Immune Reproduction Number Research Articles

Delay induced stability switch in a mathematical model of CD8 T-cell response to SARS-CoV-2 mediated by receptor ACE2.

The pathogen SARS-CoV-2 binds to the receptor angiotensin-converting enzyme 2 (ACE2) of the target cells and then replicates itself through the host, eventually releasing free virus particles. After infection, the CD8 T-cell response is triggered and appears to play a critical role in the defense against virus infections. Infected cells and their activated CD8 T-cells can cause tissue damage. Here, we established a mathematical model of within-host SARS-CoV-2 infection that incorporates the receptor ACE2, the CD8 T-cell response, and the damaged tissues. According to this model, we can get the basic reproduction number R0 and the immune reproduction number R1. We provide the theoretical proof for the stability of the disease-free equilibrium, immune-inactivated equilibrium, and immune-activated equilibrium. Finally, our numerical simulations show that the time delay in CD8 T-cell production can induce complex dynamics such as stability switching. These results provide insights into the mechanisms of SARS-CoV-2 infection and may help in the development of effective drugs against COVID-19.

Bistability and Bifurcations of Tumor Dynamics with Immune Escape and the Chimeric Antigen Receptor T-Cell Therapy

Tumor immune escape refers to the inability of the immune system to clear tumor cells, which is one of the major obstacles in designing effective treatment schemes for cancer diseases. Although clinical studies have led to promising treatment outcomes, it is imperative to design theoretical models to investigate the long-term treatment effects. In this paper, we develop a mathematical model to study the interactions among tumor cells, immune escape tumor cells, and T lymphocyte. The chimeric antigen receptor (CAR) T-cell therapy is also described by the mathematical model. Bifurcation analysis shows that there exists backward bifurcation and saddle-node bifurcation when the immune intensity is used as the bifurcation parameter. The proposed model also exhibits bistability when its parameters are located between the saddle-node threshold and backward bifurcation threshold. Sensitivity analysis is performed to illustrate the effects of different mechanisms on the backward bifurcation threshold and basic immune reproduction number. Simulation studies confirm the bifurcation analysis results and predict various types of treatment outcomes using different CAR T-cell therapy strengths. Analysis and simulation results show that the immune intensity can be used to control the tumor size, but it has no effect on the control of the immune escape tumor size. The introduction of the CAR T-cell therapy will reduce the immune escape tumor size and the treatment effect depends on the CAR T-cell therapy strength.

Stability analysis of HCV dynamic model with saturation incidence, cellular immunity and interferon effect in intrahepatic and extrahepatic tissues

An HCV model with saturation incidence and cellular immunity is constructed and analyzed. The model takes account of both intrahepatic and extrahepatic tissues and the effect of antiviral drugs especially the interferon. The basic reproduction number R01 and CTL immune reproduction number R02 are calculated, respectively. The existence of three equilibria, namely infection-free, immune response-free, infection-immune coexistence equilibrium is analyzed with a novel method. Stability analysis is also given with corresponding reproduction number, and the results show that when R01<1, there are no virus and immune response existed in the end; when R02<1<R01, the virus survives, but immune response gradually disappears; when R02>1, the system will be globally stable with both virus and immune response existing. Numerical analysis is also given which is consistent with the theoretical analysis.

Dynamic analysis of a cytokine-enhanced viral infection model with time delays and CTL immune response

In this paper, a cytokine-enhanced viral infection model with time delays and CTL immune response is proposed and analyzed. Multiple time delays, namely intracellular delay τ1, virus replication delay τ2 and immune response delay τ3 are included. Firstly, two key parameters with strong biological significance, namely the virus reproductive number R0 and the CTL immune reproductive number R1 are derived. And then the stability of equilibria of the model is investigated by constructing suitable Lyapunov functionals and using LaSalle’s invariance principle. We prove that if R0<1, then the infection-free equilibrium E0 is globally asymptotically stable for any τ1,τ2,τ3≥0; if R0>1,R1<1, then the immunity-inactivated equilibrium E1 is globally asymptotically stable for any τ1,τ2,τ3≥0; and if R1>1, then the immunity-activated equilibrium E∗ is globally asymptotically stable for any τ1,τ2≥0 and τ3=0. For τ1,τ2≥0,τ3>0, theoretical analysis and numerical simulations show that with the increase of τ3, the equilibrium E∗ loses its stability and the system generates Hopf bifurcation, which means the system produces periodic oscillation under certain conditions.

