stable. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.</p>

Stability and Hopf Bifurcation of a Generalized Chikungunya Virus Infection Model with Two Modes of Transmission and Delays

A generalized chikungunya virus (CHIKV) infection model with nonlinear incidence functions and two time delays is proposed and investigated. The model takes into account both modes of transmission that are virus-to-cell infection and cell-to-cell transmission. Furthermore, the local and global stabilities of the disease-free equilibrium and the chronic infection equilibrium are established by using the linearization and Lyapunov functional methods. Moreover, the existence of Hopf bifurcation is also analyzed. Finally, an application is presented in order to support the analytical results.

Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay.

A HIV virus-to-cell dynamical model with distributed delay and Beddington-DeAngelis functional response is proposed in this paper. Using the characteristic equations and analytical means, the principle reproduction number R0 on the local stability of infection-free and chronic-infection equilibria is established. Furthermore, by constructing suitable Lyapunov functionals and using LaSalle invariance principle, we show that if R0 ≤ 1 the infection-free equilibrium is globally asymptotically stable, while if R0 > 1 the chronic-infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Comparing the effects between discrete and distributed delays on the stability of HIV virus-to-cell dynamical models, we can see that they could be same and different even opposite.

Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment

It is well known that CTL (cytotoxic T lymphocyte) immune response could be broadly classified into lytic and nonlytic components, nonlinear functions can better reproduce saturated effects in the interaction processes between cell and viral populations, and distributed intracellular delay can realistically reflect the stochastic element in the delay effects. For these reasons, we develop an HTLV-I (Human T-cell leukemia virus type I) infection model with nonlinear lytic and nonlytic CTL immune responses, nonlinear incidence rate, distributed intracellular delay and immune impairment. Through conducting complete analysis, it is revealed that all these factors influence the concentration level of infected T-cells at the chronic-infection equilibrium, whereas intracellular distributed delay and nonlinear incidence rate may change the expression of the basic reproduction number \begin{document}$ \mathfrak{R}_0 $\end{document} in the context where the model proposed still preserves the threshold dynamics. Our analysis results obtained may improve several existing works by comparison. We also perform global sensitivity analysis for \begin{document}$ \mathfrak{R}_0 $\end{document} in order to explore the effective strategies of lowering the concentration level of infected T-cells.

Dynamique d'un modèle d'infection HBV avec transmission cellule-à-cellule et réaction immunitaire CTL

In this work, we propose a mathematical model to describe the dynamics of the hepatitis B virus (HBV) infection by taking into account the cure of infected cells, the export of precursor cytotoxic T lympho-cytes (CTL) cells from the thymus and both modes of transmission that are the virus-to-cell infection and the cell-to-cell transmission. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium. Dans ce travail, nous proposons un modèle mathématique pour décrire la dynamique du virus d'hépatite B (HBV) en prenant en compte le taux de guérison de cellules infectées, l'exportation de précurseur cytotoxic des lymphocytes T (CTL) des cellules du thymus et les deux modes de transmission qui sont l'infection virus-à-cellule et la transmission cellule-à-cellule.La stabilité locale de l'équilibre libre et l'équilibre d'infection chronique est obtenue via des équations caractéristiques. En outre, la stabilité globale des deux équilibres est établie en utilisant deux techniques, la méthode directe de Lyapunov pour l'équilibre libre et l'approche géométrique pour l'équilibre d'infection chronique.

Global stability of a fractional order HIV infection model with cure of infected cells in eclipse stage

Modeling by fractional order differential equations has more advantages to describe the dynamics of phenomena with memory which exists in many biological systems. In this paper, we propose a fractional order model for human immunodeficiency virus (HIV) infection by including a class of infected cells that are not yet producing virus, i.e., cells in the eclipse stage. We first prove the positivity and bound-edness of solutions in order to ensure the well-posedness of the proposed model. By constructing appropriate Lyapunov functionals, the global stability of the disease-free equilibrium and the chronic infection equilibrium is established. Numerical simulations are presented in order to validate our theoretical results.

