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This paper presents a novel approach to designing a fixed-time fractional order observer for estimating the states of the dynamic model of human immunodeficiency virus (HIV) infection. The proposed approach combines output injection terminal sliding mode and RBF neural network strategies to achieve a robust and efficient estimation of the states of the HIV model within a fixed time frame. The main contributions of this work are the introduction of an output injection observer that ensures the stability of the error system along with a novel nonlinear sliding surface that guarantees the fixed-time error convergence to the neighborhood of zero. Moreover, the closed-loop scheme of the observer design is proven to be bounded, and the fixed-time stability of the observer error is obtained using the fractional Lyapunov stability approach. Simulation results show that the proposed fixed-time fractional order observer design provides accurate and efficient estimation of the states of the HIV model. • Fractional fixed time output injection state estimation is achieved for HIV dynamic model. • Fractional order weight update law is developed. • Fractional Lyapunov theory is applied to guarantee fixed time convergence of estimation error. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an HIV infection model with virus-to-cell infection, cell-to-cell infection and non-cytolytic cure. Two mean-reverting Ornstein–Uhlenbeck processes are also taken into account in the model. Firstly, it is proved that the stochastic model has a unique positive global solution. The model is found to have at least one stationary distribution by constructing suitable Lyapunov functions if the critical condition R 0 s > 1. Then, the probability density function near the quasi-positive equilibrium is obtained by solving the corresponding Fokker–Planck equation. The spectral radius method is used to derive the virus extinction under a sufficient condition R 0 e < 1. Finally, some numerical simulations are carried out. • A stochastic HIV model with non-cytolytic cure and Ornstein–Uhlenbeck process is built. • A novel method is proposed to prove the existence of stationary distribution. • The probability density function near the quasi-positive equilibrium is obtained. • The spectral radius method is used to derive the virus extinction. • Investigate the effect of the noise level and cure rate on virus dynamics. [ABSTRACT FROM AUTHOR]

Abstract: Why do more complex viruses (e.g., HIV, AIDS-virus and SARS coronavirus) tend to be more fatal? The paper concludes that the cell fractal geometry of viruses is the key. This paper also suggests two possible new approaches using nanotechnology and temperature to cure or prevent virus infection. [Copyright &y& Elsevier]

The "shock and kill" strategy is being widely explored to purge Human Immunodeficiency Virus (HIV) latent reservoirs. Romidepsin, a kind of latency-reversing agents (LRAs), has been shown to induce HIV RNA transcription. However, several clinical trials testing this drug have resulted in limited effect in reducing the HIV latent reservoirs. To understand the mechanisms underlying such limited effect, we develop a multi-scale model that incorporates pharmacokinetics and considers the toxicity of romidepsin to T cells in this paper. By fitting the model to the viral load data from plasma of six patients received romidepsin, we find that the model with T cell toxicity of romidepsin can well explain the clinical data. The dynamics of latently infected cells during romidepsin administration are explored using the best-fit parameter values. The results show that latently infected cells decrease very slowly and remain very stable overall in four of the six participants under the assumption of T cell toxicity of romidepsin. This implies that the ineffectiveness of romidepsin on latent reservoirs can be explained by its toxicity to T cells. In the remaining two participants, however, latently infected cells are quite stable without T cell toxicity of LRAs. It is found that the estimated activation rate of latently infected cells by romidepsin and the estimated elimination rate of romidepsin on immune cells for these two patients are very different from those for the other four patients. Thus we speculate that the heterogeneous response to romidepsin across participants may also be a determining factor of the effectiveness of romidepsin. These results may have significant implications in the search for the control of HIV infection. • An HIV model is proposed to understand the cause of ineffectiveness of romidepsin in the "shock and kill" strategy. • The model has a good fit to patients' data when T cell toxicity of romidepsin is considered. • The toxicity of romidepsin to T cell may contribute to the ineffectiveness of "shock and kill" strategy concerning romidepsin. [ABSTRACT FROM AUTHOR]

