Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment. (English) Zbl 1159.92036

Summary: The dynamical behavior of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that the treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.


92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Hethcote, H.W.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. math. biol., 29, 271-287, (1991) · Zbl 0722.92015
[2] 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
[3] Capasso, V.; Serio, G., A generalization of the kermack C. mckendrick deterministic epidemic model, Math. biosci., 42, 43-75, (1978) · Zbl 0398.92026
[4] Feng, Z.; Thieme, H.R., Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. biosci., 128, 93-114, (1995) · Zbl 0833.92017
[5] Wu, L.; Feng, Z., Homoclinic bifurcation in an SIQR model for childhood diseases, J. diff. equat., 168, 150-167, (2000) · Zbl 0969.34042
[6] Hyman, J.M.; Li, J., Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. appl. math., 58, 912-925, (1998) · Zbl 0905.92027
[7] Wang, W.; Ruan, S., Bifurcation in an epidemic model with constant removal rate of the infectives, J. math. anal. appl., 291, 75-793, (2004) · Zbl 1054.34071
[8] Wang, W., Backward bifurcation of an epidemic model with treatment, Math. biosci., 201, 58-71, (2006) · Zbl 1093.92054
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