Journal of Applied Mathematics

Permanence and Extinction for a Nonautonomous Malaria Transmission Model with Distributed Time Delay

Xiaohong Zhang, Jianwen Jia, and Xinyu Song

Full-text: Open access

Abstract

We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 139046, 15 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305744

Digital Object Identifier
doi:10.1155/2014/139046

Mathematical Reviews number (MathSciNet)
MR3208615

Citation

Zhang, Xiaohong; Jia, Jianwen; Song, Xinyu. Permanence and Extinction for a Nonautonomous Malaria Transmission Model with Distributed Time Delay. J. Appl. Math. 2014 (2014), Article ID 139046, 15 pages. doi:10.1155/2014/139046. https://projecteuclid.org/euclid.jam/1425305744


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References

  • P. M. Hao, D. J. Fan, and J. J. Wei, “Dynamic behaviors of a delayed HIV model with stage-structure,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 4753–4766, 2012.
  • H. Y. Shu, D. J. Fan, and J. J. Wei, “Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1581–1592, 2012.
  • L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010.
  • J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi, “A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1953–1965, 2007.
  • J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi, “A host-vector model for malaria with infective immigrants,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 139–149, 2010.
  • J. Crawley and B. Nahlen, “Prevention and treatment of malaria in young African children,” Seminars in Pediatric Infectious Diseases, vol. 15, no. 3, pp. 169–180, 2004.
  • M. H. Craig, R. W. Snow, and D. Sueur, “A climate-based distribution model of malaria transmission in sub-Saharan Africa,” Parasitology Today, vol. 15, no. 3, pp. 105–111, 1999.
  • L. Wang, Z. D. Teng, and T. L. Zhang, “Threshold dynamics of a malaria transmission model in periodic environment,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 1288–1303, 2013.
  • Y. Lou and X.-Q. Zhao, “A climate-based malaria transmission model with structured vector population,” SIAM Journal on Applied Mathematics, vol. 70, no. 6, pp. 2023–2044, 2010.
  • L. Liu, X. Zhao, and Y. Zhou, “Atuberculosis model with seasonality,” Bulletin of Mathematical Biology, vol. 72, pp. 931–952, 2010.
  • Y. Nakata and T. Kuniya, “Global dynamics of a class of SEIRS epidemic models in a periodic environment,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 230–237, 2010.
  • Y. Yang and Y. Xiao, “Threshold dynamics for an HIV model in periodic environments,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 59–68, 2010.
  • W. Wang and X.-Q. Zhao, “Threshold dynamics for compartmental epidemic models in periodic environments,” Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 699–717, 2008.
  • G. P. Samanta, “Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1794–1811, 2010.
  • D. W. Jordan and P. Smith, Nonlinear Ordinary Diffrential Equations, Oxford University Press, New York, NY, USA, 2004.
  • T. L. Zhang and Z. D. Teng, “Permanence and extinction for a nonautonomous SIRS epidemic model with time delay,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 1058–1071, 2009.
  • G. P. Samanta, “Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1163–1177, 2011. \endinput