Mathematical Biosciences and Engineering, 2013, 10(5&6): 1335-1349. doi: 10.3934/mbe.2013.10.1335.

Primary: 92D30; Secondary: 35B35, 35Q92.

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Dynamics of an age-of-infection cholera model

1. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2
2. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3R4
3. Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.
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Keywords Lyapunov functional; Cholera model; final size.; global stability; age-of-infection

Citation: Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1335-1349. doi: 10.3934/mbe.2013.10.1335

References

  • 1. Nonlinear Anal. Real World Appl., 12 (2011), 3483-3498.
  • 2. Math. Biosci. Eng., 5 (2008), 681-690.
  • 3. Second edition, Springer, New York, 2012.
  • 4. Math. Biosc. Eng., 7 (2010), 1-15.
  • 5. Lecture Notes in Math., Vol. 1945, Springer, Berlin, 2008.
  • 6. Proc. Natl. Acad. Sci. USA, 108 (2011), 7081-7085.
  • 7. CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, 1998.
  • 8. Science, 330 (2010), 738-739.
  • 9. PLOS Med., 3 (2006), 63-69.
  • 10. Math. Biosci. Eng., 9 (2012), 297-312.
  • 11. SIAM J. Appl. Math., 72 (2012), 25-38.
  • 12. Math. Biosci., 155 (1999), 77-109.
  • 13. Science, 323 (2009), 860-861.
  • 14. Bull. Math. Biol., 68 (2006), 679-702.
  • 15. Appl. Anal., 89 (2010), 1109-1140.
  • 16. Appl. Math. Comput., 217 (2010), 3046-3049.
  • 17. Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772.
  • 18. Math. Biosci., 234 (2010), 118-126.
  • 19. Graduate Studies in Mathematics, Vol. 118, American Mathematical Society, Providence, 2011.
  • 20. SIAM J. Appl. Math., 53 (1993), 1447-1479.
  • 21. Bull. Math. Biol., 72 (2010), 1506-1533.
  • 22. Ann. Internal Med., 154 (2011), 593-601.
  • 23. Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985.
  • 24. Weekly Epidemiological Record, 82 (2007), 273-284.
  • 25. Weekly Epidemiological Record, 84 (2008), 309-324.
  • 26. August 2011. Available from: http://www.who.int.

 

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