Abstract :

Seasonality of infectious disease is an important factor in disease incidence, outbreaks, control and prevention. Many mathematical models that incorporate seasonality in the transmission were formulated and analyzed. In this essay a qualitative analysis is given in terms of the effective reproduction number R₀, the existence and stability of the disease-free equilibrium and endemic equilibrium of both the SEIR model and seasonal SEIR model. We perform numerical simulations  to validate the model formulation.

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Keywords :

Cyclic, Epidemic diseases, SEIR-model, Simulation.

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References :

1. AL-AJAM, R., BIZRI, A. R., MOKHBAT, J., WEEDON, J., and LUTWICK, L. (2006). Mu cormycosis in the eastern Mediterranean: a seasonal disease. Epidemiology and Infection, 134(2):34-46.
2. Al-Sheikh, A. (2012). Modeling and analysis of a seir epidemic model with a limited resource for treatment. Global Journal of Science Frontier Research: Mathematics and Decision Sciences, 12(14):57–66.
3. Altizer, Sonia, e. a. (2006a). Seasonality and the dynamics of infectious diseases. Ecology letters, 4:467–484.
4. Altizer, Sonia, e. a. (2006b). “Seasonality and the dynamics of infectious diseases.”. pages 467–484.
5. Anderson, M. and May, R. M. (1979). Population biology of infectious diseases: Part i. Nature 280.5721, (7).
6. Aron, J. L. and Schwartz, I. B. (1984). Seasonality and period-doubling bifurcations in an epidemic Journal of theoretical biology, 110(4):665–679.
7. , I. and H. L. Smith., S. (1983). Infinite subharmonic bifurcation in an seir epidemic model. Journal of mathematical biology, 18.3:233–253.
8. Bauch, C. and Earn, D. J. (2003). Interepidemic intervals in forced and unforced seir models. Fields Commun, 36:33–44.
9. Beard, B., Pye, G., Steurer, F. J., Rodriguez, R., Campman, R., Peterson, A. T., Ramsey, J., Wirtz, R. A., and Robinson, L. E. (2003). Chagas disease in a domestic transmission cycle in southern Texas, usa. Emerging infectious diseases, 9(1):103.
10. Bernoulli, D. (1760). Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Histoire de l’Acad., Roy. Sci.(Paris) avec Mem, pages 1–45.
11. Diekmann, , Heesterbeek, J. A. P., and Metz, J. A. (1990). On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations. Journal of mathematical biology, 28(4):365–382.
12. Dietz, (1976). The incidence of infectious diseases under the influence of seasonal fluctuations. Mathematical models in medicine, pages 1–15.
13. Driessche, P. and Watmough, J. (2008). Mathematical epidemiology.
14. Earn, , Dushoff, J., and Levin, S. (2002). Ecology and evolution of the flu. Trends in Ecology and Evolution, 17(7):334–340.
15. Fares, (2011). Seasonality of tuberculosis. Journal of global infectious diseases, 3(1):46.
16. Feng, , Xu, D., and Zhao, H. (2009). The uses of epidemiological models in the study of disease control. In Modeling and Dynamics of Infectious Diseases, pages 150–166. World Scientific.
17. Fine, P. E. and Clarkson, J. (1982). The recurrence of whooping cough: possible implications for assessment of vaccine efficacy. The Lancet, 319(8273):666–669.
18. Grassly, N. C. and Fraser., C. (2006). Seasonal infectious disease epidemiology. Proceedings of the Royal Society B: Biological Sciences, 1600:2541–2550.
19. Grassly, N. C. and Fraser, C. (2008). Mathematical models of infectious disease transmission. Nature Reviews Microbiology, 6(6):477.
20. Greenhalgh, and Moneim, I. (2014). Periodicity in general seasonally driven epidemic models. Heffernan, J. M., Smith, R. J., and Wahl, L. M. (2005). Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface, 2(4):281–293.
21. Hethcote, W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4):599–653. Keeling, M. J. and Rohani, P. (2011). Modeling infectious diseases in humans and animals. Princeton University Press.
22. Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epi- Proceedings of the royal society of London. Series A, Containing papers of a mathematical and physical character, 115(772):700–721.
23. Li, J. (2012). Discrete-time models with mosquitoes carrying genetically-modified bacteria. Mathe- matical biosciences, 240(1):35–44.
24. London, W. P. and Yorke, J. A. (1973). Recurrent outbreaks of measles, chickenpox and mumps: I. seasonal variation in contact American journal of epidemiology 98, (6):453–468.
25. LONDON, W. P. and YORKE, J. A. (1973). RECURRENT OUTBREAKS OF MEASLES, CHICK- ENPOX AND MUMPS: SEASONAL VARIATION IN CONTACT RATES1. American Journal of Epidemiology, 98(6):453–468.
26. Lynch, and Smith, G. D. (2005). A life course approach to chronic disease epidemiology. Annu. Rev. Public Health, 26:1–35.
27. Y. Li, J. S. M. (1995). Global stability for the seir model in epidemiology. math. biosci. 5:155–164. Organization, W. H. et al. (2002). Scaling up the response to infectious diseases. Accessed February, 3:2010.
28. Rohani, Pejman, J. K. and Grenfell., B. T. (2002). The interplay between determinism and stochas- ticity in childhood diseases. American journal of epidemiology 98, pages 469–481.
29. Ross, (1911). The prevention of malaria. John Murray; London.
30. Shah, N. and Jyoti, G. (2013). Seir model and simulation for vector borne diseases. Applied Mathe- matics, 4:13 –
31. Shah, N. H. and Gupta, J. (2013). Seir model and simulation for vector borne diseases. Applied Mathematics, 8:13.
32. Smith, (1983). Multiple stable subharmonics for periodic epidemic model.  17:179âA˘ S¸ 190.
33. Stone, L., Olinky, R., and Huppert, (2007). Seasonal dynamics of recurrent epidemics. Nature, 446(7135):533.
34. Turner, (2010). Introduction to infectious disease modelling. Sexually transmitted infections, 87.
35. Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1- 2):29–48.