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Epidemic Models

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Book cover Mathematical Models in Epidemiology

Part of the book series: Texts in Applied Mathematics ((TAM,volume 69))

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.

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References

  1. 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.

    MathSciNet  MATH  Google Scholar 

  2. Andrews, J.R. & and S. Basu (2011) Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet 377: 1248–1255.

    Google Scholar 

  3. 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.

    Google Scholar 

  4. 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.

    Google Scholar 

  5. 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.

    MathSciNet  MATH  Google Scholar 

  6. 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.

    MathSciNet  MATH  Google Scholar 

  7. Brauer, F., C. Castillo-Chavez, and Z. Feng (2010) Discrete epidemic models, Math. Biosc. & Eng. 7: 1–15.

    MathSciNet  MATH  Google Scholar 

  8. Brauer, F., Z. Shuai, & P. van den Driessche (2013) Dynamics of an age-of-infection cholera model, Math. Biosc. & Eng. 10:1335–1349.

    MathSciNet  MATH  Google Scholar 

  9. 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.

    Google Scholar 

  10. 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.

    Google Scholar 

  11. Codeço, C.T. (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases 1: 1.

    Google Scholar 

  12. Diekmann,O. & J.A.P. Heesterbeek (2000) Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester.

    MATH  Google Scholar 

  13. Erdös, P. & A. Rényi (1959) On random graphs, Publicationes Mathematicae 6: 290–297.

    MathSciNet  MATH  Google Scholar 

  14. Erdös, P. & A. Rényi (1960) On the evolution of random Pub. Math. Inst. Hung. Acad. Science 5: 17–61.

    MATH  Google Scholar 

  15. Erdös, P. & A. Rényi (1961) On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12: 261–267.

    MathSciNet  MATH  Google Scholar 

  16. Feng, Z. (2007) Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosc. & Eng. 4: 675–686.

    MathSciNet  MATH  Google Scholar 

  17. 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.

    MathSciNet  MATH  Google Scholar 

  18. Fine, P.E.M. (2003) The interval between successive cases of an infectious disease, Am. J. Epid. 158: 1039–1047.

    Google Scholar 

  19. 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.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. 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.

    Google Scholar 

  22. Heffernan, J.M., R.J. Smith? & L.M. Wahl (2005) Perspectives on the basic reproductive ratio, J. Roy. Soc. Interface 2: 281–293.

    Google Scholar 

  23. 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  Google Scholar 

  24. Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London. 115: 700–721.

    MATH  Google Scholar 

  25. King, A.A. E.L. Ionides, M. Pascual, & M.J. Bouma (2008) Inapparent infectious and cholera dynamics, Nature 454: 877–890.

    Google Scholar 

  26. Lloyd, A.L. (2001) Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Pop. Biol., 60: 59–71.

    Google Scholar 

  27. Meyers, L.A. (2007) Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc., 44: 63–86.

    MathSciNet  MATH  Google Scholar 

  28. Meyers, L.A., M.E.J. Newman and B. Pourbohloul(2006) Predicting epidemics on directed contact networks, J. Theor. Biol., 240: 400–418.

    MathSciNet  Google Scholar 

  29. 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  Google Scholar 

  30. Miller, J.C. (2010) A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. https://doi.org/10.1007/s00285-010-0337-9.

    MATH  Google Scholar 

  31. Miller, J.C. and E. Volz (2011) Simple rules govern epidemic dynamics in complex networks, to appear.

    Google Scholar 

  32. Newman, M.E.J. (2002) The spread of epidemic disease on networks, Phys. Rev. E 66, 016128.

    MathSciNet  Google Scholar 

  33. Newman, M.E.J. (2003) The structure and function of complex networks. SIAM Review, 45: 167–256.

    MathSciNet  MATH  Google Scholar 

  34. Newman, M.E.J., S.H. Strogatz & D.J. Watts (2001) Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64.

    Google Scholar 

  35. 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.

    Google Scholar 

  36. 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.

    Google Scholar 

  37. 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.

    MathSciNet  MATH  Google Scholar 

  38. Shuai, Z. & P. van den Driessche (2011) Global dynamics of cholera models with differential infectivity, Math. Biosc. 234: 118–126.

    MathSciNet  MATH  Google Scholar 

  39. Strogatz, S.H. (2001) Exploring complex networks, Nature, 410: 268–276.

    MATH  Google Scholar 

  40. Svensson, A. (2007) A note on generation time in epidemic models, Math. Biosc. 208: 300–311.

    MathSciNet  MATH  Google Scholar 

  41. 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.

    MathSciNet  MATH  Google Scholar 

  42. 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.

    MathSciNet  MATH  Google Scholar 

  43. Volz, E. (2008) SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56: 293–310.

    MathSciNet  MATH  Google Scholar 

  44. Wallinga, J. & M. Lipsitch (2007) How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Royal Soc. B 274: 599–604.

    Google Scholar 

  45. 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,

    Google Scholar 

  46. Wearing, H.J., P. Rohani & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLOS Medicine, 2: 621–627.

    Google Scholar 

  47. 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  Google Scholar 

  48. Yang, C.K. and F. Brauer (2009) Calculation of \(\mathcal {R}_0\) for age-of-infection models, Math. Biosc. & Eng. 5: 585–599

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Fred Brauer

  2. Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, USA

    Carlos Castillo-Chavez

  3. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Zhilan Feng

Authors
  1. Fred Brauer
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  2. Carlos Castillo-Chavez
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  3. Zhilan Feng
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Brauer, F., Castillo-Chavez, C., Feng, Z. (2019). Epidemic Models. In: Mathematical Models in Epidemiology. Texts in Applied Mathematics, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9828-9_4

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