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Stability and Hopf bifurcation for an epidemic disease model with delay. (English) Zbl 1165.34048
The authors consider the following predator-prey system with disease in the prey \begin{aligned} \dot S(t)&=rS(t)\left(1-\frac{S(t)+I(t)}{K}\right)-\beta S(t)I(t),\\ \dot I(t)&=\beta S(t)I(t)-cI(t)-pI(t)y(t),\\ \dot y(t)&=-dy(t)+kpI(t-\tau)y(t-\tau), \end{aligned} where $$S(t), I(t), y(t)$$ denote the susceptible prey, the infected prey and the predator population at time $$t$$, respectively; $$r>0$$ is the intrinsic growth rate of the prey, $$K>0$$ is the carrying capacity of the prey, $$\beta>0$$ is the transmission coefficient, $$d>0$$ is the death rate of the predator, $$c>0$$ is the death rate of the infected prey, $$k>0$$ is the conversing rate of the predator by consuming prey, $$\tau>0$$ is a time delay due to the gestation of predator.
By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium and the existence of Hopf bifurcations are established. By using the normal form theory and center manifold argument, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional-differential equations 92D30 Epidemiology 34K17 Transformation and reduction of functional-differential equations and systems, normal forms 34K19 Invariant manifolds of functional-differential equations 34K13 Periodic solutions to functional-differential equations
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##### References:
 [1] Kermack, W.; McKendrick, A., Contributions to the mathematical theory of epidemic, Proc roy soc A, 115, 700, (1927) · JFM 53.0517.01 [2] Bailey, N., The mathematical theory of infectious disease and its application, (1975), Griffin London [3] Hethcote, H.W.; Stech, H.W.; Van Den Driessche, P., Nonlinear oscillations in epidemic models, SIAM J appl math, 40, 1-9, (1981) · Zbl 0469.92012 [4] Liu, W.; Lerin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model, J math biol, 23, 187-204, (1986) · Zbl 0582.92023 [5] Hethcote H. A thousand and one epidemic models. In: Levin SA, editor. Frontiers in mathematical biology, Lecture notes in biomathematics, vol. 100. 1994. p. 504. · Zbl 0819.92020 [6] Chattopadhyay, J.; Arino, O., A predator-prey model with disease in the prey, Nonlinear anal, 36, 747-766, (1999) · Zbl 0922.34036 [7] Venturino, E., The influence of disease on Lotka-Volterra systems, Rockymount J math, 24, 381-402, (1994) · Zbl 0799.92017 [8] Zhou, L.; Tang, Y., Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, solitons & fractals, 14, 1201-1225, (2002) · Zbl 1038.35147 [9] Krise, S.; Choudhury, S.R., Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, solitons & fractals, 16, 59-77, (2003) · Zbl 1033.37048 [10] Yuan, S.; Han, M.; Ma, Z., Competition in the chemostat: convergence of a model with delayed response in growth, Chaos, solitons & fractals, 17, 659-667, (2003) · Zbl 1036.92037 [11] Saito, Y., The necessary and sufficient condition for global stability of a Lotka-Volterra cooperative or competition system with delays, J math anal appl, 268, 109-124, (2002) · Zbl 1012.34072 [12] Hale, J.; Lunel, S., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002 [13] Murray, T.D., Mathematical biology, (1989), Springer-Verlag New York · Zbl 0682.92001 [14] Chen, L., Mathematical ecological model and the research methods, (1988), Scientific Press [15] Yang, H.; Tian, Y., Hopf bifurcation in REM algorithm with communication delay, Chaos, solitons & fractals, 25, 1093-1105, (2005) · Zbl 1198.93099 [16] Song, Y.; Wei, J., Bifurcation analysis for chen’s system with delayed feedback and its application to control of chaos, Chaos, solitons & fractals, 22, 75-91, (2004) · Zbl 1112.37303 [17] Hassard, B.; Kazarino, D.; Wan, Y., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002 [18] Brauer, F., Absolute stability in delay equations, J differen equat, 69, 185-191, (1987) · Zbl 0636.34063 [19] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [20] Song, Y.; Han, M.; Wei, J., Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D, 200, 185-204, (2005) · Zbl 1062.34079 [21] Dieuonné, J., Foundations of modern analysis, (1960), Academic
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