Mathematical Biosciences and Engineering, 2014, 11(3): 449-469. doi: 10.3934/mbe.2014.11.449.

Primary: 34D23; Secondary: 34G20, 35B35, 92D30.

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The global stability of an SIRS model with infection age

1. Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi

Infection age is an important factor affecting the transmission ofinfectious diseases. In this paper, we consider an SIRS modelwith infection age, which is described by a mixed system ofordinary differential equations and partial differentialequations. The expression of the basic reproduction number$\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then themodel only has the disease-free equilibrium, while if$\mathscr{R}_0>1$ then besides the disease-free equilibrium themodel also has an endemic equilibrium. Moreover, if$\mathscr{R}_0<1 then="" the="" disease-free="" equilibrium="" is="" globally="" asymptotically="" stable="" otherwise="" it="" is="" unstable="" if="" mathscr="" r="" _0="">1$ then the endemicequilibrium is globally asymptotically stable under additional conditions. The local stabilityis established through linearization. The global stability of thedisease-free equilibrium is shown by applying the fluctuationlemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
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Keywords global stability; persistence.; SIRS model; infection age

Citation: Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences and Engineering, 2014, 11(3): 449-469. doi: 10.3934/mbe.2014.11.449

References

  • 1. Math. Models Methods Appl. Sci., 21 (2011), 693-718.
  • 2. J. Math. Bioi., 27 (1989), 233-258.
  • 3. Appl. Math. Comput., 217 (2010), 4010-4016.
  • 4. Nonlineaity, 24 (2011), 2891-2911.
  • 5. SIAM. J. Appl. Math., 62 (2002), 1634-1656.
  • 6. Lancet, 2 (1984), 1276-1277.
  • 7. Springer-Verlag, Berlin, 1984.
  • 8. Comm. Pure Appl. Math., 38 (1985), 733-753.
  • 9. J. Biol. Dyn., 1 (2007), 109-131.
  • 10. Math. Biosci., 190 (2004), 39-69.
  • 11. Appl. Math. Comput., 218 (2012), 6519-6525.
  • 12. British Medical J., 293 (1986), 1459-1462.
  • 13. Chaos Solitons Fractals, 40 (2009), 145-158.
  • 14. Z. Angew. Math. Phys., 62 (2011), 191-222.
  • 15. Electron. J. Differntial Equations, 2001 (2001), 35 pp.
  • 16. Appl. Anal., 89 (2010), 1109-1140.
  • 17. SIAM J. Mah. Anal., 37 (2005), 251-275.
  • 18. SIAM J. Appl. Math., 66 (2006), 843-872.
  • 19. Math. Biosci. Eng., 6 (2009), 603-610.
  • 20. British Medical J., 295 (1987), 567-569.
  • 21. Lancet, 2 (1984), 1418-1420.
  • 22. Differential Integral Equations, 3 (1990), 1035-1066.
  • 23. in Advances in Mathematical Population Dynamics-Molecules, Cells and Man (eds. O. Arino, D. Axelrod and M. Kimmel), World Sci. Publ., (1997), 691-711.
  • 24. SIAM J. Appl. Math., 53 (1993), 1447-1479.
  • 25. J. Math. Anal. Appl., 385 (2012), 655-671.
  • 26. Int. J. Biomath., 4 (2011), 329-346.
  • 27. J. Math. Anal. Appl., 331 (2007), 1396-1414.
  • 28. Math. Biosci. Eng., 5 (2008), 403-418.

 

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