Mathematical Model for an Effective Management of HIV Infection

Qualitative analysis of HIV and AIDS disease transmission: impact of awareness, testing and effective follow up

Background: Since the early 1980s, human immunodeficiency virus (HIV) and its accompanying acquired immunodeficiency syndrome (AIDS) have spread worldwide, becoming one of the world's major global health issues. From the beginning of the epidemic until 2020, about 79.3 million people became infected, with 36.3 million deaths due to AIDS illnesses. This huge figure is a result of those unaware of their status due to stigmatization and invariably spreading the virus unknowingly. Methods: Qualitative analysis through a mathematical model that will address HIV unaware individuals and the effect of an increasing defaulter on the dynamics of HIV/AIDS was investigated. The impact of treatment and the effect of inefficient follow-up on the transmission of HIV/AIDS were examined. The threshold for the effective reduction of the unaware status of HIV through testing, in response to awareness, and the significance of effective non-defaulting in treatment commonly called defaulters loss to follow-up as these individuals contribute immensely to the spread of the virus due to their increase in CD4+ count was determined in this study. Stability analysis of equilibrium points is performed using the basic reproduction number $R_0$, an epidemiological threshold that determines disease eradication or persistence in viral populations.  We tested the most sensitive parameters in the basic reproduction numbers. Results: The results portray that early identification and treatment only are inadequate for the illness to be eradicated, but effectively used of condom, strict adherence to treatment and counseling of and testing of pregnant women contribute to a decrease in infected HIV individuals. Conclusions: Other control techniques, such as treatment adherence and effective condom usage, and reduction in vertical transmission cannot be over-emphasis to lessen the disease's burden. Policymakers must address these strategies through a series of public awareness campaigns about the dangers of not adhering to treatment procedures and patterns.

Qualitative analysis of HIV and AIDS disease transmission: impact of awareness, testing and effective follow up

Background: Since the early 1980s, human immunodeficiency virus (HIV) and its accompanying acquired immunodeficiency syndrome (AIDS) have spread worldwide, becoming one of the world's major global health issues. From the beginning of the epidemic until 2020, about 79.3 million people became infected, with 36.3 million deaths due to AIDS illnesses. This huge figure is a result of those unaware of their status due to stigmatization and invariably spreading the virus unknowingly. Methods: Qualitative analysis through a mathematical model that will address HIV unaware individuals and the effect of an increasing defaulter on the dynamics of HIV/AIDS was investigated. The impact of treatment and the effect of inefficient follow-up on the transmission of HIV/AIDS were examined. The threshold for the effective reduction of the unaware status of HIV through testing, in response to awareness, and the significance of effective non-defaulting in treatment commonly called defaulters loss to follow-up as these individuals contribute immensely to the spread of the virus due to their increase in CD4+ count was determined in this study. Stability analysis of equilibrium points is performed using the basic reproduction number $R_0$, an epidemiological threshold that determines disease eradication or persistence in viral populations. We tested the most sensitive parameters in the basic reproduction numbers. The model of consideration in this study is based on the assumption that information (awareness) and non-stigmatization can stimulate change in the behaviours of infected individuals, and can lead to an increase in testing and adherence to treatment. This will in turn reduce the basic reproduction number, and consequently, the spread of the virus. Results: The results portray that the early identification and treatment are inadequate for the illness to be eradicated. Conclusions: Other control techniques, such as treatment adherence and effective condom usage, should be investigated in order to lessen the disease's burden.

Simulation Model for Hashimoto Autoimmune Thyroiditis Disease

Hashimoto thyroiditis (HT) is a pathology that often causes a gradual thyroid insufficiency in affected patients due to the autoimmune destruction of this gland. The cellular immune response mediated by T helper lymphocytes TH1 and TH17 can induce the HT disease. In this pathologic condition, there is an imbalance between the TH17 and Treg lymphocytes as well as a gut microbiota dysfunction. The objective of this work was to describe the interactions of the cell subpopulations that participate in HT. To achieve this goal, we generated a mathematical model that allowed the simulation of different scenarios for the dynamic interaction between thyroid cells, the immune system, and the gut microbiota. We used a hypothetical-deductive design of mathematical modeling based on a system of ordinary differential equations, where the state variables are the TH1, TH17, and Treg lymphocytes, the thyrocytes, and the bacteria from gut microbiota. This work generated a compartmental model of the cellular immune response occurring in the thyroid gland. It was observed that TH1 and TH17 lymphocytes could increase the immune cells' activity, as well as activate effector cells directly and trigger the apoptosis and inflammation processes of healthy thyrocytes indirectly. Likewise, the model showed that a reduction in Treg lymphocytes could increase the activity of TH17 lymphocytes when an imbalance of the gut microbiota composition occurred. The numerical results highlight the TH1, TH17, and bacterial balance of the gut microbiota activities as important factors for the development of HT disease.

