Dynamics of an HIV Infection Model with Two Infection Routes and Evolutionary Competition between Two Viral Strains

The existing combination therapy of HIV antiretroviral drugs can lead to the emergence of drug-resistant viruses, and cannot effectively block direct cell-to-cell infections, these factors results in incomplete virus suppression and increased risk of disease progression. In this paper, we formulate an HIV model with two strains representing a drug-sensitive virus and a drug-resistant virus to study the joint mechanism of drug resistance. We first reduce the infection-age model to a system of integro-differential equations with infinite delays. Then the stability of the equilibria and the dynamics of competition between two viruses are studied to illuminate the joint effects of infection-age and two infection routes on the evolution of both drug-sensitive and drug-resistant strains before and during drug treatment. Applying a persistence theorem for infinite dimensional systems, we obtain that the disease is always present when the basic reproduction number is larger than unity. Numerical simulations confirm that the basic reproduction numbers and mutation coefficient are the key threshold parameters for determining the competition results of the two viral strains and indicate the cell-to-cell transmission increases the likelihood that HIV breaks out within the host. Finally, sensitivity analyses suggest that the available combination therapy should be taken once symptoms of resistance appear during drug treatment, and demonstrate that the presence of cell-to-cell transmission attenuates the efficacy of the existing antiretroviral drug treatments.     

Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission

Recent studies reveal that cell-to-cell transmission via formation of virological synapses can contribute significantly to virus spread, and hence, may play a more important role than virus-to-cell infection in some situations. Age-structured models can be employed to study the variations w.r.t. infection age in modeling the death rate and virus production rate of infected cells. Considering the above characteristics for within-host dynamics of HIV, in this paper, we formulate an age-structured hybrid model to explore the effects of the two infection modes in viral production and spread. We offer a rigorous analysis for the model, including addressing the relative compactness and persistence of the solution semiflow, and existence of a global attractor. By subtle construction and estimates of Lyapunov functions, we show that the global attractor actually consists of an singleton, being either the infection free steady state if the basic reproduction number is less than one, or the infection steady state if the basic reproduction number is larger than one.     

The global dynamics of a model about HIV-1 infection in vivo

A four dimension ODE model is built to study the infection of human immunodeficiency virus (HIV) in vivo. We include in this model four components: the healthy T cells, the latent-infected T cells, the active-infected T cells and the HIV virus. Two types of HIV transmissions in vivo are also included in the model: the virus-to-cell transmission, and the cell-to-cell HIV transmission. There are two possible equilibriums: the healthy equilibrium, and the infected steady state. The basic reproduction number R ₀ is introduced. When R ₀ < 1, the healthy equilibrium is globally stable and when R ₀ > 1, the infected equilibrium exists and is globally stable. Through simulations, we find that, the cell-to-cell HIV transmission is very important for the final outcome of the HIV attacking. Some important clinical observations about the HIV infection situation in lymph node are also verified.      

HIV dynamics: Analysis and robust multirate MPC-based treatment schedules

Analysis and control of human immunodeficiency virus (HIV) infection have attracted the interests of mathematicians and control engineers during the recent years. Several mathematical models exist and adequately explain the interaction of the HIV infection and the immune system up to the stage of clinical latency, as well as viral suppression and immune system recovery after treatment therapy. However, none of these models can completely exhibit all that is observed clinically and account the full course of infection. Besides model inaccuracies that HIV models suffer from, some disturbances/uncertainties from different sources may arise in the modelling. In this paper we study the basic properties of a 6-dimensional HIV model that describes the interaction of HIV with two target cells, CD4⁺ T cells and macrophages. The disturbances are modelled in the HIV model as additive bounded disturbances. Highly Active AntiRetroviral Therapy (HAART) is used. The control input is defined to be dependent on the drug dose and drug efficiency. We developed treatment schedules for HIV infected patients by using robust multirate Model Predictive Control (MPC)-based method. The MPC is constructed on the basis of the approximate discrete-time model of the nominal model. We established a set of conditions, which guarantee that the multirate MPC practically stabilizes the exact discrete-time model with disturbances. The proposed method is applied to the stabilization of the uninfected steady state of the HIV model. The results of simulations show that, after initiation of HAART with a strong dosage, the viral load drops quickly and it can be kept under a suitable level with mild dosage of HAART. Moreover, the immune system is recovered with some fluctuations due to the presence of disturbances.     

Dynamics of an HIV infection model with virus diffusion and latently infected cell activation

Effective combination therapy usually reduces the plasma viral load of HIV to below the detection limit, but it cannot eradicate the virus. The latently infected cell activation is considered to be the main obstacle to completely eradicating HIV infection. In this paper, we consider an HIV infection model with latently infected cell activation, virus diffusion and spatial heterogeneity under Neumann boundary condition. The basic reproduction ratio is characterized by the principal eigenvalue of the related elliptic eigenvalue problem. Besides, by constructing Lyapunov functionals and using Green’s first identity, the global threshold dynamics of the system are completely established. Numerical simulations are carried out to illustrate the theoretical results, in particular, the influence of virus diffusion rate on the basic reproduction ratio is addressed.     

