Analysis of a model for the pathogenesis of AIDS

A mathematical model for the within-host (re)infection dynamics of SARS-CoV-2

Interactions between SARS-CoV-2 and the immune system during infection are complex. However, understanding the within-host SARS-CoV-2 dynamics is of enormous importance for clinical and public health outcomes. Current mathematical models focus on describing the within-host SARS-CoV-2 dynamics during the acute infection phase. Thereby they ignore important long-term post-acute infection effects. We present a mathematical model, which not only describes the SARS-CoV-2 infection dynamics during the acute infection phase, but extends current approaches by also recapitulating clinically observed long-term post-acute infection effects, such as the recovery of the number of susceptible epithelial cells to an initial pre-infection homeostatic level, a permanent and full clearance of the infection within the individual, immune waning, and the formation of long-term immune capacity levels after infection. Finally, we used our model and its description of the long-term post-acute infection dynamics to explore reinfection scenarios differentiating between distinct variant-specific properties of the reinfecting virus. Together, the model's ability to describe not only the acute but also the long-term post-acute infection dynamics provides a more realistic description of key outcomes and allows for its application in clinical and public health scenarios.

Optimal control of a fractional order model for the COVID - 19 pandemic

In this paper a fractional optimal control problem was formulated for the outbreak of COVID-19 using a mathematical model with fractional order derivative in the Caputo sense. The state and co-state equations were given and the best strategy to significantly reduce the spread of COVID-19 infections was found by introducing two time-dependent control measures, u 1 ( t ) (which represents the awareness campaign, lockdown, and all other measures that reduce the possibility of contacting the disease in susceptible human population) and u 2 ( t ) (which represents quarantine, monitoring and treatment of infected humans). Numerical simulations were carried out using RK-4 to show the significance of the control functions. The exposed population in susceptible population is reduced by the factor ( 1 - u 1 ( t ) ) due to the awareness and all other measures taken. Likewise, the infected population is reduced by a factor of ( 1 - u 2 ( t ) ) due to the monitoring and treatment by health professionals.

Optimal control of a fractional-order model for the HIV/AIDS epidemic

In this paper, we present a general formulation for a fractional optimal control problem (FOCP), in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the forward–backward sweep method (FBSM) using the Adams-type predictor–corrector method to solve the FOCP. We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) with treatment and incorporate three control efforts (effective use of condoms, ART treatment and behavioral change control) into the model aimed at controlling the spread of HIV/AIDS epidemic. The necessary conditions for fractional optimal control of the disease are derived and analyzed. The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people. Also, the maximum levels of the controls and the value of objective functional decrease when the derivative order [Formula: see text] limits to 1 ([Formula: see text]). In addition, the effect of the fractional derivative order [Formula: see text] ([Formula: see text]) on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated. The results show that the derivative order [Formula: see text] can play the role of using ART treatment in the model.

Optimal control analysis of malaria-schistosomiasis co-infection dynamics

This paper presents a mathematical model for malaria--schistosomiasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagin's Maximum Principle to the model in order to determine optimal strategies for control of both diseases.

Mathematical Analysis of the Effects of HIV-Malaria Co-infection on Workplace Productivity

In this paper, a nonlinear dynamical system is proposed and qualitatively analyzed to study the dynamics and effects of HIV-malaria co-infection in the workplace. Basic reproduction numbers of sub-models are derived and are shown to have LAS disease-free equilibria when their respective basic reproduction numbers are less than unity. Conditions for existence of endemic equilibria of sub-models are also derived. Unlike the HIV-only model, the malaria-only model is shown to exhibit a backward bifurcation under certain conditions. Conditions for optimal control of the co-infection are derived using the Pontryagin's maximum principle. Numerical experimentation on the resulting optimality system is performed. Using the incremental cost-effectiveness ratio, it is observed that combining preventative measures for both diseases is the best strategy for optimal control of HIV-malaria co-infection at the workplace.

A co-infection model of malaria and cholera diseases with optimal control

In this paper we formulate a mathematical model for malaria-cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagin's Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control.

Optimal control of HIV/AIDS in the workplace in the presence of careless individuals

A nonlinear dynamical system is proposed and qualitatively analyzed to study the dynamics of HIV/AIDS in the workplace. The disease-free equilibrium point of the model is shown to be locally asymptotically stable if the basic reproductive number, ℛ0, is less than unity and the model is shown to exhibit a unique endemic equilibrium when the basic reproductive number is greater than unity. It is shown that, in the absence of recruitment of infectives, the disease is eradicated when ℛ0 < 1, whiles the disease is shown to persist in the presence of recruitment of infected persons. The basic model is extended to include control efforts aimed at reducing infection, irresponsibility, and nonproductivity at the workplace. This leads to an optimal control problem which is qualitatively analyzed using Pontryagin's Maximum Principle (PMP). Numerical simulation of the resulting optimal control problem is carried out to gain quantitative insights into the implications of the model. The simulation reveals that a multifaceted approach to the fight against the disease is more effective than single control strategies.

Computational study to determine when to initiate and alternate therapy in HIV infection

HIV is a widespread viral infection without cure. Drug treatment has transformed HIV disease into a treatable long-term infection. However, the appearance of mutations within the viral genome reduces the susceptibility of HIV to drugs. Therefore, a key goal is to extend the time until patients exhibit resistance to all existing drugs. Current HIV treatment guidelines seem poorly supported as practitioners have not achieved a consensus on the optimal time to initiate and to switch antiretroviral treatments. We contribute to this discussion with predictions derived from a mathematical model of HIV dynamics. Our results indicate that early therapy initiation (within 2 years postinfection) is critical to delay AIDS progression. For patients who have not received any therapy during the first 3 years postinfection, switch in response to virological failure may outperform proactive switching strategies. In case that proactive switching is opted, the switching time between therapies should not be larger than 100 days. Further clinical trials are needed to either confirm or falsify these predictions.

Modeling the three stages in HIV infection

A typical HIV infection response consists of three stages: an initial acute infection, a long asymptomatic period and a final increase in viral load with simultaneous collapse in healthy CD4+T cell counts. The majority of existing mathematical models give a good representation of either the first two stages or the last stage of the infection. Using macrophages as a long-term active reservoir, a deterministic model is proposed to explain the three stages of the infection including the progression to AIDS. Simulation results illustrate how chronic infected macrophages can explain the progression to AIDS provoking viral explosion. Further simulation studies suggest that the proposed model retains its key properties even under moderately large parameter variations. This model provides important insights on how macrophages might play a crucial role in the long term behavior of HIV infection.

Analysis of recruitment and industrial human resources management for optimal productivity in the presence of the HIV/AIDS epidemic

The aim of this paper is to analyze the recruitment effects of susceptible and infected individuals in order to assess the productivity of an organizational labor force in the presence of HIV/AIDS with preventive and HAART treatment measures in enhancing the workforce output. We consider constant controls as well as time-dependent controls. In the constant control case, we calculate the basic reproduction number and investigate the existence and stability of equilibria. The model is found to exhibit backward and Hopf bifurcations, implying that for the disease to be eradicated, the basic reproductive number must be below a critical value of less than one. We also investigate, by calculating sensitivity indices, the sensitivity of the basic reproductive number to the model's parameters. In the time-dependent control case, we use Pontryagin's maximum principle to derive necessary conditions for the optimal control of the disease. Finally, numerical simulations are performed to illustrate the analytical results. The cost-effectiveness analysis results show that optimal efforts on recruitment (HIV screening of applicants, etc.) is not the most cost-effective strategy to enhance productivity in the organizational labor force. Hence, to enhance employees' productivity, effective education programs and strict adherence to preventive measures should be promoted.

