Skip to main content
SpringerLink
Log in

Stochastic Multigroup Epidemic Models: Duration and Final Size

  • Chapter
  • First Online:
Modeling, Stochastic Control, Optimization, and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 164))

Abstract

The epidemic duration, the final epidemic size, and the probability of an outbreak are studied in stochastic multigroup epidemic models. Two models are considered, where the transmission rate for each group either depends on the infectious individuals or on the susceptible individuals, referred to as Model 1 and Model 2, respectively. Such models are applicable to emerging and re-emerging infectious diseases. Applying a multitype branching process approximation, it is shown for Model 1 that an outbreak is dependent primarily on group reproduction numbers, whereas for Model 2, this dependence is due to group recovery rates. The probability distributions for epidemic duration and for final size are a mixture of two distributions, that depend on whether an outbreak occurs. Given there is an outbreak, it is shown that the mean final size of the stochastic multigroup model agrees well with the final size obtained from the underlying deterministic model. These methods can be extended to more general stochastic multigroup models and to other stochastic epidemic models with multiple stages, patches, hosts, or pathogens.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. J. S Allen. A primer on stochastic epidemic models: formulation, numerical simulation, and analysis. Infectious Disease Modelling, 2(1):3–10, 2017.

    Article  Google Scholar 

  2. L. J. S. Allen and P. van den Driessche. Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models. Mathematical Biosciences, 243:99–108, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Arino, F. Brauer, P. van den Driessche, J. Watmough, and J. Wu. A final size relation for epidemic models. Mathematical Biosciences and Engineering, 4(2):159–175, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. B. Athreya and P. E. Ney. Branching Processes. Springer-Verlag, New York, 1972.

    Chapter  MATH  Google Scholar 

  5. N. T. J. Bailey. The Mathematical Theory of Infectious Diseases and Its Applications. Charles Griffin & Company, Ltd., London, 2nd edition, 1975.

    Google Scholar 

  6. F. Ball and D. Clancy. The final size and severity of a generalised stochastic multitype epidemic model. Advances in Applied Probability, 25(4):721–736, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. D. Barbour. The duration of the closed stochastic epidemic. Biometrika, 62:477–482, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. J. Black and J. V. Ross. Computation of epidemic final size distribution. Journal of Theoretical Biology, 367:159–165, 2015.

    Article  MATH  Google Scholar 

  9. F. Brauer. Epidemic models with heterogeneous mixing and treatment. Bulletin of Mathematical Biology, 70(7):1869–1885, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. C. Cullen and H. C. Frey. Probabilistic Techniques in Exposure Assessment: A Handbook for Dealing with Variability and Uncertainty in Models and Inputs. Plenum Press, New York, 2nd edition, 1999.

    Google Scholar 

  11. D. J. Daley and J. Gani. Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  12. O. Diekmann, J.A.P. Heesterbeek, and J.A.J. Metz. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4):365–382, 1990.

    Google Scholar 

  13. K. S. Dorman, J. S. Sinsheimer, and K. Lange. In the garden of branching processes. SIAM Review, 46(2):202–229, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. J. Edholm, B. O. Emerenini, A. L. Murillo, O. Saucedo, N. Shakiba, X.Wang, L. J. S. Allen, and A. Peace. Searching for superspreaders: Identifying epidemic patterns associated with superspreading events in stochastic models. In Understanding Complex Biological Systems with Mathematics, pages 1–29. Springer, 2018.

    Google Scholar 

  15. C. P. Farrington, M. N. Kanaan, and N. J. Gay. Branching process models for surveillance of infectious diseases controlled by mass vaccination. Biostatistics, 4:279–295, 2003.

    Article  MATH  Google Scholar 

  16. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25):2340–2361, 1977.

    Article  Google Scholar 

  17. N. C. Grassly and C. Fraser. Seasonal infectious disease epidemiology. Proceedings of the Royal Society of London B: Biological Sciences, 273(1600):2541–2550, 2006.

