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In this paper, an HIV infection model with eclipse phase, humoral immune response and immunological delay has been discussed. By studying the characteristic equations of the model, the local stability analysis of various equilibrium points has been explored. Treating the delay as the bifurcation parameter, it has been shown that the delay can destabilize the stability of the infected steady-state leading to Hopf bifurcation and periodic solutions. By using Lyapunov functionals and LaSalle's invariance principle, the global stability analysis of the boundary equilibrium points has also been explored. It has also been shown numerically that the inclusion of the drug therapy in the model generates sporadic outbursts of virus called viral blips. In the end, numerical simulations have been employed to justify the analytical results proved in the paper. Biologically, the proposed model can explain the presence of viral blips in the system, during the introduction of HAART, as observed in the HIV-infected patient. These blips could mark the viral advancement in the system, thus resulting in complete immunological failure. One of the reasons for these viral blips can be the presence of delay in the activation of immunological response. But the development of drug-resistive virus could also be the reason for this sudden rise in viral loads. Moreover, the incorporation of the delay in the model generates oscillations and periodicity in the model, thus validating the long latency period seen in most HIV-infected patients. Also, a longer delay in the activation of immune response marks a viral advancement in the viral timeline resulting in viral blips, which signify the evolution of the virus. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic model describing the dynamics of the in-host HIV infection of CD4 + T-cells is proposed. The model incorporates the presence of the CD 8 + T-cells and two types of antiretroviral drugs, for disrupting new infection and for inhibiting virus production, respectively. First, the existence, boundedness and positivity of the model solutions are shown, the basic reproduction number R 0 being then derived and shown to be a threshold value as far as the stability of the equilibria is concerned. When R 0 < 1 , the infection-free equilibrium point is globally stable, whereas when R 0 > 1 , the system is uniformly persistent and the infected equilibrium point is globally asymptotically stable. Further, we develop an optimal control model by taking the effect of the antiretroviral drugs to be control variables in order to minimize the HIV infection in different scenarios. By using Pontryagin's Minimum Principle and solving the model numerically, the results show that each antiretroviral drug in isolation can play a key role in reducing the count of both infected CD4 + T-cells and HIV viruses. However, a combination of both drugs could reduce the in-host HIV infection more significantly. [ABSTRACT FROM AUTHOR]

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. We examine the impact on the spread of disease caused by allowing addicts to progress through three distinct stages of variable infectivity prior to the onset of full blown AIDS. We first state a three stage infectivity model of the transmission of HIV and perform an equilibrium and stability analysis on this model. We find that there is a critical threshold parameter R[sub 0] which determines its behaviour. If R[sub 0] ≤ 1 then there is a unique disease-free equilibrium which is globally stable. If R[sub 0] > 1 then there is a unique endemic equilibrium which is locally stable. Simulations indicate that if R[sub 0] > 1, then provided disease is initially present it will tend to the endemic equilibrium. We also extend this model to include recruitment and mortality from AIDS, and find that its long-term behaviour is similar to that of the simple model. We then look briefly at a stochastic version of the model and simulations indicate that again the disease dies out if R[sub 0] < 1 and if R[sub 0] > 1 the long-term conditional mean prevalence of disease is approximately the same as in the simple model. We examine the implications of our results for the prevention of the spread of HIV and AIDS amongst drug users. A short discussion concludes the paper. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for the effect of drug therapy on the in-host dynamics of HIV is considered and analyzed. As the process of reverse transcription is highly error prone, it causes mutation of virus which results in the emergence of drug resistant virus. This is also accounted in the model and corresponding model with both drug resistant and drug sensitive viral strains is studied. We found that, if reproductive ratios for both the strains are less than one, the virus population goes to extinction. If the reproductive ratio of either strain is greater than one and the reproductive ratio of drug resistant virus is smaller than that of drug sensitive virus then both the virus strains persist and infection is not cleared. However if reproductive ratio of drug resistant virus is greater than that of drug sensitive virus then the drug resistant virus out-competes the drug sensitive virus and only drug resistant virus survives. Hence the ratio of two reproduction ratios works as invading capacity threshold value for drug resistant strain. We also noted that by increasing the effective efficacy of the drug, virus may be cleared. Numerical simulations are performed to support and elaborate the analytical findings. [ABSTRACT FROM AUTHOR]

