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Published online by Cambridge University Press:  12 January 2011

National Institute for Mathematical and Biological Synthesis, The University of Tennessee, Knoxville, TN 37996, USA (email:
Department of Mathematical Sciences, Federal University of Technology Akure, P.M.B. 704, Akure, Nigeria (email:
For correspondence; e-mail:
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We present an application of optimal control theory to a simple SIR disease model of avian influenza transmission dynamics in birds. Basic properties of the model, including the epidemic threshold, are obtained. Optimal control theory is adopted to minimize the density of infected birds subject to an appropriate system of ordinary differential equations. We conclude that an optimally controlled seasonal vaccination strategy saves more birds than when there is a low uniform vaccination rate as in resource-limited places.

Research Article
Copyright © Australian Mathematical Society 2011


[1]Agusto, F. B. and Gumel, A. B., “Theoretical assessment of avian influenza vaccine”, DCDS Ser. B 13 (2010) 125.CrossRefGoogle Scholar
[2]Agusto, F. B. and Okosun, K. O., “Optimal seasonal biocontrol for Eichhornia crassipes”, Int. J. Biomath. 3 (2010) 383397.Google Scholar
[3]Anderson, R. M. and May, R. M., Infectious diseases of humans (Oxford University Press, Oxford, 1991).Google Scholar
[4]Brauer, F. and Castillo-Chavez, C., Mathematical models in population biology and epidemiology, Volume 40 of Texts in Applied Mathematics Series (Springer, New York, 2001).Google Scholar
[5]Brauer, F. and van den Driessche, P., “Models for transmission of disease with immigration of infectives”, Math. Biosci. 171 (2001) 143154.Google Scholar
[6]Breiman, R. F., Nasidi, A., Katz, M. A., Njenga, M. K. and Vertefeuille, J., “Preparedness for highly pathogenic avian influenza pandemic in Africa”, Emerg. Infect. Dis. 13 (2007) 14531458.CrossRefGoogle ScholarPubMed
[7]Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A. P., “On the definition and computation of the basic reproduction ratio ℛ0 in models for infectious diseases in heterogeneous populations”, J. Math. Biol. 28 (1990) 365382.Google Scholar
[8]Earn, D. J. D., Dushoff, J. and Levin, S. A., “Ecology and evolution of the flu”, Trends Ecol. Evol. 17 (2002).Google Scholar
[9]Flahault, A., Vergu, E., Coudeville, L. and Grais, R. F., “Strategies for containing a global influenza pandemic”, Vaccine 24 (2006) 67516755.Google Scholar
[10]Fleming, W. H. and Rishel, R. W., Deterministic and stochastic optimal control (Springer, New York, 1975).Google Scholar
[11]Gumel, A. B., “Global dynamics of a two-strain avian influenza model”, Int. J. Comput. Math. 86 (2009) 85108.CrossRefGoogle Scholar
[12]Hethcote, H. W., “The mathematics of infectious diseases”, SIAM Rev. 42 (2000) 599653.CrossRefGoogle Scholar
[13]Joshi, H. R., “Optimal control of an HIV immunology model”, Optim. Control Appl. Math. 23 (2002) 199213.Google Scholar
[14]Kermack, W. O. and McKendrick, A. G., “Contributions to the mathematical theory of epidemics I”, Proc. Roy. Soc. A 115 (1927) 700721; Reprint in Bull. Math. Biol. 53 (1991) 33–55.Google Scholar
[15]Kern, D., Lenhart, S., Miller, R. and Yong, J., “Optimal control applied to native-invasive population dynamics”, J. Biol. Dyn. 1 (2007) 413426.Google Scholar
[16]Kirschner, D., Lenhart, S. and Serbin, S., “Optimal control of the chemotherapy of HIV”, J. Math. Biol. 35 (1997) 775792.Google Scholar
[17]Le Menach, A., Vergu, E., Grais, R. F., Smith, D. L. and Flahault, A., “Key strategies for reducing transmission during avian influenza epidemics: lessons for control of human avian influenza”, Proc. Biol. Sci. 273 (2006) 24672475.Google Scholar
[18]Lenhart, S. and Workman, J. T., Optimal control applied to biological models (Chapman & Hall, Boca Raton, FL, 2007).CrossRefGoogle Scholar
[19]Malcolm, R., “Bird flu may have entered Nigeria 3 times”, Washington Post, July 5, 2006, Associated Press. Scholar
[20]Nũno, M., Chowell, G. and Gumel, A. B., “Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: scenarios for the US, UK, and the Netherlands”, J. R. Soc. Interf. 4 (2007) 505521.CrossRefGoogle ScholarPubMed
[21]Nũno, M., Feng, Z., Martcheva, M. and Castillo-Chavez, C., “Dynamics of two-strain influenza with isolation and partial cross-immunity”, SIAM J. Appl. Math. 65 (2005) 964982.CrossRefGoogle Scholar
[22]Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., The mathematical theory of optimal processes (Wiley, New York, 1962).Google Scholar
[23]Qiu, Z., Yu, J. and Zou, Y., “The asymptotic behaviour of a chemostat model”, Discrete Contin. Dyn. Syst. B 4 (2004) 721727.Google Scholar
[24]Ross, R., The prevention of malaria (John Murray, London, 1911).Google Scholar
[25]Stegeman, A., Bouma, A., Elbers, A. R., de Jong, M. C., Nodelijk, G., de Klerk, F., Koch, G. and van Boven, M., “Avian influenza A virus (H7N7) epidemic in The Netherlands in 2003: course of the epidemic and effectiveness of control measures”, J. Infect. Dis. 190 (2004) 20882095.Google Scholar
[26]Thanawat, al., “Transmission of the highly pathogenic avian influenza virus H5N1 within flocks during the 2004 epidemic in Thailand”, J. Infect. Dis. 196 (2007) 16791684.Google Scholar
[27]van den Driessche, P. and Watmough, J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci. 180 (2002) 2948.Google Scholar
[28]Webster, R. G., “Influenza: an emerging disease”, Emerg. Infect. Dis. 4 (1998) 436441.CrossRefGoogle ScholarPubMed