Mathematical Biosciences and Engineering, 2013, 10(2): 463-481. doi: 10.3934/mbe.2013.10.463.

Primary: 92D25, 92D30; Secondary: 37G99.

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On latencies in malaria infections and their impact on the disease dynamics

1. Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7
2. Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

In this paper, we modify the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1 the="" disease="" free="" equilibrium="" e_0="" is="" globally="" asymptotically="" stable="" meaning="" that="" the="" malaria="" disease="" will="" eventually="" die="" out="" and="" if="" mathcal="" r="" _0="">1$, $E_0$ becomes unstable.When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions.For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Lyapunov function/functional; stability; latency; Malaria; persistence.; basicreproduction number; delay

Citation: Yanyu Xiao, Xingfu Zou. On latencies in malaria infections and their impact on the disease dynamics. Mathematical Biosciences and Engineering, 2013, 10(2): 463-481. doi: 10.3934/mbe.2013.10.463

References

  • 1. Oxford University Press, Oxford, 1991.
  • 2. in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179.
  • 3. Bull. Math. Biol., 73 (2011), 639-657.
  • 4. in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50.
  • 5. J. Math. Biol., 35 (1990), 503-522.
  • 6. J. R. Soc. Interface, 7 (2011), 873-885.
  • 7. Can. Appl. Math. Q., 14 (2006), 259-284.
  • 8. Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
  • 9. Springer-Verlag, New York, 1993.
  • 10. J. R. Soc. Interface, 2 (2005), 281-293.
  • 11. Comm. Pure Appl. Math., 38 (1985), 733-753.
  • 12. Math. Med. Biol., 21 (2004), 75-83.
  • 13. Math. Biosci. Eng., 1 (2004), 57-60.
  • 14. J. Math. Biol., 62 (2011), 543-568.
  • 15. Benjamin, Menlo Park, California, 1971.
  • 16. Trop. Dis. Bull., 49 (1952), 569-585.
  • 17. Bull. WHO, 15 (1956), 613-626.
  • 18. Oxford University Press, London, 1957.
  • 19. J. Murray, London, 1910.
  • 20. Bull. Math. Biol., 70 (2008), 1098-1114.
  • 21. 41. AMS, Providence, 1995.
  • 22. Malaria Journal, 3 (2004), 24 pp.
  • 23. Princeton University Press, Princeton, NJ, 2003.
  • 24. SIAM J. Math. Anal., 24 (1993), 407-435.
  • 25. Math. Biosci. Eng., 4 (2007), 205-219.
  • 26. Math. Biosci., 180 (2002), 29-48.

 

This article has been cited by

  • 1. Yan Zhang, Sanyang Liu, Zhenguo Bai, A periodic malaria model with two delays, Physica A: Statistical Mechanics and its Applications, 2019, 123327, 10.1016/j.physa.2019.123327

Reader Comments

your name: *   your email: *  

Copyright Info: 2013, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved