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In several publications, the dynamical system of HIV and HTLV mono-infections taking into account diffusion, as well as latently infected cells in cellular transmission has been mathematically analyzed. However, no work has been conducted on HTLV/HIV co-infection dynamics taking both factors into consideration. In this paper, a partial differential equations (PDEs) model of HTLV/HIV dual infection was developed and analyzed, considering the cells' and viruses' spatial mobility. CD 4 + T cells are the primary target of both HTLV and HIV. For HIV, there are three routes of transmission: free-to-cell (FTC), latent infected-to-cell (ITC), and active ITC. In contrast, HTLV transmits horizontally through ITC contact and vertically through the mitosis of active HTLV-infected cells. In the beginning, the well-posedness of the model was investigated by proving the existence of global solutions and the boundedness. Eight threshold parameters that determine the existence and stability of the eight equilibria of the model were obtained. Lyapunov functions together with the Lyapunov–LaSalle asymptotic stability theorem were used to investigate the global stability of all equilibria. Finally, the theoretical results were verified utilizing numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a Middle East respiratory syndrome coronavirus (MERS-CoV) infection model with cytotoxic T lymphocyte (CTL) immune response and intracellular delay. This model includes five compartments: uninfected cells, infected cells, viruses, dipeptidyl peptidase 4 (DPP4), and CTL immune cells. We obtained an immunity-inactivated reproduction number R 0 and an immunity-activated reproduction number R 1 . By analyzing the distributions of roots of the corresponding characteristic equations, the local stability results of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium were obtained. Moreover, by constructing suitable Lyapunov functionals and combining LaSalle's invariance principle and Barbalat's lemma, some sufficient conditions for the global stability of the three types of equilibria were obtained. It was found that the infection-free equilibrium is globally asymptotically stable if R 0 ≤ 1 and unstable if R 0 > 1 ; the immunity-inactivated equilibrium is locally asymptotically stable if R 0 > 1 > R 1 and globally asymptotically stable if R 0 > 1 > R 1 and condition (H1) holds, but unstable if R 1 > 1 ; and the immunity-activated equilibrium is locally asymptotically stable if R 1 > 1 and is globally asymptotically stable if R 1 > 1 and condition (H1) holds. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R 01 and immune reproduction number R 02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R 01 < 1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually; when R 02 > 1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given. [ABSTRACT FROM AUTHOR]

In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected CD 4 + T cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an in-host model for the viral dynamics of HIV-1 infection and its interaction with the CTL immune response. The model is sufficiently general to allow nonlinear forms for both viral infection and CTL response. Threshold parameters are identified that completely determine the global dynamics and outcomes of the virus-target cell-CTL interactions. Impacts of key parameter values for CTL functions and viral budding rate on the HIV-1 viral load and CD4 count are investigated using numerical simulations. Results support clinical evidence for important differences between HIV-1 nonprogressors and progressors. [ABSTRACT FROM AUTHOR]

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters ℜ0$$ {\mathfrak{\Re}}_0 $$ (basic reproduction number) and ℜ1$$ {\mathfrak{\Re}}_1 $$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ℜ0≤1$$ {\mathfrak{\Re}}_0\le 1 $$; the immune‐free infected steady state is globally asymptotically stable when ℜ1≤1<ℜ0$$ {\mathfrak{\Re}}_1\le 1<{\mathfrak{\Re}}_0 $$; and the immune‐present steady state is globally asymptotically stable when ℜ0>1,ℜ1>1$$ {\mathfrak{\Re}}_0>1,{\mathfrak{\Re}}_1>1 $$, and φ(I)=I$$ \varphi (I)=I $$. We perform the sensitivity and uncertainly analysis on ℜ0$$ {\mathfrak{\Re}}_0 $$ with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic HTLV-I infection model with CTL immune response, and the corresponding deterministic model has two basic reproduction numbers. We consider the nonlinear CTL immune response for the interaction between the virus and the CTL immune cells. Firstly, for the theoretical needs of system dynamical behavior, we prove that the stochastic model solution is positive and global. In addition, we obtain the existence of ergodic stationary distribution by stochastic Lyapunov functions. Meanwhile, sufficient condition for the extinction of the stochastic system is acquired. Reasonably, the dynamical behavior of deterministic model is included in our result of stochastic model when the white noise disappears. [ABSTRACT FROM AUTHOR]

