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Threshold conditions for a nonautonomous epidemic model with vaccination. (English) Zbl 1144.34032
A non-autonomous epidemic model with vaccination is studied. Sufficient conditions for the permanence and boundedness of the solutions are derived. Sufficient conditions for the disease free solution (DFS i.e. $$I(t)=0$$) to exist and be globally stable are found. Numerical examples with periodic coefficients are given. The reviewer has two comments: i) Equation after (3.1) has a misprint lim inf $$I(t)>l$$ (not one). ii) The biological meaning of the conditions derived is not clear.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34C11 Growth and boundedness of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
##### Keywords:
epidemiology; vaccination; permanence; threshold condition
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##### References:
 [1] DOI: 10.1098/rspa.1927.0118 · JFM 53.0517.01 [2] DOI: 10.1038/280361a0 [3] Anderson RM, Infectious Diseases of Humans, Dynamics and Control (1991) [4] Diekmann O, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000) [5] DOI: 10.1007/BF00160539 · Zbl 0722.92015 [6] DOI: 10.1007/BF00173265 · Zbl 0774.92018 [7] DOI: 10.1007/s00285-006-0006-1 · Zbl 1098.92044 [8] DOI: 10.1007/s002850000032 · Zbl 0961.92029 [9] DOI: 10.1007/BF00178324 · Zbl 0726.92018 [10] DOI: 10.1016/0025-5564(91)90012-8 · Zbl 0748.92010 [11] DOI: 10.1016/S0025-5564(02)00108-6 · Zbl 1015.92036 [12] Ma Z, Mathematical Modeling and Research of Epidemic Dynamical Systems (2004) [13] DOI: 10.1016/j.mbs.2007.07.005 · Zbl 1133.92023 [14] DOI: 10.1007/s11538-006-9166-9 · Zbl 1298.92093 [15] Iwasa Y, Mathematics for Life Science and Medicine pp 1– (2007) [16] DOI: 10.1016/S0169-5347(02)02502-8 [17] London W, American Journal of Epidemiology 98 pp 453– (1973) [18] Dowell SF, Emerging Infectious Diseases 7 pp 369– (2001) [19] DOI: 10.1017/S0950268805004930 [20] DOI: 10.1007/s00285-006-0015-0 · Zbl 1098.92056 [21] DOI: 10.1007/s11538-006-9108-6 · Zbl 1296.92226 [22] DOI: 10.1016/S1468-1218(02)00075-5 · Zbl 1067.92053 [23] DOI: 10.1016/S0025-5564(00)00018-3 · Zbl 0970.37061 [24] DOI: 10.1090/S0002-9939-99-05034-0 · Zbl 0918.34053 [25] Abdurahman X, Journal of Biomathematics 21 pp 167– (2006) [26] DOI: 10.1007/s11538-007-9231-z · Zbl 1245.34040 [27] Ma J, Mathematical Biosciences and Engineering 3 pp 161– (2006) · Zbl 1089.92048 [28] DOI: 10.1016/j.jmaa.2006.01.085 · Zbl 1101.92046
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