×

zbMATH — the first resource for mathematics

Periodicity in an epidemic model with a generalized non-linear incidence. (English) Zbl 1073.92040
Summary: We develop and analyze a simple susceptible, infected, vaccinated (SIV) epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized nonlinear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the Poincaré index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate.
Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.

MSC:
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
65C20 Probabilistic models, generic numerical methods in probability and statistics
Keywords:
bistability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, R.M.; May, R.M., Infectious diseases of humans, (1991), Oxford University London/New York
[2] Bolker, B.M.; Grenfell, B.T., Chaos and biological complexity in measles dynamics, Philos. trans. roy. soc. lond. B, 251, 75, (1993)
[3] Capasso, V., ()
[4] Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math. biosci., 42, 43, (1978) · Zbl 0398.92026
[5] Earn, D.J.D.; Rohani, P.; Bolker, B.M.; Grenfell, B.T, A simple model for complex dynamical transitions epidemics, Science, 287, 667, (2000)
[6] Glendinning, P., Stability, instability and chaos, (1994), Cambridge University NY
[7] Grenfell, B.T.; Keczkowski, A.; Gilligan, C.A.; Bolker, B.M., Spatial heterogeneity, nonlinear dynamics and chaos in infectious diseases, Stat. meth. med. res., 4, 160, (1995)
[8] Greenhalgh, D.; Doyle, M.; Lewis, F., A mathematical treatment of AIDS and condom use, IMA J. math. appl. med. biol., 18, 3, 225, (2001) · Zbl 0998.92032
[9] Gumel, A.B.; Moghadas, S.M., A qualitative study of a vaccination model with nonlinear incidence, Appl. math. comput., 143, 409, (2003) · Zbl 1018.92029
[10] Hadeler, K.P.; Castillo-Chavez, C., A core group model for disease transmission, Math. biosci., 128, 41, (1994) · Zbl 0832.92021
[11] Hadeler, K.P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. biosci., 146, 15, (1997) · Zbl 0904.92031
[12] Hethcote, H.W.; Lewis, M.A.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. math. biol., 29, 271, (1991) · Zbl 0722.92015
[13] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.-H., ()
[14] Ivlev, V.S., Experimental ecology of the feeding of fishes, (1961), Yale University New Haven, CT
[15] Janaszek, W.; Gay, N.J.; Gut, W., Measles vaccine efficacy during an epidemic in 1998 in the highly vaccinated population of Poland, Vaccine, 21, 473, (2003)
[16] Jansen, H.; Twizell, E.H., An unconditionally convergent discretization of the SEIR model, Math. comput. simulation, 58, 147, (2002) · Zbl 0983.92025
[17] Keeling, M.J.; Rohani, P.; Grenfell, B.T., Seasonally forced disease dynamics explored as switching between attractors, Physica D, 148, 317, (2001) · Zbl 1076.92511
[18] Kribs-Zaleta, C.M., Center manifold and normal forms in epidemic models, Ima, 125, 269, (2002) · Zbl 1021.92036
[19] Kribs-Zaleta, C.M.; Velasco-Hernández, J.X., A simple vaccination model with multiple endemic states, Math. biosci., 164, 183, (2000) · Zbl 0954.92023
[20] Kuznetsov, Y.A., ()
[21] Levin, S.A.; Hallam, T.G.; Gross, L.J., Applied mathematical ecology, (1989), Springer New York
[22] Lin, J.; Andreasen, V.; Levin, S.A., Dynamics of influenza A drift: the linear three-strain model, Math. biosci., 162, 33, (1999) · Zbl 0947.92017
[23] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359, (1987) · Zbl 0621.92014
[24] 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, (1986) · Zbl 0582.92023
[25] Moghadas, S.M., Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. math. comput., 149, 3, 689, (2004) · Zbl 1034.92030
[26] S.M. Moghadas, Modelling the effect of imperfect vaccines on disease epidemiology, Discr. Contin. Dyn. Syst. Ser. B, in press · Zbl 1052.92038
[27] Moghadas, S.M.; Gumel, A.B., Global stability of a two-stage epidemic model with generalized nonlinear incidence, Math. comput. simulation, 60, 107, (2002) · Zbl 1005.92031
[28] Perko, L., Differential equations and dynamical systems, (1996), Springer New York · Zbl 0854.34001
[29] Rohani, P.; Keeling, M.J.; Grenfell, B.T., The interplay between determinism and stochasticity in childhood diseases, Am. nat., 159, 4, 469, (2002)
[30] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. diff. equ., 188, 135, (2003) · Zbl 1028.34046
[31] Scherer, A.; McLean, A.R., Mathematical models of vaccination, Brit. med. bull., 62, 187, (2002)
[32] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J. math. biol., 40, 525, (2000) · Zbl 0961.92029
[33] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer New York · Zbl 0701.58001
[34] Yorke, J.A.; London, W.P., Recurrent outbreaks of measles, chickenpox and mumps II, Am. J. epidemiol., 98, 469, (1973)
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.