Abstract
For parasites with a clearly defined life-cycle we give threshold quantities that determine the stability of the parasite-free steady state for autonomous and periodic deterministic systems formulated in terms of mean parasite burdens. We discuss the biological interpretations of the quantities, how to deal with heterogeneity in both parasite and host populations, how to incorporate the effects of periodic discontinuities, and the relation of the threshold quantities to the basic reproduction ratio R 0. Examples from the literature are given. The analysis of the periodic case extends easily to ‘micro-parasitic’ systems.
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References
Anderson, R. M., May, R. M.: Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, 1991
Barbour, A. D.: Threshold phenomena in epidemic theory. To appear in: Probability, Statistics and Optimization. F. P. Kelly (ed.), Wiley, Chichester 1994
Barbour, A. D., Kafetzaki, M.: A host-parasite model yielding heterogeneous parasite loads. J. Math. Biol. 31, 157–176 (1993)
Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979
Coddington, E. A., Levinson, N.: Theory of ordinary Differential Equations. McGraw-Hill, New York, 1955
Diekmann, O., Dietz, K., Heesterbeek, J. A. P.: The basic reproduction ratio for sexually transmitted diseases, Part 1: Theoretical considerations. Math. Biosc. 107, 325–339 (1991)
Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)
Grenfell, B. T., Smith, G., Anderson, R. M.: A mathematical model of the population biology of Ostertagia ostertagi in calves and yearlings. Parasitology 95, 389–406 (1987)
Grimshaw, R.: Nonlinear Ordinary Differential Equations. Blackwell Scientific Publications, Oxford, 1990
Hadeler, K. P., Dietz, K.: Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations. Comp. Math. Appl. 9, 415–430 (1983)
Hasibeder, G.: Heterogeneous disease transmission: estimating the basic reproduction number from prevalences. To appear in: Epidemic Models, Their Structure and Relation to Data, D. Mollison (ed.), Cambridge University Press 1993
Heesterbeek, J. A. P.: R 0. PhD-Thesis, University of Leiden 1992
Jagers, P., Nerman O.: Branching processes in periodically varying environments. Ann. Prob. 13, 254–268 (1985)
Kretzschmar, M.: A renewal equation with a birth-death process as a model for parasitic infections. J. Math. Biol. 27, 191–221 (1989)
Metz, J. A. J. & O. Diekmann (eds.): Dynamics of Physiologically Structured Populations. Lect. Notes in Biomath., Vol. 68, Springer-Verlag, Berlin 1986
Minc, H.: Nonnegative Matrices. Wiley, New York 1988
Roberts, M. G.: Stability in cyclic epidemic models. J. Math. Biol. 22, 303–311 (1985)
Roberts, M. G.: The population dynamics of nematode infections of ruminants. To appear in proceedings of the 3rd International Conference on Population Dynamics, Pau, 1992
Roberts, M. G., Grenfell, B. T.: The population dynamics of nematode infections of ruminants: periodic perturbations as a model for management. IMA J. Math. Appl. Med. Biol. 8, 83–93 (1991)
Roberts, M. G., Grenfell, B. T.: The population dynamics of nematode infections of ruminants: the effect of seasonality in the free-living stages. IMA J. Math. Appl. Med. Biol. 9, 29–41 (1992)
Rudin, W.: Functional Analysis. McGraw-Hill, New York 1973
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Heesterbeek, J.A.P., Roberts, M.G. Threshold quantities for helminth infections. J. Math. Biol. 33, 415–434 (1995). https://doi.org/10.1007/BF00176380
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DOI: https://doi.org/10.1007/BF00176380