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The basic reproduction number in epidemic models with periodic demographics. (English) Zbl 1342.92287
Summary: Patterns of contact in social behaviour and seasonality due to environmental influences often affect the spread and persistence of diseases. Models of epidemics with seasonality and patterns in the contact rate include time-periodic coefficients, making the systems nonautonomous. No general method exists for calculating the basic reproduction number, the threshold for disease extinction, in nonautonomous epidemic models. However, for some epidemic models with periodic coefficients and constant population size, the time-averaged basic reproduction number has been shown to be a threshold for disease extinction. We extend these results by showing that the time-averaged basic reproduction number is a threshold for disease extinction when the population demographics are periodic. The results are shown to hold in epidemic models with periodic demographics that include temporary immunity, isolation, and multiple strains.

MSC:
92D30 Epidemiology
92D25 Population dynamics (general)
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References:
[1] DOI: 10.1007/s00285-003-0207-9 · Zbl 1023.92022 · doi:10.1007/s00285-003-0207-9
[2] DOI: 10.3934/dcdsb.2005.5.175 · Zbl 1080.34034 · doi:10.3934/dcdsb.2005.5.175
[3] DOI: 10.1016/j.mbs.2004.01.003 · Zbl 1073.92040 · doi:10.1016/j.mbs.2004.01.003
[4] DOI: 10.1007/s11538-005-9034-4 · Zbl 1334.92387 · doi:10.1007/s11538-005-9034-4
[5] Allen L. J.S., J. Differ. Equ. Appl. (2008)
[6] DOI: 10.1016/S0022-5193(84)80150-2 · doi:10.1016/S0022-5193(84)80150-2
[7] Chu Y. K., Amer. J. Trop. Med. Hyg. 69 pp 263– (2003)
[8] Cushing J. M., An Introduction to Structured Population Dynamics (1998) · Zbl 0939.92026 · doi:10.1137/1.9781611970005
[9] DOI: 10.1007/BF00178324 · Zbl 0726.92018 · doi:10.1007/BF00178324
[10] Dietz, K. Proceedings of aWorkshop on Mathematical Modeling in Medicine. Mainz, Berlin, Germany. The incidence of infectious diseases under the influence of seasonal fluctuations, Vol. 11, pp.1–15. Lecture Notes in Biomathematics
[11] Dowell F., Emerg. Infect. Dis. 7 pp 369– (2001)
[12] DOI: 10.1073/pnas.0407293101 · doi:10.1073/pnas.0407293101
[13] DOI: 10.1098/rspb.2006.3604 · doi:10.1098/rspb.2006.3604
[14] DOI: 10.1080/023929021000008813 · Zbl 1057.92046 · doi:10.1080/023929021000008813
[15] Grossmann Z., Nonlinear Systems and Applications to Life Sciences pp 525– (1977) · doi:10.1016/B978-0-12-434150-0.50049-7
[16] Hartman P., Ordinary Differential Equations (1964) · Zbl 0125.32102
[17] DOI: 10.1016/j.tpb.2007.04.004 · Zbl 1122.92057 · doi:10.1016/j.tpb.2007.04.004
[18] DOI: 10.1007/s002850050098 · Zbl 0890.92023 · doi:10.1007/s002850050098
[19] DOI: 10.1137/S0036144500371907 · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[20] DOI: 10.1016/S0025-5564(02)00111-6 · Zbl 1019.92030 · doi:10.1016/S0025-5564(02)00111-6
[21] DOI: 10.1016/j.mbs.2006.08.028 · Zbl 1124.92045 · doi:10.1016/j.mbs.2006.08.028
[22] DOI: 10.1006/jtbi.1996.0042 · doi:10.1006/jtbi.1996.0042
[23] DOI: 10.1128/JVI.01680-06 · doi:10.1128/JVI.01680-06
[24] Ma J., Math. Biosci. Eng. 3 pp 161– (2006) · Zbl 1089.92048 · doi:10.3934/mbe.2006.3.161
[25] DOI: 10.1093/imammb/dql023 · Zbl 1118.92053 · doi:10.1093/imammb/dql023
[26] DOI: 10.1016/j.compbiomed.2007.02.007 · doi:10.1016/j.compbiomed.2007.02.007
[27] Moneim I. A., Math. Biosci. Eng. 13 pp 131– (2005)
[28] DOI: 10.1017/CBO9780511546341 · doi:10.1017/CBO9780511546341
[29] Pascual M., PLoS Med. 2 pp 18– (2004) · doi:10.1371/journal.pbio.0020018
[30] DOI: 10.1098/rspb.1998.0537 · doi:10.1098/rspb.1998.0537
[31] DOI: 10.1017/S0950268806006595 · doi:10.1017/S0950268806006595
[32] DOI: 10.1046/j.1365-2656.2003.00675.x · doi:10.1046/j.1365-2656.2003.00675.x
[33] DOI: 10.1007/BF00276232 · Zbl 0558.92013 · doi:10.1007/BF00276232
[34] DOI: 10.1007/BF00160532 · Zbl 0745.92026 · doi:10.1007/BF00160532
[35] DOI: 10.1007/BF00276090 · Zbl 0523.92020 · doi:10.1007/BF00276090
[36] DOI: 10.1371/journal.pmed.0020006 · doi:10.1371/journal.pmed.0020006
[37] DOI: 10.1016/S0025-5564(02)00108-6 · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[38] DOI: 10.1080/08898480600788584 · Zbl 1107.92050 · doi:10.1080/08898480600788584
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