Global stability for an HIV-1 infection model including an eclipse stage of infected cells

On a two-strain epidemic mathematical model with vaccination

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R 0 1 and the second reproduction number R 0 2 . We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases.

Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2

The emergence of novel RNA viruses like SARS-CoV-2 poses a greater threat to human health. Thus, the main objective of this article is to develop a new mathematical model with a view to better understand the evolutionary behavior of such viruses inside the human body and to determine control strategies to deal with this type of threat. The developed model takes into account two modes of transmission and both classes of infected cells that are latently infected cells and actively infected cells that produce virus particles. The cure of infected cells in latent period as well as the lytic and non-lytic immune response are considered into the model. We first show that the developed model is well-posed from the biological point of view by proving the non-negativity and boundedness of model's solutions. Our analytical results show that the dynamical behavior of the model is fully determined by two threshold parameters one for viral infection and the other for humoral immunity. The effect of antiviral treatment is also investigated. Furthermore, numerical simulations are presented in order to illustrate our analytical results.

Stability and Hopf bifurcation of HIV-1 model with Holling II infection rate and immune delay

This paper aims to analyse stability and Hopf bifurcation of the HIV-1 model with immune delay under the functional response of the Holling II type. The global stability analysis has been considered by Lyapunov-LaSalle theorem. And stability and the sufficient condition for the existence of Hopf Bifurcation of the infected equilibrium of the HIV-1 model with immune response are also studied. Some numerical simulations verify the above results. Finally, we propose a novel three dimension system to the future study.

Nonlinear Sub-optimal Control Design for Suppressing HIV Replication

Acquired Immunodeficiency Syndrome is a deadly viral disease caused by the Human Immunodeficiency Virus in vivo, and its purpose is to destroy the immune system of the human body. The disease does not currently have a definitive vaccine or treatment, but treatment with pharmaceutical interventions (antiretroviral therapy, or ART) can slow down the progression of HIV. Daily use of prophylaxis measures may also have serious side effects for the patient, so the dosage and regimen of drugs should be constantly controlled. The dynamic models formulated for HIV infection are nonlinear differential equations. Therefore, nonlinear optimal control methods can be effective in increasing the efficiency of treatment. In this study, a sub-optimal controller based on the state-dependent Riccati equation (SDRE) approach to the dynamic model of HIV is introduced. One of the advantages of the SDRE approach is that the nonlinear properties of the system are preserved in the design control procedure. Furthermore, the specific conditions of infected individuals can be considered via choosing appropriate coefficients in the cost function and limiting the amount of drug administered. In the procedure of control design, all the state variables must be available for feedback in order to use the SDRE controller. In this regard, the Extended Kalman Filter observer is also implemented. The effect of different weighting matrices on these states is examined. In addition, to assess the effectiveness of the proposed control strategy, the well-known performance indicator root mean square error is also considered. Numerical simulations confirm the high efficiency and flexibility of the proposed approach.

Threshold dynamics of a viral infection model with defectively infected cells

In this paper, we investigate the global dynamics of a viral infection model with defectively infected cells. The explicit expression of the basic reproduction number of virus is obtained by using the next generation matrix approach, where each term has a clear biological interpretation. We show that the basic reproduction number serves as a threshold parameter. The virus dies out if the basic reproduction number is not greater than unity, otherwise the virus persists and the viral load eventually approaches a positive number. The result is established by Lyapunov's direct method. Our novel arguments for the stability of the infection equilibrium not only simplify the analysis (compared with some traditional ones in the literature) but also demonstrate some correlation between the two Lyapunov functions for the infection-free and infection equilibria.

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 R0 and the basic immunity reproduction number R1. 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.

