Abstract and Applied Analysis

Global Stability of Multigroup SIRS Epidemic Model with Varying Population Sizes and Stochastic Perturbation around Equilibrium

Xiaoming Fan

Full-text: Open access

Abstract

We discuss multigroup SIRS (susceptible, infectious, and recovered) epidemic models with random perturbations. We carry out a detailed analysis on the asymptotic behavior of the stochastic model; when reproduction number 0>1, we deduce the globally asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time average. Numerical methods are employed to illustrate the dynamic behavior of the model and simulate the system of equations developed. The effect of the rate of immunity loss on susceptible and recovered individuals is also analyzed in the deterministic model.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 154725, 14 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277375

Digital Object Identifier
doi:10.1155/2014/154725

Mathematical Reviews number (MathSciNet)
MR3166570

Zentralblatt MATH identifier
07021823

Citation

Fan, Xiaoming. Global Stability of Multigroup SIRS Epidemic Model with Varying Population Sizes and Stochastic Perturbation around Equilibrium. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 154725, 14 pages. doi:10.1155/2014/154725. https://projecteuclid.org/euclid.aaa/1412277375


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References

  • C. Ji, D. Jiang, and N. Shi, “Multigroup SIR epidemic model with stochastic perturbation,” Physica A, vol. 390, no. 10, pp. 1747–1762, 2011.
  • A. Lajmanovich and J. A. Yorke, “A deterministic model for gonorrhea in a nonhomogeneous population,” Mathematical Biosciences, vol. 28, no. 3-4, pp. 221–236, 1976.
  • E. Beretta and V. Capasso, “Global stability results for a multigroup SIR epidemic model,” in Mathematical Ecology, T. G. Hallam, L. J. Gross, and S. A. Levin, Eds., pp. 317–342, World Scientific, Teaneck, NJ, USA, 1988.
  • H. W. Hethcote, “An immunization model for a heterogeneous population,” Theoretical Population Biology, vol. 14, no. 3, pp. 338–349, 1978.
  • H. W. Hethcote and H. R. Thieme, “Stability of the endemic equilibrium in epidemic models with subpopulations,” Mathematical Biosciences, vol. 75, no. 2, pp. 205–227, 1985.
  • H. R. Thieme, “Local stability in epidemic models for heterogeneous populations,” in Mathematics in Biology and Medicine, V. Capasso, E. Grosso, and S. L. Paveri-Fontana, Eds., vol. 57 of Lecture Notes in Biomathematics, pp. 185–189, Springer, Berlin, Germany, 1985.
  • H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003.
  • H. Guo, M. Y. Li, and Z. Shuai, “A graph-theoretic approach to the method of global Lyapunov functions,” Proceedings of the American Mathematical Society, vol. 136, no. 8, pp. 2793–2802, 2008.
  • M. Y. Li, Z. Shuai, and C. Wang, “Global stability of multi-group epidemic models with distributed delays,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 38–47, 2010.
  • H. Guo, M. Y. Li, and Z. Shuai, “Global stability of the endemic equilibrium of multigroup SIR epidemic models,” Canadian Applied Mathematics Quarterly, vol. 14, no. 3, pp. 259–284, 2006.
  • Y. Muroya, Y. Enatsu, and T. Kuniya, “Global stability for a multi-group SIRS epidemic model with varying population sizes,” Nonlinear Analysis: Real World Applications, vol. 14, no. 3, pp. 1693–1704, 2013.
  • Y. Nakata, Y. Enatsu, and Y. Muroya, “On the global stability of an SIRS epidemic model with distributed delays,” Discrete and Continuous Dynamical Systems A, vol. 2011, pp. 1119–1128, 2011.
  • Y. Enatsu, Y. Nakata, and Y. Muroya, “Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2120–2133, 2012.
  • C. C. McCluskey, “Complete global stability for an SIR epidemic model with delay–-distributed or discrete,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 55–59, 2010.
  • N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model of AIDS and condom use,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 36–53, 2007.
  • C. Ji, D. Jiang, and N. Shi, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 482–498, 2009.
  • C. Ji, D. Jiang, and X. Li, “Qualitative analysis of a stochastic ratio-dependent predator-prey system,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1326–1341, 2011.
  • E. Tornatore, S. M. Buccellato, and P. Vetro, “Stability of a stochastic SIR system,” Physica A, vol. 354, no. 1–4, pp. 111–126, 2005.
  • E. Beretta, V. Kolmanovskii, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269–277, 1998.
  • L. Shaikhet, “Stability of predator-prey model with aftereffect by stochastic perturbation,” Stability and Control, vol. 1, no. 1, pp. 3–13, 1998.
  • M. Carletti, “On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,” Mathematical Biosciences, vol. 175, no. 2, pp. 117–131, 2002.
  • M. Bandyopadhyay and J. Chattopadhyay, “Ratio-dependent predator-prey model: effect of environmental fluctuation and stability,” Nonlinearity, vol. 18, no. 2, pp. 913–936, 2005.
  • R. R. Sarkar and S. Banerjee, “Cancer self remission and tumor stability–-a stochastic approach,” Mathematical Biosciences, vol. 196, no. 1, pp. 65–81, 2005.
  • M. Bandyopadhyay, T. Saha, and R. Pal, “Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 3, pp. 958–970, 2008.
  • L. Shaikhet, “Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations,” Dynamic Systems and Applications, vol. 17, no. 1, pp. 235–253, 2008.
  • N. Bradul and L. Shaikhet, “Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: numerical analysis,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 92959, 25 pages, 2007.
  • B. Mukhopadhyay and R. Bhattacharyya, “A nonlinear mathematical model of virus-tumor-immune system interaction: deterministic and stochastic analysis,” Stochastic Analysis and Applications, vol. 27, no. 2, pp. 409–429, 2009.
  • C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2501–2516, 2012.
  • J. Yu, D. Jiang, and N. Shi, “Global stability of two-group SIR model with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 235–244, 2009.
  • L. Imhof and S. Walcher, “Exclusion and persistence in deterministic and stochastic chemostat models,” Journal of Differential Equations, vol. 217, no. 1, pp. 26–53, 2005.
  • X. Fan and Z. Wang, “Stability analysis of an SEIR epidemic model with stochastic perturbation and numerical simulation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 14, no. 2, pp. 113–121, 2013.
  • X. Fan, Z. Wang, and X. Xu, “Global stability of two-group epidemic models with distributed delays and random perturbation,” Abstract and Applied Analysis, vol. 2012, Article ID 132095, 12 pages, 2012.
  • Z. Wang, X. Fan, and Q. Han, “Global stability of deterministic and stochastic multigroup SEIQR models in computer network,” Applied Mathematical Modelling, vol. 37, no. 20-21, pp. 8673–8686, 2013.
  • X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 2nd edition, 2008. \endinput