Search Advanced

Mathematical Biosciences and Engineering

2016, Volume 13, Issue 5: 999-1010. doi: 10.3934/mbe.2016027
Previous Article Next Article

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

  • 1.
    Science College, Air Force Engineering University, Xi'an 710051
  • 2.
    College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062
  • 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.



    加载中
    [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.
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(2500) PDF downloads(542) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog