The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

Generals die in friendly fire,or modeling immune response to HIV

We develop a kinetic model for CD8 T lymphocytes (CTL) whose purpose is to kill cells infected with viruses and intracellular parasites. Using a set of first-order nonlinear differential equations, the model predicts how numbers of different cell types involved in CTL response depend on time. The model postulates that CTL response requires continuous presence of professional antigen-presenting cells (APC) comprised of macrophages and dendritic cells. It assumes that any virus present in excess of a threshold level activates APC that, in turn, activate CTL that expand in number and become armed “effector” cells. In the end, APC are deactivated after virus is cleared. The lack of signal from APC causes effector cells to differentiate, by default, into “transitory cells” that either die, or, in a small part, become long-lived memory cells. Viruses capable of infecting APC will cause premature retirement of effector CTL. If transitory cells encounter virus, which takes place after the premature depletion, CTL become anergic (unresponsive to external stimuli). The model is designed to fit recent experiments on primary CTL response to simian immunodeficiency virus closely related to HIV and lymphocytic choriomeningitis virus. The two viruses are known to infect APC and make them targets for CTL they are supposed to control. Both viruses cause premature depletion and anergy of CTL and persist in the host for life.     

Dynamics of an HIV-1 infection model with cell mediated immunity

In this paper, we study the dynamics of an improved mathematical model on HIV-1 virus with cell mediated immunity. This new 5-dimensional model is based on the combination of a basic 3-dimensional HIV-1 model and a 4-dimensional immunity response model, which more realistically describes dynamics between the uninfected cells, infected cells, virus, the CTL response cells and CTL effector cells. Our 5-dimensional model may be reduced to the 4-dimensional model by applying a quasi-steady state assumption on the variable of virus. However, it is shown in this paper that virus is necessary to be involved in the modeling, and that a quasi-steady state assumption should be applied carefully, which may miss some important dynamical behavior of the system. Detailed bifurcation analysis is given to show that the system has three equilibrium solutions, namely the infection-free equilibrium, the infectious equilibrium without CTL, and the infectious equilibrium with CTL, and a series of bifurcations including two transcritical bifurcations and one or two possible Hopf bifurcations occur from these three equilibria as the basic reproduction number is varied. The mathematical methods applied in this paper include characteristic equations, Routh–Hurwitz condition, fluctuation lemma, Lyapunov function and computation of normal forms. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.     

Dynamics of an HIV‐1 virus model with both virus‐to‐cell and cell‐to‐cell transmissions,general incidence rate,intracellular delay,and CTL immune responses

Direct cell‐to‐cell transmission of HIV‐1 is a more efficient means of virus infection than virus‐to‐cell transmission. In this paper, we incorporate both these transmissions into an HIV‐1 virus model with nonlinear general incidence rate, intracellular delay, and cytotoxic T lymphocyte (CTL) immune responses. This model admits three types of equilibria: infection‐free equilibrium, CTL‐inactivated equilibrium, and CTL‐activated equilibrium. By using Lyapunov functionals and LaSalle invariance principle, it is verified that global threshold dynamics of the model can be explicitly described by the basic reproduction numbers.     

HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission

HIV infection persists despite long-term administration of antiretroviral therapy. The mechanisms underlying HIV persistence are not fully understood. Direct viral transmission from infected to uninfected cells (cell-to-cell transmission) may be one of them. During cell-to-cell transmission, multiple virions are delivered to an uninfected cell, making it possible that at least one virion can escape HIV drugs and establish infection. In this paper, we develop a mathematical model that includes cell-to-cell viral transmission to study HIV persistence. During cell-to-cell transmission, it is assumed that various number of virus particles are transmitted with different probabilities and antiretroviral therapy has different effectiveness in blocking their infection. We analyze the model by deriving the basic reproduction number and investigating the stability of equilibria. Sensitivity analysis and numerical simulation show that the viral load is still sensitive to the change of the treatment effectiveness in blocking cell-free virus infection. To reduce this sensitivity, we modify the model by including density-dependent infected cell death or HIV latent infection. The model results suggest that although cell-to-cell transmission may have reduced susceptibility to HIV drugs, HIV latency represents a major reason for HIV persistence in patients on suppressive treatment.     