Modelling the course of an HIV infection: insights from ecology and evolution

The Human Immunodeficiency Virus (HIV) is one of the most threatening viral agents. This virus infects approximately 33 million people, many of whom are unaware of their status because, except for flu-like symptoms right at the beginning of the infection during the acute phase, the disease progresses more or less symptom-free for 5 to 10 years. During this asymptomatic phase, the virus slowly destroys the immune system until the onset of AIDS when opportunistic infections like pneumonia or Kaposi’s sarcoma can overcome immune defenses. Mathematical models have played a decisive role in estimating important parameters (e.g., virion clearance rate or life-span of infected cells). However, most models only account for the acute and asymptomatic latency phase and cannot explain the progression to AIDS. Models that account for the whole course of the infection rely on different hypotheses to explain the progression to AIDS. The aim of this study is to review these models, present their technical approaches and discuss the robustness of their biological hypotheses. Among the few models capturing all three phases of an HIV infection, we can distinguish between those that mainly rely on population dynamics and those that involve virus evolution. Overall, the modeling quest to capture the dynamics of an HIV infection has improved our understanding of the progression to AIDS but, more generally, it has also led to the insight that population dynamics and evolutionary processes can be necessary to explain the course of an infection.

Effectiveness of a 'hunter' virus in controlling human immunodeficiency virus type 1 infection

Engineered therapeutic viruses provide an alternative method for treating infectious diseases, and mathematical models can clarify the system's dynamics underlying this type of therapy. In particular, this study developed models to evaluate the potential to contain human immunodeficiency virus type 1 (HIV-1) infection using a genetically engineered 'hunter' virus that kills HIV-1-infected cells. First, we constructed a novel model for understanding the progression of HIV infection that predicted the loss of the immune system's CD4(+) T cells across time. Subsequently, it determined the effects of introducing hunter viruses in restoring cell population. The model implemented direct and indirect mechanisms by which HIV-1 may cause cell depletion and an immune response. Results suggest that the slow progression of HIV infection may result from a slowly decaying CTL immune response, leading to a limited but constant removal of uninfected CD4 resting cells through apoptosis - and from resting cell proliferation that reduces the rate of cell depletion over time. Importantly, results show that the hunter virus does restrain HIV infection and has the potential to allow major cell recovery to 'functional' levels. Further, the hunter virus persisted at a reduced HIV load and was effective either early or late in the infection. This study indicates that hunter viruses may halt the progression of the HIV infection by restoring and sustaining high CD4(+) T-cell levels.

The dual role of dendritic cells in the immune response to human immunodeficiency virus type 1 infection

Many aspects of the complex interaction between human immunodeficiency virus type 1 (HIV-1) and the human immune system remain elusive. Our objective was to study these interactions, focusing on the specific roles of dendritic cells (DCs). DCs enhance HIV-1 infection processes as well as promote an antiviral immune response. We explored the implications of these dual roles. A mathematical model describing the dynamics of HIV-1, CD4+ and CD8+ T-cells, and DCs interacting in a human lymph node was analysed and is presented here. We have validated the behaviour of our model against non-human primate simian immunodeficiency virus experimental data and published human HIV-1 data. Our model qualitatively and quantitatively recapitulates clinical HIV-1 infection dynamics. We have performed sensitivity analyses on the model to determine which mechanisms strongly affect infection dynamics. Sensitivity analysis identifies system interactions that contribute to infection progression, including DC-related mechanisms. We have compared DC-dependent and -independent routes of CD4+ T-cell infection. The model predicted that simultaneous priming and infection of T cells by DCs drives early infection dynamics when activated T-helper cell numbers are low. Further, our model predicted that, while direct failure of DC function and an indirect failure due to loss of CD4+ T-helper cells are both significant contributors to infection dynamics, the former has a more significant impact on HIV-1 immunopathogenesis.

