zbMATH — the first resource for mathematics

Infinite subharmonic bifurcation in an SEIR epidemic model. (English) Zbl 0523.92020

92D25 Population dynamics (general)
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI
[1] Ascher, U., Christiansen, J., Russell, R. D.: A collocation solver for mixed order systems of boundary value problems. Math. Comp. 33, 659-679 (1979) · Zbl 0407.65035
[2] Anderson, M., May, R. M: Directly transmitted infectious diseases: Control by Vaccination. Science, vol. 215, 1982 · Zbl 1225.37099
[3] Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications. London: C. Griffin & Co., Ltd., 1975 · Zbl 0334.92024
[4] Berger, M. S.: Nonlinearity and functional analysis. New York: Academic Press, 1977 · Zbl 0368.47001
[5] Chow, S. N., Hale, J. K., Mallet-Paret, J.: An example of bifurcation to homoclinic orbits. Journal differential equations 37, 351-373 (1980) · Zbl 0439.34035
[6] Coppel, N. A.: Stability and asymptotic behavior of differential equations. Boston: D. C. Heath and Co., 1965 · Zbl 0154.09301
[7] Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. Lecture Notes Biomathematics vol. 11, pp. 1-15. Berlin-Heidelberg-New York; Springer (1976) · Zbl 0333.92014
[8] Doedel, E.: The numerical computation of branches of periodic branches. Preprint · Zbl 0482.65048
[9] Emerson, H.: Measles and whooping cough. Am. J. Public Health 27, 1-153 (1937)
[10] Fine, P., Clarkson, J.: Measles in England and Wales. 1. An analysis of factors underlying seasonal patterns. Intenational J. of Epidemiology 11, 5-14 (1982)
[11] Grossman, Z., Gumowski, I., Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. Analytic approach. In: Nonlinear Systems and Applications to Life Sciences, pp. 525-546. New York: Academic Press (1977)
[12] Grossman, Z.: Oscillatory Phenomena in a model of infectious diseases. Theor. Population Biol. 18, (No. 2) 204-243 (1980) · Zbl 0457.92020
[13] Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience, 1969 · Zbl 0186.40901
[14] Hale, J. K., Taboas, P.: Interaction of damping and forcing in a second order equation, nonlinear analysis. T.M.A. 2, 77-84 (1978) · Zbl 0369.34014
[15] Hethcote, H. W.: Private Communication, 1982
[16] Keller, H.: Numerical solution of two-point boundary value problems. SIAM, Philadephia, 1976
[17] London, W. P., Yorke, J. A.: Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. Amer. J. Epidemiol 98, 453-468 (1973)
[18] Rheinboldt, W. C.: Numerical methods for a class of finite dimensional bifurcation problems. SIAM J. Num. Anal. 15, 59-80 (1978) · Zbl 0389.65024
[19] Schwartz, I.: Ph.D. Thesis, University of Maryland, 1980
[20] Schwartz, I.: Estimating regions of existence of unstable periodic orbits using computer-based techniques. SIAM J. Num. Anal. 20, 106-120 (1983) · Zbl 0518.34035
[21] Smith, H. L.: Subharmonic bifucrcation in an S-I-R epidemic model. Preprint (1982)
[22] Smith, H. L.: Multiple stable subharmonics for a periodic epidemic model. Preprint (1982)
[23] Soper, H. E.: The interpretation of periodicity in diseases prevalence. J. Royal Statist. Soc. 92, 34-73 (1929) · JFM 55.0941.13
[24] Yorke, J. A., Nathanson, N., Pianigiani, G., Martin, J.: Seasonality and the requirements for perpetuation and eradication of viruses in populations. American J. Epidemiology 109, 103-123 (1979)
[25] Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. Journal differential equations 31, 53-98 (1979) · Zbl 0476.34034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.