Quantifying the long-term impact of Zidovudine and other anti-HIV therapies on the survivability of those with clinical AIDS

Objective: To determine the long-term behavior of the clinical AIDS survivability curve for a cohort undergoing anti-HIV therapy and for a comparison cohort undergoing no such therapy.Results: Zidovudine therapy is shown to increase the average longevity of a cohort with clinical AIDS a maximum of 5.7 months. Not only does zidovudine eventually fail to prevent death, evidence is presented that suggests that long-term use of the drug may be counterproductive and actually decrease survivability. The analysis herein suggests that to maximize longevity in those with clinical AIDS, zidovudine therapy should be administered for a short period (on the order of several months or less) and then ended.Conclusions: The usual practice in controlled anti-HIV drug trials of prematurely ending the experiment as soon as the drug is shown to significantly increase survivability in those with AIDS is shown to be a mistake. Anti-HIV drugs typically have a negative impact on the immune system, and long-term use of these drugs can actually shorten survivability rather than prolong it. Evidence is presented showing that short-term zidovudine therapy increases longevity in those with AIDS, but long-term continuation of the therapy may actually shorten it.     

Accurately simulating the growth in the size of the HIV infected population in any AIDS epidemic country: Computing the U.S.A. HIV infection curve

A reliable approach to the simulation of the time-dependent growth of the size of a country's HIV population is described in detail and applied to the USA epidemic. The simulation depends on a knowledge of AIDS incidence data and the HIV incubation period distribution but is independent of any model regarding how the disease was spread. Using the Centers for Disease Control's December 31, 1991 update of the reported AIDS incidence data, a cumulative total of 645,445 Americans was calculated to be HIV infected as of January 1, 1991.The HIV infection curves for the USA risk groups were separately computed, and they indicate that the current rates of the spread of the infection in all of the risk groups are small fractions of what they were in the early phase of the epidemic. In fact, the calculated increase in the cumulative number of USA HIV infecteds from January 1, 1990 to January 1, 1991 was only 1.44%. These results suggest that the annual number of AIDS cases to be obtained in the next few years will not be substantially different from what it was in 1991. Since the calculated HIV infection curves for the transfusion and hemophiliac risk groups are currently growing at a particularly low rate, the modelling results confirm the great safety of the nation's blood and blood product supplies.     

Bifurcation analysis of the HIV‐1 within host model

In this paper, a bifurcation solution's analysis is proposed for an HIV‐1 within the host model around its chronic equilibrium point, this is carried out based on Lyapunov–Schmidt approach. It is shown that the coefficient b, which represents the healthy CD4⁺ T‐cells growth rate, is a bifurcation parameter; this means that the rate of multiplication of healthy cells can have serious effects on the qualitative dynamical properties and structural stability of the infection evolution dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling treatment effects in the HIV/AIDS epidemic

Advances in recent treatments for HIV/AIDS patients have shown dramatic outcomes in extending the incubation period and AIDS survival time, while also providing significant improvements in the quality of patients' lives. A compartmental model is proposed to analyse the effects of the various treatment regimens which have been introduced. The results produced are in good agreement with routinely collected data relating to levels of HIV/AIDS incidence and prevalence in the UK homosexual population. Some parameter values within the model are obtained from surveys, census results, etc, but others are derived using a maximum likelihood estimation procedure. Finally, the model is used to project levels of incidence and prevalence over the next few years, and to investigate several possible scenarios.     

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.     

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.     

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.     

A nested model on HIV/AIDs, antiretroviral therapy and drug resistance

A coupled within- (immunological) and between-host (epidemiological) dynamic model was developed which is about the spreading of drug-sensitive HIV strain and drug-resistant HIV strain in men who have sex with men (MSM) population. The within-host model was nested within the between-host model by linking the dynamics of the within-host model to the additional host mortality and transmission rate of the infection. The existences of equilibria and their stabilities were found, as well as the thresholds $\mathcal {R}_S$ and $\mathcal {R}_R$ for the two different strains of the nested model. Some simulations about the spreading of the two HIV strains in Beijing MSM population were given. Our results show that the drug-resistant strain will increase quite fast in this population and both strains can coexist, which will make a big pressure for China''s ``Four-Free-One-Care Policy''.     

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.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

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.     

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.     

A fractional-order model of HIV infection with drug therapy effect

In this paper, the fractional-order model that describes HIV infection of CD4⁺ T cells with therapy effect is given. Generalized Euler Method (GEM) is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.     

Application du contrôle optimal à l'amélioration des trithérapies

Highly Active Anti-Retroviral Therapies (HAART) have proven to be extremely effective in improving and prolonging the patient's life. Though, a concern arises since a long term drug intake induces many strong sides effects and reduces reactivity of the virus to any therapy. The purpose of the paper is to use numerical analysis and optimization tools to suggest improved therapies to handle HIV infection. The evolution of the infection is modelled by an ordinary differential equation system which includes both immune response and multi-drug effects. For a fixed time, one looks for a two drugs control strategy based on Pontryagine's minimum principle with an objective function which takes into account three contributions: the viral load, the transient evolution of infection and the quantities of drug used. Simulations are carried out using an indirect optimization method along with Runge–Kutta adaptative stepsize algorithm. Numerical solutions to the optimality system are obtained and related histories are shown. The possibility of Scheduled Treatment Interruption is also examined.     

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.     

Understanding the time delay between HIV infection and AIDS from a model of HIV's impact on the T-helper cell density

The HIV incubation distribution curve leading to AIDS is derived from the hematic T-Helper cell density distribution for the seronegative population. After the HIV acute infection stage, the T-Helper cell density distribution curve is shown to begin uniformly translating towards zero density at the constant rate of 70.9 T-Cells/μL per year leading to AIDS. The future values of the HIV incubation period curve can now be credibly calculated, and it is projected that 90% of infecteds will develop AIDS 18 years after infection. HIV is postulated to lower the hematic T-Helper cell density equilibrium set-point to zero, causing the immune system to collapse.     

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.