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Population size dependent incidence in models for diseases without immunity. (English) Zbl 0823.92027
Summary: Epidemiological models of SIS type are analyzed to determine the thresholds, equilibria, and stability. The incidence term in these models has a contact rate which depends on the total population size. The demographic structures considered are recruitment-death, generalized logistic, decay and growth. The persistence of the disease combined with disease-related deaths and reduced reproduction of infectives can greatly affect the population dynamics. For example, it can cause the population size to decrease to zero or to a new size below its carrying capacity or it can decrease the exponential growth rate constant of the population.

92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
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[1] Anderson, R. M., Jackson, H. C., May, R. M., Smith, A. D. M.: Population dynamics of fox rabies in Europe. Nature 289, 765-777 (1981) · doi:10.1038/289765a0
[2] Anderson, R. M., May, R. M.: Population biology of infectious diseases I. Nature 280, 361-367 (1979) · doi:10.1038/280361a0
[3] Anderson, R. M., May, R. M.: The population dynamics of microparasites and their invertebrate hosts. Phil. Trans. Roy. Soc. London B291, 451-524 (1981)
[4] Anderson, R. M. and May, R. M. (eds).: Population Biology of Infectious Diseases. Berlin Heidelberg New York Springer (1982) · Zbl 1225.37099
[5] Anderson, R. M., May, R. M.: Infectious Diseases of Humans: Dynamics and Control. Oxford, Oxford University Press, (1991)
[6] Anderson, R. M., May, R. M., McLean, A. R.: Possible demographic consequences of AIDS in developing countries. Nature 332, 228-234 (1988) · doi:10.1038/332228a0
[7] Anderson, R. M., Medley, G. P., May, R. M., Johnson, A. M.: A preliminary study of the transmission dynamics of the human immunodeficiency virus, the causative agent of AIDS. IMA J. Math. Appl. Med. Biol. 3, 229-263 (1986) · Zbl 0609.92025 · doi:10.1093/imammb/3.4.229
[8] Bailey, N. T. J.: The Mathematical Theory of Infectious Diseases (2nd ed.). New York: Hafner (1975) · Zbl 0334.92024
[9] Becker, N.: Analysis of Infectious Disease Data. Chapman and Hall, New York (1989)
[10] Beverton, R. J. H., Holt, S. J.: On the dynamics of exploited fish populations. Fishery Investigations, Series 2, No.19, H.M.S.O., London (1957)
[11] Brauer, F.: Epidemic models in populations of varying size. In: Mathematical Approaches to Problems in Resource Management and Epidemiology. Castillo-Chavez, C. C., Levin, S. A. and Shoemaker, C. (eds.). Lecture Notes in Biomathematics 81. pp. 109-123, Berlin Heidelberg New York: Springer (1989)
[12] Brauer, F.: Models for the spread of universally fatal diseases. J. Math. Biol. 28, 451-462 (1990) · Zbl 0718.92021 · doi:10.1007/BF00178328
[13] Brauer, F.: Models for universally fatal diseases, II. In: Differential Equations Models in Biology, Epidemiology and Ecology. Busenberg, S., Martelli, M. (eds.). Lecture Notes in Biomathematics 92, pp 57-69, Berlin Heidelberg New York: Springer (1991) · Zbl 0737.92014
[14] Bremermann, H. J., Thieme, H. R.: A competitive exclusion principle for pathogen virulence. J. Math. Biol. 27, 179-190 (1989) · Zbl 0715.92027
[15] Busenberg, S., Cooke, K. L.: Vertically Transmitted Diseases. Biomathematics Vol. 23. Berlin Heidelberg New York: Springer (1993) · Zbl 0837.92021
[16] Busenberg, S., Cooke, K. L., Pozio, A.: Analysis of a model of a vertically transmitted disease. J. Math. Biol. 17, 305-329 (1983) · Zbl 0518.92024 · doi:10.1007/BF00276519
[17] Busenberg, S., Cooke, K. L., Thieme, H. R.: Demographic change and persistence of HIV/AIDS in a heterosexual population. SIAM J. Appl. Math. 51: 1030-1051 (1991) · Zbl 0739.92014 · doi:10.1137/0151052
[18] Busenberg, S. N. and Hadeler, K. P.: Demography and epidemics. Math. Biosci. 101, 41-62 (1990) · Zbl 0751.92012 · doi:10.1016/0025-5564(90)90102-5
[19] Busenberg, S. N., van den Driessche, P.: Analysis of a disease transmission model in a population with varying size. J. Math. Biol. 28, 257-270 (1990) · Zbl 0725.92021 · doi:10.1007/BF00178776
[20] Castillo-Chavez, C. C. (ed.): Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomathematics 83. Berlin Heidelberg New York: Springer (1989) · Zbl 0682.00023