The basic models of within-host viral infection, proposed by Nowak and May2 and Perelson and Nelson,5 have been widely used in the studies of HBV and HIV infections. The basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection. In this paper, we formulate an amended Perelson and Nelson's model with standard incidence. The basic reproduction number of the amended model is independent of total cells of the host's organ. If the basic reproduction number R0 < 1, then the infection-free equilibrium is globally asymptotically stable and the virus is cleared; if R0 > 1, then the virus persists in the host, and solutions approach either an endemic equilibrium or a periodic orbit. Numerical simulations of this model agree well with the clinical HBV infection data. This can provide a possible interpretation for the viral oscillation behaviors, which were observed in chronic HBV infection patients. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years or more, then the number of infected adults may actually increase in the next 20 years or so. This creates a dilemma: to treat or not to treat? [ABSTRACT FROM AUTHOR]

In this paper, we present a dynamical model to study the spread of HIV-1 in vivo. Our goal is to better understand the interaction between HIV-1 and the human immune system. Making use of the Hill function, we describe two kinds of processions occurring in the immune response: the activation interactions or inhibitory interactions occurring between different components in the immune response, and the autocatalytic maintenance of the CD4+ T cells and CD8+ T cells populations. We also consider the impact of the cytokine interleukin-2 (IL-2) and the CD8 antiviral factor (CAF) on HIV-1 infection. Through numerical simulations we get several results. First, we find that the effects of IL-2 and CAF in the treatment for the infected are limiting. Namely, the curative effect will not always increase along with the dose of IL-2 or CAF or both. The increasing trend will stagnate at a certain dose that we used. Second, we find some possible reasons for the collapse of the lymph system in HIV-1 infection — the loss of these restraining functions, and/or the genetic variability of the virus due to immune escape that enhances the virulence, which then bring the collapse of the immune system. In some conditions the system will produce a Hopf bifurcation. We also simulate the theoretical warrant of the feasibility of the combined chemotherapy strategies for the HIV-1 infected patient. [ABSTRACT FROM AUTHOR]

We present a review and comparison of several recent differential equations models of treatment of HIV-1 infection. We seek to clarify the role of the natural anti-HIV immune response and determine its effect upon optimal treatment schemes. In this paper, we consider systems in which treatment is expressed as a forcing function, as well as those in which we determine optimal treatment via control theoretic techniques. The primary goal of this study is to compare treatment schemes for systems in which a natural nonconstant immune response of the patient is considered explicitly with those that consider implicitly a constant non-specific immune response. We find that when the natural immune response can be boosted sufficiently, drug levels may not need to be as high as previously supposed. This implies that a treatment scenario in which intervals of drug treatment are alternated with some form of immune-boosting therapy may be highly beneficial in terms of reducing toxicity to the patient. Additionally, in developing countries where HIV infection is widespread and sufficient funds are not available to supply rigourous drug regimens, the implications of these models are profound, as they suggest methods of treating HIV at a minimal cost. [ABSTRACT FROM AUTHOR]

In this paper, neglecting the effects of free virus, we consider a simple model of cell-to-cell spread of HIV-1. We discuss the impact of the CD8+ cell non-cytotoxic anti-viral response (CNAR) and cytotoxic T lymphocyte (CTL) activity on infection by HIV-1. Two types of models are considered: the ordinary differential equation (ODE) model and the discrete time delay differential equation (DDE) system. The steady states of the ODE model are globally asymptotically stable respectively under two threshold criteria. In the DDE model, the global stability of the infected steady state of the ODE model becomes only ultimately stable. Moreover, at a certain interval of the time delay, the DDE model will produce Hopf bifurcation or periodic solutions. The introduction of CTL and CNAR will change the values of these steady states and induce fluctuations in the cell concentration. It also affects the critical value of the time delay and is of utility in the interpretation of typical HIV-dynamics data resulting from clinical trials. The DDE model produces sustained infective oscillations in some real parameter values, which is different from the result of the similar cell-free viral spread model with delay. [ABSTRACT FROM AUTHOR]

