Disease regulation of age-structured host populations. (English) Zbl 0688.92009

Summary: A lethal, contagious disease can generate a density-dependent regulation of its host, provided the hosts’ contact rate grows with population size. The condition for disease-induced population control is that the expected number of offspring of an infected new-born be less than one. In vertebrates that acquire immunity if they survive infection, the disease changes the age structure of its host population.
The steady-state age structure of a disease-regulated host with age- dependent fecundity is computed. Local stability analysis indicates that the equilibrium age structure is always stable. However, when the usual exponentially distribued duration of the disease is replaced by a constant duration, the population can exhibit oscillations with a long period.


92D25 Population dynamics (general)
34C99 Qualitative theory for ordinary differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI


[1] Anderson, R.M., Directly transmitted viral and bacterial infections of man, ()
[2] Anderson, R.M., Transmission dynamics and control of infectious disease agents, () · Zbl 0492.92019
[3] Anderson, R.M.; Jackson, H.C.; May, R.M.; Smith, A.M., Population dynamics of fox rabies in Europe, Nature (London), 289, 765-771, (1981)
[4] Anderson, R.M.; May, R.M., Population biology of infectious diseases, I, Nature (London), 280, 361-367, (1979)
[5] Anderson, R.M.; May, R.M., The population dynamics of microparasites and their invertebrate hosts, Philos. trans. roy. soc. London, B, 291, 451-524, (1981)
[6] Anderson, R.M.; May, R.M., Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes, J. hyg, 94, 365-436, (1985)
[7] Anderson, R.M.; May, R.M.; McLean, A.R., Possible demographic consequences of AIDS in developing countries, Nature (London), 332, 228-234, (1988)
[8] Bailey, N.T.J., The mathematical theory of infectious diseases, (1975), Giffin London · Zbl 0115.37202
[9] Barbour, A.D., Macdonald’s model and the transmission of bilharzia, Trans. roy. soc. trop. med. hyg, 72, 6-15, (1978) · Zbl 0377.92013
[10] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0118.08201
[11] Blythe, S.P.; Anderson, R.M., Distributed incubation and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV), IMA J. math. appl. med. biol, 5, 1-19, (1988) · Zbl 0686.92015
[12] Busenberg, S.N.; Cooke, K.L., The effect of integral conditions in certain equations modeling epidemics and population growth, J. math. biol, 10, 13-32, (1980) · Zbl 0464.92022
[13] Busenberg, S.N.; Cooke, K.L.; Iannelli, M., Stability and thresholds in some age-structured epidemics, (), in press · Zbl 0686.92014
[14] Busenberg, S.N.; 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
[15] Busenberg, S.N.; Travis, C.C., Epidemic models with spatial spread due to population migration, J. math. biol, 16, 181-198, (1983) · Zbl 0522.92020
[16] Castillo-Chavez, C.; Cooke, K.L.; Huang, W.; Levin, S.A., The role of long periods of infectiousness in the dynamics of acquired immunodeficiency syndrome (AIDS), (), in press · Zbl 0682.92013
[17] Castillo-Chavez, C.; Hethcote, H.W.; Andreasen, V.; Levin, S.A.; Liu, W.-M., Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. math. biol, (1989), in press · Zbl 0715.92028
[18] Dietz, K., Transmission and control of arbovirus diseases, (), 104-121 · Zbl 0322.92023
[19] Dietz, K.; Schenzle, D., Proportionate mixing models for age-dependent infection transmission, J. math. biol, 22, 117-120, (1985) · Zbl 0558.92014
[20] Dwyer, G.; Levin, S.A.; Butttel, L., A simulation model of the population dynamics and evolution of myxomatosis, submitted, Ecol. monogr, (1989)
[21] El’sgol’ts, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[22] Fenner, F., Biological control, as exemplified by smallpox eradication and myxomatosis, (), 259-285
[23] Fenner, F.; Myers, K., Myxoma virus and myxomatosis in retrospect: the first quarter century of a new disease, (), 539-570
[24] Fenner, F.; Ratcliffe, R.N., Myxomatosis, (1965), Cambridge Univ. Press London
[25] Getz, W.M.; Pickering, J., Epidemic models: thresholds and population regulation, Amer. nat, 121, 892-898, (1983)
[26] Greenhalgh, D., Analytical results on the stability of age-structured recurrent epidemic models, IMA J. math. appl. med. biol, 4, 109-144, (1987) · Zbl 0661.92023
