Mathematical Biosciences and Engineering, 2015, 12(1): 99-115. doi: 10.3934/mbe.2015.12.99.

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Global dynamics of a general class of multi-group epidemic models with latency and relapse

1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046
2. Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseasesin heterogeneous populations.In each group, the population is divided intosusceptible, exposed, infectious, and recovered subclasses. A generalnonlinear incidence rate is used in the model. The results show that the global dynamics are completelydetermined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to provethe global stability of the disease-free equilibrium when $R_0\leq1,$while a new combinatorial identity (Theorem 3.3 in Shuai and vanden Driessche [29]) in graph theory is applied to provethe global stability of the endemic equilibrium when $R_0>1.$We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted intoa graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.
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Keywords global stability; Multigroup epidemic model; nonlinear incidence; Lyapunov function.

Citation: Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences and Engineering, 2015, 12(1): 99-115. doi: 10.3934/mbe.2015.12.99

References

  • 1. Academic Press, New York, 1979.
  • 2. Lecture Notes in Math. 35, Springer, Berlin, 1967.
  • 3. SIAM J. Appl. Math., 56 (1996), 494-508.
  • 4. SIAM J. Appl. Math., 59 (1999), 1790-1811.
  • 5. J. Math. Biol., 28 (1990), 365-382.
  • 6. Math. Biosci., 246 (2013), 105-112.
  • 7. J. Dynam. Diff. Equat., 6 (1994), 583-600.
  • 8. Addison-Wesley, Reading, MA, 1969.
  • 9. J. Theor. Biol., 300 (2012), 39-47.
  • 10. SIAM J. Appl. Math., 72 (2012), 819-841.
  • 11. Tuber Lung Dis., 75 (1994), 341-347.
  • 12. Can. Appl. Math. Q., 14 (2006), 259-284.
  • 13. SIAM J. Appl. Math., 69 (2009), 1205-1227.
  • 14. PLoS Med., 3 (2006), e492.
  • 15. Bull. Math. Biol., 71 (2009), 75-83.
  • 16. Math. Biosci., 28 (1976), 221-236.
  • 17. Lancet Infect. Dis., 3 (2003), 282-287.
  • 18. Reginal Conf. Ser. Appl., SIAM, Philadelphia, 1976.
  • 19. Math. Biosci., 160 (1999), 191-213.
  • 20. J. Math. Anal. Appl., 361 (2010), 38-47.
  • 21. J. Diff. Equat., 248 (2010), 1-20.
  • 22. Math. Biosci. Eng., 7 (2010), 675-685.
  • 23. Nonlinear Anal. Real World Appl., 12 (2011), 119-127.
  • 24. Liver Transpl., 11 (2005), 402-409.
  • 25. Nonlinear Anal. Real World Appl., 14 (2013), 1693-1704.
  • 26. Nonlinear Anal. Real World Appl., 13 (2012), 1581-1592.
  • 27. Math. Biosci., 234 (2011), 118-126.
  • 28. Math. Biosci. Eng., 9 (2012), 393-411.
  • 29. SIAM J. Appl. Math., 73 (2013), 1513-1532.
  • 30. Cambridge University Press, Cambridge, UK, 1995.
  • 31. Lancet, 358 (2001), 1687-1693.
  • 32. Appl. Math. Comput., 218 (2011), 280-286.
  • 33. Ann. Internal Med., 154 (2011), 593-601.
  • 34. Math. Biosci., 180 (2002), 29-48.
  • 35. Math. Biosci. Eng., 4 (2007), 205-219.
  • 36. Math. Biosci., 207 (2007), 89-103.
  • 37. J. Math. Biol., 69 (2014), 875-904.
  • 38. Bull. Math. Biol., 74 (2012), 1226-1251.

 

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