Successful immunologic control of HIV infection can be achieved in long-term non-progressors or HIV-1 controllers. Dendritic cells (DCs) are required for specific antigen presentation to naïve T lymphocytes and for antiviral, type I interferon secretion. To understand this mechanism, we develop a mathematical model that describes the role of direct presentation (replicating virus-infected DCs or other CD 4 + T cells directly) and cross presentation (DCs obtain antigen processed in other infected cells such as CD 4 + T lymphocytes) during HIV-1 infection. We find equilibria and determine stability in the case of no vaccination, and then, when vaccination is taken, we determine analytical thresholds for the strength and frequency of the vaccine to ensure the disease-free equilibrium remains stable. Our theoretical results suggest that the restoration of DC numbers may be predictive of immune restoration and may be a goal for immunotherapy to enhance viral control in a larger proportion of patients. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate the qualitative behavior of an Human Immunodeficiency Virus (HIV) dynamics model with two types of cocirculating target cells and under the effect of anti-viral drug therapy. The model takes into account both short-lived infected cells and long-lived chronically infected cells. In the two types of target cells, the drug efficacy is assumed to be different. The incidence rate of virus infection is given by general functional response. We have derived the basic reproduction number which determines the global dynamics of the model. We have established a set of conditions which are sufficient to investigate the global stability of the equilibria of the model. The global stability analysis of the model has been established using the Lyapunov method. Numerical simulations have been performed for the model with a specific form of the incidence rate function. We have shown that the numerical and theoretical results are consistent. [ABSTRACT FROM AUTHOR]

We develop a new a coinfection model for hepatitis C virus (HCV) and the human immunodeficiency virus (HIV). We consider treatment for both diseases, screening, unawareness and awareness of HIV infection, and the use of condoms. We study the local stability of the disease-free equilibria for the full model and for the two submodels (HCV only and HIV only submodels). We sketch bifurcation diagrams for different parameters, such as the probabilities that a contact will result in a HIV or an HCV infection. We present numerical simulations of the full model where the HIV, HCV and double endemic equilibria can be observed. We also show numerically the qualitative changes of the dynamical behavior of the full model for variation of relevant parameters. We extrapolate the results from the model for actual measures that could be implemented in order to reduce the number of infected individuals. [ABSTRACT FROM AUTHOR]

Analysis of changes in viral load after initiation of treatment with potent antiretroviral agents has provided substantial insights into the dynamics of human immunodeficiency virus type 1. We built a simple mathematics model to study the effect of latent-infected resting memory CD4+ T cells during the HIV infection and highly active anti-retroviral therapy (HAART). Through analysis of eight patients who received HAART in China, we have an insight into the mechanisms of resting memory CD4+ T cells in HIV infection. Simulations show that new infections still exist in the eight patients even under the HAART. Also, because of the long half-life of resting infected memory CD4+ T cells, removal of HIV from patient could take considerably longer time or be unattainable. [ABSTRACT FROM AUTHOR]

We investigate a delayed HIV-1 infection model with general nonlinear incidence functions and two classes of target cells: CD4+ T-cells and macrophages. To account for the time lags between viruses' entry into the corresponding two types of target cells and the production of new virus particles, we incorporate four distributed intracellular delays into the model. We show that the basic reproduction number ℜ0 is the sum of the basic reproduction numbers of HIV-1 infection with CD4+ T-cells and that with macrophages; moreover, if ℜ0 is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population and macrophages; whereas if ℜ0 is larger than one, then the viral concentration maintains at some constant level. It is shown, from both our analytic and numeric results, that ignoring the contributions of macrophages to HIV-1 infection and production will underestimate both the risk of HIV-1 infection and the viral load when persisting. This highlights the important effects of multiple target cells on HIV-1 infection. [ABSTRACT FROM AUTHOR]