Mathematical modeling of biological systems is an interesting research topic that attracted the attention of many researchers. One of the main goals in this area is the design of mathematical models that more accurately illustrate the characteristics of the real-world phenomena. Among the existing research projects, modeling of immune systems has given a growing attention due to its natural capabilities in identifying and destroying abnormal cells. The main objective of this paper is to investigate the pathological behavior of HIV-infection using a new model in fractional calculus. The proposed model is examined through three different operators of fractional derivatives. An efficient numerical method is also presented to solve these fractional models effectively. In fact, we believe that the new models presented on the basis of these three operators show various asymptomatic behaviors that do not appear during the modeling with the integer-order derivatives. Therefore, the fractional calculus provides more precise models of biological systems that help us to make more realistic judgments about their complex dynamics. Finally, simulations results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

• In this paper, a delayed HIV-TB co-infection model has been formulated. • The existence and stability of equilibrium points have been shown. • The condition under which Hopf-bifurcation takes place has been determined. • Ideal timing of the combination of treatment for both diseases has been computed. In this paper, a non-linear mathematical model for HIV-TB co-infection has been proposed. A constant time delay parameter is introduced into the model which accounts for the delay in detection and the commencement of appropriate treatment for co-infected individuals. The basic reproduction number along with the location and existence of equilibrium points is computed. Local stability analysis of all the equilibrium points in the presence of delay parameter is performed, which gives certain conditions, under which Hopf-bifurcation takes place. Numerical simulations are performed with different set of parameters for each equilibrium point to explore the effect of delay in detection and treatment of the infected individuals. The numerical results have also been used to describe the rich dynamics generated by the system in the presence of delay. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic HIV/AIDS model with standard incidence rate to investigate the impact of random fluctuations on the HIV viral dynamics. Firstly, the sufficient conditions for the stochastic extinction of the disease are obtained. Secondly, after defining the threshold parameter R s , we prove that the model has a unique ergodic stationary distribution when R s > 1 , which reveals that the disease will persist almost surely. Subsequently, we give the exact expression of the probability density function around a quasi-equilibrium point by applying four-dimensional Fokker–Planck equation. Finally, through the numerical simulations of several examples, we not only verify the correctness of the theoretical results, but also explore how the disease goes for the case of not giving clear results theoretically. Epidemiologically, we find that increasing the noise intensities may be conducive to control the spread of the HIV. • A stochastic SICA model for HIV/AIDS transmission is developed. • The sufficient conditions for the stochastic extinction of the disease is established. • The existence of a unique ergodic stationary distribution of the model is displayed. • The probability density function of the stationary distribution is given. • The noise intensity may be conducive to control the spread of the HIV. [ABSTRACT FROM AUTHOR]

Type 1 diabetes (T1D), previously known as juvenile diabetes or insulin-dependent diabetes, is an autoimmune disease characterized by the insufficient (or lack of) production of insulin by the pancreas. Insulin is crucial to maintain blood sugar at healthy levels. High blood sugar damages the body and causes a variety of symptoms, ranging from severe thirst, fatigue, to urinary infections. The cells responsible for the production of insulin are the β -cells. In T1D, these are killed by an abnormal response of the immune system. Specific clones of cytotoxic T-cells invade the pancreatic islets of Langerhans, and eliminate them. T1D diabetes may develop in human immunodeficiency virus (HIV)-infected patients, though in rare situations. In this paper, we propose a cell model for the development of T1D in these patients, after immune restoration, during highly active antiretroviral therapy (HAART). The study includes the derivation of the qualitative properties of the model, and its comprehensive investigation via path-following methods, using the continuation platform COCO. In this way, the main theoretical predictions are verified in detail. Furthermore, this numerical part establishes accurate parameter thresholds to ensure an effective disease treatment in HIV-infected persons to prevent the development of T1D. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling. By analyzing the Lyapunov exponent, singular boundary and probability density, some new criteria ensuring stochastic stability, D-bifurcation and P-bifurcation for stochastic internal HIV model are obtained, respectively. Numerical simulation results are given to support the theoretical predictions. [Copyright &y& Elsevier]

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD 4 + T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host's blood is considered a separate class. It is reported that when HIV enters a human's body, it infects a large amount of CD 4 + T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD 4 + T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2 × 2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems. [ABSTRACT FROM AUTHOR]