[27] Grossman, Z., Oscillatory phenomena in a model of infectious diseases, Theor. pop. biol, 18, 204-243, (1980) · Zbl 0457.92020
[28] Guckenheimer, J.; Homes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer-Verlag New York/Berlin
[29] Hale, J., Functional differential equations, (1971), Springer-Verlag New York/Berlin · Zbl 0222.34003
[30] Hassel, M.P., Impact of infectious diseases on host populations. group report, (), 15-35
[31] Hethcote, H.W.; Levin, S.A., Periodicity in epidemiological models, (), in press
[32] Hethcote, H.W.; Yorke, J.A., Gonorrhea: transmission dynamics and control, Lecture notes in biomath, 56, 1-105, (1984) · Zbl 0542.92026
[33] Hethcote, H.W.; van Ark, J.W., Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. biosci, 84, 85-118, (1987) · Zbl 0619.92006
[34] Holmes, J.C., Impact of infectious disease agents on the population growth and geographical distribution of animals, (), 37-51
[35] Hoppensteadt, F., An age dependent epidemic model, J. franklin inst, 297, 325-333, (1974) · Zbl 0305.92010
[36] Kermack, W.O.; McKendrick, A.G., A contribution to the mathematical theory of epidemics, (), 700-721 · JFM 53.0517.01
[37] Levin, S.A.; Pimentel, D., Selection of intermediate rates of increase in parasite-host systems, Amer. nat, 117, 308-315, (1981)
[38] Liu, W-m.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol, 23, 187-204, (1986) · Zbl 0582.92023
[39] McKendrick, A.G., Applications of mathematics to medical problems, (), 98-130 · JFM 52.0542.04
[40] Marshal, I.D.; Fenner, F., Studies in the epidemiology of infectious myxomatosis of rabbits. VII. the virulence of strains of myxoma virus recovered from Australian wild rabbits between 1951 and 1959, J. hyg, 58, 485-488, (1960)
[41] May, R.M., Parasitic infections as regulators of animal populations, Amer. sci, 71, 36-45, (1983)
[42] May, R.M., Population biology of microparasitic infections, (), 405-442
[43] May, R.M.; Anderson, R.M., Spatial heterogeneity and the design of immunization programs, Math. biosci, 72, 83-111, (1984) · Zbl 0564.92016
[44] (), 1-511
[45] Nisbet, R.M.; Gurney, W.S.C., Modelling fluctuating populations, (1982), Wiley New York · Zbl 0593.92013
[46] Nold, A., Heterogeneity in disease-transmission modeling, Math. biosci, 52, 227-240, (1980) · Zbl 0454.92020
[47] Parer, I., The population ecology of the wild rabbit, oryctolagus cuniculus (I.), in a Mediterranean-type climate in new south wales, Aust. widl. res, 4, 171-205, (1977)
[48] Pearl, R., The rate of living, (1928), Knopf New York
[49] Post, W.M.; DeAngelis, D.L.; Travis, C.C., Endemic disease in environments with spatially heterogeneous populations, Math. biosci, 63, 289-302, (1983) · Zbl 0528.92018
[50] Price, P.W.; Westoby, M.; Rice, B.; Atsatt, P.R.; Fritz, R.S.; Thompson, J.N.; Mobley, K., Parasite mediation in ecological interactions, Ann. rev. ecol. syst, 17, 487-505, (1986)
[51] Saunders, I.W., A model for myxomtosis, Math. biosci, 48, 1-15, (1980) · Zbl 0422.92024
[52] Schenzle, D.; Dietz, K., Critical population sizes for endemic virus transmission, (), 31-52
[53] Soper, H.E., Interpretation of periodicity in disease prevalence, J. roy. statist. soc, 92, 34-73, (1929) · JFM 55.0941.13
[54] Travis, C.C.; Lenhart, S.M., Eradication of infectious diseases in heterogeneous populations, Math. biosci, 81, 191-198, (1987) · Zbl 0613.92022
[55] Vance, R.R.; Newman, W.I.; Sulsky, D., The demographic meanings of the classical population growth models of ecology, Theor. pop. biol, 33, 199-225, (1988) · Zbl 0659.92021
[56] von Foerster, H., Some remarks on changing populations, (), 308-407
[57] Williams, R.T.; Fullagar, P.J.; Kogon, C.; Davey, C., Observations on a naturally occurring winter epizootic of myxomatosis at Canberra, Australia, in the presence of the rabbit fleas (spilopsyllus cuniculi dale) and virulent myxoma virus, J. appl. ecol, 10, 417-427, (1973)
[58] Wilson, E.B.; Worcester, J., The law of mass action in epidemiology, (), 24-34
[59] Wilson, E.B.; Worcester, J., The law of mass action in epidemiology, II, (), 109-116
[60] Yorke, J.A.; Hethcote, H.W.; Nold, A., Dynamics and control of the transmission of gonorrhea, J. sex. trans. dis, 5, 2, 51-56, (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.