Infected Equilibrium Research Articles

STABILITY ANALYSIS AND OPTIMAL CONTROL OF AN INTRACELLULAR HIV INFECTION MODEL WITH ANTIRETROVIRAL TREATMENT

In this paper, a deterministic model describing the dynamics of the in-host HIV infection of CD4[Formula: see text] T-cells is proposed. The model incorporates the presence of the CD[Formula: see text] T-cells and two types of antiretroviral drugs, for disrupting new infection and for inhibiting virus production, respectively. First, the existence, boundedness and positivity of the model solutions are shown, the basic reproduction number [Formula: see text] being then derived and shown to be a threshold value as far as the stability of the equilibria is concerned. When [Formula: see text] the infection-free equilibrium point is globally stable, whereas when [Formula: see text] the system is uniformly persistent and the infected equilibrium point is globally asymptotically stable. Further, we develop an optimal control model by taking the effect of the antiretroviral drugs to be control variables in order to minimize the HIV infection in different scenarios. By using Pontryagin’s Minimum Principle and solving the model numerically, the results show that each antiretroviral drug in isolation can play a key role in reducing the count of both infected CD4[Formula: see text] T-cells and HIV viruses. However, a combination of both drugs could reduce the in-host HIV infection more significantly.

Spatiotemporal dynamics of a fractional model for hepatitis B virus infection with cellular immunity

In this paper, we propose and investigate a fractional diffusive model for hepatitis B virus (HBV) infection with capsids and immune response presented by cytotoxic T lymphocyte (CTL) cells. We derive the conditions for global asymptotic stability of the steady states of the model in terms of the basic reproduction numberR0and the immune response reproduction numberR1. By constructing appropriate Lyapunov functionals, it is shown that the infection-free equilibrium is globally asymptotically stable whenR0≤ 1, the immune-free infection equilibrium is globally asymptotically stable whenR1≤ 1 <R0and the infection equilibrium with CTL immune response is globally asymptotically stable whenR1> 1. Numerical simulations are performed to illustrate the analytical results.

Global stability of a delayed and diffusive virus model with nonlinear infection function

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of by applying Lyapunov method. The results showed that if is less than 1, then the infection-free equilibrium is globally asymptotically stable. If is greater than 1, then the infection equilibrium is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results.

Global stability of age-of-infection multiscale HCV model with therapy.

In order to treat the diseases caused by hepatitis C virus (HCV) more efficiently, the concentration of HCV in blood, cells, tissues and the body has attracted widespread attention from related scholars. This paper studies a dynamic dependent HCV model (more specifically, including age structure and treatment methods model) that concludes states of infection-free and infected equilibrium. Through eigenvalue analysis and Volterra integral formula, it proves that $E_0$ is globally asymptotically stable when $\mathcal{R}<1$. After explaining the existence, uniqueness and positive properties of the solution of the system, we have proved the global asymptotic stability of $E^*$ when $\mathcal{R}>1$ by constructing a suitable Lyapunov function. Through the above proofs, it can be concluded that effective treatment measures can significantly reduce the number of HCVs, so many related researchers are aware of the importance of highly efficient nursing methods and are committed to applying relevant methods to practice.

Global Stability and Hopf Bifurcation for a Virus Dynamics Model with General Incidence Rate and Delayed CTL Immune Response

In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium E*0, CTL-inactivated infection equilibrium E*1 and CTL-activated infection equilibrium E*2. We prove that in the absence of CTL immune delay, the model has exactly the basic behaviour model, for all positive intracellular delays, the global dynamics are determined by two threshold parameters R0 and R1, if R0 ≤ 1, E*0 is globally asymptotically stable, if R1 ≤ 1 R0, E*1 is globally asymptotically stable and if R1 >1, E*2 is globally asymptotically stable. But if the CTL immune response delay is different from zero, then the behaviour of the model at E*2 changes completely, although R1 > 1, a Hopf bifurcation at E*2 is established. In the end, we present some numerical simulations.

A DYNAMIC MODEL FOR COVID-19 THERAPY WITH DEFECTIVE INTERFERING PARTICLES AND ARTIFICIAL ANTIBODIES

<abstract> In this paper, we use ordinary differential equations to propose a mathematical model for COVID-19 therapy with both defective interfering particles and artificial antibodies. For this model, the basic reproduction number <italic>R</italic>₀ is given and its threshold properties are discussed. We investigate the global asymptotic stability of disease-free equilibrium <italic>E</italic>₀ and infection equilibrium without defective interfering particles <italic>E</italic>₁ by utilizing Lyapunov function and LaSalleos invariance principle. For infection equilibrium with defective interfering particles <italic>E</italic>₂, stability and Hopf bifurcation results are presented. Numerical simulation is also given to demonstrate the applicability of the theoretical predictions. </abstract>

Time Optimal Control Studies on COVID-19 Incorporating Adverse Events of the Antiviral Drugs

