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Hopf bifurcation in two SIRS density dependent epidemic models. (English) Zbl 1065.92042
Summary: This paper uses two SIRS type epidemiological models to examine the impact on the spread of disease caused by vaccination when the immunity gained from such an intervention is not life long. This occurs, for example, in vaccination against influenza. We assume that susceptible individuals become immune immediately after vaccination and that immune individuals become susceptible to infection after a sufficient lapse of time.
In our first model, we consider a constant contact rate between infectious and susceptible individuals, whereas in our second model this depends on the current size of the population. The death rate in both models depends on population density. We examine the different types of dynamic and long term behaviour possible in our models and in particular examine the existence and stability of equilibrium solutions. We find that Hopf bifurcation is theoretically possible but appears not to occur for realistic parameter values. Numerical simulations confirm the analytical results. The paper concludes with a brief discussion.

MSC:
92D30 Epidemiology
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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[1] Benenson, A.S., Control of communicable diseases in man, (1990), American Public Health Association Oxford
[2] Anderson, R.M.; May, R.M., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)
[3] Park, J.E.; Park, K., ()
[4] Dietz, K.; Schenzle, D., Mathematical models for infectious disease statistics, (), 167-204 · Zbl 0586.92017
[5] Fine, P.E.M.; Clarkson, J.A., Measles in england and wales I. an analysis of factors underlying seasonal patterns, Int. J. epidemiol., 11, 5-14, (1982)
[6] Fine, P.E.M.; Clarkson, J.A., Measles in england and wales II. the impact of the measles vaccination programme on the distribution of immunity in the population, Int. J. epidemiol., 11, 15-25, (1982)
[7] Fine, P.E.M.; Clarkson, J.A., Measles in england and wales III. assessing published predictions of the impact of vaccination of incidence, Int. J. epidemiol., 12, 332-339, (1983)
[8] Greenhalgh, D., Deterministic models for common childhood diseases, Int. J. systems science, 21, 1-20, (1990) · Zbl 0695.92009
[9] Anderson, R.M.; May, R.M., Vaccination against rubella and measles, (), 259-325
[10] Anderson, R.M.; May, R.M., Age-related changes in the rate of disease transmission: implication for the design of vaccination programmes, Camb. J. hyg., 94, 365-436, (1985)
[11] Katzmann, W.; Dietz, K., Evaluation of age-specific vaccination strategies, Theor. popn. biol., 25, 125-137, (1984) · Zbl 0544.92023
[12] McLean, A.R.; Anderson, R.M., Measles in developing countries, part I. epidemiological parameters and patterns, Epidem. inf., 100, 111-133, (1988)
[13] McLean, A.R.; Anderson, R.M., Measles in developing countries, part II. the predicted impact of mass vaccination, Epidem. inf., 100, 419-442, (1988)
[14] Greenhalgh, D., Analytical threshold and stability results on age-structured epidemic models with vaccination, Theor. popn. biol., 33, 266-290, (1988) · Zbl 0657.92008
[15] Greenhalgh, D., Vaccination campaigns for common childhood diseases, Math. biosci., 100, 201-240, (1990) · Zbl 0721.92023
[16] Greenhalgh, D., Existence, threshold and stability results for an age-structured epidemic model with vaccination and a non-separable transmission coefficient, Int. J. systems sci., 24, 641-668, (1993) · Zbl 0780.92022
[17] Pugliese, A., Population models for diseases with no recovery, J. math. biol., 28, 65-82, (1990) · Zbl 0727.92023
[18] Greenhalgh, D., Vaccination in density-dependent epidemic models, Bull. math. biol., 54, 733-758, (1992) · Zbl 0766.92020
[19] Greenhalgh, D.; Das, R., Modelling epidemics with variable contact rates, Theoret. popn. biol., 47, 129-179, (1995) · Zbl 0833.92018
[20] Greenhalgh, D.; Das, R., An SIR epidemic model with a contact rate depending on population density, (), 79-101
[21] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Mathl. comput. modelling, 25, 2, 85-107, (1997) · Zbl 0877.92023