On the intra-host dynamics of HIV-1 infections

An extension of a previously proposed theory for the pathogenesis of AIDS is presented and analyzed using a mathematical modelling approach. This theory is based on the observation that human immunodeficiency virus type 1 (HIV-1) predominantly infects and replicates in (CD4+)-T cells, and that the infection process within an infected individual is characterized by ongoing generation and selection of HIV variants with increasing reproductive capacity. This evolutionary process is considered to be the reason for the gradual loss of immunocompetence and the final destruction of the immune system observed in most patients. The extension presented here incorporates the effect of the permanently increasing susceptibility of (CD4+)-T cell clones, as a result of the evolutionary process. The presented model reproduces and possibly explains a wide variety of findings about the HIV infection process. Numerical results indicate that the effect of the initial dose is minimal, and restricted to the primary phase of infection. According to the model predictions the impact of the HIV evolutionary speed is crucial for the progression to disease. An important progression determinant is the initial infection rate, being a component of the viral reproductive capacity. An influential role in disease progression seems to be played by the initial (CD4+)-T cell count.

A parameter sensitivity methodology in the context of HIV delay equation models

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

Optimizing within-host viral fitness: infected cell lifespan and virion production rate

We explore how an infected cell's virion production rate can affect the relative fitness of a virus within a host. We perform an invasion analysis, based on an age-structured model of viral dynamics, to derive the within-host relative viral fitness. We find that for chronic infections, in the absence of trade-offs between viral life history stages, natural selection favors viral strains whose virion production rate maximizes viral burst size. We then show how various life history trade-offs such as that between virion production and immune system recognition and clearance of virally infected cells can lead to natural selection favoring production rates lower than the one that maximizes burst size. Our findings suggest that HIV replication rates should vary between cells with different life spans, as has been suggested by recent observation.

Modeling the long-term control of viremia in HIV-1 infected patients treated with antiretroviral therapy

Highly active antiretroviral therapy (HAART), administered to a HAART-naïve patient, perturbs the steady state of chronic infection. This perturbation provides an opportunity to investigate the existence and dynamics of different sources of viral production. Models of HIV dynamics can be used to make a comparative analysis of the efficacies of different drug regimens. When HAART is administered for long periods of time, most patients achieve 'undetectable' viral loads (VLs), i.e., below 50 copies/ml. Use of an ultrasensitive VL assay demonstrates that some of these patients obtain a low steady state VL in the range 5-50 copies/ml, while others continue to exhibit VL declines to below 5 copies/ml. Interestingly, when patients exhibit continued declines below 50 copies/ml the virus has a half-life of approximately 6 months, consistent with some estimates of the rate of latent cell decline. Some patients, despite having sustained undetectable VLs, show periods of transient viremia (blips). We present a statistical characterization of the blips observed in a set of 123 patients, suggesting that patients have different tendencies to show blips during the period of VL suppression, that intermittent episodes of viremia have common amplitude profiles, and that VL decay from the peak of a blip may have two phases.

A delay-differential equation model of HIV infection of CD4(+) T-cells

A.S. Perelson, D.E. Kirschner and R. De Boer (Math. Biosci. 114 (1993) 81) proposed an ODE model of cell-free viral spread of human immunodeficiency virus (HIV) in a well-mixed compartment such as the bloodstream. Their model consists of four components: uninfected healthy CD4(+) T-cells, latently infected CD4(+) T-cells, actively infected CD4(+) T-cells, and free virus. This model has been important in the field of mathematical modeling of HIV infection and many other models have been proposed which take the model of Perelson, Kirschner and De Boer as their inspiration, so to speak (see a recent survey paper by A.S. Perelson and P.W. Nelson (SIAM Rev. 41 (1999) 3-44)). We first simplify their model into one consisting of only three components: the healthy CD4(+) T-cells, infected CD4(+) T-cells, and free virus and discuss the existence and stability of the infected steady state. Then, we introduce a discrete time delay to the model to describe the time between infection of a CD4(+) T-cell and the emission of viral particles on a cellular level (see A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak [Proc. Nat. Acad. Sci. USA 93 (1996) 7247]). We study the effect of the time delay on the stability of the endemically infected equilibrium, criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Numerical simulations are presented to illustrate the results.

