Skip to main content
Top

2019 | OriginalPaper | Chapter

4. Epidemic Models

Authors : Fred Brauer, Carlos Castillo-Chavez, Zhilan Feng

Published in: Mathematical Models in Epidemiology

Publisher: Springer New York

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we describe models for epidemics, acting on a sufficiently rapid time scale that demographic effects, such as births, natural deaths, immigration into and emigration out of a population may be ignored. The prototype epidemic model is the simple Kermack–McKendrick model studied in Sect. 2.​4.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Alexanderian, A., M.K. Gobbert, K.R. Fister, H. Gaff, S. Lenhart, and E. Schaefer (2011) An age-structured model for the spread of epidemic cholera: Analysis and simulation, Nonlin. Anal. Real World Appl. 12: 3483–3498.MathSciNetMATH Alexanderian, A., M.K. Gobbert, K.R. Fister, H. Gaff, S. Lenhart, and E. Schaefer (2011) An age-structured model for the spread of epidemic cholera: Analysis and simulation, Nonlin. Anal. Real World Appl. 12: 3483–3498.MathSciNetMATH
2.
go back to reference Andrews, J.R. & and S. Basu (2011) Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet 377: 1248–1255. Andrews, J.R. & and S. Basu (2011) Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet 377: 1248–1255.
3.
go back to reference Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2006) Simple models for containment of a pandemic, J. Roy. Soc. Interface, 3: 453–457. Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2006) Simple models for containment of a pandemic, J. Roy. Soc. Interface, 3: 453–457.
4.
go back to reference Arino, J. F. Brauer, P. van den Driessche, J. Watmough& J. Wu (2007) A final size relation for epidemic models, Math. Biosc. & Eng. 4: 159–176. Arino, J. F. Brauer, P. van den Driessche, J. Watmough& J. Wu (2007) A final size relation for epidemic models, Math. Biosc. & Eng. 4: 159–176.
5.
go back to reference Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2008) A model for influenza with vaccination and antiviral treatment, Theor. Pop. Biol. 253: 118–130.MathSciNetMATH Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2008) A model for influenza with vaccination and antiviral treatment, Theor. Pop. Biol. 253: 118–130.MathSciNetMATH
6.
go back to reference Bansal, S., J. Read, B. Pourbohloul, and L.A. Meyers (2010) The dynamic nature of contact networks in infectious disease epidemiology, J. Biol. Dyn., 4: 478–489.MathSciNetMATH Bansal, S., J. Read, B. Pourbohloul, and L.A. Meyers (2010) The dynamic nature of contact networks in infectious disease epidemiology, J. Biol. Dyn., 4: 478–489.MathSciNetMATH
7.
go back to reference Brauer, F., C. Castillo-Chavez, and Z. Feng (2010) Discrete epidemic models, Math. Biosc. & Eng. 7: 1–15.MathSciNetMATH Brauer, F., C. Castillo-Chavez, and Z. Feng (2010) Discrete epidemic models, Math. Biosc. & Eng. 7: 1–15.MathSciNetMATH
8.
go back to reference Brauer, F., Z. Shuai, & P. van den Driessche (2013) Dynamics of an age-of-infection cholera model, Math. Biosc. & Eng. 10:1335–1349.MathSciNetMATH Brauer, F., Z. Shuai, & P. van den Driessche (2013) Dynamics of an age-of-infection cholera model, Math. Biosc. & Eng. 10:1335–1349.MathSciNetMATH
9.
go back to reference Brauer, F., P. van den Driessche and J. Wu, eds. (2008) Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences subseries 1945, Springer, Berlin - Heidelberg - New York. Brauer, F., P. van den Driessche and J. Wu, eds. (2008) Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences subseries 1945, Springer, Berlin - Heidelberg - New York.
10.
go back to reference Callaway, D.S., M.E.J. Newman, S.H. Strogatz, D.J. Watts (2000) Network robustness and fragility: Percolation on random graphs, Phys. Rev. Letters, 85: 5468–5471. Callaway, D.S., M.E.J. Newman, S.H. Strogatz, D.J. Watts (2000) Network robustness and fragility: Percolation on random graphs, Phys. Rev. Letters, 85: 5468–5471.
11.
go back to reference Codeço, C.T. (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases 1: 1. Codeço, C.T. (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases 1: 1.
12.
go back to reference Diekmann,O. & J.A.P. Heesterbeek (2000) Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester.MATH Diekmann,O. & J.A.P. Heesterbeek (2000) Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester.MATH
13.
14.
go back to reference Erdös, P. & A. Rényi (1960) On the evolution of random Pub. Math. Inst. Hung. Acad. Science 5: 17–61.MATH Erdös, P. & A. Rényi (1960) On the evolution of random Pub. Math. Inst. Hung. Acad. Science 5: 17–61.MATH
15.
