Preparations for the Study of the Stationary Distribution p(1) of the SIS Model

2011 ◽  
pp. 73-91
Author(s):  
Ingemar Nåsell
Keyword(s):  
Stationary Distribution ◽  
Sis Model

The quasi-stationary distribution of the closed endemic sis model

1996 ◽  
Vol 28 (3) ◽  
pp. 895-932 ◽  
Author(s):  
Ingemar Nåsell
Keyword(s):  
Stationary Distribution ◽  
Geometric Distribution ◽  
Threshold Value ◽  
Basic Reproduction Ratio ◽  
Sis Model ◽  
Reproduction Ratio ◽  
Time To Extinction ◽  
Extreme Distributions ◽  
Stochastic Sis Model ◽  
Quasi Stationary Distribution

The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio R0 passes the deterministic threshold value 1. Approximations are derived that describe these changes. The quasi-stationary distribution is approximated by a geometric distribution (discrete!) for R0 distinctly below 1 and by a normal distribution (continuous!) for R0 distinctly above 1. Uniformity of the approximation with respect to R0 allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.

The asymptotic behavior of a stochastic multigroup SIS model

2018 ◽  
Vol 11 (03) ◽  
pp. 1850037 ◽  
Author(s):  
Chunyan Ji ◽  
Daqing Jiang
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Endemic Equilibrium ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Sis Model ◽  
Stochastic Perturbations ◽  
Long Time ◽  
Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

The quasi-stationary distribution of the closed endemic sis model

1996 ◽  
Vol 28 (03) ◽  
pp. 895-932 ◽  
Author(s):  
Ingemar Nåsell
Keyword(s):  
Stationary Distribution ◽  
Geometric Distribution ◽  
Threshold Value ◽  
Basic Reproduction Ratio ◽  
Sis Model ◽  
Reproduction Ratio ◽  
Time To Extinction ◽  
Extreme Distributions ◽  
Stochastic Sis Model ◽  
Quasi Stationary Distribution

The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio R 0 passes the deterministic threshold value 1. Approximations are derived that describe these changes. The quasi-stationary distribution is approximated by a geometric distribution (discrete!) for R 0 distinctly below 1 and by a normal distribution (continuous!) for R 0 distinctly above 1. Uniformity of the approximation with respect to R 0 allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.

scholarly journals Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Can Chen ◽  
Yanmei Kang
Keyword(s):  
Exponential Stability ◽  
Stationary Distribution ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Threshold Value ◽  
Sis Model ◽  
Almost Sure Exponential Stability ◽  
Saturated Incidence ◽  
The Mean ◽  
Moment Exponential Stability

We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and itspth moment exponential stability, which indicates that thepth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

scholarly journals Survival and Stationary Distribution in a Stochastic SIS Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanli Zhou ◽  
Weiguo Zhang ◽  
Sanling Yuan
Keyword(s):  
Stationary Distribution ◽  
Deterministic Model ◽  
Deterministic System ◽  
Initial Value ◽  
Sis Model ◽  
Sis Epidemic Model ◽  
Stochastic Sis Epidemic Model ◽  
Theoretical Results ◽  
Disease Free

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document