The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics

Bulletin of Mathematical Biology

Volumn 73, Issue 11, 2011, Pages 2552-2574

  Chaffee, J ^a   Kuske, R ^b  

^a JOHNS HOPKINS UNIVERSITY   (United States)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Coherence resonance; Multiple scales; SIRS; Stochastic averaging; Sustained oscillations

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; BIOLOGICAL RHYTHM; COMMUNICABLE DISEASE; EPIDEMIC; HUMAN; IMMUNITY; IMMUNOLOGY; MATHEMATICAL PHENOMENA; NONLINEAR SYSTEM; STATISTICS;

BIOLOGICAL CLOCKS; COMMUNICABLE DISEASES; EPIDEMICS; HUMANS; IMMUNITY; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS; STOCHASTIC PROCESSES;

EID: 81255149550     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-011-9635-7     Document Type: Article

References (30)

1. 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.
2. Alonso, D., McKane, A. J., & Pascual, M. (2007). Stochastic amplification in epidemics. J. R. Soc. Interface, 4, 575-582.
3. Aparicio, J. P., & Solari, H. G. (2001). Sustained oscillations in stochastic systems. Math. Biosci., 169, 15-25.
4. Bailey, N. T. (1975). The mathematical theory of infectious diseases and its applications (2nd ed.). London: Griffin.
5. Baxendale, P. H. (2004). Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation. Stoch. Process. Appl., 113, 235-272.
6. 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.
7. Brauer, F., van den Driessche, P., & Wang, L. (2008). Oscillations in a patchy environment disease model. Math. Biosci., 215, 1-10.
8. Buckwar, E., Kuske, R., L'Esperance, B., & Soo, T. (2006). Noise-sensitivity in machine tool vibrations. Int. J. Bifurc. Chaos, 16, 2407-2416.
9. Codeco, C. T., & Coelho, F. (2006). Trends in cholera epidemiology. PLoS Med., 3, 42.
10. 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.
11. Gardiner, C. W. (1983). Handbook of stochastic methods for physics, chemistry, and the natural sciences. Berlin: Springer.
12. Hagenaars, T. J., Donnelly, C. A., & Ferguson, N. M. (2004). J. Theor. Biol., 229, 349-359.
13. He, D. H., & Earn, D. J. D. (2007). Epidemiological effects of seasonal oscillations in birth rates. Theor. Popul. Biol., 72, 274-291.
14. Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Rev., 42, 599-653.
15. Hu, G., Ditzinger, T., Ning, C. Z., & Haken, H. (1993). Phys. Rev. Lett., 71, 807-810.
16. Kevorkian, J., & Cole, J. D. (1996). Applied mathematical sciences: Vol. 114. Multiple scale and singular perturbation methods. New York: Springer.
17. Klosek, M. M., & Kuske, R. (2005). Multiscale analysis of stochastic delay differential equations. Multiscale Model. Simul., 3, 706-729.
18. Kuske, R., Greenwood, P., & Gordillo, L. F. (2007). Sustained oscillations via coherence resonance in SIR. J. Theor. Biol., 245, 459-469.
19. Lindner, B., Garcia-Ojalvo, J., Neiman, A., & Schimansky-Geier, L. (2004). Effects of noise in excitable systems. Phys. Rep., 392, 321-424.
20. 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.
21. 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.
22. McKane, A. J., & Newman, T. J. (2005). Predator-prey cycles from resonant amplification of demographic stochasticity. Phys. Rev. Lett., 94, 218102.
23. Nasell, I. (2002). Stochastic models of some endemic infections. Math. Biosci., 179, 1-19.
24. Nguyen, H. T. H., & Rohani, P. (2008). Noise, nonlinearity, and seasonality: the epidemics of whooping cough revisited. J. R. Soc. Interface, 5, 403-413.
25. Oksendal, B. (1985). Stochastic differential equations. Berlin: Springer.
26. Pikovsky, A. S., & Kurths, J. (1997). Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett., 78, 775-778.
27. Simoes, M., Telo da Gama, M. M., & Nunes, A. (2008). Stochastic fluctuations in epidemics on networks. J. R. Soc. Interface, 5, 555-566.
28. Suel, G. M., Garcia-Ojalvo, J., & Liberman, L. M. (2006). An excitable gene regulatory circuit induces transient cellular differentiation. Nature, 440, 545-550.
29. Turner, T. E., Schnell, S., & Burrage, K. (2004). Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem., 28, 165-178.
30. Yu, N., Kuske, R., & Li, Y.-X. (2008). Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators. Chaos, 18, 015112.