Abstract COVID -19 pandemic has resulted in more than 257 million infections and 5.15 million deaths worldwide. Several drug interventions targeting multiple stages of the pathogenesis of COVID -19 can significantly reduce induced infection and thus mortality. In this study, we first develop SIV model at within-host level by incorporating the intercellular time delay and analyzing the stability of equilibrium points. The model dynamics admits a disease-free equilibrium and an infected equilibrium with their stability based on the value of the basic reproduction number R 0. We then formulate an optimal control problem with antiviral drugs and second-line drugs as control measures and study their roles in reducing the number of infected cells and viral load. The comparative study conducted in the optimal control problem suggests that if the first-line antiviral drugs show adverse effects, considering these drugs in reduced amounts along with the second-line drugs would be very effective in reducing the number of infected cells and viral load in a COVID-19 infected patient. Later, we formulate a time-optimal control problem with the goal of driving the system from any initial state to the desired infection-free equilibrium state in finite minimal time. Using Pontryagin’s Minimum Principle, it is shown that the optimal control strategy is of the bang-bang type, with the possibility of switching between two extreme values of the optimal controls. Numerically, it is shown that the desired infection-free state is achieved in a shorter time when the higher values of the optimal controls. The results of this study may be very helpful to researchers, epidemiologists, clinicians and physicians working in this field.

Stability and bifurcation analysis of an HIV-1 infection model with a general incidence and CTL immune response

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number and the basic immunity reproduction number . The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters.

Accounting for multi-delay effects in an HIV-1 infection model with saturated infection rate, recovery and proliferation of host cells

Biological models inherently contain delay. Mathematical analysis of a delay-induced model is, however, more difficult compare to its non-delayed counterpart. Difficulties multiply if the model contains multiple delays. In this paper, we analyze a realistic HIV-1 infection model in the presence and absence of multiple delays. We consider self-proliferation of CD4+T cells, nonlinear saturated infection rate and recovery of infected cells due to incomplete reverse transcription in a basic HIV-1 in-host model and incorporate multiple delays to account for successful viral entry and subsequent virus reproduction from the infected cell. Both of delayed and non-delayed system becomes disease-free if the basic reproduction number is less than unity. In the absence of delays, the infected equilibrium is shown to be locally asymptotically stable under some parametric space and unstable otherwise. The system may show unstable oscillatory behaviour in the presence of either delay even when the non-delayed system is stable. The second delay further enhances the instability of the endemic equilibrium which is otherwise stable in the presence of a single delay. Numerical results are shown to be in agreement with the analytical results and reflect quite realistic dynamics observed in HIV-1 infected individuals.

Within-host mathematical modeling on crucial inflammatory mediators and drug interventions in COVID-19 identifies combination therapy to be most effective and optimal

The unprecedented Covid-19 pandemic has resulted in more than 14.75 million infections and 6, 10, 839 deaths in 212 countries. Appropriate interventions can decrease the rate of Covid-19 related mortality. Fast track clinical trials around the world are addressing the efficacy of individual pharmaceutical agent acting at various stages of pathogenesis. However, lessons learnt while dealing with past viral epidemics mandates, simultaneous use of such drugs in combination amongst different populations. Mathematical modelling studies can be extremely helpful in understanding the efficacy of drugs both individually and in combination. The current within-host mathematical model studies the natural history of Covid-19 in terms of complex interplay of virus replication and behaviour of host immune response. Additionally it studies the role of various drugs at various stages of pathogenesis. The model was validated by generating two-parameter heat plots, representing the characteristics of Covid-19, the sensitivity analysis identified the crucial parameters. The efficacy of interventions was assessed by optimal control problem. The model dynamics exhibited disease-free equilibrium and the infected equilibrium with their stability, based on the reproduction number R0, the transcritical bifurcation observed at R0=1. The burst rate and the natural death rate of the virus were observed as most significant parameters in the life-threatening Covid-19 pneumonia. The antiviral drugs affecting viral replication and those modulating the immune response, reduce the infected cells and viral load significantly. However, the yield was optimal and most effective when the combination therapy involving one or more antiviral and one or more immunomodulating drugs were administered together. These findings may help physicians with early decision making in treatment of life-threatening Covid-19 infection.

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.

Stability Analysis of a Mathematical Model in Inflammatory Response System Due to SARS Coronavirus Infection

Severe Acute Respiratory Syndrome (SARS) Coronavirus infection in a human body indicated by cytokine response due to an inflammation. The purpose of this research is to construct and analyze a mathematical model of interaction between inflammatory pro-response and anti-response cytokine to predict the dynamic on inflammatory response system, so that the treatment can be optimized. The results obtained in this research describe some dynamics which happen on the cytokines, i.e. the disease-free equilibrium point is asymptotically stable when the basic reproduction number is less than one. In this condition, a patient with initial concentrations of the cytokines around the disease-free equilibrium point will be free of viral infection. The infection equilibrium point is asymptotically stable when the basic reproduction number is greater than one. In this condition, a patient with initial concentrations of the cytokines around the infection equilibrium point will be infected by the virus. Probability of a patient being free of viral infection can increase if the production rate of the cytokines are decreased or the degradation rate of the cytokines are increased.

On dynamics of an HIV pathogenesis model with full logistic target cell growth and cure rate

The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R 0 ≤1, the infection is cleared from T cells and if R 0 >1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.

Chimeric Tobamoviruses With Coat Protein Exchanges Modulate Symptom Expression and Defence Responses in Nicotiana tabacum.

In the pathogen infection and host defence equilibrium, plant viruses have evolved to efficiently replicate their genomes, to resist the attack from host defence responses and to avoid causing severe negative effect on growth and metabolism of the hosts. In this study, we generated chimeric tobacco mosaic virus (TMV) variants, in which the coat protein (CP) sequences were substituted with that of cucumber green mottle mosaic virus (CGMMV) or pepper mild mottle virus (PMMoV) to address the role of these in virus infection and host symptomology. The results showed that the chimeric viruses (TMV-CGCP or TMV-PMCP) induce stunting and necrotic symptoms in tobacco plants. We analyzed the transcriptomic changes in tobacco plants after infection of TMV and its chimeras using a high-throughput RNA sequencing approach and found that infection of the chimeric TMV induced significant up-regulation of host defence responsive genes together with salicylic (SA) or abscisic acid (ABA) responsive genes, but down-regulation of auxin (Aux) responsive genes. We further confirmed the increase in the levels of SA and ABA, together with the reduced levels of Aux after infection of chimeric TMV in tobacco plants. These data suggest novel roles of tobamovirus CP in induction of host symptoms and defence responses.

Some stability results for a model of Hepatitis C including alanine aminotransferase and immune system

In clinical practice, many cirrhosis scores based on alanine aminotransferase (ALT) levels exist. Although the most recent direct acting antivirals (DAAs) reduce fibrosis and ALT levels, the Hepatitis C virus (HCV) is not always removed. In this paper, we study a mathematical model of the HCV virus, which takes into account the role of the immune system, to investigate the ALT behavior during therapy. We find five equilibrium points and analyze their stability. A sufficient condition for global asymptotical stability of the infection-free equilibrium is obtained and local asymptotical stability conditions are given for the immune-free infection and cytotoxic T lymphocytes (CTL) response equilibria. The stability of the infection equilibrium with the full immune response is numerically performed.

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.

Stability Analysis of Mathematical Model on HIV Infection with the Effects of Antiretroviral therapy

HIV is a retrovirus, a virus which has enzymes and can convert genetic material from RNA to DNA. Antiretroviral therapies are the treatment to make the activity of the virus slow. The purpose of this article is to develop a mathematical model of HIV infection by reviewing antiretroviral therapy, analyze the equilibrium point, and determine the effectiveness of antiretroviral therapy. There are two equilibrium points in this HIV infection model, namely infection-free equilibrium and infected equilibrium. Numerical simulations are carried out based on selected parameters showed that infection free equilibrium is reached when the effectiveness of antiretroviral therapy is 0,4 for RT inhibitor and 0,3 for Protease Inhibitor. This means that antiretroviral therapy may change infected conditions to infection free conditions.

Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion

In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number [Formula: see text] and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if [Formula: see text], and the infection equilibrium is globally asymptotically stable if [Formula: see text]. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number [Formula: see text], while virus will extinct if [Formula: see text]. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

Ultimate tumor dynamics and eradication using oncolytic virotherapy

In this paper we study ultimate dynamics of one three-dimensional model for tumor growth under oncolytic virotherapy which describes interactions between cytotoxic T-cells, uninfected tumor cells and infected tumor cells. Using the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates convergence dynamics to equilibrium points located in invariant planes corresponding cases of the absence of uninfected or infected tumor cells. These assertions mean global eradication of uninfected or infected tumor cell populations and are presented as algebraic inequalities respecting virus replication rate θ. In particular, we find in Theorems 4 and 5the following curious phenomenon. Namely, when we vary θ from the instability range of the infected tumor free equilibrium point to the stability range we obtain in the latter range convergence dynamics to one of tumor free equilibrium points; this means that the local eradication of infected tumor cells implies their global eradication. Besides, we give conditions under which the infected tumor cell population persists. Our theoretical studies are supplied by results of numerical simulation.

Stability and bifurcation analysis for a delayed viral infection model with full logistic proliferation

In this paper, we study a delayed viral infection model with cellular infection and full logistic proliferations for both healthy and infected cells. The global asymptotic stabilities of the equilibria are studied by constructing Lyapunov functionals. Moreover, we investigated the existence of Hopf bifurcation at the infected equilibrium by regarding the possible combination of the two delays as bifurcation parameters. The results show that time delays may destabilize the infected equilibrium and lead to Hopf bifurcation. Finally, numerical simulations are carried out to illustrate the main results and explore the dynamics including Hopf bifurcation and stability switches.