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In this paper, a dynamic HIV model with cell-to-cell transmission, two immune responses, and induced apoptosis is proposed and studied. First, the non-negativity and boundedness of the solutions of the model are given, and then the exact expression of the basic reproduction number R0 is obtained by using the next generation matrix method. Second, criteria are obtained for the local stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses. Furthermore, the threshold conditions are also derived for the global asymptotic stability of the disease-free equilibrium, immune responsefree equilibrium, and the infected equilibrium with both humoral and cellular immune responses by constructing the suitable Lyapunov function. Finally, some numerical simulations are conducted to verify the theoretical results; the numerical simulation results show that the increase of apoptosis rate had a positive role in the control of viral infection. [ABSTRACT FROM AUTHOR]

In this paper, a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response is established. Two cases are considered: constant control and optimal control. For the constant control system, the existence and uniqueness of the positive solutions are proved, and then the sufficient conditions for the existence and stability of five equilibriums are obtained. For the second case, the Pontryagin's Maximum Principle is used to analyze the optimal control, and the formula of the optimal solution are derived. After that, some numerical simulations are performed to validate the theoretical prediction. Numerical simulations indicate that in the case of HIV/HTLV co-infection, the concentration of CD4+T cells is no longer suitable as an effective reference data for understanding the development process of the disease. On the contrary, the number of HIV virus particles should be used as an important indicator for reference. [ABSTRACT FROM AUTHOR]

The immune response is a crucial factor in controlling HIV infection. However, oxidative stress poses a significant challenge to the HIV-specific immune response, compromising the body's ability to control viral replication. In this paper, we develop an HIV infection model to investigate the impact of immune impairment on virus dynamics. We derive the basic reproduction number (R0) and threshold (Rc). Utilizing the antioxidant parameter as a bifurcation parameter, we establish that the system exhibits saddle-node bifurcation backward and forward bifurcations. Specifically, when R0 > Rc, the virus will rebound if the antioxidant parameter falls below the post-treatment control threshold. Conversely, when the antioxidant parameter exceeds the elite control threshold, the virus remains under elite control. The region between the two thresholds represents a bistable interval. These results can explain why some HIV-infected patients experience rapid viral rebound after treatment cessation while others achieve post-treatment control for a longer time. [ABSTRACT FROM AUTHOR]

