Contact between susceptible and infected individuals is one of the major reasons for the spread of contagious viral disease, for example, the severe acute respiratory syndrome, SARS, and is a major public health problem in the world. The present study aims to assess via a mathematical model, the role of contact rate in the control of the spread of contagious disease like SARS. In this article, we have induced an effective contact rate in the mathematical model as a periodic function of time due to the seasonal occurrence of SARS which was considered as a parameter earlier. The spread of the disease also depends on the time taken to initiate preventive measures by the authorities which have been described and explained by a new term, action time, in the present study. Numerical simulations have been performed with the help of fourth-order Runge-Kutta method to illustrate our results. With the help of simulation, the control of the spread of diseases has been explained with varying periodic effective contact rate and action time.

1. Anderson, R. M. and May, R. M. 1991. Infectious Disease of humans, Oxford University Press Inc. , New York.
2. Bjornstad, O. N. , Finkenstadt, B. F. , and Grenfell, B. T. 2002. Dynamics of measles epidemics: Estimating scaling of transmission rates using a time series SIR model. Ecol Monogr. , 72(2), 169-184.
3. Bradie, B. 2006 A Friendly Introduction to Numerical Analysis. Prentice Hall, New Jersey.
4. Braun, M. 1983. Differential equations and their applications: An introduction to applied mathematics, Springer-Verlag, New York.
5. Brannan, J. R. , and Boyce, W. E. 2009. Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, John Wiley and Sons, New York.
6. Dowell, S. F. 2001. Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerge. Infect. Dis. 7(3), 369-374.
7. Gumel, A. B. , Mccluskey, C. C. , and Watmough, J. 2006. A SVEIR model for assessing the potential impact of an imperfect anti-SARS vaccine. Math. Biosci. and Eng. 3, 485-494.
8. Hamer, W. H. 1906. Epidemic Disease in England. Lancet, 733-739.
9. Karim, S. A. A. , and Razali, R. A. 2011. Proposed mathematical model of influenza A, H1N1 for Malaysia. Journal of Appl. Sci. 11(8), 1457-1460.
10. Kermack, W. O. and McKendrick, A. G. 1927. A contribution to the mathematical theory of Epidemics. Proc. R. Soc. Lond. A 115(772), 700-721.
11. London, W. and Yorke, J. A. 1973. Recurrent outbreaks of measles, chickenpox, and mumps. i. Seasonal variation in contact rates. Am. J. Epidemiology. 98 (6), 453-468.
12. Michael, Y. Li, Graef, John R. , Wang, Liancheng, and Karsai, Janos 1999. Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191-21.
13. Michael, Y. Li, Muldowney, J. S. and Driessche, P. V. N. 1999. Global stability of SEIRS model in epidemiology. Canadian Applied Mathematics Quarterly 7, 409-425.
14. World Health Organization. http://www. who. int/csr/resources/publications/WHO_CDS_CSR_GAR_2004_16/en/.
15. World Health Organization.
16. http://www. who. int/csr/sarsarchive/2003_05_07a/en/.
17. Zhang, J. , Zhen Jin Gui- Quan Sun, Xiang-Dong Sun and Shigui Ruan 2012. Modeling Seasonal Rabies Epidemic in China. Bull Math Biol. 74, 1226-1251.

Sars Seir Model Effective Contact Rate Function Simulation Action Time Control Of The Disease.