Dynamics Analysis of a Delayed HIV Infection Model with CTL Immune Response and Antibody Immune Response

In this paper, dynamics analysis of a delayed HIV infection model with CTL immune response and antibody immune response is investigated. The model involves the concentrations of uninfected cells, infected cells, free virus, CTL response cells, and antibody antibody response cells. There are three delays in the model: the intracellular delay, virus replication delay and the antibody delay. The basic reproductive number of viral infection, the antibody immune reproductive number, the CTL immune reproductive number, the CTL immune competitive reproductive number and the antibody immune competitive reproductive number are derived. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions for the stability of each equilibrium is established. The results show that the intracellular delay and virus replication delay do not impact upon the stability of each equilibrium, but when the antibody delay is positive, Hopf bifurcation at the antibody response and the interior equilibrium will exist by using the antibody delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.

Threshold dynamics of a HCV model with virus to cell transmission in both liver with CTL immune response and the extrahepatic tissue

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 and immune reproduction number 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 , the virus will be cleared out eventually and the CTL immune response will also disappear; when , the virus persists within the host, but the CTL immune response disappears eventually; when , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.

Hopf Bifurcation and Stability Switches Induced by Humoral Immune Delay in Hepatitis C

The role of humoral immune delay on the dynamics of HCV infection incorporating both the modes of infection transmission, namely, viral and cellular transmissions with a non-cytolytic cure of infected hepatocytes is studied. The local and global asymptotic stability of the boundary equilibria, namely, infection-free and immune-free equilibrium are analyzed theoretically as well as numerically under the conditions on the basic reproduction number and the humoral immune reproduction number. The existence of Hopf bifurcation and consequent occurrence of bifurcating periodic orbits around the humoral immune activated equilibrium are illustrated. The findings show that Hopf bifurcation and stability switches occur under certain conditions as the bifurcation parameter crosses the critical values. Furthermore, the dynamical effect of the development rate of B cells is investigated numerically. The results obtained show that the system becomes unstable from stable and regains stability from instability depending on the development rate of B cells for a fixed delay value. Further, the results suggest that a high antigenic stimulation in humoral immunity is beneficial for uninfected hepatocytes with a significant reduction in virions density.

Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence

In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.

Global stability of a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response

In this paper, a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response is investigated. By constructing the Lyapunov functionals, the threshold conditions on the global stability of the infection-free, immune-free and interior equilibria are established if the space is assumed to be homogeneous. We show that the infection-free equilibrium is globally asymptotically stable if the basic reproductive number R_{0}leq1; the immune-free equilibrium is globally asymptotically stable if the immune reproduction number and the basic reproduction number satisfy R_{1}leq1< R_{0}; the interior equilibrium is globally asymptotically stable if R_{1}>1.

Global dynamics of an age‐structured virus model with saturation effects

An infection‐age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values R0, the basic reproductive number for viral infection, and R1, the viral reproductive number at the immune‐free infection steady state (R1&lt;R0). If R0&lt;1, the uninfected steady state E0 is globally asymptotically stable; if R0&gt;1 &gt; R1, the immune‐free infected steady state E∗ is globally asymptotically stable; while if R1&gt;1, the antibody immune infected steady state is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Global stability of a diffusive and delayed virus dynamics model with Beddington–DeAngelis incidence function and CTL immune response

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.

Global properties of a delayed HIV infection model with CTL immune response

In this paper, we study a delayed six-dimensional human immunodeficiency virus (HIV) model with Cytotoxic T Lymphocytes (CTLs) immune response. Our model describes the interaction of HIV with two target cells: CD4+ T cells and macrophages. We derive that the global asymptotic attractivity of the model is completely determined by the basic reproduction number R0 and the immune reproduction number R0∗ for the viral infection. By constructing Lyapunov functionals, we have shown that the infection-free equilibrium E0, the immune-free equilibrium E1 and the chronic-infection equilibrium E2 are globally asymptotically attractive when R0⩽1,R0∗⩽1<R0 and R0>R0∗>1, respectively.