## Dynamical behavior of an epidemic model with a nonlinear incidence rate.(English)Zbl 1028.34046

Here, an SIRS model of disease spread with nonlinear incidence rate is studied, assuming that the total population is constant ($$S+I+R=N_0>0$$). The rescaled system ${{dI}\over{dt}} = {{I^2}\over{1 + p I^2}} (A-I-R) - mI,\quad {{dR}\over{dt}} = q I - R,$ is analyzed by means of qualitative analysis. The authors provide conditions under which the disease can persist (with one or two positive equilibria apart from the origin) or vanishes. Moreover, they give conditions for stability of the equilibria and existence and stability of periodic orbits. They show that for some parameters at least two periodic orbits appear, and examine cases under which the system undergoes Bogdanov-Takens bifurcation, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation.

### MSC:

 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34D05 Asymptotic properties of solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
Full Text:

### References:

 [1] Bogdanov, R., Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta math. soviet., 1, 373-388, (1981) [2] Bogdanov, R., Versal deformations of a singular point on the plan in the case of zero eigen-values, Selecta math. soviet., 1, 389-421, (1981) [3] Busenberg, S.; Cooke, K.L., The population dynamics of two vertically transmitted infections, Theoret. popul. biol., 33, 181-198, (1988) · Zbl 0638.92009 [4] Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math. biosci., 42, 43-61, (1978) · Zbl 0398.92026 [5] Capasso, V.; Wilson, R.E., Analysis of a reaction – diffusion system modeling man – environment – man epidemics, SIAM J. appl. math., 57, 327-346, (1997) · Zbl 0872.35053 [6] Chow, S.N.; Hale, J.K., Methods of bifurcation, (1982), Springer New York [7] Derrick, W.R.; van den Driessche, P., A disease transmission model in a nonconstant population, J. math. biol., 31, 495-512, (1993) · Zbl 0772.92015 [8] Dushoff, J.; Huang, W.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. biol., 36, 227-248, (1998) · Zbl 0917.92022 [9] Guckenheimer, J.; Holmes, P.J., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1996), Springer New York [10] Hadeler, K.; van den Driessche, P., Backward bifurcation in epidemic control, Math. biosci., 146, 15-35, (1997) · Zbl 0904.92031 [11] Hethcote, H.W., The mathematics of infectious disease, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033 [12] Hethcote, H.W.; Lewis, M.A.; van den Driessche, P., Stability analysis for models of diseases without immunity, J. math. biol., 13, 185-198, (1981) · Zbl 0475.92014 [13] Hethcote, H.W.; Lewis, M.A.; van den Driessche, P., An epidemiological model with a delay and a nonlinear incidence rate, J. math. biol., 27, 49-64, (1989) · Zbl 0714.92021 [14] Hethcote, H.W.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. math. biol., 29, 271-287, (1991) · Zbl 0722.92015 [15] Kuznetsov, Y.A., Elements of applied bifurcation theory, (1998), Springer New York · Zbl 0914.58025 [16] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014 [17] 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 [18] Lizana, M.; Rivero, J., Multiparametric bifurcations for a model in epidemiology, J. math. biol., 35, 21-36, (1996) · Zbl 0868.92024 [19] Perko, L., Differential equations and dynamical systems, (1996), Springer New York · Zbl 0854.34001 [20] Ruan, S.; Xiao, D., Global analysis in a predator – prey system with nonmonotonic functional response, SIAM. J. math. appl., 61, 1445-1472, (2001) · Zbl 0986.34045 [21] F. Takens, Forced oscillations and bifurcation, in: Applications of Global Analysis I, Comm. Math. Inst. Rijksuniversitat Utrecht, Vol. 3, 1974, pp. 1-59. [22] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J. math. biol., 40, 525-540, (2000) · Zbl 0961.92029 [23] Wu, L.; Feng, Z., Homoclinic bifurcation in an SIQR model for childhood diseases, J. differential equations, 168, 150-167, (2000) · Zbl 0969.34042 [24] Y.Q. Ye et al., Theory of Limit Cycles, Transactions of Mathematical Monographs, Vol. 66, American Mathematical Society, Providence, RI, 1986.
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.