Seasonal Treatment of an Infectious Disease is a Social Driver of Sustained Oscillations in the Disease incidence

O. Osuna, J. G. Villavicencio Pulido


We analyze a seasonal $SIR$ model that assumes a periodic treatment rate. Using the Leray-Schauder degree theory, we prove that model shows periodic solutions. This result shows that sustained oscillations in the incidence of the disease are related to the periodic application of a treatment against the disease. So, we can say that the periodic application of treatment can be considered a seasonal driver of the sustained oscillations.


Seasonal treatment rate; periodic orbit; Leray-Schauder degree; SIRS models; reproduction number

Full Text:



G. Pappas, I. J. Kiriaze, and M. E. Falagas, “Insights into infectious disease in the era of hippocrates,” International Journal of Infectious Diseases, vol. 12, pp. 347–350, 2008.

. Altizer et al., “Seasonal and the dynamics of infectious diseases,” Ecology Letters, vol. 9, pp. 467–484, 2006.

S. F. Dowell, “Seasonal variation in host susceptibility and cycles of certain infectious diseases,” Emerg. Infect, vol. 7, pp. 369–373, 2001.

N. C. Grassly and C. Fraser, “Seasonal infectious diseases epidemiology,” Proceedings of the Royal Society B, vol. 273, pp. 2541–2550, 2006.

J. Dushoff, J. B. Plotkin, S. A. Levin, and D. J. D. Earn, “Dynamical resonance can account for seasonality of influenza epidemics,” Proc. Natl Acad. Sci. USA, vol. 101, pp. 16915–16916, 2004.

D. Fishman, “Seasonality of infectious diseases,” Ann. Rev. Public Health, vol. 28, pp. 127–143, 2007.

D. G. Williams and C. Dye, “Infectious disease persistence when transmission varies seasonally,” Mathematical Biosciences, vol. 145, pp. 77–88, 2012.

G. Katriel, “Existence of periodic solutions for the periodically forced sir

model,” Journal of Mathematical Sciences, vol. 201,3, pp. 335–342, 2014.

R. J. Nelson, G. E. Demas, S. L. Kelin, and L. J. Kriegsfeld, Seasonal patterns of stress, immune function and disease. New York: Cambridge University Press, 2002.

I. of Medicine, “The threat of pandemic influenza: Are we ready?,” 2005.

R. H. Borse, S. S. Shrestha, A. E. Fiore, C. Y. Atkins, J. A. Singleton, C. Furlow, and M. I. Meltzer, “Effects of vaccine program against pandemic influenza a(h1n1) virus, united states,” Emerging Infectious Diseases, vol. 19(3), pp. 439–448, 2013.

P. V. den Driessche and J. Watmough, “Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.

R. M. Anderson and R. M. May, Infectious diseases of humans: dynamics and control. Oxford: Oxford University Press, 1991.

D. C. Lee, J. D. Chidambaram, T. C. Porco, and T. M. Lietman, “Seasonal effects in the elimination of trachoma,” Am. J. Trop. Med. Hyg., vol. 72(4), pp. 468–470, 2005.

W. P. Glezen, “Prevention and treatment of seasonal influenza,” N. Engl. Med., vol. 359, pp. 2579–2585, 2008.


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:



Desenvolvido por:

Logomarca da Lepidus Tecnologia