Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment

Kheiri, Hossein; Jafari, Mohsen A.

doi:10.1016/j.cam.2018.06.055

Cited by 75 publications

(35 citation statements)

References 32 publications

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“…At the end of 2016, approximately 36.7 million people were living with HIV, and approximately 1.8 million new infections were occurring globally each year [2]. According to UNAIDS data reported in June 2016 [3], around 18.2 million people living with HIV were receiving antiretroviral therapy (ART). The US Center for Disease control and Prevention (CDC) [4] reported in 2017 that without treatment of HIV/AIDS with antiretroviral medicine, HIV infection advances through several stages and individuals with AIDS typically survive about 3 years.…”

Section: Introductionmentioning

confidence: 99%

“…Although antiretroviral therapies have been successful in decreasing the mortality rate in some regions, it is still necessary to increase antiretroviral therapies in other regions. Many HIV/AIDS epidemic models have been proposed to predict and control the spread of the disease (see, e.g., [3,[8][9][10][11] and the references cited therein).…”

Section: Introductionmentioning

confidence: 99%

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A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment

2019

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In recent years, many new definitions of fractional derivatives have been proposed and used to develop mathematical models for a wide variety of real-world systems containing memory, history, or nonlocal effects. The main purpose of the present paper is to develop and analyze a Caputo-Fabrizio fractional derivative model for the HIV/AIDS epidemic which includes an antiretroviral treatment compartment. The existence and uniqueness of the system of solutions of the model are established using a fixed-point theorem and an iterative method. The model is shown to have a disease-free and an endemic equilibrium point. Conditions are derived for the existence of the endemic equilibrium point and for the local asymptotic stability of the disease-free equilibrium point. The results confirm that the disease-free equilibrium point becomes increasingly stable as the fractional order is reduced. Numerical simulations are carried out using a three-step Adams-Bashforth predictor method for a range of fractional orders to illustrate the effects of varying the fractional order and to support the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment

2019

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“…The biological models for the study of HIV, which is highly valued for the AIDS, have been extensively studied in the last two decades using the fractional order derivatives . It has been studied that the virus‐infected cells of the immune system destroy or impair their function; the immune system gets weaker, and the person is further disposed to infections.…”

Section: Introductionmentioning

confidence: 99%

“…3,4 The biological models for the study of HIV, which is highly valued for the AIDS, have been extensively studied in the last two decades using the fractional order derivatives. 5,6 It has been studied that the virus-infected cells of the immune system destroy or impair their function; the immune system gets weaker, and the person is further disposed to infections. HIV is transmitted by body fluids such as blood, semen, seminal fluid, rectal fluid, and breast milk; therefore, it is largely spread by a subject through unsafe sex, sharing of needles or syringes and, less commonly, by blood transfusion or from a mother to child during pregnancy or breastfeeding.…”

mentioning

confidence: 99%

A fractional order HIV‐TB coinfection model with nonsingular Mittag‐Leffler Law

Khan

Gómez‐Aguilar

Alkhazzan

et al. 2020

Math Methods in App Sciences

119

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The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV‐related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV‐TB coinfected model in the Atangana‐Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers‐Ulam (HU) stability and numerical simulations with assumption of specific parameters.

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“…Because of various applications of fractional differential equations in the field of science and engineering, they have significantly grown in recent years . Some works have been done on analytical solution of these equations .…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of variable fractional order advection‐dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative

et al. 2020

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In this article, the Bernoulli wavelet method is used to solve the space‐time variable fractional order advection‐dispersion equation. The equation contains Coimbra time fractional derivatives with variable order of γ1false(xfalse) as well as the Riemann‐Liouville space fractional derivatives with variable orders of γ2false(x,tfalse) and γ3false(x,tfalse). In fact, first, using the new operational matrices, we study the relationship between Bernoulli wavelets and piecewise functions. Then, according to the properties of piecewise functions and computing operational matrices of their fractional derivatives, we obtain operational matrices of the Bernoulli wavelet fractional derivatives. Using new operational matrices furnished from Caputo and Riemann‐Liouville and also suitable collocation points, the advection‐dispersion equation would be converted to a system of algebraic equations. Then, we would solve the equation numerically by utilizing a common method. Finally, the upper bound of the errors of the defined operational matrices and convergence analysis of the proposed method would be discussed. We would also reveal high accuracy of the method using some numerical samples.

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Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment

Cited by 75 publications

References 32 publications

A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment

A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment

A fractional order HIV‐TB coinfection model with nonsingular Mittag‐Leffler Law

Numerical solution of variable fractional order advection‐dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative

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