Dynamical Analysis of  a Fractional Order HIV/AIDS Model

This article discusses a dynamical analysis of the fractional-order model of HIV/AIDS. Biologically, the rate of subpopulation growth also depends on all previous conditions/memory effects. The dependency of the growth of subpopulations on the past conditions is considered by applying fractional derivatives. The model is assumed to consist of susceptible, HIV infected, HIV infected with treatment, resistance, and AIDS. The fractional-order model of HIV/AIDS with Caputo fractional-order derivative operators is constructed and then, the dynamical analysis is performed to determine the equilibrium points, local stability and global stability of the equilibrium points. The dynamical analysis results show that the model has two equilibrium points, namely the disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable when the basic reproduction number is less than one. The endemic equilibrium point exists if the basic reproduction number is more than one and is globally asymptotically stable unconditionally. To illustrate the dynamical analysis, we perform some numerical simulation using the Predictor-Corrector method. Numerical simulation results support the analytical results.

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DYNAMICAL ANALYSIS OF FRACTIONAL ORDER ZIKV MODEL

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.5.13 ◽

2021 ◽

Vol 10 (5) ◽

pp. 2469-2481

Author(s):

N.A. Hidayati ◽

A. Suryanto ◽

W.M. Kusumawinahyu

Keyword(s):

Numerical Simulation ◽

Equilibrium Point ◽

Fractional Order ◽

Equilibrium Points ◽

Dynamical Analysis ◽

Stability Conditions ◽

Order Model ◽

Fractional Order Model ◽

The Stability ◽

S Model

The ZIKV model presented in this article is developed by modifying \cite{Bonyah2016}’s model. The classical order is changed into fractional order model. The equilibrium points of the model are determined and the stability conditions of each equilibrium point have been done using Routh-Hurwitz conditions. Numerical simulation is presented to verify the result of stability analysis result. Numerical simulation is also used to shows the effect of the order $\alpha$ to the stability of the model’s equilibrium point.

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THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS

Journal of Biological System ◽

10.1142/s0218339004001294 ◽

2004 ◽

Vol 12 (04) ◽

pp. 399-417 ◽

Cited By ~ 17

Author(s):

M. KGOSIMORE ◽

E. M. LUNGU

Keyword(s):

Equilibrium Point ◽

Reproduction Number ◽

Equilibrium Points ◽

Treatment Programs ◽

Asymptotically Stable ◽

Reproduction Numbers ◽

Threshold Conditions ◽

Existence And Stability ◽

Disease Free ◽

Hiv Aids

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α) are satisfied.

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Dynamics of COVID-19 Epidemic Model with Asymptomatic Infection, Quarantine, Protection and Vaccination

Communication in Biomathematical Sciences ◽

10.5614/cbms.2021.4.2.3 ◽

2021 ◽

Vol 4 (2) ◽

pp. 106-124

Author(s):

Raqqasyi Rahmatullah Musafir ◽

Agus Suryanto ◽

Isnani Darti

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Asymptomatic Infection ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Proposed Model ◽

Disease Free ◽

The Basic Reproduction Number

We discuss the dynamics of new COVID-19 epidemic model by considering asymptomatic infections and the policies such as quarantine, protection (adherence to health protocols), and vaccination. The proposed model contains nine subpopulations: susceptible (S), exposed (E), symptomatic infected (I), asymptomatic infected (A), recovered (R), death (D), protected (P), quarantined (Q), and vaccinated (V ). We first show the non-negativity and boundedness of solutions. The equilibrium points, basic reproduction number, and stability of equilibrium points, both locally and globally, are also investigated analytically. The proposed model has disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable if basic reproduction number is less than one. The endemic equilibrium point exists uniquely and is globally asymptotically stable if the basic reproduction number is greater than one. These properties have been confirmed by numerical simulations using the fourth order Runge-Kutta method. Numerical simulations show that the disease transmission rate of asymptomatic infection, quarantine rates, protection rate, and vaccination rates affect the basic reproduction number and hence also influence the stability of equilibrium points.

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A fractional-order love dynamical model with time delay for synergic couple : Stability analysis and Hopf bifurcation

10.22541/au.163671666.66187281/v1 ◽

2021 ◽

Author(s):

SANTOSHI PANIGRAHI ◽

Sunita Chand ◽

S Balamuralitharan

Keyword(s):

Stability Analysis ◽

Hopf Bifurcation ◽

Time Delay ◽

Fractional Order ◽

Reproduction Number ◽

Equilibrium Points ◽

Order Model ◽

Fractional Order Model ◽

The Stability ◽

The Impact

We investigate the fractional order love dynamic model with time delay for synergic couples in this manuscript. The quantitative analysis of the model has been done where the asymptotic stability of the equilibrium points of the model have been analyzed. Under the impact of time delay, the Hopf bifurcation analysis of the model has been done. The stability analysis of the model has been studied with the reproduction number less than or greater than 1. By using Laplace transformation, the analysis of the model has been done. The analysis shows that the fractional order model with a time delay can sufficiently improve the components and invigorate the outcomes for either stable or unstable criteria. In this model, all unstable cases are converted to stable cases under neighbourhood points. For all parameters, the reproduction ranges have been described. Finally, to illustrate our derived results numerical simulations have been carried out by using MATLAB. Under the theoretical outcomes from parameter estimation, the love dynamical system is verified.

