scholarly journals Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Xiao-Hong Zhang ◽  
Aatif Ali ◽  
Muhammad Altaf Khan ◽  
Mohammad Y. Alshahrani ◽  
Taseer Muhammad ◽  
...  
Keyword(s):  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Interpolation Polynomial ◽  
Disease Spread ◽  
Bacterial Disease ◽  
Estimation Of Parameters ◽  
Treatment Rate ◽  
Incidence Data ◽  
Numerical Result

In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction number ℛ 0 < 1 . Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.

Dynamical Analysis of a Fractional-Order Hantavirus Infection Model

2020 ◽  
Vol 21 (2) ◽  
pp. 171-181 ◽  
Author(s):  
Mahmoud Moustafa ◽  
Mohd Hafiz Mohd ◽  
Ahmad Izani Ismail ◽  
Farah Aini Abdullah
Keyword(s):  
Fractional Order ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Logistic Growth ◽  
Hantavirus Infection ◽  
Infection Model ◽  
The Impact

AbstractThis paper considers a Hantavirus infection model consisting of a system of fractional-order ordinary differential equations with logistic growth. The fractional-order model describes the spread of Hantavirus infection in a system consisting of a population of susceptible and infected mice. The existence, uniqueness, non-negativity and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional order system and the basic reproduction number are studied. The impact of basic reproduction number and carrying capacity on the stability of the fractional order system are also theoretically and numerically investigated.

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.

scholarly journals Estimation of the Reproduction Number of Influenza A(H1N1)pdm2009 in South Korea Using Heterogeneous Models

2020 ◽  
Author(s):  
Yunjeong Lee ◽  
Dong Han Lee ◽  
Hee-Dae Kwon ◽  
Changsoo Kim ◽  
jeehyun lee
Keyword(s):  
South Korea ◽  
Influenza A ◽  
Reproduction Number ◽  
Time Estimation ◽  
Estimation Of Parameters ◽  
Transmission Potential ◽  
Incidence Data ◽  
Reproduction Numbers ◽  
Number Of Patients ◽  
Influenza A H1n1

Abstract Background: The reproduction number is one of the most crucial parameters in determining disease dynamics, providing a summary measure of the transmission potential. However, estimating this value is particularly challenging owing to the characteristics of epidemic data, including non-reproducibility and incompleteness.Methods: In this study, we propose mathematical models with different population structures; each of these models can produce data on the number of cases of the influenza A(H1N1)pdm09 epidemic in South Korea. These structured models incorporating the heterogeneity of age and region are used to estimate the reproduction numbers at various terminal times. Subsequently, the age- and region-specific reproduction numbers are also computed to analyze the differences illustrated in the incidence data. Results: The basic SIR fails to provide a reasonable estimation of the reproduction numbers. The estimated values demonstrate a large variation and remains outside of the feasible range for the influenza, regardless of the time period for data. Real-time estimation using age- and region-structured models demonstrated that the effective reproduction number rose sharply during mid-October when the number of patients increased dramatically. The reproduction number fell below unity at the end of October and stayed lower than unity indicating that the epidemic starts decreasing, which is consistent with the incidence data. Conclusions: Numerical results reveal that the introduction of heterogeneity into the population to represent the general characteristics of dynamics is essential for the robust estimation of parameters.

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

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.

scholarly journals Modeling the impact of temperature on fractional order dengue model with vertical transmission

2020 ◽  
Vol 10 (1) ◽  
pp. 85-93
Author(s):  
Ozlem Defterli
Keyword(s):  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Effect Of Temperature ◽  
Dengue Disease ◽  
Host Interaction ◽  
The Stability ◽  
The Impact ◽  
Disease Free ◽  
Dengue Epidemic

A dengue epidemic model with fractional order derivative is formulated to investigate the effect of temperature on the spread of the vector-host transmitted dengue disease. The model consists of system of fractional order differential equations formulated within Caputo fractional operator. The stability of the equilibrium points of the considered dengue model is studied. The corresponding basic reproduction number R_0 is derived and it is proved that if R_0 < 1, the disease-free equilibrium (DFE) is locally asymptotically stable. L1 method is applied to solve the dengue model numerically. Finally, numerical simulations are also presented to illustrate the analytical results showing the influence of thetemperature on the dynamics of the vector-host interaction in dengue epidemics.

scholarly journals Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yakui Xue ◽  
Junfeng Wang
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Disease Spread ◽  
Treatment Rate ◽  
Stability Of Equilibria ◽  
Existence And Stability ◽  
The Basic Reproduction Number

An epidemic model with infectious force in infected and immune period and treatment rate of infectious individuals is proposed to understand the effect of the capacity for treatment of infective on the disease spread. It is assumed that treatment rate is proportional to the number of infective below the capacity and is constant when the number of infective is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infective. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.

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 Analysis of a Multiscale Model of Ebola Virus Disease

2020 ◽  
pp. 53-69
Author(s):  
Duncan O. Oganga ◽  
George O. Lawi ◽  
Colleta A. Okaka
Keyword(s):  
Ebola Virus ◽  
Reproduction Number ◽  
Virus Disease ◽  
Equilibrium Points ◽  
Vaccination Strategy ◽  
Multiscale Model ◽  
Ebola Virus Disease ◽  
Disease Spread ◽  
The Impact ◽  
Disease Free

Multiscale models are ones that link the epidemiological processes dealing with the transmission between hosts and the immunological processes dealing with the dynamics within one host. In this study, a multiscale model of Ebola Virus Disease linking epidemiological and immunological processes has been developed and analysed. The model has considered two infectious classes ; the exposed and the infected individuals. Local and global stability analyses of the Disease Free Equilibrium and the Endemic Equilibrium points of the model show that the disease dies out if the basic reproduction number Rc0 < 1 and persists in the population when Rc0 > 1 respectively. Sensitivity analysis shows that the rate of vaccination, v , is the most sensitive parameter. This indicates that effort should be directed towards implementing an effective vaccination strategy to control the spread of the disease. It has also been established that when treatment efficacy is scaled up, the viral load goes down and consequently, the transmission between hosts is also reduced. The impact of treatment on the disease spread has also been established through the coupling function (L∗) . The study indicates that a higher percentage of the exposed and the infected individuals should be treated to control the spread of the disease within the population.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

scholarly journals Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Getachew Teshome Tilahun ◽  
Woldegebriel Assefa Woldegerima ◽  
Aychew Wondifraw
Keyword(s):  
Mathematical Model ◽  
Stochastic Model ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Invariant Solution ◽  
Stochastic Approach ◽  
Disease Dynamics ◽  
Treatment Rate ◽  
Invariant Region ◽  
Contact Transmission

AbstractIn this paper we develop a stochastic mathematical model of cholera disease dynamics by considering direct contact transmission pathway. The model considers four compartments, namely susceptible humans, infectious humans, treated humans, and recovered humans. Firstly, we develop a deterministic mathematical model of cholera. Since the deterministic model does not consider the randomness process or environmental factors, we converted it to a stochastic model. Then, for both types of models, the qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction numbers are obtained for both models and compared. From the comparison, we obtained that the basic reproduction number of the stochastic model is much smaller than that of the deterministic one, which means that the stochastic approach is more realistic. Finally, we performed sensitivity analysis and numerical simulations. The numerical simulation results show that reducing contact rate, improving treatment rate, and environmental sanitation are the most crucial activities to eradicate cholera disease from the community.

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