Abstract
An epidemic model of an infectious phenomenon is analyzed. The model allows for an age-dependency to describe the phases of incubation, recovery, and relapse, and for a spatial dependency to describe diffusion of the population in geographical space.
Similar content being viewed by others
References
Bailey, N. T. J.: The mathematical theory of infectious diseases. London: Griffin 1975
Bailey, N. T. J.: Spatial models in the epidemiology of infectious diseases. Biological growth and spread. Lecture notes in biomathematics, Vol. 38, pp. 233–261. Berlin-Heidelberg-New York: Springer 1980
Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math.35, 274–294 (1978)
Capasso, V., Fortunato, D.: Asymptotic behavior for a class of nonautonomous semilinear evolution systems and application to a deterministic epidemic model (to appear)
Crandall, M., Pazy, A.: Nonlinear evolution equations in Banach spaces. Israel J. Math.11, 57–94 (1972)
Di Blasio, G.: Nonlinear age-dependent population growth with history-dependent birth rate. Math. Biosci.46, 279–291 (1979)
Di Blasio, G.: Non-linear age-dependent population diffusion. J. Math. Biol.8, 265–284 (1979)
Di Blasio, G.: A problem arising in the mathematical theory of epidemics (to appear)
Di Blasio, G., Lamberti, L.: An initial-boundary value problem for age-dependent population diffusion. SIAM J. Appl. Math.35, 593–615 (1978)
Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol.6, 109–130 (1978)
Diekmann, O.: Limiting behavior in an epidemic model. Nonl. Anal. Theory, Methods, Appl.1, 459–470 (1977)
Diekmann, O.: On the bounded solutions of nonlinear convolution equation. Nonl. Anal. Theory, Methods, Appl.2, 721–737 (1978)
Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs, N. J.: Prentice-Hall, Inc. 1964
Gurtin, M.: A system of equations for age-dependent population diffusion. J. Theoret. Biol.40, 389–392 (1973)
Gurtin, M., MacCamy, R.: Non-linear age-dependent population dynamics. Arch. Rat. Mech. Anal.54, 281–300 (1974)
Gurtin, M., MacCamy, R.: On the diffusion of biological populations. Math. Biosci.38, 35–49 (1977)
Gurtin, M., MacCamy, R.: Population dynamics with age-dependence. Nonl. Anal. Mech.: Heriot-Watt Symposium, Vol. III, Pitman 1979
Gurtin, M., MacCamy, R.: Some simple models for nonlinear age-dependent population dynamics. Math. Biosci.43, 199–211 (1979)
Gurtin, M., Murphy, L.: On optimal harvesting with an application to age-structured populations. Math. Biosci.55, 115–136 (1981)
Gurtin, M., Murphy, L.: On the optimal harvesting of age-structured populations: Some simple models (to appear)
Haimovici, A.: On the growth of a population dependent on ages and involving resources and pollution. Math. Biosci.43, 213–237 (1979)
Haimovici, A.: On the age-dependent growth of two interacting populations. Boll. Un. Mat. Ital.5, 405–429 (1979)
Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics, and epidemics. SIAM Regional Conference Series in Applied Mathematics, Philadelphia, 1975
MacCamy, R.: A population model with nonlinear diffusion. J. Diff. Eqs.39, 52–72 (1981)
Marcati, P.: Asymptotic behaviour in age dependent population dynamics with hereditary renewal law (to appear)
Marcati, P., Pozio, M.: Global asymptotic stability for a vector disease model with spatial spread. J. Math. Biol.9, 179–187 (1980)
Marcati, P., Serafini, R.: Asymptotic behavior in age-dependent population dynamics with spatial spread. Boll. Un. Mat. Ital.16-B, 734–753 (1979)
De Mottoni, P., Orlandi, E., Tesei, A.: Asymptotic behaviour for a system describing epidemics with migration and spatial spread of infection. Nonl. Anal. Theory, Methods, Appl.3, 663–675 (1979)
Murray, J.: Lectures on nonlinear-differential-equation models in biology. Oxford: Clarendon Press 1977
Pozio, M.: Behaviour of solution of some abstract functional differential equations and application to predator-prey dynamics (to appear)
Pozio, M.: Some conditions for global asymptotic stability of equilibria of integrodifferential equations (to appear)
Prüß, J.: Equilibrium solutions of age-specific population dynamics of several species. J. Math. Biol.11, 65–84 (1981)
Sinestrari, E.: Non-linear age-dependent population growth. J. Math. Biol.9, 331–345 (1980)
Thieme, H.: A model for the spatial spread of an epidemic. J. Math. Biol.4, 337–351 (1977)
Thieme, H.: Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds of the spread of populations. J. Reine Angew. Math.306, 94–121 (1979)
Waltman, P.: Deterministic threshold models in the theory of epidemics. Berlin-Heidelberg-New York: Springer 1974
Webb, G.: A reaction-diffusion model for a deterministic diffusive epidemic. J. Math. Anal. Appl.84, 150–161 (1981)
Webb, G.: A deterministic diffusive epidemic model with an incubation period. The Proceedings of the Functional Differential Equations and Integral Equations Conference at West Virginia University, pp. 119–135. New York: Marcel Dekker 1981
Webb, G.: An age-dependent epidemic model with spatial diffusion. Arch. Rat. Mech. Anal.75, 91–102 (1980)
Weinberger, H.: Some deterministic models for the spread of genetic and other alterations. Biological growth and spread. Lecture notes in biomathematics, Vol. 38, pp. 320–350. Berlin-Heidelberg-New York: Springer 1980
Yosida, K.: Functional analysis. Fourth Ed. Berlin-Heidelberg-New York: Springer 1974
Author information
Authors and Affiliations
Additional information
Supported in part by the National Science Foundation Grant NSF MCS 7903047
Rights and permissions
About this article
Cite this article
Webb, G.F. A recovery-relapse epidemic model with spatial diffusion. J. Math. Biology 14, 177–194 (1982). https://doi.org/10.1007/BF01832843
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01832843