Stability of a fractional HIV/AIDS model

Transmission dynamics of symptom-dependent HIV/AIDS models

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

Studying of COVID-19 fractional model: Stability analysis

This article focuses on the recent epidemic caused by COVID-19 and takes into account several measures that have been taken by governments, including complete closure, media coverage, and attention to public hygiene. It is well known that mathematical models in epidemiology have helped determine the best strategies for disease control. This motivates us to construct a fractional mathematical model that includes quarantine categories as well as government sanctions. In this article, we prove the existence and uniqueness of positive bounded solutions for the suggested model. Also, we investigate the stability of the disease-free and endemic equilibriums by using the basic reproduction number (BRN). Moreover, we investigate the stability of the considering model in the sense of Ulam-Hyers criteria. To underpin and demonstrate this study, we provide a numerical simulation, whose results are consistent with the analysis presented in this article.

Optimal Solution of a Fractional HIV/AIDS Epidemic Mathematical Model

This article presents a fractional mathematical model of the human immunodeficiency virus (HIV)/AIDS spread with a fractional derivative of the Caputo type. The model includes five compartments corresponding to the variables describing the susceptible patients, HIV-infected patients, people with AIDS but not receiving antiretroviral treatment, patients being treated, and individuals who are immune to HIV infection by sexual contact. Moreover, it is assumed that the total population is constant. We construct an optimization technique supported by a class of basis functions, consisting of the generalized shifted Jacobi polynomials (GSJPs). The solution of the fractional HIV/AIDS epidemic model is approximated by means of GSJPs with coefficients and parameters in the matrix form. After calculating and combining the operational matrices with the Lagrange multipliers, we obtain the optimization method. The theorems on the existence, unique, and convergence results of the method are proved. Several illustrative examples show the performance of the proposed method. Mathematics Subject Classification: 97M60; 41A58; 92C42.

Caputo fractional-order SEIRP model for COVID-19 Pandemic

We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 pandemic. The newly proposed nonlinear fractional order model is an extension of a recently formulated integer-order COVID-19 mathematical model. Using basic concepts such as continuity and Banach fixed-point theorem, existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrix was used to compute the basic reproduction number R0, a number that determines the spread or otherwise of the disease into the general population. We also investigated the local asymptotic stability for the derived disease-free equilibrium point. Numerical simulation of the constructed epidemic model was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

Model-free based control of a HIV/AIDS prevention model

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.

A fractional order HIV-TB co-infection model in the presence of exogenous reinfection and recurrent TB

In this article, a novel fractional order model has been introduced in Caputo sense for HIV-TB co-infection in the presence of exogenous reinfection and recurrent TB along with the treatment for both HIV and TB. The main aim of considering the fractional order model is to incorporate the memory effect of both diseases. We have analyzed both sub-models separately with fractional order. The basic reproduction number, which measures the contagiousness of the disease, is determined. The HIV sub-model is shown to have a locally asymptotically stable disease-free equilibrium point when the corresponding reproduction number, R H , is less than unity, whereas, for R H > 1 , the endemic equilibrium point comes into existence. For the TB sub-model, the disease-free equilibrium point has been proved to be locally asymptotically stable for R T < 1 . The existence of TB endemic equilibrium points in the presence of reinfection and recurrent TB for R T < 1 justifies the existence of backward bifurcation under certain restrictions on the parameters. Further, we numerically simulate the fractional order model to verify the analytical results and highlight the role of fractional order in co-infection modeling. The fractional order derivative is shown to have a crucial role in determining the transmission dynamics of HIV-TB co-infection. It is concluded that the memory effect plays a significant role in reducing the infection prevalence of HIV-TB co-infection. An increment in the number of recovered individuals can also be observed when the memory effect is taken into consideration by introducing fractional order model.

Estimating the impact of antiretroviral therapy on HIV-TB co-infection: Optimal strategy prediction

In this paper, a nonlinear population model for HIV-TB co-infection has been proposed. The model is incorporated with the effect of early and late initiation of HIV treatment in co-infectives already on TB treatment, on the occurrence of Immune Reconstitution Inflammatory syndrome (IRIS). A 15-dimensional (15D) mathematical model has been developed in this study. We begin with considering constant treatment rates and thereafter, proceed to time-dependent treatment rates for co-infectives as control parameters. The basic reproduction number, a threshold quantity, corresponding to each HIV and TB sub-model has been computed in case of constant controls. With constant values of control parameters, mathematical analysis shows the existence and local stability of the disease-free equilibrium point and the endemic equilibrium point for the model. Together with time-dependent parameters, an optimal control problem is introduced and solved using Pontryagin’s maximum principle with an objective to minimize the number of infectives and disease induced deaths along with the cost of treatment. Numerical simulations are performed to examine the effect of reproduction numbers on control profiles and to identify, the ideal combination of treatment strategies which provides minimum burden on a society. Numerical results imply that if both HIV and TB are endemic in the population, then in order to bring in minimum burden from the co-infection, optimal control efforts must be enforced rather than constant treatment rate.

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.