go back to reference Erdös, P. & A. Rényi (1961) On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12: 261–267.MathSciNetMATH Erdös, P. & A. Rényi (1961) On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12: 261–267.MathSciNetMATH
16.
go back to reference Feng, Z. (2007) Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosc. & Eng. 4: 675–686.MathSciNetMATH Feng, Z. (2007) Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosc. & Eng. 4: 675–686.MathSciNetMATH
17.
go back to reference Feng, Z., D. Xu & W. Zhao (2007) Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. Biol. 69: 1511–1536.MathSciNetMATH Feng, Z., D. Xu & W. Zhao (2007) Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. Biol. 69: 1511–1536.MathSciNetMATH
18.
go back to reference Fine, P.E.M. (2003) The interval between successive cases of an infectious disease, Am. J. Epid. 158: 1039–1047. Fine, P.E.M. (2003) The interval between successive cases of an infectious disease, Am. J. Epid. 158: 1039–1047.
19.
go back to reference Gumel, A., S. Ruan, T. Day, J. Watmough, P. van den Driessche, F. Brauer, D. Gabrielson, C. Bowman, M.E. Alexander, S. Ardal, J. Wu & B.M. Sahai (2004) Modeling strategies for controlling SARS outbreaks based on Toronto, Hong Kong, Singapore and Beijing experience, Proc. Roy. Soc. London 271: 2223–2232. Gumel, A., S. Ruan, T. Day, J. Watmough, P. van den Driessche, F. Brauer, D. Gabrielson, C. Bowman, M.E. Alexander, S. Ardal, J. Wu & B.M. Sahai (2004) Modeling strategies for controlling SARS outbreaks based on Toronto, Hong Kong, Singapore and Beijing experience, Proc. Roy. Soc. London 271: 2223–2232.
20.
go back to reference Diekmann, O., J.A.P. Heesterbeek, & J.A.J. Metz (1995) The legacy of Kermack and McKendrick, in Epidemic Models: Their Structure and Relation to Data, D. Mollison, ed., Cambridge University Press, pp. 95–115. Diekmann, O., J.A.P. Heesterbeek, & J.A.J. Metz (1995) The legacy of Kermack and McKendrick, in Epidemic Models: Their Structure and Relation to Data, D. Mollison, ed., Cambridge University Press, pp. 95–115.
21.
go back to reference Hartley, D.M., J.G. Morris Jr. & D.L. Smith (2006) Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics? PLOS Med. 3:63–69. Hartley, D.M., J.G. Morris Jr. & D.L. Smith (2006) Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics? PLOS Med. 3:63–69.
22.
go back to reference Heffernan, J.M., R.J. Smith? & L.M. Wahl (2005) Perspectives on the basic reproductive ratio, J. Roy. Soc. Interface 2: 281–293. Heffernan, J.M., R.J. Smith? & L.M. Wahl (2005) Perspectives on the basic reproductive ratio, J. Roy. Soc. Interface 2: 281–293.
23.
go back to reference Hyman, J.M., J. Li & E. A. Stanley (1999) The differential infectivity and staged progression models for the transmission of HIV, Math. Biosc. 155: 77–109.MATH Hyman, J.M., J. Li & E. A. Stanley (1999) The differential infectivity and staged progression models for the transmission of HIV, Math. Biosc. 155: 77–109.MATH
24.
go back to reference Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London. 115: 700–721.MATH Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London. 115: 700–721.MATH
25.
go back to reference King, A.A. E.L. Ionides, M. Pascual, & M.J. Bouma (2008) Inapparent infectious and cholera dynamics, Nature 454: 877–890. King, A.A. E.L. Ionides, M. Pascual, & M.J. Bouma (2008) Inapparent infectious and cholera dynamics, Nature 454: 877–890.
26.
go back to reference Lloyd, A.L. (2001) Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Pop. Biol., 60: 59–71. Lloyd, A.L. (2001) Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Pop. Biol., 60: 59–71.
27.
go back to reference Meyers, L.A. (2007) Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc., 44: 63–86.MathSciNetMATH Meyers, L.A. (2007) Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc., 44: 63–86.MathSciNetMATH
28.
go back to reference Meyers, L.A., M.E.J. Newman and B. Pourbohloul(2006) Predicting epidemics on directed contact networks, J. Theor. Biol., 240: 400–418.MathSciNet Meyers, L.A., M.E.J. Newman and B. Pourbohloul(2006) Predicting epidemics on directed contact networks, J. Theor. Biol., 240: 400–418.MathSciNet
29.
go back to reference Meyers, L.A., B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, and R.C. Brunham (2005) Network theory and SARS: predicting outbreak diversity, J. Theor. Biol., 232: 71–81.MathSciNet Meyers, L.A., B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, and R.C. Brunham (2005) Network theory and SARS: predicting outbreak diversity, J. Theor. Biol., 232: 71–81.MathSciNet
31.
