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December  2016, 21(10): 3515-3550. doi: 10.3934/dcdsb.2016109

A multi-group SIR epidemic model with age structure

Toshikazu Kuniya 1, , Jinliang Wang 2, and  Hisashi Inaba 3,

1. 

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-0067, Japan
2. 

School of Mathematical Science, Heilongjiang University, Harbin 150080
3. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received  September 2013 Revised  August 2016 Published  November 2016

This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations. The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant. To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.
Keywords: SIR epidemic model, age structure, multi-group model, the basic reproduction number..
Mathematics Subject Classification: Primary: 35Q92, 92D30; Secondary: 37N2.
Citation: Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
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show all references

References:
[1]

in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Wuerz Publ., (1995), 3-14. Google Scholar

[2]

in Mathematical Ecology (eds. T.G. Hallam, L.J. Gross and S.A. Levin), World Scientific, (1988), 317-342.  Google Scholar

[3]

SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069.  Google Scholar

[4]

Dynam. Syst. Appl., 9 (2000), 361-376.  Google Scholar

[5]

J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[6]

Princeton University Press, Princeton and Oxford, 2013.  Google Scholar

[7]

in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), SIAM, (1975), 104-121. Google Scholar

[8]

J. Diff. Equat., 218 (2005), 292-324. doi: 10.1016/j.jde.2004.10.009.  Google Scholar

[9]

Math. Biosci. Eng., 9 (2012), 577-599. doi: 10.3934/mbe.2012.9.577.  Google Scholar

[10]

IMA J. Math. Appl. Med. Biol., 4 (1987), 109-144. doi: 10.1093/imammb/4.2.109.  Google Scholar

[11]

J. reine angew. Math., 341 (1983), 54-67. doi: 10.1515/crll.1983.341.54.  Google Scholar

[12]

Canadian Appl. Math. Quart., 14 (2006), 259-284.  Google Scholar

[13]

in The Dynamics of Physiologically Structured Populations (eds. J.A.J. Metz and O. Diekmann), Springer, 68 (1986), 185-202. doi: 10.1007/978-3-662-13159-6_5.  Google Scholar

[14]

Theor. Popul. Biol., 14 (1978), 338-349. doi: 10.1016/0040-5809(78)90011-4.  Google Scholar

[15]

SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[16]

J. Franklin Inst., 297 (1974), 325-333. doi: 10.1016/0016-0032(74)90037-4.  Google Scholar

[17]

Math. Popul. Studies, 1 (1988), 49-77. doi: 10.1080/08898488809525260.  Google Scholar

[18]

J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326.  Google Scholar

[19]

Math. Biosci., 201 (2006), 15-47. doi: 10.1016/j.mbs.2005.12.017.  Google Scholar

[20]

Math. Model. Nat. Phenom., 3 (2008), 194-228. doi: 10.1051/mmnp:2008050.  Google Scholar

[21]

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[22]

J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.  Google Scholar

[23]

2nd edition, Springer, Berlin, 1995.  Google Scholar

[24]

Nonlinear Analysis RWA., 9 (2008), 1989-2028. doi: 10.1016/j.nonrwa.2007.06.004.  Google Scholar

[25]

Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z.  Google Scholar

[26]

1st edition, Noordhoff, Groningen, 1964.  Google Scholar

[27]

Am. Math. Soc. Transl., 1950 (1950), 128pp.  Google Scholar

[28]

Nonlinear Analysis RWA., 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar

[29]

2nd edition, SIAM, Philadelphia, 1976.  Google Scholar

[30]

Math. Biosci. Eng., 7 (2010), 123-147. doi: 10.3934/mbe.2010.7.123.  Google Scholar

[31]

Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.  Google Scholar

[32]

SIAM J. Appl. Math., 19 (1970), 607-628. doi: 10.1137/0119060.  Google Scholar

[33]

Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819.  Google Scholar

[34]

Proc. Edinburgh Math. Soc., (1925), 98-130. doi: 10.1017/S0013091500034428.  Google Scholar

[35]

Math. Biosci. Eng., 10 (2013), 369-378. doi: 10.3934/mbe.2013.10.369.  Google Scholar

[36]

1st edition, Springer, Berlin, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[37]

Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64.  Google Scholar

[38]

1st edition, Amer. Math. Soc., Providence, 2011.  Google Scholar

[39]

Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[40]

in Differential Equations Models in Biology, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158. doi: 10.1007/978-3-642-45692-3_10.  Google Scholar

[41]

Math. Biosci., 73 (1985), 131-147. doi: 10.1016/0025-5564(85)90081-1.  Google Scholar

[42]

Theor. Popul. Biol., 40 (1991), 322-353. doi: 10.1016/0040-5809(91)90059-O.  Google Scholar

[43]

Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[44]

1st edition, Plenum Press, New York and London, 1980.  Google Scholar

[45]

J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X.  Google Scholar

[46]

1st edition, Marcel Dekker, New York and Basel, 1985.  Google Scholar

[47]

6th edition, Springer, Berlin, 1980.  Google Scholar

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