Backward bifurcation in epidemic control. (English) Zbl 0904.92031

Summary: For a class of epidemiological SIRS models that include public health policies, the stability at the uninfected state and the prevalence at the infected state are investigated. Backward bifurcation from the uninfected state and hysteresis effects are shown to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. An explicit expression is given for this quantity. The phenomenon of subcritical bifurcation in epidemic modeling is also discussed in terms of group models, pair formation, and macroparasite infection.


92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
Full Text: DOI


[1] Heesterbeek, J. A.P., \((R_0\). Dissertation (1992), University of Leiden)
[2] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1991) · Zbl 0726.92018
[3] Hadeler, K. P.; Castillo-Chavez, C., A core group model for disease transmission, Math. Biosci., 128, 41-55 (1995) · Zbl 0832.92021
[4] Hethcote, H.; Yorke, J., Gonorrhea: Transmission Dynamics and Control, (Lecture Notes in Biomathematics, 56 (1984), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0542.92026
[5] Fineberg, H. V., Education to prevent AIDS: prospects and obstacles, Science, 239, 592-596 (1988)
[6] Gupta, S.; Anderson, R. M.; May, R. M., Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV, Nature, 350, 356-359 (1991)
[7] Scalia-Tomba, P., The effects of structural behavior change on the spread of HIV in one sex populations, Math. Biosci., 91, 547-555 (1991)
[8] Hadeler, K. P.; Müller, J., The effects of vaccination on sexually transmitted disease in heterosexual populations, (Arino, O.; Axelrod, D.; Kimmel, M.; Langlois, M., Mathematical Population Dynamics. Mathematical Population Dynamics, 3d Int. Conf. 1992, Vol. 1 (1995), Wuerz: Wuerz Berlin), 251-278
[9] Hadeler, K. P.; Dietz, K., Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations, Comput. Math. Appl., 9, 415-430 (1983) · Zbl 0518.92021
[10] Hadeler, K. P.; Dietz, K., Population dynamics of killing parasites which reproduce in the host, J. Math. Biol., 21, 45-65 (1984) · Zbl 0554.92015
[11] Hadeler, K. P., Vector models for infectious diseases, (Sleeman, B.; Jarvis, R., Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Proc. Dundee, 1984 Lecture Notes in Math., 1151 (1985), Springer-Verlag: Springer-Verlag Winnipeg), 204-216 · Zbl 0572.92024
[12] Hadeler, K. P., Hysteresis in a model for parasitic infection, (Küpper, T.; Mittelmann, H. D.; Weber, H., Numerical Methods for Bifurcation Problems (1984), Birkhäuser: Birkhäuser Berlin), 171-180, ISNM 70 · Zbl 0554.92014
[13] Hadeler, K. P., Integral equations with discrete parasites: hosts with Lotka birth law, (Levin, S. A.; Hallam, T. G., Mathematical Ecology. Mathematical Ecology, Lecture Notes in Biomathematics, 54 (1984), Springer-Verlag: Springer-Verlag Basel), 356-365 · Zbl 0535.92022
[14] Kretzschmar, M., A renewal equation with a birth-death-process as a model for parasitic infections, J. Math. Biol., 27, 191-221 (1989) · Zbl 0715.92026
[15] Kretzschmar, M., Persistent solutions in a model for parasitic diseases, J. Math. Biol., 27, 549-573 (1989) · Zbl 0716.92022
[16] Kretzschmar, M., Comparison of an infinite-dimensional model for parasitic diseases with a related 2-dimensional system, J. Math. Anal. Appl., 176, 235-260 (1993) · Zbl 0774.92019
[17] Anderson, R. M.; May, R. M., Regulation and stability of host-parasite population interactions I: Regulatory processes; II: Destabilizing processes, J. Anim. Ecol., 47, 249-267 (1978)
[18] Diekmann, O.; Kretzschmar, M., Pattern in effects of infectious diseases on population growth, J. Math. Biol., 29, 539-570 (1991) · Zbl 0732.92024
[19] Doyle, M. T., A constrained mixing two-sex model for the spread of HIV, (Arino, O.; Axelrod, D.; Kimmel, M.; Langlois, M., Mathematical Population Dynamics. Mathematical Population Dynamics, 3d Int. Conf., Vol. 1 (1995), Wuerz: Wuerz Berlin), 209-228
[20] Huang, W.; Cooke, K. L.; Castillo-Chavez, C., Stability and bifurcation for a multigroup model for the dynamics of HIV transmission, SIAM J. Appl., 52, 554-835 (1992) · Zbl 0769.92023
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.