scholarly journals The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Jinqing Zhao ◽  
Maoxing Liu ◽  
Wanwan Wang ◽  
Panzu Yang
Keyword(s):  
Asymptotic Behavior ◽  
Complex Networks ◽  
Stochastic System ◽  
Epidemic Model ◽  
Noise Intensity ◽  
Stochastic Perturbation ◽  
Deterministic System ◽  
Global Positive Solution ◽  
The Stability ◽  
Disease Free

We investigate a stochastic SI epidemic model in the complex networks. We show that this model has a unique global positive solution. Then we consider the asymptotic behavior of the model around the disease-free equilibrium and show that the solution will oscillate around the disease-free equilibrium of deterministic system whenR0≤1. Furthermore, we derive that the disease will be persistent whenR0>1. Finally, a series of numerical simulations are presented to illustrate our mathematical findings. A new result is given such that, whenR0≤1, with the increase of noise intensity the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system.

Stability Analysis of an SEIR Epidemic Model with Stochastic Perturbation and Numerical Simulation

2013 ◽  
Vol 14 (2) ◽  
Author(s):  
Xiaoming Fan ◽  
Zhigang Wang
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Behavior ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Random Fluctuation ◽  
Deterministic System ◽  
Stochastic Version ◽  
Special Case ◽  
The Basic Reproduction Number

AbstractAn SEIR epidemic model with constant immigration and random fluctuation around the endemic equilibrium is considered. As a special case, a deterministic system discussed by Li et al. will be incorporated into the stochastic version given by us. We carry out a detailed analysis on the asymptotic behavior of the stochastic model, also regarding of the basic reproduction number ℛ

scholarly journals A Stochastic TB Model for a Crowded Environment

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Sibaliwe Maku Vyambwera ◽  
Peter Witbooi
Keyword(s):  
Compartmental Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Equation Model ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
The Stability ◽  
Crowded Environments ◽  
Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

Stability analysis and optimal control of an epidemic model with vaccination

2015 ◽  
Vol 08 (03) ◽  
pp. 1550030 ◽  
Author(s):  
Swarnali Sharma ◽  
G. P. Samanta
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Epidemic Model ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
The Cost ◽  
Objective Functional ◽  
Disease Free

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R0) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0when R0< 1. When R0> 1 endemic equilibrium E1exists and the system becomes locally asymptotically stable at E1under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

scholarly journals Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mouhcine Naim ◽  
Fouad Lahmidi
Keyword(s):  
Epidemic Model ◽  
Deterministic Model ◽  
Sufficient Condition ◽  
Nonlinear Incidence Rate ◽  
Local Asymptotic Stability ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Stochastic Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

scholarly journals Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

2021 ◽  
Vol 6 (11) ◽  
pp. 12359-12378
Author(s):  
Yuhuai Zhang ◽  
◽  
Xinsheng Ma ◽  
Anwarud Din ◽  
◽  
...  
Keyword(s):  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Noise Intensity ◽  
Epidemic Spreading ◽  
General Incidence Rate ◽  
Global Positive Solution ◽  
Simulation Results ◽  
Theoretical Results ◽  
Temporary Immunity

<abstract><p>In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.</p></abstract>

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0&lt;1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0&gt;1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

scholarly journals Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jiying Ma ◽  
Qing Yi
Keyword(s):  
Stationary Distribution ◽  
Stochastic System ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Deterministic System ◽  
Stochastic Epidemic ◽  
Lyapunov Function Method ◽  
Deterministic Framework ◽  
Theoretical Results

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.

Media effects on the dynamics of a stochastic SIRI epidemic model with relapse and Lévy noise perturbation

2019 ◽  
Vol 12 (03) ◽  
pp. 1950037 ◽  
Author(s):  
Badr-Eddine Berrhazi ◽  
Mohamed El Fatini ◽  
Roger Pettersson ◽  
Aziz Laaribi
Keyword(s):  
Epidemic Model ◽  
Media Coverage ◽  
Deterministic Model ◽  
Dynamic Properties ◽  
Lévy Noise ◽  
Noise Perturbation ◽  
Global Positive Solution ◽  
Levy Noise ◽  
Theoretical Results ◽  
Disease Free

In this paper, we study the dynamic properties of an SIRI epidemic model incorporating media coverage, and stochastically perturbed by a Lévy noise. We establish the existence of a unique global positive solution. We investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model depending on the basic reproduction number under some noise excitation. Furthermore, we present some numerical simulations to support the theoretical results.

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