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Some epidemiological models with nonlinear incidence

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Abstract

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

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References

  • Busenberg, S., Cooke, K. L.: The population dynamics of two vertically transmitted infections. Theor. Popul. Biol. 33, 181–198 (1988)

    Google Scholar 

  • Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)

    Google Scholar 

  • Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969

    Google Scholar 

  • Hao, D.-Y., Brauer, F.: Analysis of a characteristic equation. J. Integral Equations Appl. 3, (1990). In press.

  • Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338–349 (1978)

    Google Scholar 

  • Hethcote, H. W., Levin, S. A.: Periodicity in epidemiological models. In: Gross, L., Hallam, T. G., Levin, S. A. (eds.) Applied mathematical ecology, pp. 193–211. Berlin Heidelberg New York: Springer 1989

    Google Scholar 

  • Hethcote, H. W., Lewis, M. A., van den Driessche, P.: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27, 49–64 (1989)

    Google Scholar 

  • Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40, 1–9 (1981a)

    Google Scholar 

  • Hethcote, H. W., Stech, H. W., van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol. 13, 185–198 (1981b)

    Google Scholar 

  • Hethcote, H. W., Stech, H. W., van den Driessche, P.: Periodicity and stability in epidemic models: A survey. In: Busenberg, S. N., Cooke, K. L. (eds.) Differential equations and applications in ecology, epidemics and population problems, pp. 65–82. New York: Academic Press 1981c

    Google Scholar 

  • Hethcote, H. W., Tudor, D. W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47 (1980)

    Google Scholar 

  • Hethcote, H. W., Van Ark, J. W.: Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs. Math. Biosci. 84, 85–118 (1987)

    Google Scholar 

  • Holling, C. S.: Some characteristics of simple types of predation and parasitism. Can. Ent. 91, 385–395 (1959)

    Google Scholar 

  • Liu, W. M., Hethcote, H. W., Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359–380 (1987)

    Google Scholar 

  • 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)

    Google Scholar 

  • Miller, R. K., Michel, A. N.: Ordinary differential equations. New York: Academic Press 1982

    Google Scholar 

  • van den Driessche, P.: A cyclic epidemic model with temporary immunity and vital dynamics. In: Freedman, H. I., Strobeck, C. (eds.) Population biology, (Lect. Notes Biomath., vol. 52, pp. 433–440) Berlin Heidelberg New York: Springer 1983

    Google Scholar 

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Research supported in part by Centers for Disease Control Contract 200-87-0515. Support services provided at University House Research Center at the University of Iowa

Research supported in part by NSERC A-8965 and the University of Victoria President's Committee on Faculty Research and Travel

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Hethcote, H.W., van den Driessche, P. Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287 (1991). https://doi.org/10.1007/BF00160539

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  • DOI: https://doi.org/10.1007/BF00160539

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