In this paper, we propose an improved human T‐cell leukemia virus type 1 infection model with mitotic division of actively infected cells and delayed cytotoxic T lymphocyte immune response. By constructing suitable Lyapunov functional and using LaSalle invariance principle, we investigate the global stability of the infection‐free equilibrium of the system. Our results show that the time delay can change stability behavior of the infection equilibrium and lead to the existence of Hopf bifurcations. Finally, numerical simulations are conducted to illustrate the applications of the main results. [ABSTRACT FROM AUTHOR]

A modified mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with cytotoxic T-lymphocytes (CTL) and infected cells in eclipse phase is presented and studied in this paper. The model under consideration also includes a saturated rate describing viral infection. First, the positivity and boundedness of solutions for nonnegative initial data are proved. Next, the global stability of the disease free steady state and the endemic steady states are established depending on the basic reproduction number R0 and the CTL immune response reproduction number RCTL. Moreover, numerical simulations are performed in order to show the numerical stability for each steady state and to support our theoretical findings. Our model based findings suggest that system immunity represented by CTL may control viral replication and reduce the infection. [ABSTRACT FROM AUTHOR]

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, sufiicient 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. [ABSTRACT FROM AUTHOR]

This paper studies the dynamical behavior of an HIV-1 infection model with saturated virus-target and infected-target incidences with Cytotoxic T Lymphocyte (CTL) immune response. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate and the effect of CTL immune response. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number () and the CTL immune response activation number (). By using suitable Lyapunov functionals, we show that if , then the infection-free steady state is globally asymptotically stable; if , then the CTL-inactivated infection steady state is globally asymptotically stable; if , then the CTL-activated infection steady state is globally asymptotically stable. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown. [ABSTRACT FROM AUTHOR]

In this paper, a class of virus infection model with Beddington-DeAngelis infection function and cytotoxic T-lymphocyte immune response is investigated. Time delay in the immune response term is incorporated into the model. We show that the dynamics of the model are determined by the basic reproduction number R1 and the immune response reproduction number R1 1. If R0 ≤, 1 then the infection-free equilibrium is globally asymptotically stable. If R1 > 1, then the immune-free equilibrium is globally asymptotically stable. If inline image, then the stability of the interior equilibrium is investigated. We conclude that Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion well. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability analysis of a virus dynamics model with CTL immune response and with both cell-to-cell and virus-to-cell transmissions. The model contains three types of distributed time delays. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number (R0) and the CTL immune response activation number (R1). By using suitable Lyapunov functionals, we show that if R0 ≤ 1, then the infection-free steady state E0 is globally asymptotically stable; if R1 ≤ 1 < R0, then the CTL-inactivated infection steady state E1 is globally asymptotically stable; if R1 > 1, then the CTL-activated infection steady state E2 is globally asymptotically stable. Numerical simulations are conducted to support the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some sufficient conditions for the existence of periodic oscillation of a class of in-host MERS-Cov infection model with cytotoxic T lymphocyte (CTL) immune response. A new technique is developed to obtain a lower bound of the state variable characterizing CTL immune response in the model. Our results expand on some previous works.

In this paper, we investigate the dynamical behavior of a nonlinear model for viral infection with Cytotoxic T Lymphocyte (CTL) immune response. The model is a generalization of several models presented in the literature by considering more general functions for the: (i) intrinsic growth rate of uninfected cells; (ii) incidence rate of infection; (iii) natural death rate of infected cells; (iv) rate at which the infected cells are killed by CTL cells; (v) production and removal rates of viruses; (vi) activation and natural death rates of CTLs. We derive two threshold parameters R0 (the basic infection reproduction number) and R1 (the CTL immune response activation number) and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. By using suitable Lyapunov functions and LaSalle's invariance principle, we prove the global asymptotic stability of all equilibria of the model. [ABSTRACT FROM AUTHOR]

Human T-cell Lymphotropic virus type 1(HTLV-I) causes HAM/T SP and other illnesses. HTLV-I mainly infects CD4 + T cells and activates HTLV-I-specific immune response. In this paper, we formulate a mathematical model of HTLV-I to investigate the role of selective mitotic transmission, Tax expression, and CTL response in vivo. We define two parameters (R0 and R1) to study the model dynamics. The unique infection-free equilibrium P0 is globally asymptomatic stable if R0 < 1. There exists the chronic-infection equilibrium P1 if R1 < 1 < R0. There exists a unique chronic-infection equilibrium P2 if R1 > 1. There is a backward bifurcation of chronic-infection equilibria with CTL response if R1 < 1 < R0. The numerical simulations shown that the existence of backward bifurcation may lead to the existence of periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study a diffusive and delayed virus dynamics model with Beddington–DeAngelis incidence and CTL immune response. By constructing Lyapunov functionals, we show that if the basic reproductive number is less than or equal to one, then the infection-free equilibrium is globally asymptotically stable; if the immune reproductive number is less than or equal to one and the basic reproductive number is greater than one, then the immune-free equilibrium is globally asymptotically stable; if the immune reproductive number is greater than one, then the interior equilibrium is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