In this paper, we study a model that enhances our understanding of cytokine-influenced HIV-1 infection. The impact of adaptive immune response (cytotoxic T lymphocytes (CTLs) and antibodies) and time delay on HIV-1 infection is included. The model takes into account two types of distributional delays, (i) the delay in the HIV-1 infection of CD4+T cells and (ii) the maturation delay of new virions. We first investigated the fundamental characteristics of the system, then found the system's equilibria. We derived five threshold parameters, ℜ i , i = 0, 1,..., 4, which completely determine the existence and stability of the equilibria. The Lyapunov method was used to prove the global asymptotic stability for all equilibria. We illustrate the theoretical results by performing numerical simulations. We also performed a sensitivity analysis on the basic reproduction number ℜ 0 and identified the most-sensitive parameters. We found that pyroptosis contributes to the number ℜ 0 , and then, neglecting it will make ℜ 0 underevaluated. Necrosulfonamide and highly active antiretroviral drug therapy (HAART) can be effective in preventing pyroptosis and at reducing viral replication. Further, it was also found that increasing time delays can effectively decrease ℜ 0 and, then, inhibit HIV-1 replication. Furthermore, it is shown that both CTLs and antibody immune responses have no effect on ℜ 0 , while this can result in less HIV-1 infection. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) and tuberculosis (TB) have long been known to have a synergistic relationship. This is a result of each of the diseases impacting negatively on the immune system of the infected persons. The impact of these diseases on workforce productivity is studied in this paper from the viewpoint of dynamical systems. In this paper, we present a nonlinear ordinary differential equation model to study the dynamics of HIV-TB co-infection and its effect on workforce productivity. The main model is first decoupled into two basic submodels of HIV-only and TB-only models, whose qualitative properties are presented before the qualitative properties of the main model are studied. While the HIV-only model is shown to have a globally asymptotically stable disease-free equilibrium whenever its basic reproduction number is less than unity, the TB-only model is shown to exhibit backward bifurcation under some conditions. To investigate the impact of various intervention strategies on the control of the co-infection and improvement of workforce productivity, five time-dependent controls (involving transmission prevention for the two diseases, therapy for the two diseases, and capacity building for improved workforce productivity) are incorporated into the basic model to form an optimal control problem, which is qualitatively analyzed using Pontryagin's maximum principle and numerically simulated. Incremental cost-effectiveness analysis is conducted with the results of the numerical simulations. It is observed that the most cost-effective strategy for fighting the spread of the co-infection with enhanced productivity is that of combining both preventative and curative measures along with skills training. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical analysis of the HIV/AIDS deterministic model studied in the paper called Mathematical Model of HIV/AIDS Considering Sexual Preferences Under Antiretroviral Therapy, a case study in the previous works preformed by Espitia is performed. The objective is to gain insight into the qualitative dynamics of the model determining the conditions for the persistence or effective control of the disease in the community through the study of basic properties such as positiveness and boundedness; the calculus of the basic reproduction number; stationary points such as disease-free equilibrium (DFE), boundary equilibrium (BE) and endemic equilibrium (EE); and the local stability (LAS) of disease-free equilibrium. The findings allow us to conclude that the best way to reduce contagion and consequently reach a DFE is thought to be the reduction in the rate of homosexual partners, as they are the most affected population by the virus and are therefore the most likely to become infected and spread it. Increasing the departure rate of infected individuals leads to a decrease in untreated infected heterosexual men and untreated infected women. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes two mathematical models for the human immunodeficiency virus type-1 (HIV-1) infection with Cytotoxic T Lymphocyte cell (CTL) immune impairment. These models describe the interactions between healthy CD 4 + T cells, latently and actively infected cells, HIV-1 particles, and CTLs. The healthy CD 4 + T cells might be infected when they make contact with: (i) HIV-1 particles due to virus-to-cell (VTC) contact; (ii) latently infected cells due to latent cell-to-cell (CTC) contact; and (iii) actively infected cells due to active CTC contact. Distributed time delays are considered in the second model. We show the nonnegativity and boundedness of the solutions of the systems. Further, we derive basic reproduction numbers ℜ 0 and ℜ ˜ 0 , that determine the existence and stability of equilibria of our proposed systems. We establish the global asymptotic stability of all equilibria by using the Lyapunov method together with LaSalle's invariance principle. We confirm the theoretical results by numerical simulations. The effect of immune impairment, time delay and CTC transmission on the HIV-1 dynamics are discussed. It is found that weak immunity contributes significantly to the development of the disease. Further, we have established that the presence of time delay can significantly decrease the basic reproduction number and then suppress the HIV-1 replication. On the other hand, the presence of latent CTC spread increases the basic reproduction number and then enhances the viral progression. Thus, neglecting the latent CTC spread in the HIV-1 infection model will lead to an underestimation of the basic reproduction number. Consequently, the designed drug therapies will not be accurate or sufficient to eradicate the viruses from the body. These findings may help to improve the understanding of the dynamics of HIV-1 within a host. [ABSTRACT FROM AUTHOR]

HIV infects active uninfected CD4 + + T cells, and the active CD4 + + T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4 + + T cells may play an important role in HIV infection. In this paper, we formulate a mathematical model to investigate the activation effect of CD4 + + T cells on HIV dynamics. In the model, the uninfected CD4 + + T cells are divided into two pools: quiescent and active, and the stimuli rate of quiescent cells by HIV is described by saturated form function. We derive the basic reproduction number R 0 R 0 and analyze the existence and the stability of equilibria. Numerical simulations confirm that the system may have backward bifurcation and Hopf bifurcation. The results imply that R 0 R 0 cannot completely determine the dynamics of the system and the system may have complex dynamics, which are quite different from the models without the activation effect of CD4 + + T cells. Some numerical results are further presented to assess the activation parameters on HIV dynamics. The simulation results show that the changes of the activation parameters can cause the system periodic oscillation, and activation rate by HIV may induce the supercritical Hopf bifurcation and subcritical Hopf bifurcation. Finally, we proceed to investigate the effect of activation on steady-state viral loads during antiretroviral therapy. The results indicate that, viral load may exist and remain high level even if antiretroviral therapy is effective to reduce the basic reproduction number below 1. [ABSTRACT FROM AUTHOR]