In this paper, we establish a stochastic differential equation model of the HIV/AIDS infection including four classes: susceptible individuals, undiagnosed HIV infected, diagnosed HIV infected and AIDS infected individuals. For the deterministic model, we obtain that the basic reproduction number R 0 can determine whether there is a unique disease-free equilibrium or an endemic equilibrium, and show the solution of the model is positive and global. The mathematical analysis of the stochastic model (SDE) corresponding to the deterministic model (ODE) is investigated to determine the conditions for the stability of the random equilibria. The reproduction number R 0 s of the SDE model is derived, and by applying the Markov semigroups, it shows that the threshold parameter R 0 s can be used to control the stochastic dynamics of the model. What's more, by solving the Fokker-Planck equation, we derive an expression of the probability density function of the SDE model around the quasi-endemic equilibrium of the ODE model under a certain condition. A series of numerical simulations are performed to illustrate the analytical results. Our study demonstrates that (i) reducing the recruitment rate of susceptible population and increasing diagnosis rate are the effective methods for controlling the HIV/AIDS disease; and (ii) large environmental noise intensity and small valid disease contact rate can suppress the outbreak of disease. [ABSTRACT FROM AUTHOR]

In this paper, a combined drug treatment for the Human Immunodeficiency Virus (HIV) infection is studied using fractional order differential equations. The fractional order model is considered in the governing equations to account for the memory property on the infection dynamics. The model possesses two different Anti Retro Viral (ARV) drug therapies such as Reverse Transcriptase Inhibitor (RTI) and Protease Inhibitor (PI) that influence the HIV infection spread rate. Spectral collocation method is applied to study the model computationally. Here, different scenarios of treatment regimes for HIV are analysed for three different cases such as (i) No drug therapy is administered; (ii) Single therapy is administered (either RTI and PI) and (iii) the combined therapy (both RTI and PI) is administered. It is found that single PI therapy eradicates virus particles at a higher rate compared to single RTI therapy. In the case of combined therapy, when higher dosage of PI with moderate dosage of RTI is administered, the virus particles are cleared at a higher rate compared to the reversed dosage. The detailed computational analysis and the results obtained are presented in the form of tables and figures. [ABSTRACT FROM AUTHOR]

• A CTL-inclusive HIV-1 infection model with both cell-to-cell and virus-to-cell transmissions and three time delays is designed. • The complete asymptotic behavior of the model is established, using Lyapunov-LaSalle techniques. • An in-depth bifurcation analysis is performed, where the local stability and Hopf bifurcation of the endemic equilibrium in the general case with all delays being positive are established. • An increase of the immune delay can stabilize and/or destabilize the endemic equilibrium. In this paper, we assess the viral dynamics of a HIV-1 mathematical model that incorporates virus-to-cell and cell-to-cell transmissions, CTL response immune and three delays describing intracellular delays and immune response delay. The model includes a constant production rate of CTLs export from thymus. Our focus is on the effect of three delays on the infection dynamics. The basic reproduction number, R 0 , which depends on the intracellular delay, is shown to determine stability conditions of the model steady states. Employing a suitable Lyapunov function, we show that the infection-free equilibrium is globally asymptotically stable including the three time delays when the basic reproduction number is a smaller amount than one. Additionally, for the special case where there is no immune delay in the model, the global stability of the unique endemic equilibrium is also established (using suitable Lyapunov functions) whenever R 0 is larger than one. Furthermore, we study the local and hopf bifurcation of the endemic equilibrium in general case with all delays being positive. When the three delays are positive, we determine some conditions for stability switches of the endemic equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations indicate that an increase of the immune delay can stabilize and/or destabilize the endemic equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, a new HIV CD4+T cells reaction-diffusion model in two dimensions has been introduced. Two novel and efficient positivity preserving finite difference schemes for the numerical solution has been used. The positivity property is of great importance in epidemic models because negative values have no meaning. The stability and consistency of the proposed positivity preserving schemes have been discussed briefly. A comparison of the proposed schemes with an extensively used Euler scheme has been provided. The numerical simulations of all the schemes have been presented with the help of a numerical test and found that the Euler method shows false behavior which is not a part of the continuous system. Moreover, both proposed positivity preserving schemes illustrate the behavior which is consistent with the continuous system. [ABSTRACT FROM AUTHOR]

