scholarly journals Dynamics of an SIR Epidemic Model with Information Variable and Limited Medical Resources Revisited

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Caijuan Yan ◽  
Jianwen Jia ◽  
Zhen Jin
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Asymptotical Stability ◽  
Endemic Equilibrium ◽  
Basic Reproduction Ratio ◽  
Sir Epidemic Model ◽  
Medical Resources ◽  
Reproduction Ratio ◽  
Sir Epidemic ◽  
Information Variable

The stability of the SIR epidemic model with information variable and limited medical resources was studied. When the basic reproduction ratioℛ0<1, there exists the disease-free equilibrium and when the basic reproduction ratioℛ0>1, we obtain the sufficient conditions of the existence of the endemic equilibrium. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Yao Chen ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Basic Reproduction Number

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.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 147 ◽  
Author(s):  
Toshikazu Kuniya
Keyword(s):  
Stability Analysis ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Transmission Function ◽  
Endemic Equilibrium ◽  
Dimensional System ◽  
Relevant Parameter ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Age Structured

In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of ordinary differential equations (ODEs). We show that the unique endemic equilibrium of the reduced system exists if the basic reproduction number for the original system is greater than unity. Furthermore, we perform the stability analysis of the endemic equilibrium and obtain a fourth-order characteristic equation. By using the Routh–Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings.

scholarly journals Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Stability ◽  
The Basic Reproduction Number

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Jihad Adnani ◽  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Endemic Equilibrium ◽  
Random Perturbations ◽  
Sir Epidemic Model ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sir Epidemic ◽  
Stochastic Sir ◽  
The Stability

We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results.

Global dynamics of a fractional-order SIR epidemic model with memory

2020 ◽  
pp. 2050071 ◽  
Author(s):  
Parvaiz Ahmad Naik
Keyword(s):  
Fractional Derivative ◽  
Memory Effect ◽  
Fractional Order ◽  
Epidemic Model ◽  
Memory Trace ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic

In this paper, an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate are established along the memory. The existence and stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, a threshold value, [Formula: see text], is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied L1 scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. Also, from the results, it is obvious that the memory effect is zero for [Formula: see text]. When the fractional-order [Formula: see text] is decreased from [Formula: see text] the memory trace nonlinearly increases from [Formula: see text], and its dynamics strongly depends on time. The memory effect points out the difference between the derivatives of the fractional-order and integer order.

The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps

2021 ◽  
Author(s):  
Driss Kiouach ◽  
Yassine Sabbar
Keyword(s):  
Epidemic Model ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Time Behavior ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Long Run ◽  
Long Time ◽  
Lévy Jumps

This paper reports novel theoretical and analytical results for a perturbed version of a SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô–Lévy stochastic differential equations in order to simulate sudden and unexpected external phenomena. By using some new and ameliorated mathematical approaches, we study the long-run characteristics of the perturbed delayed model. Within this scope, we give sufficient conditions for two interesting asymptotic properties: extinction and persistence of the epidemic. One of the most interesting results is that the dynamics of the stochastic model are closely related to the intensities of white noises and Lévy jumps, which can give us a good insight into the evolution of the epidemic in some unexpected situations. Our work complements the results of some previous investigations and provides a new approach to predict and analyze the dynamic behavior of epidemics with distributed delay. For illustrative purposes, numerical examples are presented for checking the theoretical study.

scholarly journals Qualitative and Bifurcation Analysis of an SIR Epidemic Model with Saturated Treatment Function and Nonlinear Pulse Vaccination

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Xiangsen Liu ◽  
Binxiang Dai
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Existence And Stability ◽  
The Stability ◽  
Saturated Treatment ◽  
Disease Free

An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.

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