In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and bound-edness of the solution. For this model, the basic reproduction number R0 and the immune reproduction number R1 are identified. Moreover, we have shown that the model has three e-quilibria, namely the infection-free equilibrium E0, the infectious equilibrium without immune response E1 and the infectious equilibrium with immune response E2. By applying uctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of E0 and E1 are only related to R0 and R1. The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to conduct an in-depth analysis of the intricacies inherent in a gender-structured model depicting the dynamics of HIV–TB coinfection. The model is articulated through a set of nonlinear fractional-order differential equations. Our focus is on elucidating the stability characteristics around equilibrium points, employing advanced fractional stability techniques. We leverage sensitivity analysis as a powerful tool to discern the specific model parameters exerting significant influence on the transmission and control of the coinfection, as gauged by the basic reproduction number. Through this, we aim to identify key factors that drive the spread or containment of the dual affliction. Furthermore, we delve into the realm of optimization by formulating a fractional optimal control problem. The objective is to design and implement two control strategies that effectively mitigate the prevalence of coinfection, demonstrating a state of perfect protection. Notably, our findings reveal that the implementation of control measures at a rate exceeding 80% suffices to eliminate the infection entirely. We anticipate that the insights garnered from this research will prove instrumental for policymakers in crafting strategic interventions to curtail the tragic impact of HIV–TB coinfection. Our scientifically grounded study serves as a foundation upon which comprehensive plans can be devised to stem the devastating consequences of this dual health challenge. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) is a serious disease that threatens and affects capital stock, population composition, and economic growth. This research paper aims to study the mathematical modeling and disease dynamics of HIV/acquired immunodeficiency syndrome (AIDS) with memory effect. We propose two fractional models in the Caputo sense for HIV/AIDS with and without migration. First, we prove the existence and positivity of both models and calculate the basic reproduction number R0$$ {\mathcal{R}}_0 $$ using the next generation method. Then, we study the local and global stability of the obtained equilibria. In addition, numerical simulations are provided for different values of the fractional order ρ$$ \rho $$ using the Adams–Bashforth–Moulton fractional scheme, to verify the theoretical results. Moreover, a sensitivity analysis of the parameters for the model with migration is carried out. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion–chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L ∞ -boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S 0 , CTL-inactivated infection steady state S 1 , and CTL-activated infection steady state S ∗. Our analyses indicate that S 0 is globally asymptotically stable if the basic reproduction number R 0 is less than 1; if R 0 is between 1 and a threshold, then S 1 is globally asymptotically stable. However, if R 0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S ∗ , and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotical behaviour of an age-structured Human Immunodeficiency Virus infection model with general non-linear infection function and logistic proliferation term is studied. Based on the existence of the equilibria and theory of operator semigroups, linearized stability/instability of the disease-free and endemic equilibria is investigated through the distribution of eigenvalues of the linear operator. Then persistence of the solution semi-flow of the considered system is studied by showing the existence of a global attractor and the obtained result shows that the solution semi-flow is persistent as long as the basic reproduction number |$R_{0}>1$|⁠. Moreover, the Hopf bifurcations problem around the endemic equilibrium is also considered for the situation with a specific infection function. Since the system has two different delays, four cases are discussed to investigate the influence of the time delays on the dynamics of system around the endemic equilibrium including stability and Hopf bifurcations. At last, some numerical examples with concrete parameters are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

