Abstract
A model is derived for the description of parasitic diseases on host populations with age structure. The parasite population develops according to a linear birth-death-process. The parasites influence mortality and fertility of the hosts and are acquired with a rate depending on the mean parasite load of the host population. The model consists of a system of partial differential equations with initial and boundary conditions. From the boundary condition a renewal equation for the host population is derived. The model is then generalized to describe a multitype process. Existence and uniqueness of solutions are proved. Results concerning persistent solutions are indicated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, R. M., May, R. M.: Regulation and stability of host-parasite population interactions. I. Regulatory processes. J. Anim. Ecol. 47, 219–247 (1978)
Bailey, N. T. J.: The elements of stochastic processes, 3 edn. New York: Wiley 1967
Bailey, N. T. J.: Mathematical theory of infectious diseases, 2nd edn. London: Griffin 1975
Bellman, R., Cooke, K. L.: Differential-difference equations. New York: Academic Press 1963
Cohen, J. E.: Mathematical models of schistosomiasis. Ann. Rev. Ecol. Syst. 8, 209–233 (1977)
Feller, W.: On the integral equation of renewal theory. Ann. Math. Statist. 12, 243–267 (1941)
Goffmann, W., Warren, K. S.: An application of the Kermack-McKendrick theory to the epidemiology of schistosomiasis. Am. J. Trop. Med. Hyg. 19, 278–283 (1970)
Hadeler, K. P.: Integral equations for infections with discrete parasites: Hosts with Lotka birth law (1982). In: Levin, S. A., Hallam, T. G. (eds.) Mathematical ecology (Lect. Notes Biomath., vol. 54, pp. 356–365) Berlin Heidelberg New York: Springer 1984
Hadeler, K. P., Dietz, K.: Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations. Comput. Math. Appl. 9, 415–430 (1983)
Hadeler, K. P., Dietz, K.: Population dynamics of killing parasites which reproduce in the host. J. Math. Biol. 21, 45–65 (1984)
Hairston, N. G.: On the mathematical analysis of schistosome populations. Bull. WHO 33, 45–62 (1965)
Hairston, N. G.: An analysis of age-prevalence data by catalytic models. Bull. WHO 33, 163–175 (1965)
Kermack, W. O., McKendrick, A. G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Ser. A 115, 700–721 (1927)
Kostitzin, V. A.: Symbiose, parasitisme et évolution (1934) In: Scudo, F., Ziegler, J. (eds.) The golden age of theoretical ecology (Lect. Notes Biomath., vol. 22, pp. 369–408) Berlin Heidelberg New York: Springer 1978
Kretzschmar, M.: Eine Erneuerungsgleichung mit einem Geburt-Tod-Prozess zur Beschreibung von Parasitosen. Dissertation, Universität Tübingen 1987
Lewis, T.: A model for the parasitic disease bilharziasis. Adv. Appl. Probab. 7, 673–704 (1975)
Linhart, H.: On some bilharzia infection and immunization models. S. Afr. Statist. J. 2, 61–66 (1968)
Lotka, A. J.: Relation between birth rates and death rates. Science N. S. 26, 21–22 (1907)
Macdonald, G.: The dynamics of helminth infections with special reference to schistosomes. Trans. Roy. Soc. Trop. Med. Hyg. 59, 489–506 (1965)
May, R. M., Anderson, R. M.: Regulation and stability of host-parasite population interactions. II. Destabilizing processes. J. Anim. Ecol. 47, 249–267 (1978)
May, R. M., Anderson, R. M.: Population biology of infectious diseases, part II. Nature 280, 455–461 (1979)
McKendrick, A. G.: Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1926)
Nåsell, I.: Hybrid models of tropical infections (Lect. Notes Biomath., vol. 59) Berlin Heidelberg New York: Springer 1980
Sharpe, F. R., Lotka, A. J.: A problem in age-distribution. Philos. Mag. Ser. 6, 21, 435–438 (1911)
Von Foerster, H.: Some remarks on changing populations. In: The kinetics of cellular proliferation 382–407. New York: Grune and Stratton 1959
Webb, G. F.: Theory of nonlinear age-dependent population dynamics. New York: Dekker 1985
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kretzschmar, M. A renewal equation with a birth-death process as a model for parasitic infections. J. Math. Biology 27, 191–221 (1989). https://doi.org/10.1007/BF00276103
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00276103