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.     

Bifurcation analysis of a multidelayed HIV model in presence of immune response and understanding of in‐host viral dynamics

In this paper, we proposed a multidelayed in‐host HIV model to represent the interaction between human immunodeficiency virus and immune response. One delay was considered to incorporate the time required by the virus for various intracellular events to make a host cell productively infective, and the second delay was introduced to take into account the time required for adaptive immune system to respond against infection. We extensively analyzed this multidelayed model analytically and numerically. We show that delay may have both destabilizing and stabilizing effects even when the system contains a single immune response delay. It happens when there exists two sequences of critical values of this delay. If the system starts with stable state in absence of delay, then the smallest value of these critical delays causes instability and the next higher value causes stability. The system may also show stability switching for different values of the virus replication factor. These results demonstrate the possible reasons of intrapatients and interpatients variability of CD4⁺ T cells and virus counts in HIV‐infected patients.     

Dynamical behaviors of a discrete HIV‐1 virus model with bilinear infective rate

We establish a discrete virus dynamic model by discretizing a continuous HIV‐1 virus model with bilinear infective rate using ‘hybrid’ Euler method. We discuss not only the existence and global stability of the uninfected equilibrium but also the existence and local stability of the infected equilibrium. We prove that there exists a crucial value similar to that of the continuous HIV‐1 virus dynamics, which is called the basic reproductive ratio of the virus. If the basic reproductive ratio of the virus is less than one, the uninfected equilibrium is globally asymptotically stable. If the basic reproductive ratio of the virus is larger than one, the infected equilibrium exists and is locally stable. Moreover, we consider the permanence for such a system by constructing a Lyapunov function v_n. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of HIV/HTLV-I co-infection model with delays

In this paper, we formulate a within-host dynamics model for HIV/HTLV-I co-infection under the influence of cytotoxic T lymphocytes (CTLs). The model incorporates silent HIV-infected CD4⁺T cells and silent HTLV-infected CD4⁺T cells. The model includes two routes of HIV transmission, virus to cell (VTC) and cell to cell (CTC). It also incorporates two modes of HTLV-I transmission, horizontal transmission via direct CTC contact and vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. The model takes into account five types of distributed-time delays. We analyze the model by proving the nonnegativity and boundedness of the solutions, calculating all possible equilibria, deriving a set of key threshold parameters, and proving the global stability of all equilibria. The global asymptotic stability of all equilibria is established by utilizing Lyapunov function and LaSalle's invariance principle. We present numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we discuss the effect of HTLV-I infection on the HIV dynamics and vice versa.     

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.     

The population‐level impact of HBV and its vaccination on HIV transmission dynamics

Hepatitis B virus (HBV) and its vaccination strategy may affect human immunodeficiency virus (HIV) transmission dynamics because both viruses have synergistic effects. To quantitatively assess the potential impact of HBV and its vaccination strategy on HIV transmission dynamics at the population level, in this paper, we formulate a deterministic compartmental model that describes the joint dynamics of HBV and HIV. We first derive the explicit expressions for the basic reproduction numbers of HIV and HBV and analyze the dynamics of HIV and HBV subsystems, respectively. Then a systematic qualitative analysis of the full system is also provided, which includes the local and global behavior. By using a set of reasonable parameter values, the full system is numerically investigated to assess the potential impact of HBV and its vaccination strategy on HIV transmission. The direct and indirect population level impact of HBV on HIV is demonstrated by calculating the fraction of HIV infections attributable to HBV and the difference between HIV prevalence in the presence and absence of HBV, respectively. The findings imply that the increase of HBV vaccination rate may unusually accelerate HIV epidemics indirectly, although the direct effect of HBV on HIV transmission decreases as HBV vaccination rate increases. Finally, the potential impact of HIV on HBV transmission dynamics is investigated by way of parenthesis. Copyright © 2016 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.      

Optimal control and sensitivity analysis of SIV→HIV dynamics with effects of infected immigrants in sub‐Saharan Africa

Regional migration has become an underlying factor in the spread of HIV transmission. In addition, immigrants with HIV status has contributed with high‐risk of sexually transmitted infection to its “destination” communities and promotes dissemination of HIV. Efforts to address HIV/AIDS among conflict‐affected populations should be properly addressed to eliminate potential role of the spread of the disease and risk of exposure to HIV. Motivated from this situation, HIV‐infected immigrants factor to HIV/SIV transmission link will be investigated in this research and examine its potential effect using optimal control method. Nonlinear deterministic mathematical model is used which is a multiple host model comprising of humans and chimpanzees. Some basic properties of the model such as invariant region and positivity of the solutions will be examined. The local stability of the disease‐free equilibrium was examined by computing the basic reproduction number, and it was found to be locally asymptotically stable when ?₀<1 and unstable otherwise. Sensitivity analysis was conducted to determine the parameters that help most in the spread of the virus. Pontryagin's maximum principle is used to obtain the optimality conditions for controlling the disease spread. Numerical simulation was conducted to obtain the analytical results. The results shows that combination of public health awareness, treatment, and culling help in controlling the HIV disease spread.     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number $R_{0}$ to characterize the viral dynamics. We prove that when $R_{0}<1$, the uninfected equilibrium of the proposed model is globally asymptotically stable. When $R_{0}>1$, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control.     

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.     

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.     

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.     

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.     

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.