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Analysis of a delayed SIR model with nonlinear incidence rate. (English) Zbl 1159.92037
Summary: An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value \(\operatorname{Re} _{0}\) determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value \(\operatorname{Re} _{0}\) and the time delay (i.e., incubation time length). If \(\operatorname{Re} _{0}\) is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one it will be endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.

MSC:
92D30 Epidemiology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:
[1] DOI: 10.1007/BF00276956 · Zbl 0582.92023 · doi:10.1007/BF00276956
[2] DOI: 10.1007/BF00277162 · Zbl 0621.92014 · doi:10.1007/BF00277162
[3] DOI: 10.1007/BF00160539 · Zbl 0722.92015 · doi:10.1007/BF00160539
[4] DOI: 10.1093/imammb/dql011 · Zbl 1098.92058 · doi:10.1093/imammb/dql011
[5] DOI: 10.1093/imammb/dqi001 · Zbl 1076.92048 · doi:10.1093/imammb/dqi001
[6] DOI: 10.1016/0025-5564(78)90006-8 · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8
[7] DOI: 10.1016/S0022-0396(02)00089-X · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[8] DOI: 10.1016/j.mbs.2006.09.025 · Zbl 1119.92042 · doi:10.1016/j.mbs.2006.09.025
[9] DOI: 10.1016/S0898-1221(99)00206-0 · Zbl 0939.92031 · doi:10.1016/S0898-1221(99)00206-0
[11] The Rocky Mountain Journal of Mathematics 9 (1) pp 31– (1979) · Zbl 0423.92029 · doi:10.1216/RMJ-1979-9-1-31
[12] Quarterly of Applied Mathematics 59 (1) pp 159– (2001) · Zbl 1035.34084 · doi:10.1090/qam/1811101
[13] DOI: 10.2307/3933 · doi:10.2307/3933
[14] DOI: 10.1016/j.vaccine.2006.05.018 · doi:10.1016/j.vaccine.2006.05.018
[15] DOI: 10.1016/j.apm.2007.06.005 · Zbl 1182.92064 · doi:10.1016/j.apm.2007.06.005
[16] DOI: 10.1016/j.amc.2006.07.124 · Zbl 1111.92049 · doi:10.1016/j.amc.2006.07.124
[17] DOI: 10.1155/2008/746951 · Zbl 1149.92329 · doi:10.1155/2008/746951
[18] DOI: 10.1016/j.bulm.2005.01.002 · Zbl 1334.92392 · doi:10.1016/j.bulm.2005.01.002
[19] Canadian Medical Association Journal 168 (11) pp 1443– (2003)
[20] DOI: 10.1001/jama.290.4.524 · doi:10.1001/jama.290.4.524
[21] DOI: 10.1016/0898-1221(96)00129-0 · Zbl 0873.34061 · doi:10.1016/0898-1221(96)00129-0
[22] Mathematics in Science and Engineering 191 pp xii+398– (1993)
[23] DOI: 10.1016/S0025-5564(01)00049-9 · Zbl 0978.92031 · doi:10.1016/S0025-5564(01)00049-9
[24] DOI: 10.1137/0520025 · Zbl 0692.34053 · doi:10.1137/0520025
[25] CBMS-NSF Regional Conference Series in Applied Mathematics 45 pp v+77– (1983)
[26] DOI: 10.1137/0512047 · Zbl 0471.92021 · doi:10.1137/0512047
[27] DOI: 10.1016/0025-5564(88)90049-1 · Zbl 0642.92016 · doi:10.1016/0025-5564(88)90049-1
[28] London Mathematical Society Lecture Note Series 41 pp v+311 pp– (1981)
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