Abstract and Applied Analysis

Analysis of a Mathematical Model of Emerging Infectious Disease Leading to Amphibian Decline

Muhammad Dur-e-Ahmad, Mudassar Imran, and Adnan Khan

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Abstract

We formulate a three-dimensional deterministic model of amphibian larvae population to investigate the cause of extinction due to the infectious disease. The larvae population of the model is subdivided into two classes, exposed and unexposed, depending on their vulnerability to disease. Reproduction ratio 0 has been calculated and we have shown that if 0<1, the whole population will be extinct. For the case of 0>1, we discussed different scenarios under which an infected population can survive or be eliminated using stability and persistence analysis. Finally, we also used Hopf bifurcation analysis to study the stability of periodic solutions.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 145398, 13 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412605855

Digital Object Identifier
doi:10.1155/2014/145398

Mathematical Reviews number (MathSciNet)
MR3206767

Zentralblatt MATH identifier
07021806

Citation

Dur-e-Ahmad, Muhammad; Imran, Mudassar; Khan, Adnan. Analysis of a Mathematical Model of Emerging Infectious Disease Leading to Amphibian Decline. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 145398, 13 pages. doi:10.1155/2014/145398. https://projecteuclid.org/euclid.aaa/1412605855


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References

  • R. M. Lehtinen, S. M. Galatowitsch, and J. R. Tester, “Consequences of habitat loss and fragmentation for wetland amphibian assemblages,” Wetlands, vol. 19, no. 1, pp. 1–12, 1999.
  • S. N. Stuart, J. S. Chanson, N. A. Cox et al., “Status and trends of amphibian declines and extinctions worldwide,” Science, vol. 306, no. 5702, pp. 1783–1786, 2004.
  • P. Daszak, L. Berger, A. A. Cunningham, A. D. Hyatt, D. Earl Green, and R. Speare, “Emerging infectious diseases and amphibian population declines,” Emerging Infectious Diseases, vol. 5, no. 6, pp. 735–748, 1999.
  • S. A. Diamond, G. S. Peterson, J. E. Tietge, and G. T. Ankley, “Assessment of the risk of solar ultraviolet radiation to amphibians. III. Prediction of impacts in selected northern midwestern wetlands,” Environmental Science and Technology, vol. 36, no. 13, pp. 2866–2874, 2002.
  • B. A. Pierce, “The effect of acid precipitation on amphibians,” Ecotoxicology, vol. 2, no. 1, pp. 65–77, 1993.
  • L. E. Licht and K. P. Grant, “The effects of ultraviolet radiation on the biology of amphibians,” American Zoologist, vol. 37, no. 2, pp. 137–145, 1997.
  • S. J. Hecnar and R. T. M'Closkey, “The effects of predatory fish on amphibian species richness and distribution,” Biological Conservation, vol. 79, no. 2-3, pp. 123–131, 1997.
  • K. R. Lips, F. Brem, R. Brenes et al., “Emerging infectious disease and the loss of biodiversity in a Neotropical amphibian community,” Proceedings of the National Academy of Sciences of the United States of America, vol. 103, no. 9, pp. 3165–3170, 2006.
  • H. R. Thieme, T. Dhirasakdanon, Z. Han, and R. Trevino, “Species decline and extinction: synergy of infectious disease and Allee effect?” Journal of Biological Dynamics, vol. 3, no. 2-3, pp. 305–323, 2009.
  • L. Berger, R. Speare, P. Daszak et al., “Chytridiomycosis causes amphibian mortality associated with population declines in the rain forests of Australia and Central America,” Proceedings of the National Academy of Sciences of the United States of America, vol. 95, no. 15, pp. 9031–9036, 1998.
  • S. E. Drury, R. E. Gough, and A. A. Cunningham, “Isolation of an iridovirus-like agent from common frogs (Rana temporaria),” The Veterinary Record, vol. 137, no. 3, pp. 72–73, 1995.
  • L. J. Rachowicz, R. A. Knapp, J. A. T. Morgan et al., “Emerging infectious disease as a proximate cause of amphibian mass mortality,” Ecology, vol. 87, no. 7, pp. 1671–1683, 2006.
  • A. Richards, J. Cory, M. Speight, and T. Williams, “Foraging in a pathogen reservoir can lead to local host population extinction: a case study of a Lepidoptera-virus interaction,” Oecologia, vol. 118, no. 1, pp. 29–38, 1999.
  • H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, USA, 2003.
  • J. P. Collins, “Intrapopulation variation in the body size at metamorphosis and timing of metamorphosis in the bullfrog, Rana castesbeiana,” Ecology, vol. 60, pp. 738–749, 1979.
  • P. L. Salceanu and H. L. Smith, “Persistence in a discrete-time stage-structured fungal disease model,” Journal of Biological Dynamics, vol. 3, no. 2-3, pp. 271–285, 2009.
  • P. L. Salceanu, “Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents,” Mathematical Biosciences and Engineering, vol. 8, no. 3, pp. 807–825, 2011.
  • H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1995.
  • F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations, Dover Books on Mathematics, New York, NY, USA, 1989. \endinput