Abstract
Epidemiological models with nonlinear incidence rates λI pSqshow a much wider range of dynamical behaviors than do those with bilinear incidence rates λIS. These behaviors are determined mainly by p and λ, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.
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References
Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications 2nd edn. London: Griffin 1975
Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)
Carr, J.: Applications of centre manifold theory. Berlin Heidelberg New York: Springer 1981
Cunningham, J.: A deterministic model for measles. Z. Naturforsch. 34c, 647–648 (1979)
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, pp. 150–156. Berlin Heidelberg New York Tokyo: Springer 1983
Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969
Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335–356 (1976)
Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol. 13, 185–198 (1981a)
Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Periodicity and stability in epidemic models: a survey. In: Cooke, K. L. (ed.) Differential equations and applications in ecology, epidemics, and population problems, pp. 65–82. New York London Toronto Sydney San Francisco: Academic Press 1981b
Liu, W. M., Levin, S. A.: Influenza and some related mathematical models. In: Levin, S. A., Hallam, T., Gross, L. (eds.) Applied mathematical ecology. Berlin Heidelberg New York: Springer
Liu, W. M., Levin, S. A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986)
Liu, W. M.: Dynamics of epidemiological models-recurrent outbreaks in autonomous systems. Ph.D. Thesis, Cornell University (1987)
Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications. Berlin Heidelberg New York: Springer 1976
Saunders, I. W.: A model for myxomatosis. Math. Biosci. 48, 1–15 (1980)
Wang, F. J. S.: Asymptotic behavior of some deterministic epidemic models. SIAM J. Math. Anal. 9, 529–534 (1978)
Wilson, E. B., Worcester, J.: The law of mass action in epidemiology. Proc. Natl. Acad. Sci. USA 31, 24–34 (1945)
Wilson, E. B., Worcester, J.: The law of mass action in epidemiology II. Proc. Natl. Acad. Sci. USA 31, 109–116 (1945)
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Liu, Wm., Hethcote, H.W. & Levin, S.A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biology 25, 359–380 (1987). https://doi.org/10.1007/BF00277162
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DOI: https://doi.org/10.1007/BF00277162