Optimal treatment strategies to control acute HIV infection

Various antiretroviral therapies (ART) are administered to symptomatic human immunodeficiency virus (HIV) infected individuals to improve their health. The treatment effectiveness may depend on suppressing development of drug resistance, reduce evolution of new viral strains, minimize serious side effects and the costs of drugs. This paper deals with some results concerning optimal drug administration scheme successful in improving patients' health especially in poorly resourced settings. The model under consideration describes the interaction between the uninfected cells, the latently infected cells, the productively infected cells, and the free viruses. Generally, in viral infection, the drug strategy aspects either the virus infectivity or reduce the virion production. The mathematical model proposed here, deals with both situations with the objective function based on a combination of maximizing benefit relied on T cells count (the white cells that coordinate activities of the immune system) and minimizing the systemic cost. The existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward sweep method (FBSM). Our results indicate that early initiation of treatment makes a profound impact in both improving the quality of life and reducing the economic costs of therapy.

Optimal Control of an HIV Model with Gene Therapy and Latency Reversing Agents

In this paper, we study the dynamics of HIV under gene therapy and latency reversing agents. While previous works modeled either the use of gene therapy or latency reversing agents, we consider the effects of a combination treatment strategy. For constant treatment controls, we establish global stability of the disease-free equilibrium and endemic equilibrium based on the value of R0. We then consider time-dependent controls and formulate an associated optimal control problem that emphasizes reduction of the latent reservoir. Characterizations for the optimal control profiles are found using Pontryagin’s Maximum Principle. We perform numerical simulations of the optimal control model using the fourth-order Runge–Kutta forward-backward sweep method. We find that a combination treatment of gene therapy with latency reversing agents provides better remission times than gene therapy alone. We conclude with a discussion of our findings and future work.

A mathematical model of HIV dynamics treated with a population of gene-edited haematopoietic progenitor cells exhibiting threshold phenomenon

The use of gene-editing technology has the potential to excise the CCR5 gene from haematopoietic progenitor cells, rendering their differentiated CD4-positive (CD4+) T cell descendants HIV resistant. In this manuscript, we describe the development of a mathematical model to mimic the therapeutic potential of gene editing of haematopoietic progenitor cells to produce a class of HIV-resistant CD4+ T cells. We define the requirements for the permanent suppression of viral infection using gene editing as a novel therapeutic approach. We develop non-linear ordinary differential equation models to replicate HIV production in an infected host, incorporating the most appropriate aspects found in the many existing clinical models of HIV infection, and extend this model to include compartments representing HIV-resistant immune cells. Through an analysis of model equilibria and stability and computation of $R_0$ for both treated and untreated infections, we show that the proposed therapy has the potential to suppress HIV infection indefinitely and return CD4+ T cell counts to normal levels. A computational study for this treatment shows the potential for a successful 'functional cure' of HIV. A sensitivity analysis illustrates the consistency of numerical results with theoretical results and highlights the parameters requiring better biological justification. Simulations of varying level production of HIV-resistant CD4+ T cells and varying immune enhancements as the result of these indicate a clear threshold response of the model and a range of treatment parameters resulting in a return to normal CD4+ T cell counts.

STABILITY ANALYSIS AND OPTIMAL CONTROL OF AN INTRACELLULAR HIV INFECTION MODEL WITH ANTIRETROVIRAL TREATMENT

In this paper, a deterministic model describing the dynamics of the in-host HIV infection of CD4[Formula: see text] T-cells is proposed. The model incorporates the presence of the CD[Formula: see text] 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 [Formula: see text] being then derived and shown to be a threshold value as far as the stability of the equilibria is concerned. When [Formula: see text] the infection-free equilibrium point is globally stable, whereas when [Formula: see text] 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[Formula: see text] T-cells and HIV viruses. However, a combination of both drugs could reduce the in-host HIV infection more significantly.

