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

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.

scholarly journals Analisis Kestabilan Model SEIR Penyebaran Penyakit Campak dengan Pengaruh Imunisasi dan Vaksin MR

2019 ◽  
Vol 16 (1) ◽  
pp. 107
Author(s):  
Willyam Daniel Sihotang ◽  
Ceria Clara Simbolon ◽  
July Hartiny ◽  
Desrinawati Tindaon ◽  
Lasker Pangarapan Sinaga
Keyword(s):  
Infectious Disease ◽  
Equilibrium Point ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Seir Model ◽  
Rate Of Spread ◽  
The Stability ◽  
Disease Free

Measles is a contagious infectious disease caused by a virus and has the potential to cause an outbreak. Immunization and vaccination are carried out as an effort to prevent the spread of measles. This study aims to analyze and determine the stability of the SEIR model on the spread of measles with the influence of immunization and MR vaccines. The results obtained from model analysis, namely there are two disease free and endemic equilibrium points. If the conditions are met, the measles-free equilibrium point will be asymptotically stable and the measles endemic equilibrium point will be stable. Numerical solutions show a decrease in the rate of spread of measles due to the effect of immunization and the addition of MR vaccines.

scholarly journals Forecast analysis of the epidemics trend of COVID-19 in the United States by a generalized fractional-order SEIR model

2020 ◽  
Author(s):  
Conghui Xu ◽  
Yongguang Yu ◽  
YangQuan Chen ◽  
Zhenzhen Lu
Keyword(s):  
United States ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Equilibrium Points ◽  
The United States ◽  
Endemic Equilibrium ◽  
Qualitative Properties ◽  
Prediction Ability ◽  
Seir Model ◽  
Disease Free

AbstractIn this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like COVID-19 and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number R0 is derived. When R0 < 1, the disease-free equilibrium point is unique and locally asymptotically stable. When R0 > 1, the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asymptotic stability of disease-free and endemic equilibrium points. The trend of COVID-19 spread in the United States is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model. According to the real data of the United States, it is found that our improved model has a better prediction ability for the epidemic trend in the next two weeks. Hence, the epidemic trend of the United States in the next two weeks is investigated, and the peak of isolated cases are predicted. The modified SEIQRP model successfully capture the development process of COVID-19, which provides an important reference for understanding the trend of the outbreak.

scholarly journals Dynamical Analysis of a Fractional Order HIV/AIDS Model

2021 ◽  
Vol 5 (1) ◽  
pp. 14
Author(s):  
Septiangga Van Nyek Perdana Putra ◽  
Agus Suryanto ◽  
Nur Shofianah
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
Order Model ◽  
Globally Asymptotically Stable ◽  
Fractional Order Model ◽  
Disease Free ◽  
Hiv Aids

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.

scholarly journals 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

2020 ◽  
Vol 10 (22) ◽  
pp. 8296 ◽  
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.

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

2004 ◽  
Vol 12 (04) ◽  
pp. 399-417 ◽  
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.

scholarly journals Analisis Model Matematika Penyebaran Demam Berdarah Dengue dengan Fungsi Lyapunov

2019 ◽  
Vol 1 (2) ◽  
pp. 125
Author(s):  
Syafruddin Side ◽  
Ahmad Zaki ◽  
Nurwahidah Sari
Keyword(s):  
Lyapunov Function ◽  
Equilibrium Point ◽  
Dengue Fever ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Secondary Data ◽  
Hemorrhagic Fever ◽  
Endemic Equilibrium ◽  
Sirs Model ◽  
Disease Free

Abstrak. Artike lini adalah penelitian teori dan terapan. Artikelini bertujuan untuk membahas mengenai model matematika SIRS untuk penyebaran Demam Berdarah Dengue. Data yang digunakanadalah data sekunder jumlah penderita penyakit Demam Berdarah Dengue dari Side pada tahun 2014. Pembahasan di mulai dari membangun model matematika SIRS penyakit Demam Berdarah Dengue, menentukan eksistensi model SIRS menggunakan fungsi Lyapunov, penentuan titik ekuilibrium, kemudian mencari analisis kestabilan titik ekuilibrium menggunakan fungsi Lyapunov, menentukan nilai bilangan reproduksi dasar , membuat simulasi model, dan menginterpretasikannya. Dalam artikel ini diperoleh model matematika SIRS untuk penyakit Demam Berdarah Dengue, eksistensi model SIRS, dua titik ekuilibrium bebas penyakit dan endemik dari model SIRS, kestabilan global keseimbangan bebas penyakit dan endemik dari model SIRS dengan nilai bilangan reproduksi dasar , ini menunjukkan bahwa penyakit Demam Berdarah Dengue berstatus epidemik.Kata Kunci: Model Matematika, Penyebaran Penyakit, Demam Berdarah Dengue, Model  SIRS, Fungsi LyapunovAbstract. This paper is theorethical and applied research. This paper aims to discus about SIRS mathematical models for the spread of dengue fever. The data used is a secondary data about the number of people with dengue fever disease from Side (2014). The discussion start from constructing SIRS models of dengue fever disease, determining the existence of SIRS models using Lyapunov function, determining equilibrium point, then looking for stability analysis of equilibrium point using Lyapunov function, determining reproduction number , making models simulation, and interpreting it. In this paper, we obtained mathemathical models of SIRS for dengue fever disease, existence of SIRS models, disease-free and endemic equilibrium points of SIRS models, global stability of disease-free and endemic equilibrium of SIRS models with basic reproduction number , it shows that dengue fever disease is epidemic status. , This shows that Dengue Hemorrhagic Fever is an epidemic.Keyword: Mathematical Model, Spread of Disease, Dengue Fever, SIRS Model, Lyapunov Function

scholarly journals Model matematika penyebaran COVID-19 dengan penggunaan masker kesehatan dan karantina

2021 ◽  
Vol 2 (2) ◽  
pp. 68-79
Author(s):  
Muhammad Manaqib ◽  
Irma Fauziah ◽  
Eti Hartati
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Reproduction Numbers ◽  
Geometric Picture ◽  
Disease Free ◽  
The Basic Reproduction Number

This study developed a model for the spread of COVID-19 disease using the SIR model which was added by a health mask and quarantine for infected individuals. The population is divided into six subpopulations, namely the subpopulation susceptible without a health mask, susceptible using a health mask, infected without using a health mask, infected using a health mask, quarantine for infected individuals, and the subpopulation to recover. The results obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point, and the basic reproduction number (R0). The existence of a disease-free equilibrium point is unconditional, whereas an endemic equilibrium point exists if the basic reproduction number is more than one. Stability analysis of the local asymptotically stable disease-free equilibrium point when the basic reproduction number is less than one. Furthermore, numerical simulations are carried out to provide a geometric picture related to the results that have been analyzed. The results of numerical simulations support the results of the analysis obtained. Finally, the sensitivity analysis of the basic reproduction numbers carried out obtained four parameters that dominantly affect the basic reproduction number, namely the rate of contact of susceptible individuals with infection, the rate of health mask use, the rate of health mask release, and the rate of quarantine for infected individuals.

Stability analysis and numerical solutions of fractional order HIV/AIDS model

2019 ◽  
Vol 122 ◽  
pp. 119-128 ◽  
Author(s):  
Aziz Khan ◽  
J.F. Gómez-Aguilar ◽  
Tahir Saeed Khan ◽  
Hasib Khan
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Aids Model ◽  
Hiv Aids

scholarly journals A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi
Keyword(s):  
Fractional Order ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Point Theory ◽  
Euler Method ◽  
Transmission Model ◽  
Order Model ◽  
Influenza Transmission ◽  
The Stability ◽  
Disease Free

Abstract We study the SEIR epidemic model for the spread of AH1N1 influenza using the Caputo–Fabrizio fractional-order derivative. The reproduction number of system and equilibrium points are calculated, and the stability of the disease-free equilibrium point is investigated. We prove the existence of solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. In the numerical section, we present a simulation to examine the system, in which we calculate equilibrium points of the system and examine the behavior of the resulting functions at the equilibrium points. By calculating the results of the model for different fractional order, we examine the effect of the derivative order on the behavior of the resulting functions and obtained numerical values. We also calculate the results of the integer-order model and examine their differences with the results of the fractional-order model.

scholarly journals Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
F. Talay Akyildiz ◽  
Fehaid Salem Alshammari
Keyword(s):  
Fractional Order ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Exponential Kernel ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractThis paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_{0} > 1$ R 0 > 1 ; a disease-free equilibrium $E_{0}$ E 0 and a disease endemic equilibrium $E_{1}$ E 1 . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_{0} <1$ R 0 < 1 , we show that the endemic equilibrium state is locally asymptotically stable if $R_{0} > 1$ R 0 > 1 . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

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