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Dependence of epidemic and population velocities on basic parameters. (English) Zbl 0743.92029
This paper describes the use of linear deterministic models for examining the spread of population processes, discussing their advantages and limitations. Their main advantages are that their assumptions are relatively transparent and that they are easy to analyze, yet they generally give the same velocity as more complex linear stochastic and nonlinear deterministic models. Their simplicity, especially if we use the elegant reproduction and dispersal kernel formulation of O. Diekmann [J. Math. Biol. 6, 109-130 (1978; Zbl 0415.92020)] and F. van den Bosch, J. A. J. Metz and O. Diekmann [ibid. 28, 529- 565 (1990; Zbl 0732.92026)], allows us greater freedom to choose a biologically realistic model and greatly facilitates examination of the dependence of conclusions on model components and of how these are incorporated into the model and fitted from data. This is illustrated by consideration of a range of examples, including both diffusion and dispersal models and by discussion of their application to both epidemic and population dynamic problems.

MSC:
92D30 Epidemiology
92D40 Ecology
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[1] Andral, L.; Artois, M.; Aubert, M.F.A.; Blancou, J., Radio-pistage des renards enrages, Comp. immunol. microbiol. infect. dis., 5, 284-291, (1982)
[2] Atkinson, C.; Reuter, G.E.H., Deterministic epidemic waves, Math. proc. camb. phil. soc., 80, 315-330, (1975) · Zbl 0338.92019
[3] Ball, F.G., Spatial models for the spread and control of rabies incorporating group size, (), 197-222
[4] Ball, F.G., Front-wave velocity and fox habitat heterogeneity, (), 255-289
[5] Ball, F.G., Dynamic population epidemic models, Math. biosci., 107, 299-324, (1991) · Zbl 0747.92025
[6] Berger, J., Model of rabies control, Lect. notes biomath., 11, 75-88, (1976)
[7] Bögel, K.; Moegle, H., Characteristics of the spread of a wildlife rabies epidemic in Europe, Biogeographica, 8, 251-258, (1980)
[8] Bramson, M., Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. am. math. soc., 44, (1983)
[9] Brower, R.C.; Furman, M.A.; Moshe, M., Critical exponents for the Reggeon quantum spin model, Phys. lett. B, 76, 213-219, (1978)
[10] Cox, J.T.; Durrett, R., Limit theorems for the spread of epidemics and forest fires, Stoch. processes appl., 30, 171-191, (1988) · Zbl 0667.92016
[11] Diekmann, O., Thresholds and traveling waves for the geographical spread of infection, J. math. biol., 6, 109-130, (1978) · Zbl 0415.92020
[12] Elton, C.S., The ecology of invasions by animals and plants, (1958), Methuen London
[13] Fisher, R.A., The wave of advance of advantageous genes, Ann. eugen., 7, 355-369, (1937) · JFM 63.1111.04
[14] Harris, T.E., Contact interactions on a lattice, Ann. prob., 2, 969-988, (1974) · Zbl 0334.60052
[15] Hengeveld, R., Dynamics of biological invasions, (1989), Chapman & Hall London
[16] Hughes, R., The fox in the attic, (1961), Chatto & Windus London
[17] Keyfitz, N., Introduction to the mathematics of population, (1968), Addison-Wesley Reading, Mass
[18] Kolmogoroff, A.N.; Petrovsky, I.G.; Piscounoff, N.S., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. univ. état moscou (Sér. int.) A, 1, 6, 1-25, (1937) · Zbl 0018.32106
[19] Kurtz, T.G., Relationships between stochastic and deterministic population models, Lect. notes biomath., 38, 449-467, (1980)
[20] Kuulasmaa, K., The spatial general epidemic and locally dependent random graphs, J. appl. probab., 19, 745-758, (1982) · Zbl 0509.60094
[21] Lambinet, D.; Boisvieux, J.F.; Mallet, A.; Artois, M.; Andral, L., Modele mathématique de la propagation d’une épidemie de rage vulpine, Rév. epidém. santé publ., 26, 9-28, (1978)
[22] Liggett, T., Interacting particle systems, (1985), Springer New York · Zbl 0559.60078
[23] Macdonald, D.W.; Voigt, D.R., The biological basis of rabies models, (), 71-108
[24] McKean, H.P., Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piscounov, Commun. pure appl. math., 28, 323-331, (1975) · Zbl 0316.35053
[25] Mollison, D., Possible velocities for a simple epidemic, Adv. appl. probab., 4, 233-258, (1972) · Zbl 0251.92012
[26] Mollison, D., The spatial propagation of simple epidemics, Proc. 6th Berkeley symp. math. stat. probab., 3, 579-614, (1972)
[27] Mollison, D., Spatial contact models for ecological and epidemic spread, J. roy. stat. soc. B, 39, 283-326, (1977), (with discussion) · Zbl 0374.60110
[28] Mollison, D., Simplifying simple epidemic models, Nature, 310, 224-225, (1984)
[29] Mollison, D., Sensitivity analysis of simple endemic models, (), 223-234
[30] Mollison, D., Modeling biological invasions: chance, explanation, prediction, Phil. trans. roy. soc. lond. B, 314, 675-693, (1986)
[31] Mollison, D., Population dynamics of Mammalian diseases, Symp. zool. soc. lond., 58, 329-342, (1987)
[32] Mollison, D.; Daniels, H.E., The deterministic simple epidemic unmasked, (1977), (unpublished preprint) · Zbl 0785.92026
[33] Mollison, D.; Kuulasmaa, K., Spatial endemic models: theory and simulations, (), 291-309
[34] Mollison, D.; McKendrick, I.; Moretta, B., Velocity estimates for spatial epidemics, (1991), (preprint)
[35] Murray, J.D.; Stanley, E.A.; Brown, D.L., On the spatial spread of rabies among foxes, Proc. roy. soc. lond. B, 229, 111-150, (1986)
[36] Okubo, A.; Maini, P.K.; Williamson, M.H.; Murray, J.D., On the spatial spread of the gray squirrel in britain, Proc. roy. soc. lond. B, 238, 113-125, (1989)
[37] Roughgarden, J., Theory of population genetics and evolutionary ecology: an introduction, (1979), Macmillan New York
[38] McA. Sayers, B.; Ross, A.J.; Saengcharoenrat, P.; Mansourian, B.G., Pattern analysis of the case occurrences of fox rabies in Europe, (), 235-254
[39] Skellam, J.G., Random dispersal in theoretical populations, Biometrika, 38, 196-218, (1951) · Zbl 0043.14401
[40] Smith, C.E.G., Major factors in the spread of infections, Symp. zool. soc. lond., 50, 207-235, (1982)
[41] van den Bosch, F.; Frinking, H.D.; Metz, J.A.J.; Zadoks, J.C., Focus expansion in plant disease. III. two experimental examples, Phytopathology, 78, 919-925, (1988)
[42] van den Bosch, F.; Metz, J.A.J.; Diekmann, O., The velocity of spatial population expansion, J. math. biol., 28, 529-565, (1990) · Zbl 0732.92026
[43] van den Bosch, F.; Verhaar, M.A.; Buiel, A.A.M.; Hoogkamer, W.; Zadoks, J.C., Focus expansion in plant disease. IV. expansion rates in mixtures of resistant and susceptible hosts, Phytopathology, 80, 598-602, (1990)
[44] van den Bosch, F.; Hengeveld, R.; Metz, J.A.J., Analysing the velocity of animal range expansion, (1991), (preprint)
[45] Zadoks, J.C., 25 years of botanical epidemiology, Phil. trans. roy. soc. lond. B, 321, 377-387, (1988)
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