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Article

Keywords:
epidemic model; Fredholm mapping; coincidence degree
Summary:
An SEIR model with periodic coefficients in epidemiology is considered. The global existence of periodic solutions with strictly positive components for this model is established by using the method of coincidence degree. Furthermore, a sufficient condition for the global stability of this model is obtained. An example based on the transmission of respiratory syncytial virus (RSV) is included.
References:
[1] Al-Ajam, M. R., Bizri, A. R., Mokhbat, J., Weedon, J., Lutwick, L.: Mucormycosis in the Eastern Mediterranean: a seasonal disease. Epidemiol. Infect. 134 (2006), 341-346. DOI 10.1017/S0950268805004930
[2] Anderson, R. M., May, R. M.: Population biology of infectious diseases, Part 1. Nature 280 (1979), 361. DOI 10.1038/280361a0
[3] Anderson, R. M., May, R. M.: Infectious Diseases of Humans, Dynamics and Control. Oxford University Oxford (1991).
[4] Arenas, A. J., Gonzalez, G., Jódar, L.: Existence of periodic solutions in a model of respiratory syncytial virus RSV. J. Math. Anal. Appl. 344 (2008), 969-980. DOI 10.1016/j.jmaa.2008.03.049 | MR 2426325 | Zbl 1137.92318
[5] Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28 (1990), 365-382. DOI 10.1007/BF00178324 | MR 1057044
[6] Diekmann, O., Heesterbeek, J. A. P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley & Sons Chichester (2000). MR 1882991
[7] Driessche, P. van den, Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002), 29-48. DOI 10.1016/S0025-5564(02)00108-6 | MR 1950747
[8] Earn, D. J. D., Dushoff, J., Levin, S. A.: Ecology and evolution of the flu. Trends in Ecology and Evolution 17 (2002), 334-340.
[9] Fan, M., Wang, K.: Periodicity in a delayed ratio-dependent predator-prey system. J. Math. Anal. Appl. 262 (2001), 179-190. DOI 10.1006/jmaa.2001.7555 | MR 1857221 | Zbl 0994.34058
[10] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Springer Berlin (1977). MR 0637067 | Zbl 0339.47031
[11] Hale, J. K.: Ordinary Differential Equations. Wiley-Interscience New York (1969). MR 0419901 | Zbl 0186.40901
[12] Herzog, G., Redheffer, R.: Nonautonomous SEIRS and Thron models for epidemiology and cell biology. Nonlinear Anal., Real World Appl. 5 (2004), 33-44. MR 2004085 | Zbl 1067.92053
[13] Hethcote, H. W.: The mathematics of infectious diseases. SIAM Review 42 (2000), 599-653. DOI 10.1137/S0036144500371907 | MR 1814049 | Zbl 0993.92033
[14] Jódar, L., Villanueva, R. J., Arenas, A.: Modeling the spread of seasonal epidemiological diseases: Theory and applications. Math. Comput. Modelling 48 (2008), 548-557. DOI 10.1016/j.mcm.2007.08.017 | MR 2431484 | Zbl 1145.92336
[15] Li, Y., Kuang, Y.: Periodic solutions of periodic delay Lotka-Volterra equations and Systems. J. Math. Anal. Appl. 255 (2001), 260-280. DOI 10.1006/jmaa.2000.7248 | MR 1813821 | Zbl 1024.34062
[16] Li, M. Y., Muldowney, J. S.: Global stability for the SEIR model in epidemiology. Math. Biosci. 125 (1995), 155-164. DOI 10.1016/0025-5564(95)92756-5 | MR 1315259 | Zbl 0821.92022
[17] London, W., Yorke, J. A.: Recurrent outbreaks of measles, chickenpox and mumps. 1. Seasonal variation in contact rates. Am. J. Epidemiol. 98 (1973), 453-468. DOI 10.1093/oxfordjournals.aje.a121575
[18] Ma, J., Ma, Z.: Epidemic threshold conditions for seasonally forced SEIR models. Math. Biosci. Eng. 3 (2006), 161-172. DOI 10.3934/mbe.2006.3.161 | MR 2192132 | Zbl 1089.92048
[19] Nuño, M., Feng, Z., Martcheva, M., Carlos, C. C.: Dynamics of two-strain influenza with isolation and partial cross-immunity. SIAM J. Appl. Math. 65 (2005), 964-982. DOI 10.1137/S003613990343882X | MR 2136038
[20] Teng, Z.: On the periodic solutions of periodic multi-species competitive systems with delays. Appl. Math. Comput. 127 (2002), 235-247. DOI 10.1016/S0096-3003(00)00171-5 | MR 1883850 | Zbl 1035.34078
[21] Teng, Z., Chen, L.: Permanence and extinction of periodic predator-prey systems in a patchy environment with delay. Nonlinear Anal., Real World Appl. 4 (2003), 335-364. MR 1942689 | Zbl 1018.92033
[22] Weber, A., Weber, M., Milligan, P.: Modeling epidemics caused by respiratory syncytial virus (RSV). Math. Biosci. 172 (2001), 95-113. DOI 10.1016/S0025-5564(01)00066-9 | MR 1853471 | Zbl 0988.92025
[23] Zhang, X., Chen, L.: The periodic solution of a class of epidemic models. Comput. Math. Appl. 38 (1999), 61-71. DOI 10.1016/S0898-1221(99)00206-0 | MR 1703408 | Zbl 0939.92031
[24] Zhang, T., Liu, J., Teng, Z.: Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure. Nonlinear Anal., Real World Appl. 11 (2010), 293-306. MR 2570549 | Zbl 1195.34130
[25] Zhang, J., Ma, Z.: Global dynamics of an SEIR epidemic model with saturating contact rate. Math. Biosci. 185 (2003), 15-32. DOI 10.1016/S0025-5564(03)00087-7 | MR 2003259 | Zbl 1021.92040
[26] Zhang, T., Teng, Z.: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69 (2007), 2537-2559. DOI 10.1007/s11538-007-9231-z | MR 2353845 | Zbl 1245.34040
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