Research article

Poisson integral type quarantine in a stochastic SIR system

  • Received: 13 June 2020 Accepted: 03 August 2020 Published: 14 August 2020
  • We propose a SIR system that includes a Poisson measure term to model the quarantine of infected individuals. An inequality concerning the term representing the transmission rate is given to establish the stochastic stability of the disease free equilibrium. It is further shown that if R0 > 1 then the long-run behavior the system will reside within a neighborhood of the equilibrium in the underlying deterministic version of this system.

    Citation: Andrew Vlasic, Troy Day. Poisson integral type quarantine in a stochastic SIR system[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5534-5544. doi: 10.3934/mbe.2020297

    Related Papers:

  • We propose a SIR system that includes a Poisson measure term to model the quarantine of infected individuals. An inequality concerning the term representing the transmission rate is given to establish the stochastic stability of the disease free equilibrium. It is further shown that if R0 > 1 then the long-run behavior the system will reside within a neighborhood of the equilibrium in the underlying deterministic version of this system.


    加载中


    [1] F. Rao, W. Wang, Z. Li, Stability analysis of an epidemic model with diffusion and stochastic perturbation, Comm. Nonlinear Sci. Num. Simul., 17 (2012), 2551-2563.
    [2] E. Tornatore, S. Buccellato, P. Vetro, Stability of a stochastic SIR system, Phys. A, 354 (2005), 111-126.
    [3] R. Kuske, L. F. Gordillo, P. Greenwood, Sustained oscillations via coherence resonance in SIR, J. Theor. Biol., 245 (2007), 459-469.
    [4] J. Yu, D. Jiang, N. Shi, Globaly stability of the two-group SIR model with random perturbation, J. Math. Anal. App., 360 (2009), 235-244.
    [5] A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, Stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.
    [6] Q. Liu, D. Jiang, T. Hayat, B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps, Nonlinear Anal. Hybrid Sys., 27 (2018), 29-43.
    [7] X. Zhang, K. Wang, Stochastic SIR model with jumps, Appl. Math. Letters, 26 (2013), 867-874.
    [8] Y. Zhou, W. Zhang, Threshold of a stochastic SIR epidemic model with Lévy jumps, Phys. A, 446 (2016), 204-216.
    [9] A. Vlasic, D. Troy, Modeling stochastic anomalies in an SIS system, Stoch. Anal. App., 35 (2017), 27-39.
    [10] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (200), 599-653.
    [11] D. Applebaum, Lévy processes and stochastic calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, Cambridge, 2004.
    [12] K. Sato, Lévy Processes and Infinitely Divisible Distributions, 1st edition, Cambridge University Press, Cambridge, 1999.
    [13] J. Bertoin, Lévy Processes, 1st edition, Cambridge University Press, Cambridge, 1996.
    [14] E. Tornatore, S. Buccellato, On a stochastic SIR model, Appl. Math. (Warsaw), 34 (2007), 389-400.
    [15] Q. Lu, Stability of SIR system with random perturbations, Phys. A, 388 (2009), 3677-3686.
    [16] C. Ji, D. Ji, N. Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stoch. Anal. App., 30 (2012), 755-773.
    [17] G. Chen, T. Li, Ch. Liu, Lyapunov exponent of a stochastic SIR model, Comptes Rendus Math., 351 (2013), 33-35.
    [18] J. Calatayud, J. C. Cortés, M. Jornet, Computing the density function of complex models with randomness by using polynomial expansions and the RVT technique. Application to the SIR epidemic model, Chaos Solit. Fract., 133 (2020), 109639. https://doi.org/10.1016/j.chaos.2020.109639.
    [19] J. C. Cortés, S. K. El-Labany, A. Navarro-Quiles, M. M. Selim, H. Slama, A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized DiscreteTime Markov Chain formulation with applications, Math. Methods Appl. Sci., (2020), https://doi.org/10.1002/mma.6482.
    [20] S. P. Meyn, R. L. Tweedie, Stability of Markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25 (1993), 518-548.
    [21] I. Gihman, A. V. Skorohod, Stochastic differential equations, 2nd edition, Springer-Verlag, New York, 1972.
    [22] D. Down, S. P. Meyn, R. L. Tweedie, Exponential and uniform ergodicity of Markov Processes, Ann.Appl. Prob., 23 (1995), 1671-1691.
    [23] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, 1st edition, American Mathematical Society, Moscow, 1989.
    [24] X. Mao, Stochastic Differential Equations and Applications, 1st edition, Horwood, Chichester, 2007.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2508) PDF downloads(150) Cited by(0)

Article outline

Figures and Tables

Figures(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog