scholarly journals Stochastic Extinction in an SIRS Epidemic Model Incorporating Media Coverage

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Liyan Wang ◽  
Huilin Huang ◽  
Ancha Xu ◽  
Weiming Wang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Epidemic Model ◽  
Media Coverage ◽  
Control Strategies ◽  
Disease Dynamics ◽  
Unique Positive Solution ◽  
Sirs Epidemic Model ◽  
Deterministic Framework ◽  
Stochastic Extinction

We extend the classical SIRS epidemic model incorporating media coverage from a deterministic framework to a stochastic differential equation (SDE) and focus on how environmental fluctuations of the contact coefficient affect the extinction of the disease. We give the conditions of existence of unique positive solution and the stochastic extinction of the SDE model and discuss the exponentialp-stability and global stability of the SDE model. One of the most interesting findings is that if the intensity of noise is large, then the disease is prone to extinction, which can provide us with some useful control strategies to regulate disease dynamics.

scholarly journals Extinction and persistence of a stochastic SIRV epidemic model with nonlinear incidence rate

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ramziya Rifhat ◽  
Zhidong Teng ◽  
Chunxia Wang
Keyword(s):  
Epidemic Model ◽  
Control Strategies ◽  
Random Perturbations ◽  
Disease Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Analysis Theory ◽  
Lyapunov Function Method ◽  
Random Fluctuations ◽  
Theoretical Results

AbstractIn this paper, a stochastic SIRV epidemic model with general nonlinear incidence and vaccination is investigated. The value of our study lies in two aspects. Mathematically, with the help of Lyapunov function method and stochastic analysis theory, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. In other words, neglecting random perturbations overestimates the ability of the disease to spread. The numerical simulations are given to illustrate the main theoretical results.

scholarly journals A Stochastic Differential Equation SIS Epidemic Model

2011 ◽  
Vol 71 (3) ◽  
pp. 876-902 ◽  
Author(s):  
A. Gray ◽  
D. Greenhalgh ◽  
L. Hu ◽  
X. Mao ◽  
J. Pan
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Epidemic Model ◽  
Sis Epidemic Model ◽  
Sis Epidemic

Hopf bifurcation of a delay SIRS epidemic model with novel nonlinear incidence: Application to scarlet fever

2018 ◽  
Vol 11 (07) ◽  
pp. 1850091 ◽  
Author(s):  
Yong Li ◽  
Xianning Liu ◽  
Lianwen Wang ◽  
Xingan Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Scarlet Fever ◽  
Reproduction Number ◽  
Control Strategies ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Intervention Measures ◽  
Theoretical Results

An [Formula: see text] epidemic model incorporating incubation time delay and novel nonlinear incidence is proposed and analyzed to seek for the control strategies of scarlet fever, where the contact rate which can reflect the regular behavior and habit changes of children is non-monotonic with respect to the number of susceptible. The model without delay may exhibit backward bifurcation and bistable states even though the basic reproduction number is less than unit. Furthermore, we derive the conditions for occurrence of Hopf bifurcation when the time delay is considered as a bifurcation parameter. The data of scarlet fever of China are simulated to verify our theoretical results. In the end, several effective preventive and intervention measures of scarlet fever are found out.

scholarly journals On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

2020 ◽  
Vol 61 ◽  
pp. C1-C14
Author(s):  
Hidekazu Yoshioka ◽  
Yumi Yoshioka
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Growth Rates ◽  
Epidemic Model ◽  
Volterra Model ◽  
Sis Epidemic Model ◽  
Sis Epidemic

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

scholarly journals Dynamics of a Stochastic SIRS Epidemic Model with Regime Switching and Specific Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Amine EL Koufi ◽  
Abdelkrim Bennar ◽  
Noura Yousfi
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Epidemic Model ◽  
Regime Switching ◽  
Unique Positive Solution ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Incident Rate ◽  
Stochastic Sirs Epidemic Model

The purpose of this work is to investigate the dynamic behaviors of the SIRS epidemic model with nonlinear incident rate under regime switching. We establish the existence of a unique positive solution of our system. Furthermore, we obtain the conditions for the extinction of diseases, and we show the existence of the stationary distribution for our stochastic SIRS model under regime switching. Numerical simulations are employed to illustrate our theoretical analysis.

scholarly journals Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1122
Author(s):  
Yanlin Ding ◽  
Jianjun Jiao ◽  
Qianhong Zhang ◽  
Yongxin Zhang ◽  
Xinzhi Ren
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Stationary Distribution ◽  
Dynamic Characteristics ◽  
Epidemic Model ◽  
Regime Switching ◽  
Media Coverage ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Theoretical Results

This paper is concerned with the dynamic characteristics of the SIQR model with media coverage and regime switching. Firstly, the existence of the unique positive solution of the proposed system is investigated. Secondly, by constructing a suitable random Lyapunov function, some sufficient conditions for the existence of a stationary distribution is obtained. Meanwhile, the conditions for extinction is also given. Finally, some numerical simulation examples are carried out to demonstrate the effectiveness of theoretical results.

Dynamic behavior of a stochastic SIRS epidemic model with media coverage

2018 ◽  
Vol 41 (14) ◽  
pp. 5506-5525 ◽  
Author(s):  
Wenjuan Guo ◽  
Qimin Zhang ◽  
Xining Li ◽  
Weiming Wang
Keyword(s):  
Dynamic Behavior ◽  
Epidemic Model ◽  
Media Coverage ◽  
Sirs Epidemic Model ◽  
Stochastic Sirs Epidemic Model

A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise

2017 ◽  
Vol 105 ◽  
pp. 60-68 ◽  
Author(s):  
Badr-eddine Berrhazi ◽  
Mohamed El Fatini ◽  
Aziz Laaribi ◽  
Roger Pettersson ◽  
Regragui Taki
Keyword(s):  
Epidemic Model ◽  
Media Coverage ◽  
Lévy Noise ◽  
Sirs Epidemic Model ◽  
Levy Noise ◽  
Stochastic Sirs Epidemic Model
Sign in / Sign up

Export Citation Format

Share Document