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Population dynamics of killing parasites which reproduce in the host. (English) Zbl 0554.92015

In this paper the authors suggest a ”density model” to study a parasitic infection in human hosts. Taking into account the density of an infection explicitly, its effects on host mortality, duration of infectious period and transmission rate, they rigorously derived a semistochastic model of partial differential equations.
Results of local and global existence for this system of equations are given. They prove that nontrivial stationary distributions exist and give results of stability. At last they compare the results obtained with another model for a parasitic infection studied by R. M. May and R. M. Anderson, Nature 280, 455-461 (1979).
Reviewer: L.Maddalena

MSC:

92D25 Population dynamics (general)
92C50 Medical applications (general)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Anderson, R. M., May, R. M.: Coevolution of hosts and parasites. Parasitology 85, 411–426 (1982) · doi:10.1017/S0031182000055360
[2] Busenberg, S., Iannelli, M.: A degenerate nonlinear diffusion problem in age structured population dynamics. J. Nonl. Anal. T.M.A. 7, 1411–1429 (1983) · Zbl 0523.92014 · doi:10.1016/0362-546X(83)90009-3
[3] Busenberg, S., Cooke, K. L., Pozio, M. A.: Analysis of a model of a vertically transmitted disease. J. Math. Biol. 17, 305–329 (1983) · Zbl 0518.92024 · doi:10.1007/BF00276519
[4] Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173–184 (1981) · Zbl 0468.92016 · doi:10.1007/BF00275212
[5] Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978) · Zbl 0415.92020 · doi:10.1007/BF02450783
[6] Hadeler, K. P., Dietz, K.: Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations. Comp. Math. Appl. 9, (Nr. 3) 415–430 (1983) · Zbl 0518.92021 · doi:10.1016/0898-1221(83)90056-1
[7] Hadeler, K. P.: An integral equation for helminthic infections: Stability of the non-infected population. In: Lakshmikanthaw, V. (ed.). Proc. Vth. Int. Conf. on Trends in Theory and Practice of Nonl. Diff. Eqn. 1982. Lecture Notes Pure and Appl. Math. 90, M. Dekker 1984, p. 231–240 · Zbl 0528.92021
[8] Hadeler, K. P.: Integral equations for infections with discrete parasites: Hosts with Lotka birth law. In: Levin, S., Hallam, T. (eds.) Mathematical Ecology, Trieste. Lecture Notes in Biomath. 54, Springer-Verlag, 1984, p. 356–365 · Zbl 0535.92022
[9] Hadeler, K. P., Dietz, K.: An integral equation for helminthic infections: Global existence of solutions. In: Conference Proceedings ”Recent Trends in Mathematics” Reinhardsbrunn (1982), Teubner Texte zur Mathematik 50, Teubner Verlag Leipzig (1982/83) p. 153–163 · Zbl 0506.92021
[10] Hadeler, K. P.: Models for endemic diseases. In: Capasso, V., Paveri-Fontana, S. (eds.) Proceedings ”Mathematics in Biology and Medicine” Bari (1983) · Zbl 0567.92015
[11] Hadeler, K. P.: Hysteresis in a model for parasitic infection. In: Mittelmann, H. D., Küpper, T., (eds.). Proceedings ”Numerical methods for bifurcation problems” Dortmund (1983) Boston: Birkhäuser, p. 171–180
[12] Hadeler, K. P.: A transmission model for multiplying parasites and killing: Stability of the endemic states. In preparation
[13] Hethcote, H. W., Tudor, D.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47 (1980) · Zbl 0433.92026 · doi:10.1007/BF00276034
[14] Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics, and epidemics, Regional Conference Series in Appl. Math. 20, SIAM Philadelphia (1975) · Zbl 0304.92012
[15] Karlin, S., Tavaré, S.: Linear birth and death processes with killing. J. Appl. Prob. 19, 477–487 (1982) · Zbl 0495.60086 · doi:10.2307/3213507
[16] May, R. M., Anderson, R. M.: Population dynamics of infectious diseases, II. Nature 280, 455–461 (1979) · doi:10.1038/280455a0
[17] Puri, P. S.: A method for studying the integral functionals of stochastic processes with applications III. Proc. Sixth Berkeley Symp. Math. Stat. Prob. Vol. III, 481–500, UCLA Press (1972) · Zbl 0257.60029
[18] Thieme, H. R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187 (1979) · Zbl 0417.92022 · doi:10.1007/BF00279720
[19] Webb, G. F.: A recovery relapse epidemic model with spatial diffusion. J. Math. Biol. 14, 177–199 (1982) · Zbl 0518.92023 · doi:10.1007/BF01832843
[20] Webb, G. F.: Theory of nonlinear age-dependent population dynamics. Vanderbilt University, Nashville, Tennessee (1983) · Zbl 0533.92013
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