This paper studies an -dimensional nonlinear HIV dynamics model that characterizes the interactions of HIV particles, susceptible CD4 T cells and -stages of infected CD4 T cells. Both virus-to-cell and cell-to-cell infection modes have been incorporated into the model. The incidence rates of viral and cellular infection as well as the production and death rates of all compartments are modeled by general nonlinear functions. We have revealed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. The basic reproduction number is determined which insures the existence of the two equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the model's equilibria. The global asymptotic stability of the two equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. We have proven that if , then the infection-free equilibrium is globally asymptotically stable, and if , then the chronic-infection equilibrium is globally asymptotically stable. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions. [ABSTRACT FROM AUTHOR]

This paper mainly targets to deduce an optimal treatment strategy for combined antiretroviral drugs, which can maximize healthy CD4+ T cells level with minimum side effects and cost. For this purpose, we consider a within-host treatment model for the HIV infection with two controls incorporating full logistic proliferation of healthy CD4+ T cells, cure rate and fusion effect. These two controls represent the effects of reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs), respectively. The model analysis begins with proving different basic properties like non-negativity, boundedness of the model solutions and calculation of the basic reproduction number of the model under consideration. Then, stability results are obtained for HIV infection-free equilibrium point and also, a critical efficacy for the combined therapies is calculated. After that, the optimal control problem is proposed and solved numerically using a forward–backward iterative method. Finally, we obtain an optimal treatment strategy that can maximize healthy CD4+ T cells count and control the viral load, and HIV-infected CD4+ T cells count to an undetectable level resulting in improved health conditions of infected individuals. [ABSTRACT FROM AUTHOR]

Taking into account the effects of the heterogeneity of the population and media coverage on disease transmission, in this paper, a hybrid HIV/AIDS model with age-structure, self-protection awareness and media coverage is formulated, which is made up of five partial differential equations (PDEs) and one ordinary differential equation (ODE). We establish the existence of the solution associated with the hybrid system and prove that the solution is unique, bounded and positive utilizing the semigroup approach. Based on the basic reproduction number R 0 , the threshold dynamics of this model are rigorously investigated, that is, there always is a unique disease-free steady state E 0 and it is globally stable when R 0 < 1 , that is, the disease dies out. Further, there exists a unique endemic steady state E * and it is locally stable when R 0 > 1 and some additional technical conditions are met. In addition, the uniform persistence of this hybrid system is demonstrated for R 0 > 1 , which means that the disease remains at the endemic level for a long time, which is not discussed in other age-structured infectious disease articles. Numerical simulations are also given to explain the main theoretical results, which suggest that age variability is a non-negligible factor in HIV/AIDS transmission, that is, the moment and scale of HIV/AIDS outbreaks are diverse for people of different ages, and media coverage can encourage people to take steps to avoid potential infection and control the spread of the disease. [ABSTRACT FROM AUTHOR]

Recent experimental evidence suggests that spatial heterogeneity plays an important role in within‐host infections caused by different viruses including hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To examine the spatial effects of viral infections, in this paper we study the asymptotic spreading in a within‐host viral infection model, which describes the spatial expansion speeds of viruses and infected cells within an infected host. We first establish the boundedness of solutions to the Cauchy problem via local Lp$L^p$‐estimates and dual arguments. Then the spreading speed is estimated when the basic reproduction number of the corresponding kinetic system is larger than one. More precisely, the upper bounds of the spreading speed are given by constructing suitable upper solutions while the lower bounds of the spreading speed are obtained by introducing an auxiliary equation with nonlocal delay. When the basic reproduction number of the corresponding kinetic system is less than or equal to one, the virus dies out uniformly. Finally, we present some numerical simulations to illustrate our theoretical findings and discuss the biological relevance of these results. [ABSTRACT FROM AUTHOR]