Stability analysis of the hiv model through incommensurate fractional-order nonlinear system

In this study, it is employed a new model of HIV infection in the form of incommensurate fractional differential equations systems involving the Caputo fractional derivative. Existence of the model's equilibrium points has been investigated. According to some special cases of the derivative-orders in the proposed model, the asymptotic stability of the infection-free equilibrium and endemic equilibrium has been proved under certain conditions. These stability conditions related to the derivative-orders depend on not only the basic reproduction rate frequently emphasized in the literature but also the newly obtained conditions in this study. Qualitative analysis results were complemented by numerical simulations in Matlab, illustrating the obtained stability result.

Analysis and optimal control of an HIV model based on CD4 count

A non-linear mechanistic model for the transmission dynamics of HIV/AIDS is developed and analyzed. The model classified the infected individuals based on their CD4 count level. Furthermore, education campaign, voluntary testing and counseling and treatment are considered as intervention strategies for controlling the disease. The analysis of the model reveals that imperfect public enlightenment campaign can induce backward bifurcation. It has been shown that when public enlightenment campaign is [Formula: see text] effective, the disease free equilibrium is globally asymptotically stable for [Formula: see text], whereas for [Formula: see text] the global stability of the endemic equilibrium is proved only in a special case. Time dependent controls of the intervention strategies mentioned above are incorporated into the model and the optimal control strategies with minimal implementation cost are identified. In addition, cost effectiveness analysis in the form of incremental cost effectiveness ratio is carried-out to identify the most cost effective strategies. The results suggest that out of the three non dominated strategies, the strategy of educating the newly entrants only or combination of newly entrants and susceptible individuals is very cost effective using per capita GDP of Nigeria as at 2018. However, the choice of which strategy to implement depends on budgetary allocation and resource availability.

Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge-Kutta method failed to solve it. Consequently, the "Adams predictor-corrector" method is used. The results reveal that the patient's immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients' treatment intervention strategies.

Modeling the effect of delay strategy on transmission dynamics of HIV/AIDS disease

In this manuscript, we investigate a nonlinear delayed model to study the dynamics of human-immunodeficiency-virus in the population. For analysis, we find the equilibria of a susceptible-infectious-immune system with a delay term. The well-established tools such as the Routh-Hurwitz criterion, Volterra-Lyapunov function, and Lasalle invariance principle are presented to investigate the stability of the model. The reproduction number and sensitivity of parameters are investigated. If the delay tactics are decreased, then the disease is endemic. On the other hand, if the delay tactics are increased then the disease is controlled in the population. The effect of the delay tactics with subpopulations is investigated. More precisely, all parameters are dependent on delay terms. In the end, to give the strength to a theoretical analysis of the model, a computer simulation is presented.

Modelling Optimal Control of In-Host HIV Dynamics Using Different Control Strategies

HIV is one of the major causes of deaths, especially in Sub-Saharan Africa. In this paper, an in vivo deterministic model of differential equations is presented and analyzed for HIV dynamics. Optimal control theory is applied to investigate the key roles played by the various HIV treatment strategies. In particular, we establish the optimal strategies for controlling the infection using three treatment regimes as the system control variables. We have applied Pontryagin's Maximum Principle in characterizing the optimality control, which then has been solved numerically by applying the Runge-Kutta forth-order scheme. The numerical results indicate that an optimal controlled treatment strategy would ensure significant reduction in viral load and also in HIV transmission. It is also evident from the results that protease inhibitor plays a key role in virus suppression; this is not to underscore the benefits accrued when all the three drug regimes are used in combination.

The In Vivo Dynamics of HIV Infection with the Influence of Cytotoxic T Lymphocyte Cells

The in vivo dynamics of HIV infection, the infection mechanism, the cell types infected, and the role played by the cytotoxic cells are poorly understood. This paper uses mathematical modelling as a tool to investigate and analyze the immune system dynamics in the presence of HIV infection. We formulate a six-dimensional model of nonlinear ordinary differential equations derived from known biological interaction mechanisms between the immune cells and the HIV virions. The existence and uniqueness as well as positivity and boundedness of the solutions to the differential equations are proved. Furthermore, the disease-free reproduction number is derived and the local asymptotic stability of the model investigated. In addition, numerical analysis is carried out to illustrate the importance of having R₀ < 1. Lastly, the biological dynamics of HIV in vivo infection are graphically represented. The results indicate that, at acute infection, the cytotoxic T-cells play a paramount role in reducing HIV viral replication. In addition, the results emphasize the importance of developing controls, interventions, and management policies that when implemented would lead to viral suppression during acute infection.