A condition for successful escape of a mutant after primary HIV infection

Cytotoxic T lymphocytes (CTLs) vigorously restrict primary human immunodeficiency virus (HIV) infection. However, the frequently erroneous process of viral replication favors the creation of mutants not recognizable by primary CTLs. Variants that tolerate the mutations may have selective advantage and may increase in abundance, until the immune system reacts against them. Therefore, such variants represent a way of propagating the viremia. With the aid of a simple mathematical model, here we estimate the intensity of CTL cross-reactivity against different strains of HIV in a typical progressor. We show that below a critical intensity of cross-reactivity, the concentration of a mutant created at primary peak grows and causes a secondary peak in viremia. Above this critical intensity, such a mutant strain is prevented from reaching a detectable level. We speculate about how this result may contribute to the design of an anti-HIV vaccine.

Modeling plasma virus concentration during primary HIV infection

During primary HIV infection the viral load in plasma increases, reaches a peak, and then declines. Phillips has suggested that the decline is due to a limitation in the number of cells susceptible to HIV infection, while other authors have suggested that the decline in viremia is due to an immune response. Here we address this issue by developing models of primary HIV-1 infection, and by comparing predictions from these models with data from ten anti-retroviral, drug-naive, infected patients. Applying nonlinear least-squares estimation, we find that relatively small variations in parameters are capable of mimicking the highly diverse patterns found in patient viral load data. This approach yields an estimate of 2.5 days for the average lifespan of productively infected cells during primary infection, a value that is consistent with results obtained by drug perturbation experiments. We find that the data from all ten patients are consistent with a target-cell-limited model from the time of initial infection until shortly after the peak in viremia. However, the kinetics of the subsequent fall and recovery in virus concentration in some patients are not consistent with the predictions of the target-cell-limited model. We illustrate that two possible immune response mechanisms, cytotoxic T lymphocyte destruction of infected target cells and cytokine suppression of viral replication, could account for declines in viral load data not predicted by the original target-cell-limited model. We conclude that some additional process, perhaps mediated by CD8+ T cells, is important in at least some patients.

A model of HIV-1 pathogenesis that includes an intracellular delay

Mathematical modeling combined with experimental measurements have yielded important insights into HIV-1 pathogenesis. For example, data from experiments in which HIV-infected patients are given potent antiretroviral drugs that perturb the infection process have been used to estimate kinetic parameters underlying HIV infection. Many of the models used to analyze data have assumed drug treatments to be completely efficacious and that upon infection a cell instantly begins producing virus. We consider a model that allows for less then perfect drug effects and which includes a delay in the initiation of virus production. We present detailed analysis of this delay differential equation model and compare the results to a model without delay. Our analysis shows that when drug efficacy is less than 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy, and the length of the delay. Thus, previous estimates of infected cell loss rates can be improved upon by considering more realistic models of viral infection.

A model of primary HIV-1 infection

We construct a model based on biological principles of the interaction of HIV-1 with the CD4+ T cells at primary infection. Most of the parameters are obtained from the literature, the remainder from fitting the output of the model to data from seven patients. On the basis of the model we find that initial viral containment is due to an effective immune response. The viral level after the initial peak, a surrogate marker of disease progression, was determined by the rate of reactivation of memory cells. Differences in this rate may occur because of inter- or intra-individual differences in the capability of memory cells to recognise and dispose of variants of HIV, either due to immune escape mutations within the virus or because the virus directly inhibits reactivation. With no choice of parameters could direct and indirect killing produce the gradual loss in CD4+ T cells with the observed viral behaviour. The loss of CD4+ T cells is perhaps due to defective expansion of activated cells of both HIV specific and nonspecific cells. As less memory cells are produced as a result then this compartment decreases and hence so do naive numbers through less reversion of memory cells to the naive phenotype.