Backward bifurcation of an epidemic model with saturated treatment function. (English) Zbl 1144.92038

Summary: An epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical conditions are limited. The global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, driving the basic reproduction number below unity is not enough to eradicate the disease. A critical value at the turning point is deduced as a new threshold. Some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained. The mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.


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
34D23 Global stability of solutions to ordinary differential equations
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[2] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. appl. math., 64, 260-276, (2003) · Zbl 1034.92025
[3] Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math. biosci., 42, 43-61, (1978) · Zbl 0398.92026
[4] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases. model building, analysis and interpretation, Wiley ser. math. comput. biol., (2000), John Wiley and Sons Chichester · Zbl 0997.92505
[5] 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
[6] Hadeler, K.P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. biosci., 146, 15-35, (1997) · Zbl 0904.92031
[7] 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
[8] Lizana, M.; Rivero, J., Multiparametric bifurcations for a model in epidemiology, J. math. biol., 35, 21-36, (1996) · Zbl 0868.92024
[9] Martcheva, M.; Thieme, H.R., Progression age enhanced backward bifurcation in an epidemic model with superinfection, J. math. biol., 46, 385-424, (2003) · Zbl 1097.92046
[10] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differential equations, 188, 135-163, (2003) · Zbl 1028.34046
[11] Takeuchi, Y.; Liu, X.; Cui, J., Global dynamics of SIS models with transport-related infection, J. math. anal. appl., 329, 1460-1471, (2007) · Zbl 1154.34353
[12] Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
[13] 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
[14] Wang, W., Backward bifurcation of an epidemic model with treatment, Math. biosci., 201, 58-71, (2006) · Zbl 1093.92054
[15] Wang, W.; Ma, Z., Global dynamics of an epidemic model with time delay, Nonlinear anal. real world appl., 3, 365-373, (2002) · Zbl 0998.92038
[16] Wang, W.; Ruan, S., Bifurcation in an epidemic model with constant removal rate of the infectives, J. math. anal. appl., 291, 775-793, (2004) · Zbl 1054.34071
[17] Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math. biosci., 208, 419-429, (2007) · Zbl 1119.92042
[18] X. Zhang, X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal. Real World Appl. (2008), doi:10.1016/j.nonrwa.2007.10.011, in press
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