Modeling and stability analysis of HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that infect the susceptible CD[Formula: see text]T cells. It is known that HIV and HTLV-I have in common a way of transmission through direct contact with certain body fluids related to infected individuals. Therefore, it is not surprising that a mono-infected person with one of these viruses can be co-infected with the other virus. In the literature, a great number of mathematical models has been presented to describe the within-host dynamics of HIV or HTLV-I mono-infection. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In this paper, we develop a new within-host HIV/HTLV-I co-infection model. The model includes the impact of Cytotoxic T lymphocytes (CTLs) immune response, which is important to control the progression of viral co-infection. The model describes the interaction between susceptible CD[Formula: see text]T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. We first show the nonnegativity and boundedness of the model’s solutions and then we calculate all possible equilibria. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle’s invariance principle. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we discuss the effect of HTLV-I infection on the HIV-infected patients and vice versa.

Analysis of a within-host HIV/HTLV-I co-infection model with immunity

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the immune cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome, while HTLV-I is the causative agent for adult T-cell leukemia and HTLV-I-associated myelopathy/tropical spastic paraparesis. Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. In the present paper, we are concerned to formulate and analyze a new HIV/HTLV co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4⁺T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell and cell-to-cell. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact, and (ii) vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Oncolytic virus therapy benefits from control theory

Oncolytic virus therapy aims to eradicate tumours using viruses which only infect and destroy targeted tumour cells. It is urgent to improve understanding and outcomes of this promising cancer treatment because oncolytic virus therapy could provide sensible solutions for many patients with cancer. Recently, mathematical modelling of oncolytic virus therapy was used to study different treatment protocols for treating breast cancer cells with genetically engineered adenoviruses. Indeed, it is currently challenging to elucidate the number, the schedule, and the dosage of viral injections to achieve tumour regression at a desired level and within a desired time frame. Here, we apply control theory to this model to advance the analysis of oncolytic virus therapy. The control analysis of the model suggests that at least three viral injections are required to control and reduce the tumour from any initial size to a therapeutic target. In addition, we present an impulsive control strategy with an integral action and a state feedback control which achieves tumour regression for different schedule of injections. When oncolytic virus therapy is evaluated in silico using this feedback control of the tumour, the controller automatically tunes the dose of viral injections to improve tumour regression and to provide some robustness to uncertainty in biological rates. Feedback control shows the potential to deliver efficient and personalized dose of viral injections to achieve tumour regression better than the ones obtained by former protocols. The control strategy has been evaluated in silico with parameters that represent five nude mice from a previous experimental work. Together, our findings suggest theoretical and practical benefits by applying control theory to oncolytic virus therapy.

Control strategy design for the anti-HBV mathematical model

Currently, the anti-viral therapy has been extensively utilised to reduce the viral burden and switch off certain infectious sources for hepatitis B virus (HBV) infected patients in clinical treatment. Several pieces of existing evidence have demonstrated that large-scale coverage with anti-viral therapy has obtained a certain great contribution in hygiene and disease control. In this study, an anti-HBV mathematical model is considered and its control strategy of the drug treatment is designed. Based on the Lyapunov theory, this study derives three main theorems to propose three different control strategies, respectively, for drug treatments [inline-formula removed] and [inline-formula removed], such that all states of the anti-HBV model can finally converge to the infection-free equilibrium point [inline-formula removed] asymptotically. Especially, the designed drug treatment [inline-formula removed] or [inline-formula removed] is not a fixed value, but it is time-varying and dependent on states. In Theorem 1, the single drug treatment [inline-formula removed] without [inline-formula removed] is synthesised. Theorem 2 considers the single drug treatment [inline-formula removed] without [inline-formula removed]. In Theorem 3, the combination therapy of [inline-formula removed] and [inline-formula removed] is designed. Finally, there are several simulations to show that the proposed combination therapy is much more effective to cure HBV infected patients than the drug treatment with solely single [inline-formula removed] or single [inline-formula removed].

Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth

In this paper, we investigate an age-structured HIV infection model with logistic growth for target cell. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analysing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al., we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our results.

Analysis of General Humoral Immunity HIV Dynamics Model with HAART and Distributed Delays

This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4 + T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The model is incorporated with distributed time delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The effect of antibody on HIV infection is analyzed. The production and removal rates of the ten compartments of the model are given by general nonlinear functions which satisfy reasonable conditions. Nonnegativity and ultimately boundedness of the solutions are proven. Using the Lyapunov method, the global stability of the equilibria of the model is proven. Numerical simulations of the system are provided to confirm the theoretical results. We have shown that the antibodies can play a significant role in controlling the HIV infection, but it cannot clear the HIV particles from the plasma. Moreover, we have demonstrated that the intracellular time delay plays a similar role as the Highly Active Antiretroviral Therapies (HAAT) drugs in eliminating the HIV particles.

Stability of a CD4+ T cell viral infection model with diffusion

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

Global stability of infectious disease models with contact rate as a function of prevalence index

In this paper, we consider a SEIR epidemiological model with information--related changes in contact patterns. One of the main features of the model is that it includes an information variable, a negative feedback on the behavior of susceptible subjects, and a function that describes the role played by the infectious size in the information dynamics. Here we focus in the case of delayed information. By using suitable assumptions, we analyze the global stability of the endemic equilibrium point and disease--free equilibrium point. Our approach is applicable to global stability of the endemic equilibrium of the previously defined SIR and SIS models with feedback on behavior of susceptible subjects.

The Effects of Time Lag and Cure Rate on the Global Dynamics of HIV-1 Model

In this research article, a new mathematical model of delayed differential equations is developed which discusses the interaction among CD4 T cells, human immunodeficiency virus (HIV), and recombinant virus with cure rate. The model has two distributed intracellular delays. These delays denote the time needed for the infection of a cell. The dynamics of the model are completely described by the basic reproduction numbers represented by R0, R1, and R2. It is shown that if R0 < 1, then the infection-free equilibrium is locally as well as globally stable. Similarly, it is proved that the recombinant absent equilibrium is locally as well as globally asymptotically stable if 1 < R0 < R1. Finally, numerical simulations are presented to illustrate our theoretical results. Our obtained results show that intracellular delay and cure rate have a positive role in the reduction of infected cells and the increasing of uninfected cells due to which the infection is reduced.

Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response

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 ([Formula: see text]) and the CTL immune response activation number ([Formula: see text]). By using suitable Lyapunov functionals, we show that if [Formula: see text], then the infection-free steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text] [Formula: see text], then the CTL-inactivated infection steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text], then the CTL-activated infection steady state [Formula: see text] 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.

GLOBAL STABILITY OF A GENERAL VIRUS DYNAMICS MODEL WITH MULTI-STAGED INFECTED PROGRESSION AND HUMORAL IMMUNITY

In this paper, we propose an [Formula: see text]-dimensional nonlinear virus dynamics model that describes the interactions of the virus, uninfected target cells, [Formula: see text]-stages of infected cells and B cells. We assume that the incidence rate of infection, the generation and removal rates of all compartments are given by general nonlinear functions. We derive two threshold parameters, the basic reproduction number, [Formula: see text] and the humoral immunity number, [Formula: see text] and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. Utilizing Lyapunov functions and LaSalle’s invariance principle, the global asymptotic stability of all steady states of the model is proved. Numerical simulations are conducted for specific forms of the general functions in order to illustrate the dynamical behavior.

Global dynamics of hepatitis C viral infection with logistic proliferation

A modified mathematical model of hepatitis C viral dynamics has been presented in this paper, which is described by four coupled ordinary differential equations. The aim of this paper is to perform global stability analysis using geometric approach to stability, based on the higher-order generalization of Bendixson’s criterion. The result is also supported numerically. An important epidemiological issue of eradicating hepatitis C virus has been addressed through the global stability analysis.

Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage

In this paper, we propose two HIV infection models with specific nonlinear incidence rate by including a class of infected cells in the eclipse phase. The first model is described by ordinary differential equations (ODEs) and generalizes a set of previously existing models and their results. The second model extends our ODE model by taking into account the diffusion of virus. Furthermore, the global stability of both models is investigated by constructing suitable Lyapunov functionals. Finally, we check our theoretical results with numerical simulations.

Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model

This study proposes a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. This model has an infection-free equilibrium point and an endemic infection equilibrium point. Using Lyapunov functions and LaSalle's invariance principle shows that if the model's basic reproductive number R0 < 1, the infection-free equilibrium point is globally asymptotically stable, otherwise the endemic infection equilibrium point is globally asymptotically stable. It is shown that a forward bifurcation will occur when R0 = 1. The basic reproductive number R0 of the modified model is independent of plasma total CD4⁺ T cell counts and thus the modified model is more reasonable than the original model proposed by Buonomo and Vargas-De-León. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two group patients' anti-HIV infection treatments. The simulation results have shown that the first 4 weeks' treatments made the two group patients' R'0 < 1, respectively. After the period, drug resistance made the two group patients' R'0 > 1. The results explain why the two group patients' mean CD4⁺ T cell counts raised and mean HIV RNA levels declined in the first period, but contrary in the following weeks.

Global properties of nonlinear humoral immunity viral infection models

In this paper, we consider two nonlinear models for viral infection with humoral immunity. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The second model is a modification of the first one by including the latently infected cells. The incidence rate, removal rate of infected cells, production rate of viruses and the latent-to-active conversion rate are given by more general nonlinear functions. We have established a set of conditions on these general functions and determined two threshold parameters for each model which are sufficient to determine the global dynamics of the models. The global asymptotic stability of all equilibria of the models has been proven by using Lyapunov theory and applying LaSalle's invariance principle. We have performed some numerical simulations for the models with specific forms of the general functions. We have shown that, the numerical results are consistent with the theoretical results.

Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment

We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number R 0 and the immune response reproduction number R 1. By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.

Global stability of an SEIQV epidemic model with general incidence rate

In this paper, a class of SEIQV epidemic model with general nonlinear incidence rate is investigated. By constructing Lyapunov function, it is shown that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number ℛ0≤ 1. If ℛ0> 1, we show that the endemic equilibrium is globally asymptotically stable by applying Li and Muldowney geometric approach.

Dynamics analysis and simulation of a modified HIV infection model with a saturated infection rate

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive number R 0 of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; if R 0 of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients' anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients' HIV RNA levels. It can be assumed that if an HIV infected individual's basic virus reproductive number R 0 < 1 then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual's R 0 < 1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual's HIV RNA load to be of unpredictable level, the time that the patient's HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

Modeling the impact on HIV incidence of combination prevention strategies among men who have sex with men in Beijing, China

OBJECTIVE

To project the HIV/AIDS epidemics among men who have sex with men (MSM) under different combinations of HIV testing and linkage to care (TLC) interventions including antiretroviral therapy (ART) in Beijing, China.

DESIGN

Mathematical modeling.

METHODS

Using a mathematical model to fit prevalence estimates from 2000-2010, we projected trends in HIV prevalence and incidence during 2011-2020 under five scenarios: (S1) current intervention levels by averaging 2000-2010 coverage; (S2) increased ART coverage with current TLC; (S3) increased TLC/ART coverage; (S4) increased condom use; and (S5) increased TLC/ART plus increased condom use.

RESULTS

The basic reproduction number based upon the current level of interventions is significantly higher than 1 (R0 = 2.09; 95% confidence interval (CI), 1.83-2.35), suggesting that the HIV epidemic will continue to increase to 2020. Compared to the 2010 prevalence of 7.8%, the projected HIV prevalence in 2020 for the five prevention scenarios will be: (S1) Current coverage: 21.4% (95% CI, 9.9-31.7%); (S2) Increased ART: 19.9% (95% CI, 9.9-28.4%); (S3) Increased TLC/ART: 14.5% (95% CI, 7.0-23.8%); (S4) Increased condom use: 13.0% (95% CI, 9.8-28.4%); and (S5) Increased TLC/ART and condom use: 8.7% (95% CI, 5.4-11.5%). HIV epidemic will continue to rise (R0 > 1) for S1-S4 even with hyperbolic coverage in the sensitivity analysis, and is expected to decline (R0 = 0.93) for S5.

CONCLUSION

Our transmission model suggests that Beijing MSM will have a rapidly rising HIV epidemic. Even enhanced levels of TLC/ART will not interrupt epidemic expansion, despite optimistic assumptions for coverage. Promoting condom use is a crucial component of combination interventions.