To control the HIV/AIDS epidemics in Guangxi Zhuang Autonomous Region in China, Guangxi government launched the 5-year Guangxi AIDS Conquering Project (GACP, Phase I: 2010-2014, Phase II: 2015-2020). In the project, three measures are implemented, such as great improvements of the coverage of HIV/AIDS education, promotion of HIV voluntary counseling and testing, and enhancement of antiretroviral treatment. In this paper, we explore the effects of the three measures of GACP by construction of a Susceptible-Infected-Diagnosed-Treated population compartments model and via evaluation of the basic reproduction number derived from the model. A computational framework is developed for estimating the model parameters based on the HIV surveillance data, with application of the Markov-Chain Monte-Carlo method and Nonlinear Least Squares method. By estimating the new infections and evaluating the basic reproduction number, we find that the implementation of the three measures of GACP has a significant effect on controlling the rise of HIV/AIDS cases and the epidemic trend. Compared with HIV voluntary counseling and testing, strengthening HIV/AIDS education and expanding the coverage of antiretroviral treatment show a greater impact on HIV/AIDS epidemic control, which provides a reference project for other provinces with a similar epidemic situation in Guangxi Zhuang Autonomous Region. At the same time, our research fills the current research gap for the evaluation of large-scale AIDS prevention and control projects in developing areas. [ABSTRACT FROM AUTHOR]

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number R 0 and the basic immunity reproduction number R 1 . The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters. [ABSTRACT FROM AUTHOR]

In this paper, we formulate an age-structured HIV model, in which the influence of humoral immunity and the infection age of the infected cells are considered. The model is governed by three ordinary differential equations and two first-ordered partial differential equations and admits three equilibria: disease-free, immune-inactivated and immune-activated equilibria. We introduce two important thresholds: the basic reproduction number R 0 and immune-activated reproduction number R 1 and further show the global stability of above three equilibria in terms of R 0 and R 1 , respectively. The numerical simulations are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mathematical model for the spread of HIV disease within two different age classes. We define a basic reproduction number R0 that depends on the characteristics of the two age classes. We prove that if R0 < 1, then the disease is extinct in both age classes. In contrast, we prove that if R0 > 1, then the disease is endemic in both age classes. [ABSTRACT FROM AUTHOR]

We construct a model to investigate HIV/AIDS dynamics in real cases and study its mathematical analysis. The study examines the qualitative outcomes and confirms the local and global asymptotic stability of both the endemic equilibrium and the disease-free equilibrium. The model's criteria for exhibiting both local and global asymptotically stable behavior are examined. We compute the endemic equilibria and obtain the existence of a unique positive endemic equilibrium. The data is fitted to the model using the idea of nonlinear least-squares fitting. Accurate parameter values are achieved by fitting the data to the model using a 95% confidence interval. The basic reproduction number is computed using parameters that have been fitted or estimated. Sensitivity analysis is performed to discover the influential parameters that impact the reproduction number and the eradication of the disease. The results show that implementing preventive measures can reduce HIV/AIDS cases. [ABSTRACT FROM AUTHOR]

HIV/AIDS has a strong impact on society, the economy, and health. Early diagnosis of cases, adherence to treatment, and prevention are important factors in controlling the epidemic in the population. In this paper, we present a new mathematical model for the study of HIV/AIDS transmission. Our model is stratified in men and women, to account for the main forms of sexual transmission homosexual and heterosexual relationships, and infectiousness in the HIV and AIDS stages. In addition, in the construction of the model, we take into account the influence of Pre-Exposure Prophylaxis (PrEP) and Post-Exposure Prophylaxis (PEP) to study the impact of these implementations, diagnosis, and effectiveness of treatment based on viral load undetectability. We study the basic reproduction number by subpopulation (men and women) and general. Working by subpopulations allows us to study men who have sex with men who have a strong impact on virus transmission. Also, we study the infection-free equilibrium points due to their relationship with the basic reproduction number and demonstrate the global stability by subpopulation and general. To explore our model, we performed computational simulations on. [ABSTRACT FROM AUTHOR]

In the present paper, we propose and analyze a nonlinear fractional-order SEIR (susceptible-exposed-infected-recovered) epidemic model to transmit HIV. The fixed points of the model and their stability results are obtained. Using the fractional derivatives, we relied on the Caputo fractional derivative. Further, we employed the homotopy analysis method (HAM) to get an approximate solution of the dynamic fractional derivatives of the model. The purpose of using HAM as a solution technique is its reliability, easy to handle, that utilizes a simple process to adjust and control the convergence region of the obtained infinite series solution. It uses an auxiliary parameter and allows to obtain a one-parametric family of explicit series solutions. Firstly, several h-curves are plotted to demonstrate the regions of convergence, then the residual and square residual errors are obtained for different values of these regions. In the end, numerical solutions are presented for various iterations to show the accuracy of the HAM. Besides, the convergence theorem of HAM is also proved. The obtained results show the effectiveness and strength of the applied HAM on the proposed fractional-order SEIR model. Also, from the sensitivity analysis results, it is seen that the parameters µ and σ are more sensitive than ϵ and ρ in disease transmission. [ABSTRACT FROM AUTHOR]

The basic reproduction number (R0) often cannot be explicitly computed when dealing with continuously age‐structured epidemic models. In this paper, we numerically compute R0 of a PDE model for a human immunodeficiency virus (HIV) infection, defined as the spectral radius of a next‐generation operator. Since R0 cannot be analytically obtained, on the one hand, we discretize the linearized PDE model into a system of linear ODEs (Euler method), and on the other hand, we discretize the eigenvalue problem (pseudo‐spectral method). In both cases, we approximate R0 by the largest eigenvalue R0, n of a next‐generation matrix, and we show the convergence of R0, n to R0 as the discretization index n increases to infinity. Finally, we present several tests to check/compare the accuracy of both numerical methods. [ABSTRACT FROM AUTHOR]

To enhance my research, I would like to share my knowledge of Herpes simplex virus-2 (HSV-2) and HIV/AIDS (Acquired Immunodeficiency Syndrome) through a mathematical model. The study's objective is to develop and examine the co-disease model for the modern era using mathematical modeling. Based on their modes of transmission, HSV-2 and HIV/AIDS are currently the leading causes of death from infectious and severe chronic diseases. Depending on the severity of HIV-2's chronic disease, the model is divided into five phases: the first two HIV-2 stages and the remaining three HIV stages. Ordinary differential equations (ODEs) are arranged differently by each individual. Investigations into the mathematical equation model have revealed the points of equilibrium between the free and endemic models. A study of the developed model was conducted using the basic reproduction numbers [ R 0 ] of HSV-2 and HIV. The results demonstrate that if R 0 < 1 , the free equilibrium of disease is asymptotically locally stable. When R 0 > 1 , equilibrium endemic states are regarded as existing. Finally, MATLAB software was used to simulate the numerical equations of the model. [ABSTRACT FROM AUTHOR]

In this study, we consider a system of nonlinear differential equations modeling the human immunodeficiency virus type-1 (HIV-1) in a variable environment. Infected cells were subdivided into two compartments describing both latently and productively infected cells. Thus, three routes of infection were considered including the HIV-to-cell contact, latently infected cell-to-cell contact, and actively infected cell-to-cell contact. The nonnegativity and boundedness of the trajectories of the dynamics were proved. The basic reproduction number was determined through an integral operator. The global stability of steady states is then analyzed using the Lyapunov theory together with LaSalle's invariance principle for the case of a fixed environment. Similarly, for the case of a variable environment, we showed that the virus-free periodic solution is globally asymptotically stable once R 0 ≤ 1 , while the virus will persist once R 0 > 1 . Finally, some numerical examples are provided illustrating the theoretical investigations. [ABSTRACT FROM AUTHOR]

In this study, a modified model of HIV with therapeutic and preventive controls is developed. Moreover, a simple evaluation of the optimal control problem is investigated. We construct the Hamiltonian function by way of integrating Pontryagin’s maximal principle to achieve the point-wise optimal solution. The effects obtained from the version analysis strengthen public health education to a conscious population, PrEP for early activation of HIV infection prevention, and early treatment with artwork for safe life after HIV infection. Moreover, numerical simulations are done using the MATLAB platform to illustrate the qualitative conduct of the HIV infection. In the end, we receive that adhering to ART protective prone people, the usage of PrEP along with different prevention control is safer control measures. [ABSTRACT FROM AUTHOR]

