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A non-autonomous SEIRS model with general incidence rate. (English) Zbl 1338.92133
Summary: For a non-autonomous SEIRS model with general incidence, that admits [Y. Nakata and T. Kuniya, J. Math. Anal. Appl. 363, No. 1, 230–237 (2010; Zbl 1184.34056)] as a very particular case, we obtain conditions for extinction and strong persistence of the infectives. Our conditions are computed for several particular settings and extend the hypothesis of several proposed non-autonomous models. Additionally we show that our conditions are robust in the sense that they persist under small perturbations of the parameters in some suitable family. We also present some simulations that illustrate our results.

MSC:
92D30 Epidemiology
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[1] Bacaër, N., Approximation of the basic reproduction number \(R_0\) for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69, 1067-1091, (2007) · Zbl 1298.92093
[2] Buonomo, B.; Lacitignola, D., On the dynamics of an SEIR epidemic model with a convex incidence rate, Ric. Mat., 57, 261-281, (2008) · Zbl 1232.34061
[3] Castillo-Chavez, C.; Thieme, H. R., Asymptotically autonomous epidemic models, (Arino, O.; Axelrod, D. E.; Kimmel, M.; Langlais, M., Mathematical Population Dynamics: Analysis and Heterogenity, (1995), Wuerz Winnipeg, Canada), 33
[4] den Driessche, P.; Hethcote, H., J. Math. Biol., 29, 271-287, (1991)
[5] van den Driessche, P.; Li, M.; Muldowney, J., Global stability of SEIRS models in epidemiology, Can. Appl. Math. Q., 7, 409-425, (1999) · Zbl 0976.92020
[6] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115, 700-721, (1927) · JFM 53.0517.01
[7] Kuniya, T.; Nakata, Y., Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363, 230-237, (2010) · Zbl 1184.34056
[8] Kuniya, T.; Nakata, Y., Permanence and extinction for a nonautonomous SEIRS epidemic model, Appl. Math. Comput., 218, 9321-9331, (2012) · Zbl 1245.34038
[9] Li, X.; Zhou, L., Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate, Chaos Solitons Fract., 40, 874-884, (2009) · Zbl 1197.34077
[10] Markus, L., Asymptotically autonomous differential systems, (Lefschetz, S., Contributions to the Theory of Nonlinear Oscillations 111, Annals of Mathematics Studies, vol. 36, (1956), Princeton University Press Princeton, NJ), 17-29 · Zbl 0075.27002
[11] Mischaikow, K.; Smith, H.; Thieme, H. R., Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Am. Math. Soc., 347, 1669-1685, (1995) · Zbl 0829.34037
[12] Pereira, E.; Silva, C. M.; Silva, J. A.L., A generalized non-autonomous SIRVS model, Math. Methods Appl. Sci., 36, 275-289, (2013) · Zbl 1257.93013
[13] Rebelo, C.; Margheri, A.; Bacaër, N., Persistence in seasonally forced epidemiological models, J. Math. Biol., 54, 933-949, (2012) · Zbl 1303.92122
[14] Shope, R., Environ. Health Perspect., 96, 171-174, (1991)
[15] Zhang, T.; Teng, Z., On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69, 2537-2559, (2007) · Zbl 1245.34040
[16] Wang, W.; Zhao, X.-Q., Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20, 699-717, (2008) · Zbl 1157.34041
[17] Zhang, T.; Teng, Z.; Gao, S., Threshold conditions for a nonautonomous epidemic model with vaccination, Appl. Anal., 87, 181-199, (2008) · Zbl 1144.34032
[18] Zhang, H.; Yingqi, L.; Xu, W., Global stability of an SEIS epidemic model with general saturation incidence, Appl. Math., (2013), Art. ID 710643 · Zbl 1298.34138
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