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HIV/AIDS-Pneumonia Coinfection Model with Treatment at Each Infection Stage: Mathematical Analysis and Numerical Simulation

Journal of Applied Mathematics ◽

10.1155/2021/5444605 ◽

2021 ◽

Vol 2021 ◽

pp. 1-21

Author(s):

Shewafera Wondimagegnhu Teklu ◽

Temesgen Tibebu Mekonnen

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Transmission Rates ◽

Reproduction Numbers ◽

Disease Free ◽

The Basic Reproduction Number ◽

Hiv Aids

In the paper, we have considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R H and R P ) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R P and R H ) are greater than unity. The full HIV/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection model R H P is less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV/AIDS-pneumonia coinfection model is max 7 , 17 = 17 . Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV/AIDS and pneumonia transmission rates β 1 and β 2 , respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV/AIDS-only, pneumonia-only, and HIV/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.

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A FRACTIONAL ORDER HIV/AIDS MODEL USING CAPUTO-FABRIZIO OPERATOR

African Journal of Infectious Diseases ◽

10.21010/ajid.v15i2s.1 ◽

2021 ◽

Vol 15 (2s) ◽

pp. 1-18

Author(s):

Ebenezar Nkemjika Unaegbu ◽

Ifeanyi Sunday Onah ◽

Moses Oladotun Oyesanya

Keyword(s):

Equilibrium Point ◽

Fractional Order ◽

Numerical Solutions ◽

Fixed Point Theory ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Existence Of A Solution ◽

Aids Model ◽

Disease Free ◽

Hiv Aids

Background: HIV is a virus that is directed at destroying the human immune system thereby exposing the human body to the risk of been affected by other common illnesses and if it is not treated, it generates a more chronic illness called AIDS. Materials and Methods: In this paper, we employed the fixed-point theory in developing the uniqueness and existence of a solution of fractional order HIV/AIDS model having Caputo-Fabrizio operator. This approach adopted in this work is not conventional when solving biological models by fractional derivatives. Results: The results showed that the model has two equilibrium points namely, disease-free, and endemic equilibrium points, respectively. We showed conditions necessitating the existence of the endemic equilibrium point and showed that the disease-free equilibrium point is locally asymptotically stable. We also tested the stability of our solution using the iterative Laplace transform method on our model which was also shown stable agreeing with the disease-free equilibrium. Conclusions: Numerical simulations of our model showed clear comparison with our analytical results. The numerical solutions show that given fractional operator like the Caputo-Fabrizio operator, it is less noisy and plays a major role in making a precise decision and gives room (‘freedom’) to use data of specific patients as the model can be easily adjusted to accommodate this, as it a better fit for the patients’ data and provide meaningful predictions. Finally, the result showed the advantage of using fractional order derivative in the analysis of the dynamics of HIV/AIDS over the classical case.

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Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models

Axioms ◽

10.3390/axioms10040238 ◽

2021 ◽

Vol 10 (4) ◽

pp. 238

Author(s):

Ricardo Almeida ◽

Natália Martins ◽

Cristiana J. Silva

Keyword(s):

Global Stability ◽

Equilibrium Point ◽

Fractional Order ◽

Epidemiological Models ◽

Order Model ◽

Globally Asymptotically Stable ◽

Incidence Function ◽

Sirs Model ◽

Fractional Differential ◽

Disease Free

In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can be easily applied to study global asymptotic stability. After proving some auxiliary lemmas involving the Mittag–Leffler function, we present the main result of the paper. Under some assumptions, we prove that the disease-free equilibrium point of a fractional differential system is globally asymptotically stable. We then exemplify the procedure with some epidemiological models: a fractional-order SEIR model with classical incidence function, a fractional-order SIRS model with a general incidence function, and a fractional-order model for HIV/AIDS.

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On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

Applied Sciences ◽

10.3390/app10228296 ◽

2020 ◽

Vol 10 (22) ◽

pp. 8296 ◽

Cited By ~ 1

Author(s):

Malen Etxeberria-Etxaniz ◽

Santiago Alonso-Quesada ◽

Manuel De la Sen

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Disease Transmission ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Impulsive Vaccination ◽

Disease Free ◽

The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

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Dynamic Analysis of a Fractional-Order Model for Hepatitis B Virus with Holling II Functional Response

Complexity ◽

10.1155/2019/1097201 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Ruiqing Shi ◽

Ting Lu ◽

Cuihong Wang

Keyword(s):

Hepatitis B Virus ◽

Hepatitis B ◽

Fractional Order ◽

Reproduction Number ◽

Sufficient Conditions ◽

Order Model ◽

Fractional Order Model ◽

B Virus ◽

The Stability ◽

Holling Ii Functional Response

In this paper, a fractional-order model is constructed to describe the transmission of Hepatitis B Virus (HBV). Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the existence of two equilibriums are obtained. Thirdly, the stability of equilibriums are analyzed. After that, some numerical simulations are performed to verify the theoretical prediction. Finally, a brief discussion is presented.

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