A multi-patch HIV/AIDS model with heterosexual transmission is formulated to investigate the impact of population migration on the spread of HIV/AIDS. We derive the basic reproduction number $ R_{0} $ R 0 and prove that if $ R_{0} \lt 1 $ R 0 < 1 , the disease-free equilibrium is globally asymptotically stable. If $ R_{0} \gt 1 $ R 0 > 1 and certain conditions are satisfied, the endemic equilibrium is globally asymptotically stable. We apply the model to two patches and conduct numerical simulations. If HIV/AIDS becomes extinct in each patch when two patches are isolated, the disease remains extinct in two patches when the population migration occurs; if HIV/AIDS spreads in each patch when two patches are isolated, the disease remains persistent in two patches when the population migration occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread or disappear in two patches if migration rates of individuals are suitably chosen. [ABSTRACT FROM AUTHOR]

South Africa has the highest number of people living with the human immunodeficiency virus (HIV) in the world, accounting for nearly one in five people living with HIV globally. As of 2021, 8 million people in South Africa were infected with HIV, which is 13% of the country's total population. Approximately 450,000 people in the country develop tuberculosis (TB) disease every year, and 270,000 of those are HIV positive. This suggests that being HIV positive significantly increases one's susceptibility to TB, accelerating the spread of the epidemic. To better understand the disease burden at the population level, a Susceptible–Infected–Recovered–Dead (SIRD) TB–HIV co-infection epidemic model is presented. Parameter values are estimated using the method of moments. The disease-free equilibrium and basic reproduction number of the model are also obtained. Finally, numeric simulations are carried out for a 30-year period to give insights into the transmission dynamics of the co-infection. [ABSTRACT FROM AUTHOR]

In this paper, we propose a six-dimensional nonlinear system of differential equations for the human immunodeficiency virus (HIV) including the B-cell functions with a general nonlinear incidence rate. The compartment of infected cells was subdivided into three classes representing the latently infected cells, the short-lived productively infected cells, and the long-lived productively infected cells. The basic reproduction number was established, and the local and global stability of the equilibria of the model were studied. A sensitivity analysis with respect to the model parameters was undertaken. Based on this study, an optimal strategy is proposed to decrease the number of infected cells. Finally, some numerical simulations are presented to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this research work, we suggest two mathematical models that take into account (i) two categories of target cells, CD 4 + T cells and macrophages, and (ii) two modes of infection transmissions, the direct virus-to-cell (VTC) method and cell-to-cell (CTC) infection transmission, where CTC is an effective method of spreading human immunodeficiency virus type-1 (HIV-1), as with the VTC method. The second model incorporates four time delays. In both models, the presence of a bounded and positive solution of the biological model is investigated. The existence conditions of all equilibria are established. The basic reproduction number R 0 that identifies a disease index is obtained. Lyapunov functions are utilized to verify the global stability of all equilibria. The theoretical findings are verified through numerical simulations. According to the outcomes, the trajectories of the solutions approach the infection-free equilibrium and infection-present equilibrium when R 0 ≤ 1 and R 0 > 1 , respectively. Further, we study the sensitivity analysis to investigate how the values of all the parameters of the suggested model affect R 0 for given data. We discuss the impact of the time delay on HIV-1 progression. We find that a longer time delay results in suppression of the HIV-1 infection and vice versa. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel multi-scale network model of two epidemics: HIV infection and opioid addiction. The HIV infection dynamics is modeled on a complex network. We determine the basic reproduction number of HIV infection, $ \mathcal{R}_{v} $, and the basic reproduction number of opioid addiction, $ \mathcal{R}_{u} $. We show that the model has a unique disease-free equilibrium which is locally asymptotically stable when both $ \mathcal{R}_{u} $ and $ \mathcal{R}_{v} $ are less than one. If $ \mathcal{R}_{u} > 1 $ or $ \mathcal{R}_{v} > 1 $, then the disease-free equilibrium is unstable and there exists a unique semi-trivial equilibrium corresponding to each disease. The unique opioid only equilibrium exist when the basic reproduction number of opioid addiction is greater than one and it is locally asymptotically stable when the invasion number of HIV infection, $ \mathcal{R}^{1}_{v_i} $ is less than one. Similarly, the unique HIV only equilibrium exist when the basic reproduction number of HIV is greater than one and it is locally asymptotically stable when the invasion number of opioid addiction, $ \mathcal{R}^{2}_{u_i} $ is less than one. Existence and stability of co-existence equilibria remains an open problem. We performed numerical simulations to better understand the impact of three epidemiologically important parameters that are at the intersection of two epidemics: $ q_v $ the likelihood of an opioid user being infected with HIV, $ q_u $ the likelihood of an HIV-infected individual becoming addicted to opioids, and $ \delta $ recovery from opioid addiction. Simulations suggest that as the recovery from opioid use increases, the prevalence of co-affected individuals, those who are addicted to opioids and are infected with HIV, increase significantly. We demonstrate that the dependence of the co-affected population on $ q_u $ and $ q_v $ are not monotone.

