Abstract
The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The analysis captures the key elements supporting the nearly regular oscillations of the infected and susceptible populations, namely, the interaction of the deterministic and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic amplitude equation describing the envelope of the oscillations yields two criteria providing explicit parameter ranges where they can be observed. These conditions are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent without noise. In this context the criteria indicate how loss of immunity and other factors can lead to a significant increase in the parameter range for prevalence of the sustained oscillations, without any external driving forces. Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of the oscillations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen, E. J., Allen, L. J. S., Arciniega, A., & Greewood, P. E. (2008). Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl., 26, 274–297.
Alonso, D., McKane, A. J., & Pascual, M. (2007). Stochastic amplification in epidemics. J. R. Soc. Interface, 4, 575–582.
Aparicio, J. P., & Solari, H. G. (2001). Sustained oscillations in stochastic systems. Math. Biosci., 169, 15–25.
Bailey, N. T. (1975). The mathematical theory of infectious diseases and its applications (2nd ed.). London: Griffin.
Baxendale, P. H. (2004). Stochastic averaging and asymptotic behavior of the stochastic Duffing–van der Pol equation. Stoch. Process. Appl., 113, 235–272.
Bertuzzo, E., Azaele, S., Maritan, A., Gatto, M., Rodriguez-Iturbe, I., & Rinaldo, A. (2008). On the space-time evolution of a cholera epidemic. Water Resour. Res., 44, W01424.
Brauer, F., van den Driessche, P., & Wang, L. (2008). Oscillations in a patchy environment disease model. Math. Biosci., 215, 1–10.
Buckwar, E., Kuske, R., L’Esperance, B., & Soo, T. (2006). Noise-sensitivity in machine tool vibrations. Int. J. Bifurc. Chaos, 16, 2407–2416.
Codeco, C. T., & Coelho, F. (2006). Trends in cholera epidemiology. PLoS Med., 3, 42.
Dushoff, J., Plotkin, J. B., SA, S. A. Levin, & Earn, D. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proc. Natl. Acad. Sci. USA, 101, 16915–16916.
Gardiner, C. W. (1983). Handbook of stochastic methods for physics, chemistry, and the natural sciences. Berlin: Springer.
Hagenaars, T. J., Donnelly, C. A., & Ferguson, N. M. (2004). J. Theor. Biol., 229, 349–359.
He, D. H., & Earn, D. J. D. (2007). Epidemiological effects of seasonal oscillations in birth rates. Theor. Popul. Biol., 72, 274–291.
Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Rev., 42, 599–653.
Hu, G., Ditzinger, T., Ning, C. Z., & Haken, H. (1993). Phys. Rev. Lett., 71, 807–810.
Kevorkian, J., & Cole, J. D. (1996). Applied mathematical sciences: Vol. 114. Multiple scale and singular perturbation methods. New York: Springer.
Klosek, M. M., & Kuske, R. (2005). Multiscale analysis of stochastic delay differential equations. Multiscale Model. Simul., 3, 706–729.
Kuske, R., Greenwood, P., & Gordillo, L. F. (2007). Sustained oscillations via coherence resonance in SIR. J. Theor. Biol., 245, 459–469.
Lindner, B., Garcia-Ojalvo, J., Neiman, A., & Schimansky-Geier, L. (2004). Effects of noise in excitable systems. Phys. Rep., 392, 321–424.
Liu, Q.-X., & Jin, Z. (2007). Formation of spatial patterns in an epidemic model with constant removal rate of the infectives. J. Stat. Mech. Theory Exp., P05002.
Liu, Q.-X., Wang, R.-H., & Jin, Z. (2009). Persistence, extinction, and spatio-temporal synchronization of SIRS spatial models. J. Stat. Mech. Theory Exp., P07007.
McKane, A. J., & Newman, T. J. (2005). Predator-prey cycles from resonant amplification of demographic stochasticity. Phys. Rev. Lett., 94, 218102.
Nasell, I. (2002). Stochastic models of some endemic infections. Math. Biosci., 179, 1–19.
Nguyen, H. T. H., & Rohani, P. (2008). Noise, nonlinearity, and seasonality: the epidemics of whooping cough revisited. J. R. Soc. Interface, 5, 403–413.
Oksendal, B. (1985). Stochastic differential equations. Berlin: Springer.
Pikovsky, A. S., & Kurths, J. (1997). Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett., 78, 775–778.
Simoes, M., Telo da Gama, M. M., & Nunes, A. (2008). Stochastic fluctuations in epidemics on networks. J. R. Soc. Interface, 5, 555–566.
Suel, G. M., Garcia-Ojalvo, J., & Liberman, L. M. (2006). An excitable gene regulatory circuit induces transient cellular differentiation. Nature, 440, 545–550.
Turner, T. E., Schnell, S., & Burrage, K. (2004). Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem., 28, 165–178.
Yu, N., Kuske, R., & Li, Y.-X. (2008). Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators. Chaos, 18, 015112.
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Chaffee previously Department of Mathematics, University of British Columbia.
Rights and permissions
About this article
Cite this article
Chaffee, J., Kuske, R. The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics. Bull Math Biol 73, 2552–2574 (2011). https://doi.org/10.1007/s11538-011-9635-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-011-9635-7