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.

The threshold for a stochastic within-host CHIKV virus model with saturated incidence rate

2021 ◽  
pp. 2150042
Author(s):  
C. Gokila ◽  
M. Sambath
Keyword(s):  
Lyapunov Function ◽  
Incidence Rate ◽  
Stationary Distribution ◽  
Reproduction Number ◽  
Threshold Value ◽  
Saturated Incidence Rate ◽  
Number Formula ◽  
Saturated Incidence ◽  
Virus Model ◽  
Global Positive Solution

This paper deals with stochastic Chikungunya (CHIKV) virus model along with saturated incidence rate. We show that there exists a unique global positive solution and also we obtain the conditions for the disease to be extinct. We also discuss about the existence of a unique ergodic stationary distribution of the model, through a suitable Lyapunov function. The stationary distribution validates the occurrence of disease; through that, we find the threshold value for prevail and disappear of disease within host. With the help of numerical simulations, we validate the stochastic reproduction number [Formula: see text] as stated in our theoretical findings.

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.

scholarly journals Almost sure exponential stability of stochastic differential delay equations

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 789-814
Author(s):  
Wei Zhang ◽  
M.H. Song ◽  
M.Z. Liu
Keyword(s):  
Exponential Stability ◽  
Differential Delay ◽  
Almost Sure Exponential Stability ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Exponentially Stable ◽  
Pth Moment Exponential Stability ◽  
Stochastic Theta Method ◽  
Moment Exponential Stability ◽  
Theta Method

This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.

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.

DEMOGRAPHICS INDUCE EXTINCTION OF DISEASE IN AN SIS MODEL BASED ON CONDITIONAL MARKOV CHAIN

2017 ◽  
Vol 25 (01) ◽  
pp. 145-171 ◽  
Author(s):  
XIAOFENG LUO ◽  
LILI CHANG ◽  
ZHEN JIN
Keyword(s):  
Markov Chain ◽  
Disease Transmission ◽  
Numerical Solutions ◽  
Deterministic Model ◽  
Moment Generating Function ◽  
Network Evolution ◽  
Stochastic Simulations ◽  
Disease Spread ◽  
Sis Model ◽  
Scale Free

Demographics have significant effects on disease spread in populations and the topological evolution of the underlying networks that represent the populations. In the context of network-based epidemic modeling, Markov chain-based approach and pairwise approximation are two powerful tools — the former can capture stochastic effects of disease transmission dynamics and the latter can characterize the dynamical correlations in each pair of connected individuals. However, to our best knowledge, the study on epidemic spreading in networks relying on these two techniques is still lacking. To fill this gap, in this paper, a deterministic pairwise susceptible–infected–susceptible (SIS) epidemic model with demographics on complex networks with arbitrary degree distributions is studied based on a continuous time conditional Markov chain. This deterministic model is rigorously derived — using the moment generating function — from the Kolmogorov differential equations for the evolution of individuals and pairs. It is found that demographics will induce the extinction of the disease by reducing the basic reproduction number or lowering the epidemic prevalence after the disease prevails. Moreover, due to the demographical effects, the resulting network tends to a homogeneous network with a degree distribution similar to Poisson distribution, irrespective of the initial network structure. Additionally, we find excellent agreement between numerical solutions and individual-based stochastic simulations using both Erdös–Renyi (ER) random and Barabási–Albert (BA) scale-free initial networks. Our results may provide new insights on the understanding of the influence of demographics on epidemic dynamics and network evolution.

scholarly journals Stochastic Dynamics of Nonautonomous Cohen-Grossberg Neural Networks

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Chuangxia Huang ◽  
Jinde Cao
Keyword(s):  
Neural Networks ◽  
Lyapunov Function ◽  
Exponential Stability ◽  
Stochastic Stability ◽  
Stochastic Dynamics ◽  
Type Theorem ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Almost Sure Exponential Stability ◽  
Moment Exponential Stability

This paper is devoted to the study of the stochastic stability of a class of Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying. With the help of Lyapunov function, Burkholder-Davids-Gundy inequality, and Borel-Cantell's theory, a set of novel sufficient conditions onpth moment exponential stability and almost sure exponential stability for the trivial solution of the system is derived. Compared with the previous published results, our method does not resort to the Razumikhin-type theorem and the semimartingale convergence theorem. Results of the development as presented in this paper are more general than those reported in some previously published papers. An illustrative example is also given to show the effectiveness of the obtained 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.

Aktivitätskonzentration der vom Personal einer Radiojodtherapiestation eingeatmeten Luft

1987 ◽  
Vol 26 (03) ◽  
pp. 143-146 ◽  
Author(s):  
H. Fill ◽  
M. Oberladstätter ◽  
J. W. Krzesniak
Keyword(s):  
Nuclear Medicine ◽  
Length Of Stay ◽  
Activity Concentration ◽  
Threshold Value ◽  
Whole Body ◽  
Therapeutic Activity ◽  
External Radiation ◽  
Oral Application ◽  
Exhaled Air ◽  
The Mean

The mean activity concentration of1311 during inhalation by the nuclear medicine personnel was measured at therapeutic activity applications of 22 GBq (600 mCi) per week. The activity concentration reached its maximum in the exhaled air of the patients 2.5 to 4 hours after oral application. The normalized maximum was between 2 • 10−5 and 2 • 10−3 Bq-m−3 per administered Bq. The mean activity concentration of1311 inhaled by the personnel was 28 to 1300 Bq-m−3 (0.8 to 35 nCi-rrf−3). From this the1311 uptake per year was estimated to be 30 to 400 kBq/a (x̄ = 250, SD = 50%). The maximum permitted uptake from air per year is, according to the German and Austrian radiation protection ordinances 22/21 µiCi/a (= 8 • 105 Bq/a). At maximum 50% and, on the average, 30% of this threshold value are reached. The length of stay of the personnel in the patient rooms is already now limited to such an extent that 10% of the maximum permissible whole-body dose for external radiation is not exceeded. Therefore, increased attention should be paid also to radiation exposure by inhalation.

scholarly journals Survival analysis of a stochastic delay single-species system in polluted environment with psychological effect and pulse toxicant input

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiangjun Dai ◽  
Suli Wang ◽  
Weizhi Xiong ◽  
Ni Li
Keyword(s):  
Sufficient Conditions ◽  
Single Species ◽  
Threshold Value ◽  
Psychological Effect ◽  
Polluted Environment ◽  
Population System ◽  
Stochastic Delay ◽  
Species Population ◽  
The Mean ◽  
Single Species Population

Abstract We propose and study a stochastic delay single-species population system in polluted environment with psychological effect and pulse toxicant input. We establish sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, and strong persistence of the single-species population and obtain the threshold value between extinction and weak persistence. Finally, we confirm the efficiency of the main results by numerical simulations.

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