一类HIV病毒动力学模型的全局稳定性与周期解

建立了一类较广泛的HIV感染CD4+T细胞病毒动力学模型,给出了一个感染细胞在其整个感染期内产生的病毒的平均数(基本再生数)R0的表达式,运用Lyapunov原理和Routh-Hurwitz判据得到了该模型的未感染平衡点与感染平衡点的存在性与稳定性条件.同时也得到了模型存在轨道渐近稳定周期解和系统持续生存的条件,并通过数值模拟验证了所得到的结果.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Optimal strategies for mitigating the HIV/AIDS epidemic in the Philippines

The human immunodeficiency virus (HIV) impairs a person's immune system against many infections and some types of cancer, leading to acquired immunodeficiency syndrome (AIDS), which is characterized by severe illnesses. The number of HIV infections in the Philippines has increased, more than doubled, within the last decade. This alarming HIV crisis in the country requires urgent actions. In this study, a mathematical model is developed to describe the disease transmission in the Philippines. Disease-free and endemic equilibria are obtained, stability analysis is performed, and the basic reproduction number is computed. Sensitivity analyses and subset selection are performed to identify influential parameters and to determine an identifiable parameter set given measurements, respectively. Available data on the number of asymptomatic aware infectious, those who are in the AIDS stage, and those under treatment are utilized to estimate key epidemiological parameters such as transmission, treatment, and screening rates. Uncertainty of these parameter estimates is quantified through bootstrapping method. Furthermore, intervention strategies are investigated in the framework of optimal control theory. Control measures include precaution, HIV screening, antiretroviral treatment, and pre-exposure prophylaxis (PrEP) treatment. These various control efforts are compared with regard to cost efficiency and effectiveness in reducing the number of infected individuals. Given limited available control measures, the PrEP-only scenario is shown to be the most cost-effective, followed by other scenarios that combine PrEP with other controls.     

Modelling and analysis of dynamics of viral infection of cells and of interferon resistance

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.     

Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.     

Global stability of delay-distributed viral infection model with two modes of viral transmission and B-cell impairment

This paper formulates a virus dynamics model with impairment of B-cell functions. The model incorporates two modes of viral transmission: cell-free and cell-to-cell. The cell-free and cell-cell incidence rates are modeled by general functions. The model incorporates both, latently and actively, infected cells as well as three distributed time delays. Nonnegativity and boundedness properties of the solutions are proven to show the well-posedness of the model. The model admits two equilibria that are determined by the basic reproduction number R₀. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions and time delays on the virus dynamics are studied. We have shown that if the functions of B-cell is impaired, then the concentration of viruses is increased in the plasma. Moreover, we have observed that increasing the time delay will suppress the viral replication.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

An age-structured within-host HIV model with T-cell competition

Recent studies demonstrate that resource competition is an essential component of T-cell proliferation in HIV progression, which can contribute instructively to the disease development. In this paper, we formulate an age-structured within-host HIV model, in the form of a hyperbolic partial differential equation (PDE) for infected target cells coupled with two ordinary differential equations for uninfected T-cells and the virions, to explore the effects of both the T-cell competition and viral shedding variations on the viral dynamics. The basic reproduction number is derived for a general viral production rate which determines the local stability of the infection-free equilibrium. Two special forms of viral production rates, which are extensively investigated in previous literature, the delayed exponential distribution and a step function rate, are further investigated, where the original system can be reduced into systems of delay differential equations. It is confirmed that there exists a unique positive equilibrium for two special viral production rates when the basic reproduction number is greater than one. However, the model exhibits the phenomenon of backward bifurcation, where two positive steady states coexist with the infection-free equilibrium when the basic reproduction number is less than one.     

关于一个脉冲用药的HIV免疫模型的研究

对一个脉冲用药的HIV免疫模型进行研究,发现在用药间隔足够小时,预防抗化剂的脉冲使用通过使T细胞水平无限接近于未被感染的免疫水平来保持个体的免疫功能,并且文中进一步给出这个合适的用药间隔估计.这对指导AIDS治疗临床实践具有参考意义.     

Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load, and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

Global properties for virus dynamics model with mitotic transmission and intracellular delay

In this paper we study the basic model of viral infections with mitotic transmission and intracellular delay discrete. The delay corresponds to the time between infection of uninfected cells and the emission of virus on a cellular level. By means of Volterra-type Lyapunov functionals, we provide the global stability for this model. Let η be the number of virus produced per infected cell. If ηcrit, the critical number, satisfies η?ηcrit, then the virus-free steady state is globally asymptotically stable. On the contrary if η>ηcrit, then the infected steady state is globally asymptotically stable if a sufficient condition is satisfied.     

一类具有饱和感染率与治愈率的HIV病理模型

将治愈率以及饱和感染率引入基本的HIV病理模型,构建一个改进的HIV病理模型.利用微分动力系统的相关理论,证明改进模型中无病平衡点和染病平衡点的全局渐近稳定性,然后执行相关的数值模拟以验证所得结论.研究结果表明:在饱和感染率的条件下,HIV感染进程变缓;同时提高治愈率能有效地控制HIV感染.