[21] Castillo-Chavez, C. C., Cooke, K., Huang, W., Levin, S. A.: On the role of long incubation periods in the dynamics of AIDS I: single population models. J. Math. Biol. 27, 373-398 (1989) · Zbl 0715.92029 · doi:10.1007/BF00290636
[22] DeJong, M. C. M., Diekmann, O., Heestebeek, J. A. P.: How does transmission depend on population size? In: Human Infectious Diseases, Isham, V., Medley, G. (eds), Proceedings of a Conference at the Isaac Newton Institute for Mathematical Sciences, March 1993
[23] Derrick, W. R., van den Driessche, P.: A disease transmission model in a nonconstant population. J. Math. Biol. 31, 495-512 (1993) · Zbl 0772.92015 · doi:10.1007/BF00173889
[24] Diekmann, O., Kretzschmar, M.: Patterns in the effects of infectious diseases on population growth. J. Math. Biol. 29, 539-570 (1991) · Zbl 0732.92024 · doi:10.1007/BF00164051
[25] Dietz, K.: Overall population patterns in the transmission cycle of infectious disease agents. In: Population Biology of Infectious Diseases, Anderson, R. M., May, R. M. (eds). Berlin Heidelberg New York: Springer (1982)
[26] Dietz, K., Schenzle, D.: Mathematical models for infectious disease statistics. In: A Celebration of Statistics, Atkinson, A. C., Fienberg, S. E. (eds.). Berlin Heidelberg New York: Springer 167-204 (1985) · Zbl 0586.92017
[27] Edelstein-Keshet, L.: Mathematical Models in Biology. New York, Random House (1988) · Zbl 0674.92001
[28] Fine, P.: Vectors and vertical transmission: an epidemiological perspective. Ann N. Y. Acad. Sci. 266, 173-194 (1975) · doi:10.1111/j.1749-6632.1975.tb35099.x
[29] Gao, L. Q., Hethcote, H. W.: Disease transmission models with density-dependent demographics. J. Math. Biol. 30, 717-731 (1992) · Zbl 0774.92018 · doi:10.1007/BF00173265
[30] Grabiner, D.: Mathematical models for vertically transmitted diseases. Technical report, Pomona College, Claremont, California (1988)
[31] Greenhalgh, D.: An epidemic model with a density-dependent death rate. IMA J. Math. Appl. Med. Biol. 7, 1-26 (1990) · Zbl 0751.92014 · doi:10.1093/imammb/7.1.1
[32] Greenhalgh, D.: Vaccination in density dependent epidemic models. Bull. Math. Biol. 54, 733-758 (1992) · Zbl 0766.92020
[33] Greenhalgh, D.: Some threshold and stability results for epidemic models with a density dependent death rate. Theor. Pop. Biol. 42, 130-151 (1992) · Zbl 0759.92009 · doi:10.1016/0040-5809(92)90009-I
[34] Greenhalgh, D.: Some results for an SEIR epidemic model with density dependence in the death rate. IMA J. Math. Appl. Med. Biol. 9, 67-106 (1992) · Zbl 0805.92025 · doi:10.1093/imammb/9.2.67
[35] Guilland, F. M. D.: The impact of infectious diseases on wild animal populations. Proceedings of the Wildlife Diseases Workshop, March 14-20, 1993, Isaac Newton Institute for the Mathematical Sciences. Cambridge: Cambridge University Press (to be published)
[36] Hale, J. K.: Ordinary Differential Equations. New York; Wiley (1969) · Zbl 0186.40901
[37] Heesterbeek, J. A. P., Metz, J. A. J.: The saturating contact rate in marriage- and epidemic models. J. Math. Biol. 31, 529-539 (1993) · Zbl 0770.92021 · doi:10.1007/BF00173891
[38] Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335-356 (1976) · Zbl 0326.92017 · doi:10.1016/0025-5564(76)90132-2
[39] Hethcote, H. W.: Three basic epidemiological models. In: Applied Mathematical Ecology, pp. 119-144, Gross, L., Hallam, T. G., Levin, S. A. (eds.). Berlin Heidelberg New York: Springer (1989)
[40] Hethcote, H. W.: A thousand and one epidemic models. In: Frontiers in Theoretical Biology, Levin, S. A. (ed.), Lecture Notes in Biomathematics 100. Berlin Heidelberg New York: Springer (to be published) · Zbl 0819.92020
[41] Hethcote, H. W., Levin, S. A.: Periodicity in epidemiological models. In: Applied Mathematical Ecology, pp. 193-211, Gross, L., Hallam, T. G., Levin, S. A. (eds.). Berlin Heidelberg New York: Springer (1989)
[42] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Periodicity and stability in epidemic models: asurvey. In: Differential Equations and Applications in Ecology, Epidemics and Population Problems, pp. 65-82, Busenberg, S. N., Cooke, K. L. (eds.). New York: Academic Press, (1981) · Zbl 0477.92014
