Abstract
We discuss a simple deterministic model for the spread, in a closed population, of an infectious disease which confers only temporary immunity. The model leads to a nonlinear Volterra integral equation of convolution type. We are interested in the bifurcation of periodic solutions from a constant solution (the endemic state) as a certain parameter (the population size) is varied. Thus we are led to study a characteristic equation. Our main result gives a fairly detailed description (in terms of Fourier coefficients of the kernel) of the traffic of roots across the imaginary axis. As a corollary we obtain the following: if the period of immunity is longer than the preceding period of incubation and infectivity, then the endemic state is unstable for large population sizes and at least one periodic solution will originate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Busenberg, S., Cooke, K. L.: The effect of integral conditions in certain equations modelling epidemics and population growth. J. Math. Biol.10, 13–32 (1980)
Cushing, J. M.: Nontrivial periodic solutions of some Volterra integral equations. In: Volterra equations, S.-O. Londen, O. J. Staffans (eds.), Vol. 737, pp. 50–66, Lecture notes in mathematics. Berlin-Heidelberg-New York: Springer 1979
Cushing, J. M.: Bifurcation of periodic solutions of nonlinear equations in age-structured population dynamics. To appear in the Proceedings of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Arlington, 1980
Cushing, J. M., Simmes, S.D.: Bifurcation of asymptotically periodic solutions of Volterra integral equations. J. Int. Eq.2, 339–361 (1980)
Diekmann, O.: Volterra integral equations and semigroups of operators. Math. Centre Report TW 197, 1980
Diekmann, O., Van Gils, S. A.: A variation of constants formula for nonlinear Volterra integral equations of convolution type. In: Nonlinear Differential Equations: Invariance, Stability and Bifurcation. P. de Mottoni, L. Salvadori (eds.), pp. 133–143, Academic Press, 1981
Diekmann, O., Van Gils, S. A.: Invariant manifolds for Volterra integral equations of convolution type. (Preprint) Math. Centre Report TW 219, 1981
Gripenberg, G.: Periodic solutions of an epidemic model. J. Math. Biol.10, 271–280 (1980)
Hale, J. K.: Behavior near constant solutions of functional differential equations. J. Diff. Eq.15, 278–294 (1974)
Hale, J. K.: Nonlinear oscillations in equations with delays. In: F. Hoppensteadt (ed.), Nonlinear oscillations in biology. Providence: AMS 1979
Hale, J. K., De Oliveira, J. C. F.: Hopf bifurcation for functional equations. J. Math. Anal. Appl.74, 41–59 (1980)
Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Applied Math.40, 1–9 (1981)
Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol.13, 185–198 (1981)
Hethcote, H. W., Tudor, D. W.: Integral equations describing endemic infectious diseases. J. Math. Biol.9, 37–48 (1980)
Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics, and epidemics. Philadelphia: SIAM 1975
Lauwerier, H. A.: Mathematische modellen voor epidemische processen, unpublished manuscript (in Dutch)
Lauwerier, H. A.: Mathematical models of epidemics. Math. Centre Tract138, 1981
Montijn, R.: Een karakteristieke vergelijking uit de mathematische epidemiologie. Math. Centre Report TN 94, 1980 (in Dutch)
Stech, H. W., Williams, M.: Stability in a class of cyclic epidemic models with delay. J. Math. Biol.11, 95–103 (1981)
Turyn, L.: Functional difference equations and an epidemic model. To appear in the Proceedings of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Arlington, 1980
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Diekmann, O., Montijn, R. Prelude to hopf bifurcation in an epidemic model: Analysis of a characteristic equation associated with a nonlinear Volterra integral equation. J. Math. Biology 14, 117–127 (1982). https://doi.org/10.1007/BF02154757
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02154757