Quantifying the strength and durability of induced immunity to HIV infection in women engaging in unprotected sexual contacts with infected men

Evidence is accumulating that exposure to human immunodeficiency virus (HIV) can lead to an increased resistance or immunity to subsequent infection. A multirisk model that permits either induced immunity or infection to develop after heterosexual inoculation with HIV is shown to be compatible with a wide spectrum of disparate male-to-female transmission data.When the model is applied to time-dependent, HIV-seroprevalence data, the probability that an unexposed woman would remain unexposed after an unprotected contact with an infected man was estimated to be greater than 0.95 on the average. Thus, it would require at least 14 unprotected sexual contacts with HIV-infected men for 50% of an unexposed cohort of women to become exposed to the virus. This suggests that there is a low probability that HIV virions will be found to have penetrated the mucosal barriers of the reproductive tract after a contact.The model also predicts, that the average woman whose mucosal barriers have been breached by HIV has a significant probability of developing immunity to the virus rather than infection. Modelling data for a cohort of unexposed Nairobi women leads to the prediction that the probability of acquiring induced immunity per contact is about 60% of the probability of acquiring the disease per contact.The modelling results also predict that those who had developed resistance to HIV run the small, but significant risk of becoming infected nonetheless by continuing high-risk behavior. For the common contact rate of ten per month, the modelling predicts that the HIV-transmission risk per contact for unexposed women in the Nairobi cohort is 1/178 while the transmission risk for the cohort's immunized women is 1/1548. These numbers suggest that HIV infection is difficult to transmit through heterosexual intercourse on the average and that male-to-female HIV-transmission risk per contact for African women lies between 1/178 and 1/1548.Direct confirmation of the predictions in the last paragraph has been subsequently observed in two completely independent studies. The Nairobi research team recently reported that a notable number of Nairobi prostitutes previously identified to be members of the HIV-resistant group became infected nonetheless. Second, in a study of 174 sexually monogamous, discordant couples in Rakai, Uganda reporting contacts rates of nine to ten per month, the male-to-female HIV-transmission risk per contact was found to be 1/769 by direct measurement, a value that falls between the above limits of 1/178 and 1/1548 predicted by the modelling. Thus, a second major prediction of this paper has been directly confirmed, and induced immunity to HIV is limited and not absolutely protective.Circumstantial evidence suggests that the induced immunity to HIV predicted by the model could be generated and/or initiated by nonspecific innate immune responses, specific immunological responses, including IgA-mediated mucosal immunity and cytotoxic T lymphocytes (CTL) immunity, or some combination of the above. It is suggested here, that a decrease in the ability of HIV virions to penetrate the protective mucus layer of the reproductive tract may be a prerequisite, cofactor, or the principle cause of the induced immunity or resistance demonstrated to exist in this paper. The value of the probability that induced immunity to HIV will develop after a contact is shown to be a sensitive function of the woman's human leucocyte antigen (HLA) supertype profile.     

Stability of a general adaptive immunity virus dynamics model with multistages of infected cells and two routes of infection

This paper studies an (n+4)-dimensional nonlinear virus dynamics model that characterizes the interactions of the viruses, susceptible host cells, n-stages of infected cells, B cells and cytotoxic T lymphocyte (CTL) cells. Both viral and cellular infections have been incorporated into the model. The infected-susceptible and virus-susceptible infection rates as well as the generation and removal rates of all compartments are described by general nonlinear functions. Five threshold parameters are computed, which insure the existence of the equilibria of the model under consideration. A set of conditions on the general functions has been established, which is sufficient to investigate the global dynamics of the model. The global asymptotic stability of all equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.     

Optimal control applied on an HIV‐1 within‐host model

The treatment of human immunodeficiency virus (HIV) remains a major challenge, even if significant progress has been made in infection treatment by ‘drug cocktails’. Nowadays, research trend is to minimize the number of pills taken when treating infection. In this paper, an HIV‐1 within host model where healthy cells follow a simple logistic growth is considered. Basic reproduction number of the model is calculated using next generation matrix method, steady states are derived; their local, as well as global stability, is discussed using the Routh–Hurwitz criteria, Lyapunov functions and the Lozinskii measure approach. The optimal control policy is formulated and solved as an optimal control problem. Numerical simulations are performed to compare several cases, representing a treatment by Interleukin2 alone, classical treatment by multitherapy drugs alone, then both treatments at the same time. Objective functionals aim to (i) minimize infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood. Copyright © 2015 John Wiley & Sons, Ltd.     