Infection with the hepatitis C virus (HCV) is the most common coinfection in people with the human immunodeficiency virus (HIV), and hepatitis C is categorized as an HIV-related illness. The study of the joint dynamics of HIV and HCV present formidable mathematical challenges in spite the fact that they share similar routes of transmission. A deterministic model for the co-interaction of HCV and HIV in a community is presented and rigorously analyzed. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than the unity. The Centre Manifold theory is used to show that the HCV only and HIV/AIDS only endemic equilibria are locally asymptotically stable when their associated reproduction numbers are greater than the unity. We compute two coexistence thresholds for the stability of boundary equilibria. Numerical results are presented to validate analytical results. [ABSTRACT FROM AUTHOR]

Primary prevention measures designed to alter susceptibility and/or reduce exposure of susceptible individuals to diseases, remain the mainstay in the fight against HIV/AIDS. A model for HIV/AIDS, that investigates the reduction in infection by advocating for sexual behavior change through public-health information campaigns and withdrawal of individuals with AIDS from sexual activity is proposed and analyzed. The contact rate is modeled using an incidence function with saturation that depends on the number of infectives. The dynamics of the model is determined using the model reproduction number ${\cal R}_0$. Numerical simulations are presented to illustrate the role of some key epidemiological parameters. The results from the study demonstrate that an increase in the rate of dissemination of effective public-health information campaigns results in a decrease in the prevalence of the disease. Similarly, an increase in the fraction of individuals with AIDS who withdraw from sexual activities reduces the burden of the disease. [ABSTRACT FROM AUTHOR]

A schistosomiasis and HIV/AIDS co-infection model is presented as a system of nonlinear ordinary differential equations. Qualitative analysis (properties) of the model are presented. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The Centre Manifold theory is used to show that the schistosomiasis only and HIV/AIDS only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. The model is numerically analyzed to assess the effects of schistosomiasis on the dynamics of HIV/AIDS. Analysis of the reproduction numbers and numerical simulations show that an increase of schistosomiasis cases result in an increase of HIV/AIDS cases, suggesting that schistosomiasis control have a positive impact in controlling the transmission dynamics of HIV/AIDS. [ABSTRACT FROM AUTHOR]

An impulsive differential model for post exposure prophylaxis (PEP) to HIV-1 exposure is proposed with presence of drugs. Two therapy strategies (the Drugs Perfect Adherence strategy and the Drugs Therapy Breaks strategy) are studied. The threshold $R_c^d$ is obtained for the model. The safe-dose threshold ($\bar{R}$) and the unsafe-dose threshold ($\hat{R}$) for PEP treatment are evaluated for the impulsive model under the Drugs Prefect Adherence strategy. The numerical simulations show that, in addition to the sufficient dosing of drug and to the sufficiently frequent dosing of drug, the basic reproduction number Rc is also responsible for the different final outcome of PEP treatment. [ABSTRACT FROM AUTHOR]

The Sub-Sahara African region is inhabited by only 11% of the global population, but is home to 67% of the total HIV infected people and accounts for more than 70% of global AIDS deaths. In this study, we construct a mathematical model to investigate the effect of heavy alcohol consumption on the transmission and progression of HIV/AIDS, and to assess the impact of heavy drinkers on HIV/AIDS related social and health problems such as TB case load and number of orphans. Using demographic data for Botswana, we have shown that if more HIV/AIDS individuals had been de-addicted from heavy alcohol consumption, the severity of the HIV/AIDS epidemic and the impact of HIV/AIDS on the number of TB cases and orphans would have been significantly less than is the case currently. The study points to the vital need for counseling and education about the evils of heavy alcohol consumption and for alcohol de-addiction programmes. [ABSTRACT FROM AUTHOR]

A mathematical model for the effect of Reverse Transcriptase (RT) Inhibitor on the dynamics of HIV is proposed and analyzed. Further, with help of numerical simulations, the relation between efficacy of administered drug, the total number of virus particles emitted from the infected cell and the transition period is also discussed. [ABSTRACT FROM AUTHOR]

