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DYNAMICAL SYSTEMS ANALYSIS OF A MODEL DESCRIBING TASMANIAN DEVIL FACIAL TUMOUR DISEASE

Part of: Local and nonlocal bifurcation theory Asymptotic theory Stability theory Approximation methods and numerical treatment of dynamical systems

Published online by Cambridge University Press:  18 March 2013

N. J. BEETON  and
L. K. FORBES
N. J. BEETON*
Affiliation:
School of Zoology, University of Tasmania, Sandy Bay, Tasmania 7005, Australia
L. K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania
*
✉[email protected]
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Abstract

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A susceptible–exposed–infectious theoretical model describing Tasmanian devil population and disease dynamics is presented and mathematically analysed using a dynamical systems approach to determine its behaviour under a range of scenarios. The steady states of the system are calculated and their stability analysed. Closed forms for the bifurcation points between these steady states are found using the rate of removal of infected individuals as a bifurcation parameter. A small-amplitude Hopf region, in which the populations oscillate in time, is shown to be present and subjected to numerical analysis. The model is then studied in detail in relation to an unfolding parameter which describes the disease latent period. The model’s behaviour is found to be biologically reasonable for Tasmanian devils and potentially applicable to other species.

Keywords

Tasmanian devilepidemic modelnonlinear dynamicsstabilityHopf bifurcation

MSC classification

Secondary: 92D25: Population dynamics (general) 34D20: Stability 34E10: Perturbations, asymptotics 37M20: Computational methods for bifurcation problems 37M05: Simulation 37G15: Bifurcations of limit cycles and periodic orbits
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 1-2 , October 2012 , pp. 89 - 107
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Anderson, R.  M. and May, R.  M., Infectious diseases of humans: dynamics and control (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
Arrowsmith, D.  K. and Place, C.  M., Dynamical systems: differential equations, maps, and chaotic behaviour (Chapman & Hall, London, 1992).CrossRefGoogle Scholar
Barlow, N.  D., “A spatially aggregated disease host model for bovine Tb in New Zealand possum populations”, J. Appl. Ecol. 28 (1991) 777–793; doi:10.2307/2404207.CrossRefGoogle Scholar
Beeton, N.  J. and McCallum, H.  I., “Models predict that culling is not a feasible strategy to prevent extinction of Tasmanian devils from facial tumour disease”, J. Appl. Ecol. 48 (2011) 1315–1323; doi:10.1111/j.1365-2664.2011.02060.x.CrossRefGoogle Scholar
Diekmann, O. and Heesterbeek, J.  A.  P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (Wiley, Chichester, 2000).Google Scholar
Edelstein-Keshet, L., Mathematical models in biology, Volume 46 of Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2005).CrossRefGoogle Scholar
Fouchet, D., Leblanc, G., Sauvage, F., Guiserix, M., Poulet, H. and Pontier, D., “Using dynamic stochastic modelling to estimate population risk factors in infectious disease: the example of FIV in 15 cat populations”, PLoS One 4 (2009) 1–12; doi:10.1371/journal.pone.0007377.CrossRefGoogle ScholarPubMed
Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, New York, 1990).Google Scholar
Hamede, R., Lachish, S., Belov, K., Woods, G., Kreiss, A., Pearse, A.-M., Lazenby, B., Jones, M. and McCallum, H., “Reduced effect of Tasmanian devil facial tumor disease at the disease front”, Conserv. Biol. 26 (2012) 124–134; doi:10.1111/j.1523-1739.2011.01747.x.CrossRefGoogle ScholarPubMed
Hawkins, C.  E., McCallum, H., Mooney, N., Jones, M. and Holdsworth, M., Sarcophilus harrisii, IUCN Red List of Threatened Species, Version 2011.2, 2011, IUCN.Google Scholar
McCallum, H., “Tasmanian devil facial tumour disease: lessons for conservation biology”, Trends Ecol. Evol. 23 (2008) 631–637; doi:10.1016/j.tree.2008.07.001.CrossRefGoogle ScholarPubMed
McCallum, H., Barlow, N. and Hone, J., “How should pathogen transmission be modelled?”, Trends Ecol. Evol. 16 (2001) 295–300; doi:10.1016/S0169-5347(01)02144-9.CrossRefGoogle ScholarPubMed
McCallum, H. and Jones, M., “To lose both would look like carelessness: Tasmanian devil facial tumour disease”, PLoS Biol. 4 (2006) 1671–1674; doi:10.1371/journal.pbio.0040342.CrossRefGoogle ScholarPubMed
McCallum, H., Jones, M., Hawkins, C., Hamede, R., Lachish, S., Sinn, D.  L., Beeton, N. and Lazenby, B., “Transmission dynamics of Tasmanian devil facial tumor disease may lead to disease-induced extinction”, Ecology 90 (2009) 3379–3392; doi:10.1890/08-1763.1.CrossRefGoogle ScholarPubMed
Murray, J.  D., Mathematical biology: I. An introduction, 3rd edn (Springer-Verlag, New York, 2002).CrossRefGoogle Scholar
R Development Core Team, R: A language and environment for statistical computing (R Foundation for Statistical Computing, Vienna, 2011).Google Scholar
Roberts, M. G., “The pluses and minuses of ${R}_{0} $”, J. Roy. Soc. Interface 4 (2007) 949–961; doi:10.1098/rsif.2007.1031.CrossRefGoogle Scholar
Roberts, M. G. and Jowett, J., “An SEI model with density-dependent demographics and epidemiology”, IMA J. Math. Appl. Med. 13 (1996) 245–257; http://imammb.oxfordjournals.org/content/13/4/245.CrossRefGoogle ScholarPubMed
Sterner, R.  T. and Smith, G.  C., “Modelling wildlife rabies: Transmission, economics, and conservation”, Biol. Conserv. 131 (2006) 163–179; doi:10.1016/j.biocon.2006.05.004.CrossRefGoogle Scholar
Wasserberg, G., Osnas, E.  E., Rolley, R.  E. and Samuel, M.  D., “Host culling as an adaptive management tool for chronic wasting disease in white-tailed deer: a modelling study”, J. Appl. Ecol. 46 (2009) 457–466; doi:10.1111/j.1365-2664.2008.01576.x.CrossRefGoogle ScholarPubMed