Epidemic characteristics of two classic models and the dependence on the initial conditions

  • Received: 01 December 2015 Accepted: 29 June 2018 Published: 01 July 2016
  • MSC : Primary: 92D30.

  • The epidemic characteristics, including the epidemic final size, peak, andturning point, of two classical SIR models with disease-induced death are investigated when asmall initial value of the infective population is released. The models have mass-action (i.e.bilinear), or density dependent (i.e. standard)  incidence, respectively. For the two models, theconditions that determining whether the related  epidemic characteristics of an epidemicoutbreak appear are explicitly determine by rigorous mathematical analysis. The dependenceof the epidemic final size on the initial values of the infective class is demonstrated. The peak,turning point (if it exists) and the corresponding time are found. The obtained results suggestthat their basic reproduction numbers are one factor determining the epidemic characteristics,but not the only one. The characteristics of the two models depend on the initial values andproportions of various compartments as well. At last, the similarities and differences of theepidemic characteristics between the two models are discussed.

    Citation: Jianquan Li, Yiqun Li, Yali Yang. Epidemic characteristics of two classic models and the dependence on the initial conditions[J]. Mathematical Biosciences and Engineering, 2016, 13(5): 999-1010. doi: 10.3934/mbe.2016027

    Related Papers:

  • The epidemic characteristics, including the epidemic final size, peak, andturning point, of two classical SIR models with disease-induced death are investigated when asmall initial value of the infective population is released. The models have mass-action (i.e.bilinear), or density dependent (i.e. standard)  incidence, respectively. For the two models, theconditions that determining whether the related  epidemic characteristics of an epidemicoutbreak appear are explicitly determine by rigorous mathematical analysis. The dependenceof the epidemic final size on the initial values of the infective class is demonstrated. The peak,turning point (if it exists) and the corresponding time are found. The obtained results suggestthat their basic reproduction numbers are one factor determining the epidemic characteristics,but not the only one. The characteristics of the two models depend on the initial values andproportions of various compartments as well. At last, the similarities and differences of theepidemic characteristics between the two models are discussed.



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    [1] J. R. Soc. Interface, 3 (2006), 453-457.
    [2] Math. Bios. Eng., 4 (2007), 159-175.
    [3] Bull. Math. Biol., 71 (2009), 1954-1966.
    [4] Math. Bios. Eng., 3 (2006), 1-15.
    [5] Math. Bios., 198 (2005), 119-131.
    [6] Bull. Math. Biol., 77 (2015), 460-469.
    [7] Springer-Verlag, New York, 2001.
    [8] Math. Bios. Eng., 13 (2016), 249-259.
    [9] J. Math. Biol., 28 (1990), 365-382.
    [10] In: Epidemic Models: Their Structure and Relation to Data, Their Structure and Relation to Data, Cambridge University Press, 1995, 95-115.
    [11] Math. Bios., 266 (2015), 10-14.
    [12] Math. Bios. Eng., 4 (2007), 675-686.
    [13] Proc. R. Soc. A, 469 (2013), 20120436, 22 pp.
    [14] Infl. Other Resp. Vir. 4 (2010), 187-197.
    [15] In: Z. Ma, J. Wu and Y. Zhou editors, Modeling and Dynamics of Infectious Diseases, Series in Contemporary Applied Mathematics (CAM), Higher Education Press, Beijing, 2009, 216-236.
    [16] Trop. Med. Int. Health., 18 (2013), 830-838.
    [17] Trop. Med. Int. Heal., 14 (2009), 628-638.
    [18] Emerg. Infect. Dis., 12 (2006), 122-127.
    [19] BMC Research Notes, 3 (2010), p283.
    [20] Emerg. Infect. Dis., 10 (2004), 1165-1167.
    [21] Am. J. Trop. Med. Hyg., 80 (2005), 66-71.
    [22] PLoS ONE, 9 (2014), e111834.
    [23] J. Math. Biol., 65 (2012), 309-348.
    [24] Proc. R. Soc. Lond. A, 115 (1927), 700-721.
    [25] Bull. Math. Biol., 68 (2006), 679-702.
    [26] Singapore, 2009.
    [27] PLoS ONE, 9 (2014), e101421.
    [28] J. Exp. Bot., 10 (1959), 290-301.
    [29] Ecol. Model., 220 (2009), 3472-3474.
    [30] Math. Biosci., 234 (2011), 108-117.
    [31] Math. Biosci., 180 (2002), 29-48.
    [32] SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.
    [33] J. Dyn. Differ. Equ., 25 (2013), 535-562.
    [34] J. Dyn. Diff. Equ., 20 (2008), 699-717.
    [35] J. Theo. Biol., 313 (2012), 12-19.
    [36] Math. Bios. Eng., 12 (2015), 1055-1063.
    [37] J. Math. Biol., 72 (2016), 343-361.
    [38] J. Dyn. Diff. Equat., (2015), 1-16.
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