Human Immunodeficiency Virus type-1 (HIV-1) fuels the pathogenesis of Mycobacterium tuberculosis (Mtb) in humans. We develop a mathematical model in an attempt to understand the immune mechanisms that are involved during the co-infection of Mtb and HIV-1. Our study reveals that infection of an Mtb infected individual with HIV-1 results in fast development of active TB. The mathematical model analysis and simulations show that Mtb infection is linked to HIV infection through macrophages and CD4+ T cells. The study shows that depletion of macrophages and CD4+ T cells by HIV-1 worsens the picture of Mtb infection and in-turn Mtb infection affects the progression of HIV-1 infection since it is also capable of inducing rapid replication of HIV. Our analytical and numerical simulations show that macrophages are a potential reservoir of HIV particles during HIV-1 infection. Co-infection simulations reveal that co-infection exacerbates more the pathogen that caused the first infection. Simulations also show that co-infection disease progression patterns converge to a similar trend after a considerable time interval irrespective of which pathogen first caused infection and the second pathogen that caused co-infection. This work suggests directions for further studies and potential treatment strategies. [ABSTRACT FROM AUTHOR]

Cancer remains a significant burden in HIV-infected individuals. We present an HIV-1 dynamical model which incorporates the AIDS-related cancer cells in vivo. The model consists of four components: cancer cells, healthy CD4+ T-lymphocytes, infected CD4+ T-lymphocytes and free HIV-1 virus. We discuss the existence, the stability properties and the biological meanings of the model's steady states, in particular the positive cancer-HIV steady state. We find that Hopf bifurcation of the positive steady state can exist, which lead to periodic solutions, sequences of period doubling bifurcations and the appearance of chaos. We also prove theoretically that HIV-1 is of help in the growth of cancer. The result fits very well with the clinical observations that several different types of cancer are observed at an increased frequency in HIV-1 infection individuals. [ABSTRACT FROM AUTHOR]

HIV/AIDS vaccination model for heterosexual transmission with explicit incubation period is presented as a system of delay differential equations. The model considers prophylactic vaccination of sexually immature (pre- and early-adolescents) and mature susceptibles in a community. We start by formulating and analyzing an HIV/AIDS vaccination model that we extend to incorporate condom use based on efficacy and compliance. Further, we extend HIV/AIDS vaccination model with condom use by incorporating treatment which allows AIDS patients to undergo amelioration. The thresholds and equilibria for the models are determined, and stabilities analyzed. The basic reproductive numbers for the models are computed and compared to assess the possible community benefits of using prophylactic vaccines, condoms and treatment with amelioration of AIDS patients. We conclude from the study that vaccination and condom use can reduce the basic reproductive number $({\mathcal R}_{0})$ to values below unity but treatment with amelioration intended to lengthen the lives of AIDS patients may result in more numbers of HIV infections and fail to reduce ${\mathcal R}_{0}$ to values less than unity as intended for disease control. [ABSTRACT FROM AUTHOR]

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t0, R0t0v and γeRVT(σ)UT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t0, R0t0v and γeRVT(σ)UT(α) are satisfied. [ABSTRACT FROM AUTHOR]

The devastating figures that recently emerged from a demographic study of the impact of HIV/AIDS in some African countries mark the return to the conditions of the XIXth century, when high birth rates were neutralized by equally high death rates. In the State of São Paulo, Brazil, AIDS is the second cause of death among men aged twenty to forty nine years and the first cause of death of women in the same age class. In this work we propose a mathematical treatment to evaluate the impact of AIDS mortality on the age structure of an affected population, namely, that of the state of São Paulo, Brazil. We propose four indicators for the estimation of the impact of AIDS mortality. The first is the age-dependent differences in ten years survival probabilities attributable to AIDS. The second is the difference in the average age of survivors after 10 years of AIDS. The next is the conventional life expectancy at birth for children born in 1996 and with AIDS prevalence assumed at its maximum value and remaining in steady-state afterwards. Finally, we calculate the differences in the life expectancy of individuals considering the effect of AIDS for only ten years. We found that, in the period between 1987 and 1996 the effects were small but very interesting. However, projecting to the future the conditions of 1996, we calculate that the population of the state of São Paulo would lose 3 years in the average life expectancy at birth. [ABSTRACT FROM AUTHOR]