Stability and bifurcation analysis in a delayed SIR model. (English) Zbl 1131.92055
Summary: A time-delayed SIR model with a nonlinear incidence rate is considered. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.
MSC:
| 92D30 | Epidemiology |
| 34K18 | Bifurcation theory of functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
| 37N25 | Dynamical systems in biology |
| 34K13 | Periodic solutions to functional-differential equations |
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
References:
| [1] | Alexander, M.; Mmoghadas, S., Periodicity in an epidemic model with a generalized non-linear incidence, Math Biosci, 198, 75-96 (2004) · Zbl 1073.92040 |
| [2] | Beretta, E.; Takeuchi, Y., Global stability of a SIR epidemic model with time delays, J Math Biol, 33, 250-260 (1995) · Zbl 0811.92019 |
| [3] | Beretta, E.; Takeuchi, Y., Convergence results in SIR epidemic models with varying population sizes, Nonlinear Anal TMA, 28, 1909-1921 (1997) · Zbl 0879.34054 |
| [4] | Beretta, E.; Kuang, Y., Geometric, stability switch criteria in delay differential systems with delay dependent parameters, SIAM J Math Anal, 33, 1144-1165 (2002) · Zbl 1013.92034 |
| [5] | Brauer, F.; Van den Diessche, P., Models for transmission of disease with immigration of infective, Math Biosci, 171, 143-154 (2001) · Zbl 0995.92041 |
| [6] | BuriImage, N.; TodoroviImage, D., Dynamics of delay-differential equations modelling immunology of tumor growth, Chaos, Solitons & Fractals, 13, 645-655 (2002) · Zbl 1029.34069 |
| [7] | Choudhury, S. R., On bifurcations and chaos in predator-prey models with delay, Chaos, Solitons & Fractals, 2, 393-409 (1992) · Zbl 0753.92022 |
| [8] | Cooke, K. L.; van den Driessche, P.; Zou, X., Interaction of maturation delay and nonlinear birth in population and epidemic models, J Math Biol, 39, 332-352 (1999) · Zbl 0945.92016 |
| [9] | Dieudonné, J., Foundations of modern analysis (1969), Academic Press: Academic Press New York · Zbl 0176.00502 |
| [10] | Gakkhar, S.; Singh, B., Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters, Chaos, Solitons & Fractals, 27, 1239-1255 (2006) · Zbl 1094.92059 |
| [11] | Gao, S.; Chen, L., The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses, Chaos, Solitons & Fractals, 24, 1013-1023 (2005) · Zbl 1061.92059 |
| [12] | Hale, J., Theory of functional differential equations (1977), Springer: Springer New York · Zbl 0352.34001 |
| [13] | Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and applications of Hopf bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002 |
| [14] | Jiang G, Lu Q, Qian L. Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos, Solitons & Fractals, in press, in press. doi:10.1016/j.chaos.2005.09.077; Jiang G, Lu Q, Qian L. Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos, Solitons & Fractals, in press, in press. doi:10.1016/j.chaos.2005.09.077 · Zbl 1203.34071 |
| [15] | Kyrychko, Y.; Blyuss, B., Global properties of a delay SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal: Real World Appl, 6, 495-507 (2005) · Zbl 1144.34374 |
| [16] | Li, S.; Liao, X.; Li, C., Hopf bifurcation in a Volterra prey-predator model with strong kernel, Chaos, Solitons & Fractals, 22, 713-722 (2004) · Zbl 1073.34086 |
| [17] | Liu, Z.; Yuan, R., Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos, Solitons & Fractals, 27, 1395-1407 (2006) · Zbl 1097.34051 |
| [18] | Meng, X.; Wei, J., Stability and bifurcation of mutual system with time delay, Chaos, Solitons & Fractals, 21, 729-740 (2004) · Zbl 1048.34122 |
| [19] | Moghadas, S.; Gumel, A., Global stability of a two-stage epidemic model with generalized non-linear incidence rate, Math Comput Simul, 60, 107-118 (2002) · Zbl 1005.92031 |
| [20] | Song, Y.; Han, M.; Peng, Y., Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays, Chaos, Solitons & Fractals, 22, 1139-1148 (2004) · Zbl 1067.34075 |
| [21] | Sun C, Han M, Lin Y, Chen Y. Global qualitative analysis for a predator-prey system with delay. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2005.11.038; Sun C, Han M, Lin Y, Chen Y. Global qualitative analysis for a predator-prey system with delay. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2005.11.038 · Zbl 1145.34042 |
| [22] | Sun, C.; Lin, Y.; Han, M., Stability and Hopf bifurcation for an epidemic disease model with delay, Chaos, Solitons & Fractals, 30, 204-216 (2006) · Zbl 1165.34048 |
| [23] | Wang, F.; Zhang, S.; Chen, L.; Sun, L., Bifurcation and complexity of Monod type predator-prey system in a pulsed chemostat, Chaos, Solitons & Fractals, 27, 447-458 (2006) · Zbl 1096.34029 |
| [24] | Wei J, Zou X. Bifurcation analysis in population and epidemic models with delay. J Comput Appl Math, in press.; Wei J, Zou X. Bifurcation analysis in population and epidemic models with delay. J Comput Appl Math, in press. · Zbl 1098.92055 |
| [25] | Zeng, G.; Chen, L.; Sun, L., Complixity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, Solitons & Fractals, 26, 495-505 (2005) · Zbl 1065.92050 |
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