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Existence of epidemic waves in a disease transmission model with two-habitat population. (English) Zbl 1160.93349

Summary: A three variable mathematical model describing the propagation of an infectious disease in a human population is proposed and analyzed. The human population is assumed to live in two distinct habitats with no inter-habitat migration. The infectious agent disperse randomly among the said habitats. Methods of upper and lower solutions are used to establish the existence of traveling wave solutions connecting the trivial with the nontrivial equilibrium. The critical wave speed required for the existence of such wave solutions has been found out and shown to depend on different system parameters together with the dispersal rate.

MSC:

93C15 Control/observation systems governed by ordinary differential equations
92D25 Population dynamics (general)
93A30 Mathematical modelling of systems (MSC2010)
92D30 Epidemiology
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[1] DOI: 10.1007/BFb0070595 · doi:10.1007/BFb0070595
[2] Capasso V, J. Math. Biol 13 pp 173– (1981) · Zbl 0468.92016 · doi:10.1007/BF00275212
[3] DOI: 10.1016/0378-4754(82)90656-5 · Zbl 0502.92018 · doi:10.1016/0378-4754(82)90656-5
[4] Capasso V, Quart. Appl. Math. 46 pp 431– (1988) · Zbl 0704.35069 · doi:10.1090/qam/963580
[5] Capasso V, Lecture Notes in Biomath 97 (1993)
[6] DOI: 10.1137/S0036139995284681 · Zbl 0872.35053 · doi:10.1137/S0036139995284681
[7] DOI: 10.1137/0146063 · Zbl 0617.92020 · doi:10.1137/0146063
[8] Fife PC, Lecture Notes in Biomath 28 (1979)
[9] DOI: 10.1007/s002850000047 · Zbl 0982.92028 · doi:10.1007/s002850000047
[10] Hadeler KP, Differential Equations with Application to Biology 21 pp 251– (1999)
[11] DOI: 10.1093/imamat/63.2.199 · Zbl 0939.35091 · doi:10.1093/imamat/63.2.199
[12] DOI: 10.1137/0146062 · Zbl 0655.35046 · doi:10.1137/0146062
[13] Krinsky VI, Mathematical Cybernetics Popular Ser. (Life Sciences) 8 pp 1– (1986)
[14] DOI: 10.1006/jdeq.2000.3846 · Zbl 0988.34053 · doi:10.1006/jdeq.2000.3846
[15] DOI: 10.1137/S0036139998328034 · Zbl 0944.34021 · doi:10.1137/S0036139998328034
[16] DOI: 10.1080/00207720500150911 · Zbl 1077.35062 · doi:10.1080/00207720500150911
[17] Murray JD, Mathematical Biology - I (2002)
[18] DOI: 10.1002/cpa.3160260402 · Zbl 0253.92004 · doi:10.1002/cpa.3160260402
[19] Perumpanani AJ, Invasion Metastasis 16 pp 209– (1996)
[20] Schaaf KW, Trans. Amer. Math. Soc. 302 pp 587– (1987)
[21] Segel (Ed.) LA, Mathematical Models in Molecular and Cellular Biology (1980) · Zbl 0448.92001
[22] DOI: 10.1137/S0036139993243746 · Zbl 0806.35080 · doi:10.1137/S0036139993243746
[23] DOI: 10.1016/0167-2789(94)00224-E · Zbl 0900.35194 · doi:10.1016/0167-2789(94)00224-E
[24] DOI: 10.1016/S0893-9659(97)00001-3 · Zbl 0883.65071 · doi:10.1016/S0893-9659(97)00001-3
[25] DOI: 10.1016/S0167-2789(97)00317-5 · Zbl 0940.35111 · doi:10.1016/S0167-2789(97)00317-5
[26] DOI: 10.1046/j.1461-0248.2001.00193.x · doi:10.1046/j.1461-0248.2001.00193.x
[27] DOI: 10.1098/rspb.2001.1890 · doi:10.1098/rspb.2001.1890
[28] DOI: 10.1137/S0036139902392483 · Zbl 1036.35093 · doi:10.1137/S0036139902392483
[29] DOI: 10.1137/S0036141098346785 · Zbl 0941.35125 · doi:10.1137/S0036141098346785
[30] Volpert AI, Amer. Math. Soc. 140 (1994)
[31] DOI: 10.1038/scientificamerican0674-82 · doi:10.1038/scientificamerican0674-82
[32] Winfree AT, The Geometry of Biological Time (1980) · doi:10.1007/978-3-662-22492-2
[33] Wu J, Theory and Applications of Partial Differential Equations (1996) · doi:10.1007/978-1-4612-4050-1
[34] DOI: 10.1023/A:1016690424892 · Zbl 0996.34053 · doi:10.1023/A:1016690424892
[35] Zhao XQ, Canad. Appl. Math. Quart. 4 pp 421– (1996)
[36] DOI: 10.3934/dcdsb.2004.4.1117 · Zbl 1097.34022 · doi:10.3934/dcdsb.2004.4.1117
[37] Zykov VS, Modelling of Wave Processes in Excitable Media (1988)
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