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Lyapunov Functionals and Stability of Stochastic Functional Differential Equations pp 283–296Cite as

Stability of SIR Epidemic Model Equilibrium Points

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Abstract

Chapter 11 deals with a mathematical model of the spread of infectious diseases, so called SIR epidemic model. Sufficient conditions for stability in probability of the degenerated equilibrium point \(E_{0}=(b\mu_{1}^{-1},0,0)\) and the positive equilibrium point E ∗=(S ∗,I ∗,R ∗) of SIR epidemic model with distributed delays and stochastic perturbations are obtained. Results of numerical simulations of stability regions and trajectories of stable solutions are shown in 3 figures. Similarly to the previous chapters numerical simulations of the processes S(t), I(t) and R(t) were obtained via the Euler–Maruyama scheme for stochastic differential equations.

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References

  1. Alonso-Quesada S, De la Sen M, Agarwal R, Ibeas A (2012) An observer-based vaccination control law for a SEIR epidemic model based on feedback linearization techniques for nonlinear systems. Adv Differ Equ 2012:161

    Article  Google Scholar 

  2. Bailey NTJ (1957) The mathematical theory of epidemics. Griffin, London

    Google Scholar 

  3. Beretta E, Takeuchi Y (1995) Global stability of an SIR epidemic model with time delays. J Math Biol 33:250–260

    MathSciNet  MATH  Google Scholar 

  4. Beretta E, Kolmanovskii V, Shaikhet L (1998) Stability of epidemic model with time delays influenced by stochastic perturbations. Math Comput Simul 45(3–4):269–277 (Special Issue “Delay Systems”)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beretta E, Hara T, Ma W, Takeuchi Y (2001) Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal 47:4107–4115

    Article  MathSciNet  MATH  Google Scholar 

  6. Buonomo B, Rionero S (2010) On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate. Appl Math Comput 217(8):4010–4016

    Article  MathSciNet  MATH  Google Scholar 

  7. Busenberg S, Cooke KL (1980) The effect of integral conditions in certain equations modelling epidemics and population growth. J Math Biol 10(1):13–32

    Article  MathSciNet  MATH  Google Scholar 

  8. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction R0 in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382

    MathSciNet  MATH  Google Scholar 

  9. Fan X, Wang Z, Xu X (2012) Global stability of two-group epidemic models with distributed delays and random perturbation. Abstr Appl Anal 2012:132095

    MathSciNet  Google Scholar 

  10. Hethcote HW (1976) Qualitative analysis of communicable disease models. Math Biol Sci 28:335–356

    MathSciNet  MATH  Google Scholar 

  11. Jovanovic M, Krstic M (2012) Stochastically perturbed vector-borne disease models with direct transmission. Appl Math Model 36:5214–5228

    Article  MathSciNet  MATH  Google Scholar 

  12. Jumpen W, Orankitjaroen S, Boonkrong P, Wiwatanapataphee B (2011) SEIQR-SIS epidemic network model and its stability. Int J Math Comput Simul 5:326–333

    Google Scholar 

  13. Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton

    MATH  Google Scholar 

  14. Khan H, Mohapatra RN, Vajravelu K, Liao SJ (2009) The explicit series solution of SIR and SIS epidemic models. Appl Math Comput 215:653–669

    Article  MathSciNet  MATH  Google Scholar 

  15. Korobeinikov A (2006) Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull Math Biol 68:615–626

    Article  MathSciNet  Google Scholar 

  16. Korobeinikov A, Maini PK (2005) Nonlinear incidence and stability of infectious disease models. Math Med Biol 22:113–128

    Article  MATH  Google Scholar 

  17. Lahrouz A, Omari L, Kiouach D (2011) Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model. Nonlinear Anal, Model Control 16:59–76

    MathSciNet  Google Scholar 

  18. Li MY, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160:191–213

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma W, Takeuchi Y, Hara T, Beretta E (2002) Permanence of an SIR epidemic model with distributed time delays. Tohoku Math J 54:581–591