[43] Hethcote, H. W., Van Ark, J. W.: Epidemiological models with heterogeneous populations: Proportionate mixing, parameter estimation and immunization programs. Math Biosci. 84, 85-118 (1987) · Zbl 0619.92006 · doi:10.1016/0025-5564(87)90044-7
[44] Hethcote, H. W., Van Ark, J. W.: Modeling HIV Transmission and AIDS in the United States. Lecture Notes in Biomathematics 95. Berlin Heideiberg New York: Springer (1992) · Zbl 0805.92026
[45] Hethcote, H. W., Yorke, J. A.: Gonorrhea Transmission Dynamics and Control. Lecture Notes in Biomathematics 56. Berlin Heidelberg New York: Springer (1984) · Zbl 0542.92026
[46] Hirsch, W. M., Hanisch, H., Gabriel, J. P.: Differential equation models for some parasitic infections; methods for the study of asymptotic behavior. Comm. Pure Appl. Math. 38, 733-753 (1985) · Zbl 0637.92008 · doi:10.1002/cpa.3160380607
[47] Hyman, J. M., Stanley, E. A.: Using mathematical models to understand the AIDS epidemic. Math. Biosci. 90, 415-473 (1988) · Zbl 0727.92025 · doi:10.1016/0025-5564(88)90078-8
[48] Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., Perry, T.: Modeling and analyzing HIV transmission: the effect of contact patterns. Math. Biosci. 92, 119-199 (1988) · Zbl 0686.92016 · doi:10.1016/0025-5564(88)90031-4
[49] Jordan, D. W., Smith, P.: Nonlinear Ordinary Differential Equations. Oxford: Clarendon Press (1987) · Zbl 0611.34001
[50] Lin, X.: On the uniqueness of endemic equilibria of an HIV/AIDS transmission model for a heterogeneous population. J. Math. Biol. 29, 779-790 (1991) · Zbl 0745.92025 · doi:10.1007/BF00160192
[51] Lin, X.: Qualitative analysis of an HIV transmission model. Math. Biosci. 104, 111-134 (1991) · Zbl 0748.92011 · doi:10.1016/0025-5564(91)90033-F
[52] May, R. M., Anderson, R. M.: Regulation and stability of host-parasite population interactions II: Destabilizing processes. J. Anim Ecol 47, 248-267 (1978)
[53] May, R. M., Anderson, R. M.: Population biology of infectious diseases II. Nature 280, 455-461 (1979) · doi:10.1038/280455a0
[54] McNeill, W. H.: Plagues and People. Blackwell, Oxford (1976)
[55] Mena-Lorca, J.: Periodicity and stability in epidemiological models with disease-related deaths. Ph.D. Thesis: University of Iowa (1988)
[56] Mena-Lorca, J., Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30, 693-716 (1992) · Zbl 0748.92012
[57] Miller, R. K., Michel, A. N.: Ordinary Differential Equations. New York: Academic Press, (1982) · Zbl 0552.34001
[58] Plowright, W.: The effects of rinderpest and rinderpest control on wildlife in Africa. Symp. Zool. Soc. Lond. 60, 175-199 (1982)
[59] Pugliese, A.: Population models for diseases with no recovery. J. Math. Biol. 28, 65-82 (1990) · Zbl 0727.92023 · doi:10.1007/BF00171519
[60] Pugliese, A.: An SEI epidemic model with varying population size. In: Differential Equation Models in Biology, Epidemiology and Ecology, pp. 121-138, Busenberg, S., Martelli, M. (eds.). Lecture Notes in Biomathematics 92. Berlin Heidelberg New York: Springer (1991) · Zbl 0735.92022
[61] Ricker, W. E.: Stock and recruitment. J. Fish. Res. Bd. Can 11, 559-623 (1954)
[62] Swart, J. H.: Hopf bifurcation and stable limit cycle behavior in the spread of infectious diesease, with special application to fox rabies. Math. Biosci. 95, 199-207 (1989) · Zbl 0687.92013 · doi:10.1016/0025-5564(89)90033-3
[63] Thieme, H. R.: Persistence under relaxed point-dissipativity (with Application to an Endemic Model), SIAM J. Math. Anal. 24, 407-435 (1993) · Zbl 0774.34030 · doi:10.1137/0524026
[64] Thieme, H. R.: Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci. 111, 99-130 (1992) · Zbl 0782.92018 · doi:10.1016/0025-5564(92)90081-7
[65] Thieme, H. R., Castillo-Chavez, C. C.: On the role of variable infectivity in the dynamics of human immunodeficiency virus. In: Mathematical and Statistical Approaches to AIDS Epidemiology, pp. 157-176. Lecture Notes in Biomathematics 83. Berlin Heidelberg New York: Springer (1989) · Zbl 0687.92009
[66] Van Riper, C. III, Van Riper, S.G., Goff, M. L, Laird, M.: The epizootiology and ecological significance of malaria in Hawaiian land birds. Ecological Monographs 56, 327-344 (1986) · doi:10.2307/1942550
[67] Zhou, J.: An epidemiological model with population size dependent incidence. Rocky Mt. J. Math. 24, No.1 (to be published)
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