Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-02T04:29:37.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical Solutions for an Avian Influenza Epidemic Model incorporating Spatial Spread as a Diffusive Process

Published online by Cambridge University Press:  28 May 2015

Phontita Thiuthad ,
Valipuram S. Manoranjan  and
Yongwimon Lenbury
Phontita Thiuthad
Affiliation:
Department of Mathematics, Faculty of Science, Mahidol University, 272 Rama 6 Road, Bangkok 10400, Thailand Centre of Excellence in Mathematics, CHE, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
Valipuram S. Manoranjan
Affiliation:
Department of Mathematics, Washington State University, Pullman WA 99164, USA
Yongwimon Lenbury*
Affiliation:
Department of Mathematics, Faculty of Science, Mahidol University, 272 Rama 6 Road, Bangkok 10400, Thailand Centre of Excellence in Mathematics, CHE, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
*
*Corresponding author. Email addresses: [email protected] (P. Thiuthad), [email protected] (V.S. Manoranjan), [email protected] (Y. Lenbury)
Get access

Abstract

We consider a theoretical model for the spread of avian influenza in a poultry population. An avian influenza epidemic model incorporating spatial spread as a diffusive process is discussed, where the infected individuals are restricted from moving to prevent spatial transmission but infection occurs when susceptible individuals come into contact with infected individuals or the virus is contracted from the contaminated environment (e.g. through water or food). The infection is assumed to spread radially and isotropically. After a stability and phase plane analysis of the equivalent system of ordinary differential equations, it is shown that an analytical solution can be obtained in the form of a travelling wave. We outline the methodology for finding such analytical solutions using a travelling wave coordinate when the wave is assumed to move at constant speed. Numerical simulations also produce the travelling wave solution, and a comparison is made with some predictions based on empirical data reported in the literature.

Keywords

34D2035C0792B05Travelling wave solutionsreaction-diffusion equationsavian influenza epidemic
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 2 , May 2015 , pp. 150 - 159
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chang, C.M. and Manoranjan, V.S., Travelling wave solutions of a contaminant transport model with nonlinear sorption, Math. Comput. Modelling 28, 1–10 (1998).Google Scholar
[2]Ferguson, N.M., Cummings, D.A.T., Cauchemez, S., Fraser, C., Riley, S., Meeyai, A., Iamsiritha-worn, S. and Burke, D.S., Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature 437, 209–214 doi:10.1038/nature04017 (2005).Google Scholar
[3]Food and Agriculture Organization of the United Nations (FAO), Avian Influenza, http://www.fao.org/avianflu/en/index.html.Google Scholar
[4]Fournié, G., Guitian, J., Desvaux, S., Cuong, V.C., Do, H. Dung, Pfeiffer, D.U., Mangtani, P. and Ghani, A.C., Interventions for avian influenza A (H5N1) risk management in live bird market networks, Proc. Nat. Acad. Sci. U.S.A. 110, 9177–9182 doi: 10.1073/pnas.1220815110 (2013).Google Scholar
[5]Fournié, G., Guitian, J., Desvaux, S., Mangtani, P., Ly, S., Cong, V.C., San, S., Dung, D.H., Holl, D., Pfeiffer, D.U., Vong, S. and Ghani, A.C., Identifying live bird markets with the potential to act as reservoirs of avian influenza A (H5N1) virus: A survey in Northern Viet Nam and Cambodia, PLoS ONE 7, doi:10.1371/journal.pone.0037986 (2012).Google Scholar
[6]Guan, Y., Chen, H., Li, K.S., Riley, S., Leung, G.M., Webster, R., Peiris, J.S.M. and Yuen, K.Y., A model to control the epidemic of H5N1 influenza at the source, BMC Infectious Diseases 7, doi:10.1186/1471-2334-7-132 (2007).Google Scholar
[7]Iwami, S., Takeuchi, Y. and Liu, X., Avian-human influenza epidemic model, Math. Biosciences 207, 1–25 (2007).Google Scholar
[8]Kermack, W.O. and Mckendrick, A.G., A contribution to the mathematical theory of epidemics, Proc. R. Soc. A. 115, 700–721 (1927).Google Scholar
[9]Kim, K.I., Lin, Z. and Zhang, L., Avian-human influenza epidemic model with diffusion, Nonlinear Anal. 11, 313–322 (2010).Google Scholar
[10]Liu, M. and Xiao, Y., Modeling and analysis of epidemic diffusion with population migration, Applied Math. 2013, 1–8 doi:10.1155/2013/583648 (2013).Google Scholar
[11]Lucchetti, J., Roy, M. and Martcheva, M., An avian influenza model and its fit to human avian influenza cases, in Advances in Disease Epidemiology, (Tchuenche, J.M. and Mukandavire, Z., Eds.), pp. 1–30, Nova Science Publishers, New York (2009).Google Scholar
[12]Manoranjan, V.S. and Lee, I., Analysis of a population model with efficient resource utilization, J. Interd. Math. 13, 41–50 (2010).Google Scholar
[13]Martin, V., Zhou, X., Marshall, E., Jia, B., Fusheng, G., FrancoDixon, M.A., Dehaan, N., Pfeiffer, D.U., Magalhaes, R.J. Soares and Gilbert, M., Risk-based surveillance for avian influenza control along poultry market chains in South China: The value of social network analysis, Preventive Veterinary Medicine 102, 196–205 (2011).Google Scholar
[14]Mubayi, A., Zaleta, C.K., Martcheva, M. and Castillo-Chavez, C., A Cost-based comparison of quarantine strategies for new emerging diseases, Math. Biosc. Eng. 7, 687–717 (2010).Google Scholar
[15]Office International des Epizooties (OIE), Update on Highly Pathogenic Avian Influenza in Animals (Type H5 and H7), http://www.oie.int/animal-health-in-the-world/update-on-avian-influenza/.Google Scholar
[16]Pappis, C.P., Rachaniotis, N.P. and Dasaklis, T., A deterministic resource scheduling model in epidemic logistics, Theory Appl. MISTA, 570–580 (2009).Google Scholar
[17]Prasertsang, P., Manoranjan, V.S. and Lenbury, Y., Analytical travelling wave solutions of a dental plaque model with nonlinear sorption, Nonlinear Studies 18, 87–97 (2011).Google Scholar
[18]Rattanakul, C. and Lenbury, Y., Stability analysis and analytical solution of a nonlinear model for controlled drug release: Travelling wave fronts, Int. J. Math. Comp. in Simulation 6, 351–359 (2012).Google Scholar
[19]Roche, B., Lebarbenchon, C., Gauthier-Clerc, M., Chang, C.M., Thomas, F., Renaud, F., Werf, S. Van Der and Guegan, J.F., Water-borne transmission drives avian influenza dynamics in wild birds: The case of the 2005-2006 epidemics in the Camargue area, Infect. Genet. Evol. 9, 800–805 (2009).Google Scholar
[20]Samanta, G.P., Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay, Math. Comp, Modeling, 52, 1794–1811 (2010).Google Scholar
[21]Magalhaes, R.J. Soares, Zhou, X., Jia, B., Guo, F., Pfeiffer, D.U. and Martin, V., Live poultry trade in Southern China provinces and HPAIV H5N1 infection in humans and poultry: The role of Chinese New Year festivities, PLoS ONE 7, doi:10.1371/journal.pone.0049712 (2012).Google Scholar
[22]Stallknecht, D.E., Shane, S.M., Kearney, M.T. and Zwank, P.J., Persistence of avian influenza viruses in water, Avian. Dis. 34), 406–411 (1990).CrossRefGoogle ScholarPubMed
[23]Tien, J.H. and Earn, D.J., Multiple transmission pathways and disease dynamics in a water-borne pathogen model, Bull. Math. Biol. 72, 1506–1533 (2010).Google Scholar
[24]Kerkhove, M.D. Van, Vong, S., Guitian, J., Holl, D., Mangtani, P., San, S. and Ghani, A.C., Poultry movement networks in Cambodia: Implications for surveillance and control of highly pathogenic avian influenza (HPAI/H5N1), Vaccine 27, ISSN:0264-410X, 6345–6352 (2009).Google Scholar
[25]Webster, R.G., Yakhno, M., Hinshaw, V.S., Bean, W.J. and Murti, K.G., Intestinal influenza: Replication and characterization of influenza viruses in ducks, Virology 84, 268–278 (1978).Google Scholar
[26]World Health Organization (WHO), Influenza at the Human-Animal Interface (HAI), http://www.who.int/csr/disease/avianinfluenza/aitimeline/en/index.html.Google Scholar