## A delayed SIRS epidemic model with pulse vaccination.(English)Zbl 1152.34379

Summary: A delayed SIRS epidemic model with pulse vaccination and saturated contact rate is investigated. By using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the system. Further, by using the comparison theorem, we prove that under the condition that $$R_{0} < 1$$ the infection-free periodic solution is globally attractive, and that under the condition that $$R^{\prime} > 1$$ the disease is uniformly persistent, which means that after some period of time the disease will become endemic.

### MSC:

 34K20 Stability theory of functional-differential equations 92D30 Epidemiology
Full Text:

### References:

 [1] Sabin, A.B., Measles: killer of millions in developing countries: strategies of elimination and continuation control, Eur J epid, 7, 1-22, (1991) [2] d’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math comput model, 36, 473-489, (2002) · Zbl 1025.92011 [3] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026 [4] Lu, Z.; Chi, X.; Chen, L., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math comput model, 36, 1039-1057, (2002) · Zbl 1023.92026 [5] Hui, J.; Chen, L., Impulsive vaccination of SIR epidemic models with nonlinear incidence rates, Discrete and continuous dynamical systems (series B), 4, 595-605, (2004) · Zbl 1100.92040 [6] Zeng, G.; Chen, L.; Sun, L., Complexity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, solitons & fractals, 26, 495-505, (2005) · Zbl 1065.92050 [7] Agur, Z.; Cojocaru, L.; Anderson, R.; Danon, Y., Pulse mass measles vaccination across age cohorts, Proc natl acad sci USA, 90, 11698-11702, (1993) [8] Bai, Y.; Jin, Z., Prediction of SARS epidemic by BP neural networks with online prediction strategy, Chaos, solitons & fractals, 26, 559-569, (2005) · Zbl 1065.92037 [9] Awad, El-Gohary, Optimal control of the genital herpes epidemic, Chaos, solitons & fractals, 12, 1817-1822, (2001) · Zbl 0979.92022 [10] Li, G.; Jin, Z., Global stability of an SEI epidemic model, Chaos, solitons & fractals, 21, 925-931, (2004) · Zbl 1045.34025 [11] Li, X.; Wang, W., A discrete epidemic model with stage structure, Chaos, solitons & fractals, 26, 947-958, (2005) · Zbl 1066.92045 [12] Engbert, R.; Drepper, F.R., Chance and chaos in population biology-models of recurrent epidemics and food chain dynamics, Chaos, solitons & fractals, 4, 1147-1169, (1994) · Zbl 0802.92023 [13] Wang, W., Global behavior of an SEIRS epidemic model with time delays, Appl math lett, 15, 423-428, (2002) · Zbl 1015.92033 [14] 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 [15] Korobeinikov, A.; Wake, G.C., Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl math lett, 15, 955-960, (2002) · Zbl 1022.34044 [16] Smith, H.L.; Williams, M., Stability in a cyclic epidemic models with delay, J math biol, 11, 95-103, (1981) · Zbl 0449.92022 [17] Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J math biol, 33, 250-260, (1995) · Zbl 0811.92019 [18] Kyrychko, Yuliya N.; Blyuss, Konstantin B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear analysis: realworld applications, 6, 495-507, (2005) · Zbl 1144.34374 [19] Cull, P., Global stability for population models, Bull math biol, 43, 47-58, (1981) · Zbl 0451.92011 [20] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.C., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.