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Modeling disease spread via transport-related infection by a delay differential equation. (English) Zbl 1194.34111
Summary: A delayed SIS model is developed to describe the effect of transport-related infection, where time delay arises very naturally and the basic reproduction number \(R_0\) can be calculated. It is shown that this number characterizes the disease transmission dynamics: if \(R_0<1\), there exists only a disease-free equilibrium which is globally asymptotically stable; and if \(R_0>1\), then there is a disease-endemic equilibrium and the disease persists. Analysis of the dependence of \(R_0\) on the transport-related infection parameters shows that an outbreak can arise purely due to this transport-related infection.

MSC:
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
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[1] J. Arino and P. Van den Driessche, A multi-city epidemic model , Math. Population Stud. 10 (2003), 175-193. · Zbl 1028.92021 · doi:10.1080/08898480306720
[2] J. Cui, Y. Takeuchi and Y. Saito, Spreading disease with transport-related infection , J. Theoret. Biol. 239 (2006), 376-390. · doi:10.1016/j.jtbi.2005.08.005
[3] O. Diekmann, S.A. Van Gils, S.M. Verduyn Lunel and H.O. Walther, Delay equations , Functional-, Complex-, and Nonlinear Analysis, Springer, New York, 1995. · Zbl 0826.34002
[4] J.K. Hale and P. Waltman, Persistence in infinite-dimensional systems , SIAM J. Math. Anal. 20 (1998), 388-395. · Zbl 0692.34053 · doi:10.1137/0520025
[5] X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening , J. Theoret. Biol. 242 (2006), 517-528. · doi:10.1016/j.jtbi.2006.03.018
[6] I. Longini, A mathematical model for predicting the geographic spread of new infectious agents , Math. Biosci. 90 (1988), 367-383. · Zbl 0651.92016 · doi:10.1016/0025-5564(88)90075-2
[7] L. Rvachev and I. Longini, A mathematical model for the global spread of influenza , Math. Biosci. 75 (1985), 3-22. · Zbl 0567.92017 · doi:10.1016/0025-5564(85)90064-1
[8] L. Sattenspiel and K. Dietz, Structured epidemic models incorporating geographic mobility among regions , Math. Biosci. 128 (1995), 71-91. · Zbl 0833.92020 · doi:10.1016/0025-5564(94)00068-B
[9] L. Sattenspiel and D.A. Herring, Structured epidemic models and the spread of influenza in the central Canada subarctic , Human Biol. 70 (1998), 91-115.
[10] ——–, Simulating the effect of quarantine on the spread of the \(1918\)-\(1919\) flu in central Canada , Bull. Math. Biol. 65 (2003), 1-26.
[11] H.L. Smith, Monotone dynamical systems, An introduction to the theory of competitive and cooperative systems , American Mathematical Society, Providence, RI, 1995. · Zbl 0821.34003
[12] Y. Takeuchi, X. Liu and J. Cui, Global dynamics of SIS models with transport-related infection , J. Math. Anal. Appl. 329 (2007), 1460-1471. · Zbl 1154.34353 · doi:10.1016/j.jmaa.2006.07.057
[13] W. Wang and G. Mulone, Threshold of disease transmission in a patch environment , J. Math. Anal. Appl. 285 (2003) 321-335. · Zbl 1021.92039 · doi:10.1016/S0022-247X(03)00428-1
[14] W. Wang and S. Ruan, Simulating the SARS outbreak in Beijing with limited data , J. Theoret. Biol. 227 (2004), 369-379. · doi:10.1016/j.jtbi.2003.11.014
[15] W. Wang and X.-Q Zhao, An epidemic model in a patchy environment , Math. Biosci. 190 (2004), 97-112. · Zbl 1048.92030 · doi:10.1016/j.mbs.2002.11.001
[16] ——–, An age-structured epidemic model in a patchy environment , SIAM J. Appl. Math. 65 (2005), 1597-1614. · Zbl 1072.92045 · doi:10.1137/S0036139903431245
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