In several publications, the dynamical system of HIV and HTLV mono-infections taking into account diffusion, as well as latently infected cells in cellular transmission has been mathematically analyzed. However, no work has been conducted on HTLV/HIV co-infection dynamics taking both factors into consideration. In this paper, a partial differential equations (PDEs) model of HTLV/HIV dual infection was developed and analyzed, considering the cells’ and viruses’ spatial mobility. CD4+T cells are the primary target of both HTLV and HIV. For HIV, there are three routes of transmission: free-to-cell (FTC), latent infected-to-cell (ITC), and active ITC. In contrast, HTLV transmits horizontally through ITC contact and vertically through the mitosis of active HTLV-infected cells. In the beginning, the well-posedness of the model was investigated by proving the existence of global solutions and the boundedness. Eight threshold parameters that determine the existence and stability of the eight equilibria of the model were obtained. Lyapunov functions together with the Lyapunov–LaSalle asymptotic stability theorem were used to investigate the global stability of all equilibria. Finally, the theoretical results were verified utilizing numerical simulations.

Abstract In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected CD 4 + T $\mathrm{CD} 4^{+}\mathrm{T}$ cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system.

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number $ R_{01} $ and immune reproduction number $ R_{02} $ are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when $ R_{01} \lt 1 $ , the virus will be cleared out eventually and the CTL immune response will also disappear; when $ R_{02} \lt 1 \lt R_{01} $ , the virus persists within the host, but the CTL immune response disappears eventually; when $ R_{02} \gt 1 $ , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.

In this paper, we propose and study a Middle East respiratory syndrome coronavirus (MERS-CoV) infection model with cytotoxic T lymphocyte (CTL) immune response and intracellular delay. This model includes five compartments: uninfected cells, infected cells, viruses, dipeptidyl peptidase 4 (DPP4), and CTL immune cells. We obtained an immunity-inactivated reproduction number R0 and an immunity-activated reproduction number R1. By analyzing the distributions of roots of the corresponding characteristic equations, the local stability results of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium were obtained. Moreover, by constructing suitable Lyapunov functionals and combining LaSalle’s invariance principle and Barbalat’s lemma, some sufficient conditions for the global stability of the three types of equilibria were obtained. It was found that the infection-free equilibrium is globally asymptotically stable if R0≤1 and unstable if R0>1; the immunity-inactivated equilibrium is locally asymptotically stable if R0>1>R1 and globally asymptotically stable if R0>1>R1 and condition (H1) holds, but unstable if R1>1; and the immunity-activated equilibrium is locally asymptotically stable if R1>1 and is globally asymptotically stable if R1>1 and condition (H1) holds.

To study the emergence and evolution of drug resistance during treatment of HIV infection, we study a mathematical model with two strains, one drug-sensitive and the other drug-resistant, by incorporating cytotoxic T lymphocyte (CTL) immune response. The reproductive numbers for each strain with and without the CTL immune response are obtained and shown to determine the stability of the steady states. By sensitivity analysis, we evaluate how the changes of parameters influence the reproductive numbers. The model shows that CTL immune response can suppress the development of drug resistance. There is a dynamic relationship between antiretroviral drug administration, the prevalence of drug resistance, the total level of viral production, and the strength of immune responses. We further investigate the scenario under which the drug-resistant strain can outcompete the wild-type strain. If drug efficacy is at an intermediate level, the drug-resistant virus is likely to arise. The slower the immune response wanes, the slower the drug-resistant strain grows. The results suggest that immunotherapy that aims to enhance immune responses, combined with antiretroviral drug treatment, may result in a functional control of HIV infection. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured HIV-1 infection model with CTL immune response is investigated. In the model, we consider the infection age (i.e. the time that has elapsed since an HIV virion has penetrated the cell) of infected $T$ cells. The asymptotic smoothness of the semi-flow generated by the system is established. By calculation, the immune-inactivated reproduction rate $\mathscr{R}_0$ and the immune-activated reproduction rate $\mathscr{R}_1$ are obtained. By analyzing the corresponding characteristic equations, the local stability of an infection-free steady state and a CTL-inactivated infection steady state of the model is established. By using the persistence theory for infinite dimensional system, the uniform persistence of the system is established when $\mathscr{R}_1>1$. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if $\mathscr{R}_0 < 1$, the infection-free steady state is globally asymptotically stable; if $\mathscr{R}_1 < 1 < \mathscr{R}_0$, sufficient conditions are derived for the global stability of the CTL-inactivated infection steady state; if $\mathscr{R}_1>1$, sufficient conditions are obtained for the global attractivity of the CTL-activated infection steady state. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.