    Article  Google Scholar 

  18. T. E. Harris. The Theory of Branching Processes. Springer-Verlag, Berlin, 1963.

    Google Scholar 

  19. J. A. P. Heesterbeek and M. G. Roberts. The type-reproduction number T in models for infectious disease control. Mathematical Biosciences, 206(1):3–10, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  20. T. House, J. V. Ross, and D. Sirl. How big is an outbreak likely to be? Methods for epidemic final-size calculation. Proceedings of the Royal Society A, 469(2150):22, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London, 115:700–721, 1927.

    Article  MATH  Google Scholar 

  22. P. Magal, O. Seydi, and G.Webb. Final size of an epidemic for a two-group SIR model. SIAM Journal on Applied Mathematics, 76:2042–2059, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Magal, O. Seydi, and G.Webb. Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission. Mathematical Biosciences, 301:59–67, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. J. Massey Jr. The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46(253):68–78, 1951.

    Article  MATH  Google Scholar 

  25. J. C. Miller. A note on the derivation of epidemic final sizes. Bulletin of Mathematical Biology, 74:2125–2141, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. M. Mood, F. A. Graybill, and D. C. Boes. Introduction to the Theory of Statistics, 3rd Ed. McGraw-Hill series in probability and statistics, 1973.

    Google Scholar 

  27. A. Nandi and L. J. S. Allen. Stochastic two-group models with transmission dependent on host infectivity or susceptibility. Journal of Biological Dynamics, 2018 (In press).

    Google Scholar 

  28. National Institutes of Health (US). Biological Sciences Curriculum Study. NIH Curriculum Supplement Series [Internet]. Bethesda (MD): National Institutes of Health (US); 2007. Understanding Emerging and Re-emerging Infectious Diseases. https://www.ncbi.nlm.nih.gov/books/NBK20370/. Accessed: 2018-11-24.

  29. S. Pénisson. Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis. PhD thesis, Universit¨at Potsdam; Universit´e Paris Sud-Paris XI, 2010.

    Google Scholar 

  30. RDocumentation. Akaike’s an information criterion. https://www.rdocumentation.org/packages/stats/versions/3.5.1/topics/AIC. Accessed: 2018-11-01.

  31. M. G. Roberts and J. A. P. Heesterbeek. A new method for estimating the effort required to control an infectious disease. Proceedings of the Royal Society of London B: Biological Sciences, 270(1522):1359–1364, 2003.

    Article  Google Scholar 

  32. Z. Shuai, J.A.P. Heesterbeek, and P. van den Driessche. Extending the type reproduction number to infectious disease control targeting contacts between types. Journal of Mathematical Biology, 67:1067–1082, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  33. N. Smirnov. Table for Estimating the Goodness of Fit of Empirical Distributions. Annals of Mathematical Statistics, 19:279–281, 1948.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. A. Stein. Super-spreaders in infectious diseases. International Journal of Infectious Diseases, 15:e510–e513, 2011.

    Article  Google Scholar 

  35. W. Tritch and L. J. S. Allen. Duration of minor epidemic. Infectious Disease Modelling, 3:60–73, 2018.

    Article  Google Scholar 

  36. P. van den Driessche and J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  37. P. van den Driessche and J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Whittle. The outcome of a stochastic epidemic: A note on Bailey’s paper. Biometrika, 42:116–122, 1955.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. B. Wilk and R. Gnanadesikan. Probability plotting methods for the analysis of data. Biometrika, 55(1):1–17, 1968.

    Google Scholar 

  40. G. Wong, W. Liu, Y. Liu, B. Zhou, Y. Bi, and G.F. Gao. MERS, SARS, and Ebola: the role of super-spreaders in infectious disease. Cell Host & Microbe, 18(4):398–401, 2015.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Texas Tech University, Lubbock, TX, 79409-1042, USA

    Aadrita Nandi & Linda J. S. Allen

Authors
  1. Aadrita Nandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linda J. S. Allen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aadrita Nandi .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, USA

    George Yin

  2. Department of Mathematics, University of Georgia, Athens, GA, USA

    Qing Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Nandi, A., Allen, L.J.S. (2019). Stochastic Multigroup Epidemic Models: Duration and Final Size. In: Yin, G., Zhang, Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its Applications, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-25498-8_20

Download citation

Publish with us

Policies and ethics

Search

Navigation