Global Stability of a Variation Epidemic Spreading Model on Complex Networks

Epidemic spreading on networks becomes a hot issue of nonlinear systems, which has attracted many researchers’ attention in recent years. A novel epidemic spreading model with variant factors in complex networks is proposed and investigated in this paper. One main feature of this model is that virus variation is investigated in the process of epidemic dynamical spreading. The global dynamics of this model involving an endemic equilibrium and a disease-free equilibrium are, respectively, discussed. Some sufficient conditions are given for the existence of the endemic equilibrium. In addition, the global asymptotic stability problems of the disease-free equilibrium and the endemic equilibrium are also investigated by the Routh-Hurwitz stability criterion and Lyapunov stability criterion. And the uniform persistence condition of the new system is studied. Finally, numerical simulations are provided to illustrate obtained theoretical results.

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Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence

International Journal of Engineering Mathematics ◽

10.1155/2014/487589 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Amine Bernoussi ◽

Abdelilah Kaddar ◽

Said Asserda

Keyword(s):

Global Stability ◽

Numerical Simulations ◽

Epidemic Model ◽

Sufficient Conditions ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Nonlinear Incidence ◽

Lasalle Theorem ◽

Nonlinear Incidence Function ◽

Disease Free

In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

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Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Mathematical Problems in Engineering ◽

10.1155/2015/745732 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Yanju Xiao ◽

Weipeng Zhang ◽

Guifeng Deng ◽

Zhehua Liu

Keyword(s):

Sufficient Conditions ◽

Global Dynamics ◽

Sis Model ◽

Delayed Treatment ◽

Sis Epidemic Model ◽

Treatment Function ◽

Lyapunov Coefficient ◽

Saturated Treatment ◽

Theoretical Results ◽

Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

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Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

Abstract and Applied Analysis ◽

10.1155/2014/109372 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Caijuan Yan ◽

Jianwen Jia

Keyword(s):

Hopf Bifurcation ◽

Epidemic Model ◽

Sufficient Conditions ◽

Endemic Equilibrium ◽

Logistic Growth ◽

Sir Epidemic Model ◽

Medical Resources ◽

Reproduction Ratio ◽

Disease Free ◽

Information Variable

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

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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 ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Disease Free ◽

The Basic Reproduction Number

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.

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Global stability of an age-structured SVEIR epidemic model with waning immunity, latency and relapse

International Journal of Biomathematics ◽

10.1142/s1793524517500383 ◽

2017 ◽

Vol 10 (03) ◽

pp. 1750038 ◽

Cited By ~ 2

Author(s):

Lili Liu ◽

Xianning Liu

Keyword(s):

Asymptotic Stability ◽

Global Asymptotic Stability ◽

Epidemic Model ◽

Uniform Persistence ◽

Global Dynamics ◽

Sharp Threshold ◽

Age Dependent ◽

Waning Immunity ◽

Age Structured ◽

Disease Free

The global dynamics of an SVEIR epidemic model with age-dependent waning immunity, latency and relapse are studied. Sharp threshold properties for global asymptotic stability of both disease-free equilibrium and endemic equilibrium are given. The asymptotic smoothness, uniform persistence and the existence of interior global attractor of the semi-flow generated by a family of solutions of the system are also addressed. Furthermore, some related strategies for controlling the spread of diseases are discussed.

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A VECTOR-HOST EPIDEMIC MODEL WITH HOST MIGRATION

International Journal of Biomathematics ◽

10.1142/s1793524511001179 ◽

2011 ◽

Vol 04 (01) ◽

pp. 93-108

Author(s):

QINGKAI KONG ◽

ZHIPENG QIU ◽

YUN ZOU

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Threshold Value ◽

Endemic Equilibrium ◽

Uniform Persistence ◽

Asymptotically Stable ◽

Host Disease ◽

Disease Free

The host migration is one of the important elements that cause the worldwide diffusion and outbreak of many vector-host diseases. In this paper, we formulate a patchy model to investigate the effect of host migration between two patches on the spread of a vector-host disease. The results of the paper show that the reproduction number R0 is a threshold value that determines the uniform persistence and extinction of the disease. If the reproduction number R0 < 1 the disease free equilibrium (DFE) is locally asymptotically stable. If the reproduction number R0 > 1 then the DFE is unstable and the system is uniformly persistent. It is also shown that a unique endemic equilibrium, which exists when R0 > 1, is locally stable if both regions are identical.

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Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

Discrete Dynamics in Nature and Society ◽

10.1155/2014/161509 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Xiaoguang Zhang ◽

Rui Song ◽

Gui-Quan Sun ◽

Zhen Jin

Keyword(s):

Complex Networks ◽

Reproduction Number ◽

Epidemic Models ◽

Infectious Period ◽

Global Dynamics ◽

Globally Asymptotically Stable ◽

Sis Epidemic Model ◽

Different Types ◽

Theoretical Results ◽

Disease Free

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

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Global Dynamics of an SIS Model on Metapopulation Networks with Demographics

Complexity ◽

10.1155/2021/8884236 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Maoxing Liu ◽

Xinjie Fu ◽

Jie Zhang ◽

Donghua Zhao

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Sufficient Conditions ◽

Active Role ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Sis Model ◽

Globally Asymptotically Stable ◽

Sis Epidemic Model ◽

The Basic Reproduction Number

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

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Global Dynamics of a Host-Vector-Predator Mathematical Model

Journal of Applied Mathematics ◽

10.1155/2014/245650 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Fengyan Zhou ◽

Hongxing Yao

Keyword(s):

Mathematical Model ◽

Predation Rate ◽

Disease Suppression ◽

Uniform Persistence ◽

Global Dynamics ◽

Vector Population ◽

Threshold Conditions ◽

Vector Theory ◽

The Impact ◽

Disease Free

A mathematical model which links predator-vector(prey) and host-vector theory is proposed to examine the indirect effect of predators on vector-host dynamics. The equilibria and the basic reproduction numberR0are obtained. By constructing Lyapunov functional and using LaSalle’s invariance principle, global stability of both the disease-free and disease equilibria are obtained. Analytical results show thatR0provides threshold conditions on determining the uniform persistence and extinction of the disease, and predator density at any time should keep larger or equal to its equilibrium level for successful disease eradication. Finally, taking the predation rate as parameter, we provide numerical simulations for the impact of predators on vector-host disease control. It is illustrated that predators have a considerable influence on disease suppression by reducing the density of the vector population.

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Global Dynamics of a Discrete-Time MERS-Cov Model

Mathematics ◽

10.3390/math9050563 ◽

2021 ◽

Vol 9 (5) ◽

pp. 563

Author(s):

Mahmoud H. DarAssi ◽

Mohammad A. Safi ◽

Morad Ahmad

Keyword(s):

Discrete Time ◽

Global Attractivity ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Proposed Model ◽

Threshold Conditions ◽

Theoretical Results ◽

Disease Free

In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. We proved that the DFE is globally asymptotically stable when R0≤1. Whenever R˜0>1, the proposed model has a unique endemic equilibrium that is globally asymptotically stable. The theoretical results are illustrated by a numerical simulation.

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