# Optimal Vaccination and Treatment Schedules in a Deterministic Avian influenza Model

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2016.48.2.7## Keywords:

genetic algorithm, host-vector model, medical treatment, optimization, vaccination,## Abstract

In this study, a transmission model of the*Avian influenza*disease was developed and analyzedin view of optimization of vaccination and medical treatment. The model is a host-vector model. We focussed on control of

*Avian influenza*, where a vaccination is given to susceptible poultry, while medical treatment is given to infected humans. In the model, the human population is divided into four compartments: susceptible humans, infected humans, recovered humans, and treated humans. Meanwhile, the poultry population is divided into three compartments: susceptible poultry, infected poultry, and vaccinated poultry. To analyze the dynamical behavior of the model, we obtained the disease-free equilibrium, the endemic equilibrium, and the basic reproduction ratio.Furthermore, a model of the optimal vaccination and medical treatment schedule was constructed to know the optimal strategy for controlling

*Avian influenza*. The model can be used to determine the minimal cost of controlling the disease. The model is solved by a genetic algorithm method. Numerical simulations showed that effective control of

*Avian influenza*can be achieved with a combination of vaccination and medical treatment. Likewise, the optimal schedule and strategy for controlling

*Avian influenza*are shown.

## References

Alexander, D.J., A Review of Avian Influenza in Different Bird Species, Veterinary Microbiology, 714(1-2), pp. 3-13, 2000.

Liu, S., Pang, L., Ruan, S. & Zhang, X., Global Dynamics of Avian Influenza Epidemic Model with Psychological Effect, Computational and Mathematical Methods in Medicine, 2015(913726), pp. 1-12, 2015.

Lim, S., J., Heo, C., Hwang, Y. & Yoon, T., Analyzing Patterns of Various Avian Influenza Virus by Decision Tree, International Journal of Computer Theory and Engineering, 7(4), pp. 302-305, 2015.

MMWR, Update: Isolation of Avian Influenza A(H5N1) Viruses From Humans-Hong Kong 1997-1998, American Medical Association, 279(5), pp. 347-352, 1998.

WHO, Cumulative Number of Confirmed Human Cases of Avian Influenza A/(H5N1) Reported to WHO, World Health Organization, http://apps.who.int/csr/disease/avian_influenza/country/cases_table_2011_08_09/en/index.html, (August 9th, 2011).

Agusto, F.B. & Gumel, A.B., Theoretical Assessment of Avian Influenza Vaccine, Discrete and Continuous Dynamical Systems, 13(1), pp. 1-25, 2010.

Manach, A.L., Vergu, E., Grais, R.F., Smith, D.L. & Flahault, L., Key Strategies for Reducing Spread of Avian Influenza Among Commercial Poultry Holdings: Lessons for Transmission to Humans, Proceedings of Royal Society, pp. 2467-2475, 2006.

Iwami, S., Takeuchi, Y. & Liu, X., Avian-Human Influenza Epidemic Model, Mathematical Biosciences, 207(2007), pp. 1-25, 2007.

Gumel, A.B., Global Dynamics of A Two-Strain Avian Influenza Model, International Journal of Computer Mathematics, 86(1), pp. 85-108, 2009.

Jung, E., Iwami, S., Takeuchi, Y. & Jo, T.C., Optimal Control Strategy for Prevention of Avian Influenza Pandemic, Journal of Theoretical Biology, 260(2), pp. 220-229, 2009.

Vaidya, K.N., Wang, F.B. & Zou, X., Avian Influenza Dynamics in Wild Birds Mobility and Spatial Heterogeneous Environment, Discrete and Continuous Dynamical Systems Series B, 17(8), pp. 2829-2848, 2012.

Martcheva, M., Avian Flu: Modeling and Implications for Control, J. Biological Systems, 22(1), pp. 151-175, 2014.

Diekmann, O. & Heesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases, Model Building, Analysis and Interpretation, John Wiley and Son, Chichester, pp. 73-95, 2000.

Hethcote, H.W. & Waltman, P., Optimal Vaccination Schedules in a Deterministic Epidemic Model, Mathematical Biosciences, 18(3-4), pp. 365-381, 1973.