[22] Hamer, W.H., Epidemic disease in england, Lancet, 1, 733-739, (1906)
[23] Bailey, N.T.J., The mathematical theory of infectious diseases and its applications, (1975), Griffin Winnipeg, Canada · Zbl 0115.37202
[24] Dietz, K., The evaluation of rubella vaccination strategies, (), 81-98 · Zbl 0502.92022
[25] Sinnecker, H., General epidemiology, (1976), Wiley London
[26] Anderson, R.M., The influence of parasite infection on the dynamics of host population growth, (), 245-281
[27] Berger, J., Model of rabies control, (), 74-88, 1976, Lecture Notes in Biomathematics
[28] Hethcote, H.W.; Yorke, J.A., (), Lecture Notes in Biomathematics
[29] Nold, A., Heterogeneity in disease transmission modelling, Math. biosci., 52, 227-240, (1980) · Zbl 0454.92020
[30] Boily, M.C.; Anderson, R.M., Sexual contact patterns between men and women and the spread of HIV-1 in urban societies in africa, IMA J. math. appl. med. biol., 8, 221-247, (1991) · Zbl 0738.92016
[31] Jacquez, J.A.; Simon, C.P.; Koopman, J.; Sattenspiel, L.; Perry, T., Modelling and analying HIV transmission: the effect of contact patterns, Math. biosci., 92, 119-199, (1988) · Zbl 0686.92016
[32] May, R.M.; Anderson, R.M.; McLean, A.R., Possible demographic consequences of HIV/AIDS epidemics I. assuming HIV infection always leads to AIDS, Math. biosci., 60, 475-505, (1988) · Zbl 0673.92008
[33] Schenzle, D.; Dietz, K., Räumliche persistenz und diffusion von krankenheiten, sonderdruck, Heidelberger geographische arbeiten, 83, 31-42, (1987)
[34] Mollison, D., Sensitivity analysis of simple epidemic models, (), 223-234
[35] Tuljapurkar, S.; John, A.M., Disease in changing populations: growth and disequilibrium, Theoret. popn. biol., 40, 322-353, (1991) · Zbl 0737.92017
[36] 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
[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
[38] Brauer, F., Models for the spread of universally fatal diseases, J. math. biol., 28, 451-462, (1990) · Zbl 0718.92021
[39] Brauer, F., Models for the spread of universally fatal diseases II, (), 57-67, Lecture Notes in Biomathematics, Proceedings of a Conference held in Claremont, CA, January 13-16, 1990 · Zbl 0718.92021
[40] Diekmann, O.; Kretzschmar, M., Patterns in the effects of infectious diseases on population growth, J. math. biol., 29, 539-570, (1991) · Zbl 0732.92024
[41] Huang, W.; Cooke, K.L.; Castillo-Chavez, C., Stability and bifurcation for a multiple group model for the dynamics of HIV/AIDS transmission, SIAM J. appl. math., 52, 835-854, (1992) · Zbl 0769.92023
[42] Busenberg, S.; Van den Driessche, P., Analysis of a disease transmission model with varying population size, J. math. biol., 29, 257-270, (1990) · Zbl 0725.92021
[43] Busenberg, S.; Van den Driessche, P., Non-existence of periodic solutions for a class of epidemiological models, (), 70-79, Lecture Notes in Biomathematics, Proceedings of a Conference held in Claremont, CA, January 13-16, 1990 · Zbl 0735.92020
[44] 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
[45] Pugliese, A., Population models for diseases with no recovery, J. math. biol., 28, 65-82, (1990) · Zbl 0727.92023
[46] Pugliese, A., An SEI epidemic model with varying population size, (), 121-138, Lecture Notes in Biomathematics, Proceedings of a Conference held in Claremont, CA, January 13-16, 1990
[47] Gao, L.Q.; Hethcote, H.W., Disease transmission models with density-dependent demographics, J. math. biol., 30, 717-731, (1992) · Zbl 0774.92018
[48] Nisbet, R.M.; Gurney, W.S.C., Modelling fluctuating populations, (1982), Wiley New York · Zbl 0593.92013
[49] Greenhalgh, D., An epidemic model with a density-dependent death rate, IMA J. math. appl. med. biol., 7, 1-26, (1990) · Zbl 0751.92014
[50] Khan, Q.J.A.; Greenhalgh, D., Hopf bifurcation in epidemic models with a time delay in vaccination, IMA J. math. appl. med. biol., 16, 113-142, (1999) · Zbl 0943.92031
[51] Marsden, J.E.; McKracken, M., The Hopf bifurcation and its applications, (1976), Springer-Verlag Chichester
[52] Chow, S.N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer-Verlag New York
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