go back to reference Miller, J.C. and E. Volz (2011) Simple rules govern epidemic dynamics in complex networks, to appear. Miller, J.C. and E. Volz (2011) Simple rules govern epidemic dynamics in complex networks, to appear.
32.
go back to reference Newman, M.E.J. (2002) The spread of epidemic disease on networks, Phys. Rev. E 66, 016128.MathSciNet Newman, M.E.J. (2002) The spread of epidemic disease on networks, Phys. Rev. E 66, 016128.MathSciNet
33.
34.
go back to reference Newman, M.E.J., S.H. Strogatz & D.J. Watts (2001) Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64. Newman, M.E.J., S.H. Strogatz & D.J. Watts (2001) Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64.
35.
go back to reference Riley, S., C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Raddad, A.J. Hedley, G.M. Leung, L-M Ho, T-H Lam, T.Q. Thach, P. Chau, K-P Chan, S-V Lo, P-Y Leung, T. Tsang, W. Ho, K-H Lee, E.M.C. Lau, N.M. Ferguson, & R.M. Anderson (2003) Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science 300: 1961–1966. Riley, S., C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Raddad, A.J. Hedley, G.M. Leung, L-M Ho, T-H Lam, T.Q. Thach, P. Chau, K-P Chan, S-V Lo, P-Y Leung, T. Tsang, W. Ho, K-H Lee, E.M.C. Lau, N.M. Ferguson, & R.M. Anderson (2003) Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science 300: 1961–1966.
36.
go back to reference Roberts, M.G. and J.A.P. Heesterbeek (2003) A new method for estimating the effort required to control an infectious disease, Pro. Roy. Soc. London B 270: 1359–1364. Roberts, M.G. and J.A.P. Heesterbeek (2003) A new method for estimating the effort required to control an infectious disease, Pro. Roy. Soc. London B 270: 1359–1364.
37.
go back to reference Scalia-Tomba, G., A. Svensson, T. Asikaiainen, and J. Giesecke (2010) Some model based considerations on observing generation times for communicable diseases, Math. Biosc. 223: 24–31.MathSciNetMATH Scalia-Tomba, G., A. Svensson, T. Asikaiainen, and J. Giesecke (2010) Some model based considerations on observing generation times for communicable diseases, Math. Biosc. 223: 24–31.MathSciNetMATH
38.
go back to reference Shuai, Z. & P. van den Driessche (2011) Global dynamics of cholera models with differential infectivity, Math. Biosc. 234: 118–126.MathSciNetMATH Shuai, Z. & P. van den Driessche (2011) Global dynamics of cholera models with differential infectivity, Math. Biosc. 234: 118–126.MathSciNetMATH
39.
go back to reference Strogatz, S.H. (2001) Exploring complex networks, Nature, 410: 268–276.MATH Strogatz, S.H. (2001) Exploring complex networks, Nature, 410: 268–276.MATH
40.
41.
go back to reference Tien, J.H. & D.J.D. Earn (2010) Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72: 1506–1533.MathSciNetMATH Tien, J.H. & D.J.D. Earn (2010) Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72: 1506–1533.MathSciNetMATH
42.
go back to reference van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc., 180: 29–48.MathSciNetMATH van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc., 180: 29–48.MathSciNetMATH
43.
go back to reference Volz, E. (2008) SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56: 293–310.MathSciNetMATH Volz, E. (2008) SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56: 293–310.MathSciNetMATH
44.
go back to reference Wallinga, J. & M. Lipsitch (2007) How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Royal Soc. B 274: 599–604. Wallinga, J. & M. Lipsitch (2007) How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Royal Soc. B 274: 599–604.
45.
go back to reference Wallinga, J. & P. Teunis (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures, Am. J. Epidem. 160: 509–516, Wallinga, J. & P. Teunis (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures, Am. J. Epidem. 160: 509–516,
46.
go back to reference Wearing, H.J., P. Rohani & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLOS Medicine, 2: 621–627. Wearing, H.J., P. Rohani & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLOS Medicine, 2: 621–627.
47.
go back to reference Yan, P. and Z. Feng (2010) Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness Math. Biosc. 224: 43–52.MATH Yan, P. and Z. Feng (2010) Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness Math. Biosc. 224: 43–52.MATH
48.
go back to reference Yang, C.K. and F. Brauer (2009) Calculation of \(\mathcal {R}_0\) for age-of-infection models, Math. Biosc. & Eng. 5: 585–599 Yang, C.K. and F. Brauer (2009) Calculation of \(\mathcal {R}_0\) for age-of-infection models, Math. Biosc. & Eng. 5: 585–599
Metadata
Title
Epidemic Models
Authors
Fred Brauer
Carlos Castillo-Chavez
Zhilan Feng
Copyright Year
2019
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-9828-9_4

Premium Partner