Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

Epidemic Model ◽

Discrete Model ◽

Reproduction Number ◽

Continuous Model ◽

Asymptotically Stable ◽

Nsfd Scheme ◽

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $\mathcal{R}_0 \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $\mathcal{R}_0 > 1$. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

Author(s):

Yanli Ma ◽

Jia-Bao Liu ◽

Haixia Li

Keyword(s):

Lyapunov Function ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Disease Free ◽

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

Dynamics of a delayed epidemic model with varying immunity period and nonlinear transmission

10.1142/s1793524515500278 ◽

2015 ◽

Vol 08 (02) ◽

pp. 1550027 ◽

Cited By ~ 3

Author(s):

Aadil Lahrouz

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Sufficient Conditions ◽

Distributed Delay ◽

Incidence Rates ◽

Asymptotically Stable ◽

Nonlinear Incidence ◽

An epidemic model with a class of nonlinear incidence rates and distributed delay is analyzed. The nonlinear incidence is used to describe the saturated or the psychological effect of certain serious epidemics on the community when the number of infectives is getting larger. The distributed delay is derived to describe the dynamics of infectious diseases with varying immunity. Lyapunov functionals are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one. Moreover, it is shown that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions under which the endemic equilibrium is locally and globally asymptotically stable are obtained.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

10.1142/s1793524518500699 ◽

2018 ◽

Vol 11 (05) ◽

pp. 1850069 ◽

Author(s):

Xia Wang ◽

Ying Zhang ◽

Xinyu Song

Keyword(s):

Steady State ◽

Epidemic Model ◽

Reproduction Number ◽

Threshold Dynamics ◽

Asymptotically Stable ◽

Waning Immunity ◽

Age Structured ◽

In this paper, a susceptible-vaccinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

THE DYNAMICS AND THERAPEUTIC STRATEGIES OF A SEIS EPIDEMIC MODEL

10.1142/s1793524513500290 ◽

2013 ◽

Vol 06 (05) ◽

pp. 1350029 ◽

Author(s):

XINZHU MENG ◽

ZHITAO WU ◽

TONGQIAN ZHANG

Keyword(s):

Epidemic Model ◽

Therapeutic Strategy ◽

Reproduction Number ◽

Asymptotically Stable ◽

Epidemic Disease ◽

Effective Therapeutic Strategy ◽

The Impact ◽

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R0 < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R0 > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

Partial Differential Equations of an Epidemic Model with Spatial Diffusion

International Journal of Partial Differential Equations ◽

10.1155/2014/186437 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 14

Author(s):

El Mehdi Lotfi ◽

Mehdi Maziane ◽

Khalid Hattaf ◽

Noura Yousfi

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Reaction Diffusion ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Sir Epidemic Model ◽

Disease Free ◽

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

Discrete Dynamics in Nature and Society ◽

10.1155/2012/530267 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Author(s):

Hai-Feng Huo ◽

Li-Xiang Feng

Keyword(s):

Global Stability ◽

Epidemic Model ◽

Reproduction Number ◽

Global Dynamics ◽

Asymptotically Stable ◽

Incomplete Treatment ◽

Disease Free ◽

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is constructed. We establish that the global dynamics are completely determined by the basic reproduction numberR0. IfR0≤1, then the disease-free equilibrium is globally asymptotically stable. IfR0>1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also given to explain our conclusions.

Global Dynamics of a Discretized Heroin Epidemic Model with Time Delay

Abstract and Applied Analysis ◽

10.1155/2014/742385 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 7

Author(s):

Xamxinur Abdurahman ◽

Ling Zhang ◽

Zhidong Teng

Keyword(s):

Time Delay ◽

Epidemic Model ◽

Reproduction Number ◽

Global Dynamics ◽

Asymptotically Stable ◽

Nonstandard Finite Difference Scheme ◽

Nonstandard Finite Difference ◽

We derive a discretized heroin epidemic model with delay by applying a nonstandard finite difference scheme. We obtain positivity of the solution and existence of the unique endemic equilibrium. We show that heroin-using free equilibrium is globally asymptotically stable when the basic reproduction numberR0<1, and the heroin-using is permanent when the basic reproduction numberR0>1.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

10.1142/s1793524515500825 ◽

2015 ◽

Vol 08 (06) ◽

pp. 1550082 ◽

Cited By ~ 4

Author(s):

Muhammad Altaf Khan ◽

Yasir Khan ◽

Qaiser Badshah ◽

Saeed Islam

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Number Formula ◽

Preventive Vaccination ◽

Disease Free ◽

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

A mathematical investigation of an "SVEIR" epidemic model for the measles transmission

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022131 ◽

2022 ◽

Vol 19 (3) ◽

pp. 2853-2875

Author(s):

Miled El Hajji ◽

◽

Amer Hassan Albargi ◽

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Global Stability ◽

Epidemic Model ◽

Reproduction Number ◽

Asymptotically Stable ◽

<abstract><p>A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} > 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} > 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} > 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.</p></abstract>

Transmission Dynamics of a Two-City SIR Epidemic Model with Transport-Related Infections

Journal of Applied Mathematics ◽

10.1155/2014/764278 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Author(s):

Yao Chen ◽

Mei Yan ◽

Zhongyi Xiang

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Sufficient Conditions ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Sir Epidemic Model ◽

Sir Epidemic ◽

A two-city SIR epidemic model with transport-related infections is proposed. Some good analytical results are given for this model. If the basic reproduction numberℜ0γ≤1, there exists a disease-free equilibrium which is globally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the basic reproduction numberℜ0γ>1. We also show the permanence of this SIR model. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium.