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Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. (English) Zbl 1111.92049

Summary: A new SEIRS epidemic disease model with two profitless delays and nonlinear incidence are proposed, and the dynamic behavior of the model under pulse vaccination is analyzed. Using a discrete dynamical system determined by the stroboscopic map, we show that there exist ‘infection-free’ periodic solutions; further we show that the ‘infection-free’ periodic solution is globally attractive when the period of impulsive effect is less than some critical value.
Using a new modeling method, we obtain sufficient conditions for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and nonlinear incidence can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show the delays are “profitless”. The main feature is to introduce two discrete time delays and impulse into the SEIRS epidemic model and to give pulse vaccination strategies.

MSC:

92D30 Epidemiology
34C25 Periodic solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34K60 Qualitative investigation and simulation of models involving functional-differential equations
37N25 Dynamical systems in biology
65L99 Numerical methods for ordinary differential equations
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[1] Li, G.; Jin, Z., Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos solitons fract., 25, 1177-1184, (2005) · Zbl 1065.92046
[2] Michael, Y.; Smith, H.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. appl. math., 62, 1, 58-69, (2001) · Zbl 0991.92029
[3] Al-Showaikh, F.N.M.; Twizell, E.H., One-dimensional measles dynamics, Appl. math. comput., 152, 169-194, (2004) · Zbl 1047.92040
[4] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. comput. model., 25, 85-107, (1997) · Zbl 0877.92023
[5] Hethcote, H.W.; Levin, S.A., Periodicity in epidemiological models, (), 193-211
[6] Hethcote, H.W.; Stech, H.W.; Van den Driessche, P., Periodicity and stability in epidemic models: a survey, (), 65-85
[7] D’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. bicsci., 179, 57-72, (2002) · Zbl 0991.92025
[8] d’Onofrio, A., Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times, Appl. math. comput., 151, 181-187, (2004) · Zbl 1043.92033
[9] Cooke, K.; van Den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. math. biol., 35, 240-260, (1996) · Zbl 0865.92019
[10] Langlais, M., A remark on a generic SEIRS model and application to cat retrovirusese and fox rabies, Math. comput. model., 31, 117-124, (2000)
[11] Wang, W., Global behavior of an SEIRS epidemic model with time delays, Appl. math. lett., 15, 423-428, (2002) · Zbl 1015.92033
[12] Pourabbas, E.; d’Onofrio, A.; Rafanelli, M., A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera, Appl. math. comput., 118, 161-174, (2001) · Zbl 1017.92032
[13] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differ. equat., 188, 135-163, (2003) · Zbl 1028.34046
[14] May, R.M.; Anderson, R.M., Regulation and stability of host parasite population interactions, II. destabilizing process, J. anim. ecol., 47, 219-267, (1978)
[15] Anderson, R.M.; May, R.M., Infectious diseases of humans, dynamics and control, (1992), Oxford University Press Oxford
[16] Fine, P.M., Vectors and vertical transmission, an epidemiological perspective, Annal. N.Y. acad. sci., 266, 173-194, (1975)
[17] Busenberg, S., Analysis of a model of a vertically transmitted disease, J. math. bwlogy., 17, 305-329, (1983) · Zbl 0518.92024
[18] Busenberg, S.N.; Cooke, K.L., Models of vertical transmitted diseases with sequential-continuous dynamics, (), 179-187
[19] Cook, K.L.; Busenberg, S.N., Vertical transmission diseases, (), 189-197
[20] 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
[21] D’Onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. lett., 18, 729-732, (2005) · Zbl 1064.92041
[22] Nokes, D.; Swinton, J., The control of childhood viral infections by pulse vaccination, IMA J. math. appl. biol. med., 12, 29-53, (1995) · Zbl 0832.92024
[23] Zeng, G.; Chen, L., Complexity and asymptotical behavior of a SIRS epidemic model with proportional impulse vaccination, Adv. complex syst., 8, 4, 419-431, (2005) · Zbl 1082.92040
[24] Takeuchi, Y.; Ma, W.; Berettac, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
[25] Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl. math. lett., 17, 1141-1145, (2004) · Zbl 1071.34082
[26] Jin, Z.; Ma, Z., The stability of an SIR epidemic model with time delays, Math. biosci. eng., 3, 1, 101-109, (2006) · Zbl 1089.92045
[27] Beretta, E.; Hara, T., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal., 47, 4107-4115, (2001) · Zbl 1042.34585
[28] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press, Inc. San Diego, CA · Zbl 0777.34002
[29] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
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