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In the last decade, hundreds of papers have been published on fractional modeling. The approach is often similar, with the motivation of incorporating a "memory effect": take the integer-order differential-equation model, and replace the ordinary derivative to the left-hand side by a fractional operator/derivative. Does this make sense? Which is the memory there? What is the physical and biological interpretation of fractional models? This paper aims at investigating these important issues. With fluxes, transitions, and concepts from survival analysis (hazard function), we give a meaning for Caputo compartmental models and their non-Markovian property. Essentially, a fractional index is connected with the instantaneous risk of leaving a compartment, conditioned on the past stay in there. We distinguish between Caputo models that are purely fractional (i.e., the usual ones) and partially fractional (i.e., a mix of ordinary and Riemann–Liouville components), and we explore initialization as well. We build new Euler-type numerical schemes, rooted in probability, and assess them for different models: a simple death process, logistic growth, SIR equations, the dynamics of an HIV infection, and the evolution of the smoking habit. A detailed discussion on the pros and cons of the fractional methodology is made along the article. • Which is the role of a fractional operator in an equation? What is memory? • The paper investigates these issues, for Caputo fractional compartmental models. • We analyze the hazard function (risk of transition), initialization, non-Markovianity. • Some Caputo models in the literature are unphysical. Critical evaluation is included. • Numerical experiments are conducted with probabilistic Euler-type discrete schemes. [ABSTRACT FROM AUTHOR]

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]

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]

• A stochastic staged progression AIDS model with staged treatment and second-order perturbation is studied. • The existence and uniqueness of the ergodic stationary distribution is obtained under the condition of. • If, we obtain that the AIDS epidemic will go to extinction in long-term. Focusing on deterministic AIDS model proposed by Hyman (2000) and the detailed data from the World Health Organization (WHO), there are three stages of AIDS process which are described as Acute infection period, Asymptomatic phase and AIDS stage. Our paper is therefore concerned with a stochastic staged progression AIDS model with staged treatment. In view of the complexity of random disturbances, we reasonably take second-order perturbation into consideration for realistic sense. By means of our creative transformation technique and stochastic Lyapunov method, a critical value R 0 H > 1 is firstly obtained for the existence and uniqueness of ergodic stationary distribution to the stochastic system. Not only does it respectively reveal the corresponding dynamical effects of the linear and second-order perturbations to the model, but the unified form of second-order and linear fluctuations is derived. Next, some sufficient conditions about extinction of stochastic system are established in view of the basic reproduction number R 0. Finally, some examples and numerical simulations are introduced to illustrate our analytical results. In addition, some advantages of our new method and theory are highlighted by comparison with other existing results at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the global dynamics of an HIV infection model, which incorporates a logistic mitosis term for the uninfected target cells and direct cell-to-cell transmission. It is assumed that beside the diffusion-limited cell-free virus, the infection directly occurs from infected cells to healthy cells. The existence of Hopf bifurcation are discussed by identifying the recruitment rate and mitosis rate of CD4 + T cells as bifurcation parameter. In the case where taking intrinsic mitosis as bifurcation parameter, the direction, stability and period of Hopf bifurcation are investigated by computing normal form. The effect of cell-to-cell transmission rate on viral dynamics are also addressed. Some numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a delayed differential equation model that describes HIV infection of CD4+ T cells is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a delayed mathematical model for the interactions of HIV infection and cells. We first investigate the existence and stability of the Equilibria. We then study the effect of the time delay on the stability of the infected equilibrium. Criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Moreover, by applying Nyquist criterion, the length of delay is estimated for which stability continues to hold. Finally by using a delay as a bifurcation parameter, the existence of Hopf bifurcation is also investigated. Numerical simulations are presented to illustrate the analytical results. [Copyright &y& Elsevier]

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 stochastic HIV model with latent infection and cell-to-cell transmission is built. • The unique positive solution and global solution are rigorously analyzed. • A novel method is proposed to prove the existence of stationary distribution. • Investigate the effect of the noise intensity on virus dynamics. • Examine the relative contribution from the two modes of viral infections. During the HIV infection process in vitro cell cultures, recent experimental and mathematical investigations have revealed that there are two fundamentally distinct viral transmission modes: cell-free infection and cell-to-cell transmission. A stochastic HIV model with latent infection, cell-free infection and cell-to-cell transmission is formulated in this paper. The existence of the unique ergodic stationary distribution is established by constructing appropriate Lyapunov function. Moreover, virus can be eradicated under certain sufficient conditions. Numerical simulations are carried out to investigate three aspects: (i) the effect of stochastic perturbation; (ii) the influence of latency; (iii) the relative contributions of cell-to-cell and cell-free infections. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an adaptive immunity HIV infection model. The model describes the interaction between healthy CD 4 + T cells, silent (latent) infected cells, active infected cells, free HIV particles, Cytotoxic T lymphocytes (CTLs) and antibodies. The healthy CD 4 + T cells can be infected when they are contacted by one of the following: (i) free HIV particles, and this is known as virus-to-cell (VTC) transmission (ii) silent infected cells, and we call this mode of infection as silent HIV-infected cell-to-cell (CTC) transmission, and (iii) active infected cells, and we call this mechanism active HIV-infected CTC transmission. The incidence rates of the healthy CD 4 + T cells with free HIV particles, silent infected cells, and active infected cells are given by general functions. Moreover, the production/proliferation and removal/death rates of all compartments are represented by general functions. The model is an improvement of the existing HIV infection models which have neglected the incidence between the silent infected cells and healthy CD 4 + T cells. We first show that the model is well-posed. Then, we show that the model has five equilibria and their existence are governed by five threshold parameters. Under a set of conditions on the general functions and the threshold parameters, we have proven the global asymptotic stability of all equilibria by using Lyapunov's method. We have illustrated the theoretical results via numerical simulations. We have studied the effect of CTC transmission on the dynamical behavior of the system. We have shown that inclusion of CTC transmission decreases the concentration of the healthy CD4 + T cells and increases the concentrations of the infected cells and free HIV particles. [ABSTRACT FROM AUTHOR]

In this paper we consider fractional order model to HIV infection whose components are plasma densities of uninfected C D 4 + T-cells, the infected such cells and the free virus. The main purpose of this work is to study the dynamic of fractional HIV infection of C D 4 + T-cells with the impact of drug treatment. An iterative scheme based on the discretization of the domain and short memory principle is proposed to solve fractional HIV infection model numerically. The main reason for using this discretization technique is low computational cost and high accuracy compared to some other methods. Numerical results are presented graphically. Figures are used to show the behaviour of densities of uninfected and infected cells before and after the drug therapy. It is also shown that how the approximate solution varies from fractional order to integer order derivative. By listing CPU time in tabular form efficiency of the iterative scheme is shown. The proposed scheme is also effective in a long time period. The validation of results are shown by comparing results among the proposed technique, Runge Kutta Fourth order method, homotopy perturbation method, homotopy analysis method, Adam-Bashforth predictor-corrector method, Legendre wavelets operational matrix method and Laplace adomian-decomposition method. [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]