zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
92D30Epidemiology
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
65C20Models (numerical methods)
WorldCat.org
Full Text: DOI
References:
[1] Benenson, A. S.: Control of communicable diseases in man. (1990)
[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.: Textbook of preventive and social medicine. (1989)
[4] Dietz, K.; Schenzle, D.: Mathematical models for infectious disease statistics. ISI centenary volume, 167-204 (1985) · 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. Camb. J. Hyg. 90, 259-325 (1983)
[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. Mathematical population dynamics: analysis of heterogeneity, 79-101 (1995)
[21] Greenhalgh, D.: Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity. Mathl. comput. Modelling 25, No. 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) · Zbl 0334.92024
[24] Dietz, K.: The evaluation of rubella vaccination strategies. The mathematical theory of the dynamics of biological populations, 81-98 (1981) · Zbl 0502.92022
[25] Sinnecker, H.: General epidemiology. (1976)
[26] Anderson, R. M.: The influence of parasite infection on the dynamics of host population growth. Population dynamics, 245-281 (1979)
[27] Berger, J.: Model of rabies control. The Proceedings of a workshop on mathematical models in medicine, mainz 11, 74-88 (1976)
[28] Hethcote, H. W.; Yorke, J. A.: Gonorrhoea transmission dynamics and control. 56 (1974)
[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. Population dynamics of rabies in wildlife, 223-234 (1985)
[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. Differential equation models in biology, epidemiology and ecology 92, 57-67 (1991) · Zbl 0737.92014
[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.; Den Driessche, P. Van: Analysis of a disease transmission model with varying population size. J. math. Biol. 29, 257-270 (1990) · Zbl 0725.92021
[43] Busenberg, S.; Den Driessche, P. Van: Non-existence of periodic solutions for a class of epidemiological models. Differential equation models in biology, epidemiology and ecology 92, 70-79 (1991) · 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. Differential equation models in biology, epidemiology and ecology 92, 121-138 (1991) · Zbl 0735.92022
[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) · 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)
[52] Chow, S. N.; Hale, J. K.: Methods of bifurcation theory. (1982) · Zbl 0487.47039