In this paper, a mathematical model is formulated to investigate the effect of cytotoxic T lymphocyte (CTL) immune response on human immunodeficiency virus (HIV) infection dynamics. The model includes latently infected cells, antiretroviral therapy, cell-free virus infection and cell-to-cell viral transmission. By constructing Lyapunov functionals, the global stability of three equilibria is obtained. More specifically, the infection-free equilibrium $E_{f}$ is globally asymptotically stable when the basic reproductive numbers $\mathcal{R}_{0} < 1$, implying that the virus can be eventually cleared; the infected equilibrium without immune response $E_{w}$ is globally asymptotically stable when the CTL immune response reproduction number $\mathcal{R}_{1}$ is less than one and $\mathcal{R}_{0}$ is greater than one, implying that the infection becomes chronic, but CTL immune response has not been established; the infected equilibrium with immune response $E_{c}$ is globally asymptotically stable when $\mathcal{R}_{1}>1$, implying that the infection becomes chronic with persistent CTL immune response. Numerical simulations confirm the above theoretical results. Moreover, the inclusion of CTL immune response can generate a higher level of uninfected CD4+ T cells, and significantly reduce infected cells and viral load. These results may help to improve the understanding of HIV infection dynamics.

In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number $R_0$. When $R_0\le 1$, there is only the infection-free steady state, which is globally asymptotically stable if $R_0 < 1$. When $R_0> 1$, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.

We consider a mathematical model that describes a viral infection of HIV-1 with both virus-tocell and cell-to-cell transmission, CTL response immune and four distributed delays, describing intracellular delays and immune response delay. One of the main features of the model is that it includes a constant production rate of CTLs export from thymus, and an immune response delay. We derive the basic reproduction number and show that if the basic reproduction number is less than one, then the infection free equilibrium is globally asymptotically stable; whereas, if the basic reproduction number is greater than one, then there exist a chronic infection equilibrium, which is globally asymptotically stable in absence of immune response delay. Furthermore, for the special case with only immune response delay, we determine some conditions for stability switches of the chronic infection equilibrium. Numerical simulations indicate that the intracellular delays and immune response delay can stabilize and/or destabilize the chronic infection equilibrium. [ABSTRACT FROM AUTHOR]

Dynamical behaviors of an HBV infection model with distributed delay are studied. The model includes the ordinary, discrete delay and Gamma distributed delay differential equation models as special cases. The model always has two equilibria: the liver failure equilibrium which is unstable and the infection-free equilibrium which is globally asymptotically stable if the basic reproduction number $$R_{0}\leqslant 1$$ . There is an immune-free equilibrium if $$R_{0}>1$$ and an endemic equilibrium if the CTL immune response reproduction number $$R_{1}>1$$ . We study the stability of these two equilibria in three models respectively. In the ODE model, they are globally asymptotically stable under some conditions. However, there are Hopf bifurcations near the equilibria in both delay models by using the (average) delay as a bifurcation parameter. Numerical simulations have been done to support the analytical findings. [ABSTRACT FROM AUTHOR]

A modified mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with cytotoxic T-lymphocytes (CTL) and infected cells in eclipse phase is presented and studied in this paper. The model under consideration also includes a saturated rate describing viral infection. First, the positivity and boundedness of solutions for nonnegative initial data are proved. Next, the global stability of the disease free steady state and the endemic steady states are established depending on the basic reproduction number R 0 and the CTL immune response reproduction number R C T L . Moreover, numerical simulations are performed in order to show the numerical stability for each steady state and to support our theoretical findings. Our model based findings suggest that system immunity represented by CTL may control viral replication and reduce the infection.