This paper investigates the exact traveling wave solutions of the fractional model of the human immunodeficiency virus (HIV-1) infection for CD4 + T-cells. This model also treats with the effect of antiviral drug therapy. These solutions calculate both the boundary and initial conditions that allow employing the septic-B-spline scheme which is one of the most recent schemes in the numerical field. We use the obtained computational solutions via the modified Khater, the extended simplest equation, and sech–tanh methods through Atangana-Baleanu derivative operator. The comparison between the exact and numerical evaluated solutions is illustrated by some distinct sketches. The functioning of our numerical method is tested under three computational obtained solutions. [ABSTRACT FROM AUTHOR]

• The epicenter of COVID-19 keeps shifting to the south. • Coexisting HIV & COVID-19 is a worry for southern hemisphere countries. • Stability analysis is provided for a HIV-COVID-19 simple model. • Existence of backward and forward bifurcations is shown. • Predicted prevalence from a generalized COVID-19 model is given. Were southern hemisphere countries right to undertake national lockdown during their summer time? Were they right to blindly follow the self-isolation wave that hit European countries in full winter? As a southern hemisphere country like South Africa stands now as the most COVID-19 and HIV affected country in Africa, we use in this paper, recent COVID-19 data to provide a statistical and comparative analysis that may alert southern hemisphere countries entering the winter season. After that, we use a generalized simple mathematical model of HIV-COVID-19 together with graphs, curves and tables to compare the pandemic situation in countries that were once the epicenter of the disease, such as China, Italy, Spain, United Kingdom (UK) and United States of America (USA). We perform stability and bifurcation analysis and show that the model contains a forward and a backward bifurcation under certain conditions. We also study different scenarios of stability/unstability equilibria for the model. The fractional (generalized) COVID-19 model is solved numerically and a predicted prevalence for the COVID-19 is provided. Recall that Brazil and South Africa share number of similar social features like Favellas (Brazil) and Townships (South Africa) with issues like promiscuity, poverty, and where social distanciation is almost impossible to observe. We can now ask the following question: Knowing its HIV situation, is South Africa the next epicenter in weeks to come when winter conditions, proven to be favorable to the spread of the new coronavirus are comfily installed? [ABSTRACT FROM AUTHOR]

• We developed a stochastic HIV model that includes latent reservoir. • We performed stochastic analysis and simulations of the model. • We considered Pharmacokinetic effects of taking different drug therapies. • Locking in latency therapy cannot eradicate the latent reservoir. • Lytic Therapy combined with drugs treatment may eradicate the virus. In this paper, we propose a stochastic HIV model with latency to describe the factors that are responsible for extinction and persistence of the infection. This model incorporates latent period of reservoirs that inhibits the eradication of the virus. It is shown that this model leads naturally to a stochastic system of differential delay equations. We derived the global existence, positivity and boundedness properties of solutions of the proposed stochastic system. Analyzing the model, sufficient conditions for the eradication and the persistence of the disease are also obtained. Furthermore, numerical simulations are performed to study the impact of several strategies of drug therapy on the persistence and the extinction of HIV virus. Our results constitute an important step towards control strategies to reduce HIV infection. [ABSTRACT FROM AUTHOR]

• Approximate analytical solution of basic HIV viral dynamic model is obtained. • HAM is applied to describe the HIV viral dynamic model. • HAM presented in the article shows better efficiency and accuracy on the obtained results. • The convergence theorem of HAM to the HIV viral dynamic model is proved. • The solution is improved by applying the least squares method. Viruses have different mechanisms in causing a disease in an organism, which largely depend on the viral species. The recent advancement, through coupling data analysis and mathematical modeling, has allowed the identification and characterization of the nature of the virus. In the present paper, the homotopy analysis method is applied to provide an approximate solution of the basic HIV viral dynamic model describing the viral dynamics in a susceptible population. The proposed method allows for the solution of the governing system of differential equations to be calculated in the form of an infinite series with components which can be easily calculated. The homotopy analysis method utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter. By using the homotopy series solutions, firstly, several β − curves using an appropriate ratio are plotted to demonstrate the regions of convergence and the optimum value of ℏ, then the residual and absolute errors are obtained for different values of these regions. Secondly, the residual error functions are applied to show the accuracy of the applied homotopy analysis method. Also, the convergence theorem of homotopy analysis method for the HIV viral dynamic model is proved. Mathematica software is used for the calculations and numerical results. The results obtained show the effectiveness and strength of the homotopy analysis method. [ABSTRACT FROM AUTHOR]