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.      

Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

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.     

Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model

Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Bonhoeffer et al. (1997) [1], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar type of approximate model incorporating time delay in the process of infection on the healthy T cells which, in turn, implies inclusion of a similar delay in the process of viral replication. The model is studied both analytically and numerically. We also include a similar delay in the killing rate of infected CD4⁺ T cells by Cytotoxic T-Lymphocyte (CTL) and in the stimulation of CTL and analyse two resulting models numerically.The models with no time delay present have two equilibria: one where there is no infection and a non-trivial equilibrium where the infection can persist. If there is no time delay then the non-trivial equilibrium is locally asymptotically stable. Both our analytical results (for the first model) and our numerical results (for all three models) indicate that introduction of a time delay can destabilize the non-trivial equilibrium. The numerical results indicate that such destabilization occurs at realistic time delays and that there is a threshold time delay beneath which the equilibrium with infection present is locally asymptotically stable and above which this equilibrium is unstable and exhibits oscillatory solutions of increasing amplitude.     

Global stability analysis and permanence for an HIV-1 dynamics model with distributed delays

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.     

System Dynamic simulation of treatment policies to address colliding epidemics of tuberculosis,drug resistant tuberculosis and injecting drug users driven HIV in Russia

The explosive increase in the number of people infected with tuberculosis (TB),multi drug resistant tuberculosis (MDRTB), and injecting drug users (IDU)HIV/AIDS has become a serious public health challenge in Russia. The WorldHealth Organization is recommending policies including simultaneous use ofhighly active antiretroviral therapy (HAART) to treat HIV/AIDS and second linedrugs to treat MDRTB. We developed a System Dynamics simulation model torepresent the dynamic transmission of TB, MDRTB and human immunodeficiency virus(HIV). The model simulated scenarios regarding MDRTB cure rate and HAARTcoverage, that is the HIV/AIDS population covered by HAART. The results over a20-year period indicate that reduction in TB and HIV-associated TB deaths wouldbe negligible for HAART coverage up to 50%. The reduction is onlysignificant for HAART coverage of 70% and above. Similarly, high MDRTBcure rate reduces significantly deaths from TB and MDRTB and this reduction ismore important as the HAART coverage is increased.     

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.     

A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law

The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV-related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV-TB coinfected model in the Atangana-Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers-Ulam (HU) stability and numerical simulations with assumption of specific parameters.     

Probability of a major infection in a stochastic within-host model with multiple stages

Establishment or spread of a viral infection within healthy individuals depends on exposure to a viral source, either through virus particles or through cells that have been infected. We assume that a potential infection has reached the site of the healthy target cells and we apply stochastic within-host models and multitype branching processes to investigate whether a major infection becomes established. The model includes multiple latent and actively infected stages. It is shown that the probability of a major infection is generally more likely after the virus has entered the target cell and the cell is actively infected. In some cases, the probability of a major infection is less likely if the burst size of actively infected cells is small.     

Deriving the HIV incubation period distribution for AIDS using immunological markers and devising infection treatment strategies

Survey data suggest that it is impossible for HIV infecteds to develop AIDs if the values of their CD4+ T-cell densities are above a critical threshold. An infected whose CD4+ T-cell density falls below 200 cells per microliter is now automatically regarded as having AIDS by the CDC. Using the CD4+ T-cell density as a surrogate marker of disease progression, a model that is consistent with the data is developed and applied to the homosexual/bisexual and IVDU risk groups. Assuming that the critical CD4+ T-cell density for these risk groups are identical, it is found that their progression towards AIDS during the incubation period is identical, suggesting that the dynamics of the HIV infection may be independent of risk group. The different incubation period distributions obtained from this modelling for these two risk groups is shown to be entirely due to their different normal seronegative CD4+ T-cell density distributions. Using IFN-γ as a surrogate marker is shown to give similar results.The impact of the HIV infection on the immune system is reviewed, and immunological infection models are developed. The data suggest to this author that Homo sapiens have generally lost the ability to generate T-cells and B-cells with the specificity necessary to neutralize HIV as they evolved from the primates. It is plausible that a legacy of primate immunity to HIV still remains in the 10% of Homo sapiens who show no immune system deterioration in the first 10 years of the HIV infection. New HIV infection treatment strategies based on this model are devised and discussed.     

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.