A model for assessing the impact of key population at risk in spreading HIV/ AIDS is designed. The model is shown to have a globally asymptotically stable (GAS) disease-free equilibrium whenever the associated reproduction number is less than unity. It has a unique GAS endemic equilibrium whenever reproduction number exceeds unity, if there is no back and forth movement in susceptible classes. It is shown that if infected individuals with high risk abandon their risky behavioral practices, then the disease burden will reduce. Furthermore, if infected individuals with low risk do not adopt risky behaviors then the transmission of the disease will decrease. Partial Rank Correlation Coefficient (PRCC) indices are assessed using all the infection-related classes and basic reproduction number of the model. Uncertainty and sensitivity analysis results show that controlling the effective contact rate, disease-induced death rate in AIDS stage with low and high risk classes will help controlling the transmission of the disease. [ABSTRACT FROM AUTHOR]

Objective: The Mediterranean island of Cyprus has a high rate of human transit as a result of tourism, human trafficking, and migration, which could contribute to the rate of HIV infection. The island is divided into 2 states - the northern portion is mainly populated by Turkish-Cypriots, and the southern portion is populated by Greek-Cypriots. The aim of this study was to assess the dynamics of HIV infection in both segments of Cyprus using a mathematical model. Materials and Methods: Data of HIV-positive individuals diagnosed during the period 1997-2018 in northern Cyprus were obtained from the Turkish Republic of Northern Cyprus Ministry of Health database, yielding a total of 129 cases, and data for the southern region, the Republic of Cyprus, were retrieved from a European Centre for Disease Prevention and Control (ECDC) 2013 report and ECDC/World Health Organization 2017 data. The total number of reported HIV infections was 1057. A mathematical model was used to evaluate the current and future HIV infection rate. Results: Stability analysis of the equilibrium point - disease-free or endemic - was conducted using the Lyapunov function. The basic reproduction number, represented as R0, is a measure of the potential for disease spread and serves as a threshold for stability. The R0 value was 0.83 in northern Cyprus and 0.040 in southern Cyprus. An R0 indicated a disease-free equilibrium. Conclusion: The calculations suggest that there is no current HIV epidemic on either part of the island; however, the model predicted a significant increase in the near future. [ABSTRACT FROM AUTHOR]

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we prove that the disease will be permanent but its impact on the population can be significantly minimized as the amount of education dissemination increases. In particular, under appropriate hypothesis on the model parameters, we establish that the size of the component of the infected population of the endemic equilibrium decreases linearly as a function of the amount of information disseminated. We also fit our model to a set of data on HIV/AIDS in order to estimate the infection, effective response, and information rates of the disease. We then use these estimates to present numerical simulations to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]