Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment.(English)Zbl 1167.34338

Summary: An SIS epidemic model with treatment is proposed. The incidence rate of the model, which can include the bilinear incidence rate and the standard incidence rate, is a general nonlinear incidence rate. The global dynamics of the model are studied and then we can understand the effect of the capacity for treatment. It is found that a backward bifurcation occurs and there exist bistable endemic equilibria if the capacity is low. Mathematical results suggest that decreasing the basic reproduction number is insufficient for disease eradication and improving the efficiency and capacity of treatment is important for this end.

MSC:

 34C23 Bifurcation theory for ordinary differential equations 37N25 Dynamical systems in biology 92D30 Epidemiology 34D23 Global stability of solutions to ordinary differential equations
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 [1] () [2] () [3] 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 [4] Castillo-Chavez, C.; Thieme, H.R., Asymptotically autonomous epidemic models, (), 33-50 [5] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases, () · Zbl 0997.92505 [6] 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 [7] Hadeler, K.P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. biosci., 146, 15-35, (1997) · Zbl 0904.92031 [8] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033 [9] Li, G.; Wang, W.; Wang, K.; Jin, Z., Dynamic behavior of a parasite – host model with general incidence, J. math. anal. appl., 331, 631-643, (2007) · Zbl 1121.34054 [10] Lizana, M.; Rivero, J., Multiparametric bifurcations for a model in epidemiology, J. math. biol., 35, 21-36, (1996) · Zbl 0868.92024 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] Thieme, H.R., Convergence results and a poincare – bendixson trichotomy for asymptotically autonomous differential equations, J. math. biol., 30, 755-763, (1992) · Zbl 0761.34039 [17] 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 [18] Wang, W., Backward bifurcation of an epidemic model with treatment, Math. biosci., 201, 58-71, (2006) · Zbl 1093.92054 [19] Wang, W.; Ma, Z., Global dynamics of an epidemic model with time delay, Nonlinear anal. RWA, 3, 365-373, (2002) · Zbl 0998.92038 [20] Ye, Y., ()
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