Stability analysis of delay integro-differential equations of HIV-1 infection model

Abstract In this paper, the analysis of an HIV-1 epidemic model is presented by incorporating a distributed intracellular delay. The delay term represents the latent period between the time that the target cells are contacted by the virus and the time the virions penetrated into the cells. To understand the analysis of our proposed model, the Rouths–Hurwiz criterion and general theory of delay differential equations are used. It is shown that the infection free equilibrium and the chronic-infection equilibrium are locally as well as globally asymptotically stable, under some conditions on the basic reproductive number R 0 {R_{0}} . Furthermore, the obtained results show that the value of R 0 {R_{0}} can be decreased by increasing the delay. Therefore, any drugs that can prolong the latent period will help to control the HIV-1 infection.

Spatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activation

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction number R0 and the CTL immune response reproduction number R1. The stability of the last equilibrium depends on R0 and R1 as well as time delay τ in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when τ passes through a certain critical value.

Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate

The aim of this paper is to study the dynamics of a new chronic HBV infection model that includes spatial diffusion, three time delays and a general incidence function. First, we analyze the well-posedness of the initial value problem of the model in the bounded domain. Then, we define a threshold parameter \begin{document}$R_{0}$\end{document} called the basic reproduction number and show that our model admits two possible equilibria, namely the infection-free equilibrium \begin{document}$E_{1}$\end{document} as well as the chronic infection equilibrium \begin{document}$E_{2}$\end{document} . Further, by constructing two appropriate Lyapunov functionals, we prove that \begin{document}$E_{1}$\end{document} is globally asymptotically stable when \begin{document}$R_{0} , corresponding to the viruses are cleared and the disease dies out; if \begin{document}$R_{0}>1$\end{document} , then \begin{document}$E_{1}$\end{document} becomes unstable and the equilibrium point \begin{document}$E_{2}$\end{document} appears and is globally asymptotically stable, which means that the viruses persist in the host and the infection becomes chronic. An application is provided to confirm the main theoretical results. Additionally, it is worth saying that, our results suggest theoretically useful method to control HBV infection and these results can be applied to a variety of possible incidence functions presented in a series of other papers.

Andronov-Hopf bifurcation and sensitivity analysis of a time-delay HIV model with logistic growth and antiretroviral treatment

A mathematical model of the infection of CD4+ T-cells by HIV that includes the effects of treatment by a reverse transcriptase inhibitor (RTI) and a protease inhibitor (PI) is studied. The model includes three populations of CD4+ T-cells (healthy cells, latently-infected cells which cannot produce virus, and productively-infected cells which can produce virus) and two populations of free virus in the blood (infectious virus and non-infectious virus). The model includes a time delay between a T-cell becoming latently infected and productively infected. The model has a virus-free and a chronic infection equilibrium. It is shown that the model has Andronov-Hopf bifurcations leading to limit cycle behavior in the chronic infection region at critical values of the time delays. For three data sets obtained from the work of previous authors, numerical simulations have given critical delay values ranging from approximately 15 days to more than 200 days. This range includes the period of approximately 50 days for intermittent viral blips reported by Rong and Perelson (Plos Comp. Biol. 5(10), 1-18 (2009)). Simple formulas are derived for the sensitivity indices of the equilibrium populations and the basic reproductive number with respect to all parameters in the model. Numerical simulations are carried out to support the analytical results. The numerical results suggest that the most effective methods of reducing both the basic reproductive number and the chronic infection CD4+ T-cell and virus populations are the following: (1) to increase the efficacy of the antiretroviral treatments and (2) to increase virus clearance rate, decrease infection rate, or decrease viral reproduction rate.

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.

A generalized virus dynamics model with cell-to-cell transmission and cure rate

Viruses can be spread and transmitted through two fundamental modes, one by virus-to-cell infection and the other by direct cell-to-cell transmission. In this paper, we propose a new generalized virus dynamics model, which incorporates both modes and takes into account the cure of infected cells. We first show mathematically and biologically the well-posedness of our model. Further, an explicit formula for the basic reproduction number $R_{0}$ of the model is determined. By analyzing the characteristic equations we establish the local stability of the disease-free equilibrium and the chronic infection equilibrium in terms of $R_{0}$ . The global behavior of the model is investigated by constructing an appropriate Lyapunov functional for disease-free equilibrium and by applying geometrical approach to chronic infection equilibrium. Moreover, mathematical virus models and results presented in many previous studies are generalized and improved.