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma W, Song M, Takeuchi Y (2004) Global stability of an SIR epidemic model with time delay. Appl Math Lett 17:1141–1145

    Article  MathSciNet  MATH  Google Scholar 

  21. Makinde OD (2007) Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Appl Math Comput 184:842–848

    Article  MathSciNet  MATH  Google Scholar 

  22. Maruyama G (1955) Continuous Markov processes and stochastic equations. Rend Circ Mat Palermo 4:48–90

    Article  MathSciNet  MATH  Google Scholar 

  23. McCluskey CC (2010) Complete global stability for an SIR epidemic model with delay—distributed or discrete. Nonlinear Anal, Real World Appl 11:55–59

    Article  MathSciNet  MATH  Google Scholar 

  24. McCluskey CC (2010) Global stability for an SIR epidemic model with delay and nonlinear incidence. Nonlinear Anal, Real World Appl 11(4):3106–3109

    Article  MathSciNet  MATH  Google Scholar 

  25. Mollison D (2003) Epidemic models: their structure and relation to data. Publications of the Newton Institute. Cambridge University Press, Cambridge

    Google Scholar 

  26. Mukhopadhyay B, Bhattacharyya R (2007) Existence of epidemic waves in a disease transmission model with two-habitat population. Int J Syst Sci 38:699–707

    Article  MathSciNet  MATH  Google Scholar 

  27. Muroya Y, Enatsu Y, Nakata Y (2011) Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate. J Math Anal Appl 377(1):1–14

    Article  MathSciNet  MATH  Google Scholar 

  28. Naresh R, Tripathi A, Tchuenche JM, Sharma D (2009) Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Comput Math Appl 58(2):348–359

    Article  MathSciNet  MATH  Google Scholar 

  29. Song XY, Jiang Y, Wei HM (2009) Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Appl Math Comput 214:381–390

    Article  MathSciNet  MATH  Google Scholar 

  30. Takeuchi Y, Ma W, Beretta E (2000) Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal 42:931–947

    Article  MathSciNet  MATH  Google Scholar 

  31. Tamizhmani KM, Ramani A, Grammaticos B, Carstea AS (2004) Modelling AIDS epidemic and treatment with difference equations. Adv Differ Equ 3:183–193

    MathSciNet  Google Scholar 

  32. Tapaswi PK, Chattopadhyay J (1996) Global stability results of a “susceptible-infective-immune-susceptible” (SIRS) epidemic model. Ecol Model 87(1–3):223–226

    Article  Google Scholar 

  33. Wang W, Xin J, Zhang F (2010) Persistence of an SEIR model with immigration dependent on the prevalence of infection. Discrete Dyn Nat Soc. doi:10.1155/2010/727168. Article ID 727168, 7 pages

    MathSciNet  Google Scholar 

  34. Yang J, Wang X (2010) Existence of a nonautonomous SIR epidemic model with age structure. Adv Differ Equ. Article ID 212858, 23 pages

    Google Scholar 

  35. Zhang J, Jin Z (2010) The analysis of epidemic network model with infectious force in latent and infected period. Discrete Dyn Nat Soc. Article ID 604329, 12 pages

    Google Scholar 

  36. Zhang F, Li Z, Zhang F (2008) Global stability of an SIR epidemic model with constant infectious period. Appl Math Comput 199:285–291

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang TL, Liu JL, Teng ZD (2009) Dynamic behaviour for a nonautonomous SIRS epidemic model with distributed delays. Appl Math Comput 214:624–631

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang Z, Wu J, Suo Y, Song X (2011) The domain of attraction for the endemic equilibrium of an SIRS epidemic model. Math Comput Simul 81:1697–1706

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine

    Leonid Shaikhet

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  1. Leonid Shaikhet
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Shaikhet, L. (2013). Stability of SIR Epidemic Model Equilibrium Points. In: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00101-2_11

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  • DOI: https://doi.org/10.1007/978-3-319-00101-2_11

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-00100-5

  • Online ISBN: 978-3-319-00101-2

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