Applying the stochastic Galerkin method to epidemic models with individualised parameter distributions

David Harman, Peter Johnston


There are many different models to help predict the likely course an epidemic will take. However, the parameters within these models are often not known with certainty. It is important for this uncertainty to be incorporated into these models to ensure accurate predictions. This article considers the stochastic Galerkin method to solve an sir model with uncertainty in its parameters. A data set from an influenza outbreak in a boarding school is then investigated. Rather than just finding the `best' values for the parameters, several possible probability distributions for the parameters in the sir model are determined. The stochastic Galerkin method is then used to determine the mean solution of the model as well as its variance.

  • Influenza in a boarding school. Br. Med. J. 1:586, 1978. doi:10.1136/bmj.1.6112.586
  • B. M. Chen-Charpentier, J. C. Cortes, J. V. Romero, and M. D. Rosello. Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Appl. Math. Comput. 219(9):4208–4218, 2013. doi:10.1016/j.amc.2012.11.007
  • B. M. Chen-Charpentier and D. Stanescu. Epidemic models with random coefficients. Math. Comput. Model. 52(7–8):1004–1010, 2010. doi:10.1016/j.mcm.2010.01.014
  • D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with uncertainty in the parameters. Math. Biosci. 277:25–37, 2016. doi:10.1016/j.mbs.2016.03.012
  • H. W. Hethcote. The mathematics of infectious diseases. SIAM Rev. 42(4):599–653, 2000. doi:10.1137/S0036144500371907
  • R. I. Hickson and M. G. Roberts. How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. J. Theor. Biol. 350:70–80, 2014. doi:10.1016/j.jtbi.2014.01.014
  • W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lon. Ser. A, 115(772):700–721, August 1927. doi:10.1098/rspa.1927.0118
  • M. G. Roberts. A two-strain epidemic model with uncertainty in the interaction. ANZIAM J. 54:108–115, 2012. doi:10.1017/S1446181112000326
  • M. G. Roberts. Epidemic models with uncertainty in the reproduction number. J. Math. Biol. 66(7):1463–1474, 2013. doi:10.1007/s00285-012-0540-y
  • F. Santonja and B. Chen-Charpentier. Uncertainty quantification in simulations of epidemics using polynomial chaos. Comput. Math. Method. Med. 2012:742086, 2012. doi:10.1155/2012/742086
  • B. Sudret. Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Safe. 93(7):964–979, 2008. doi:10.1016/j.ress.2007.04.002
  • D. Xiu. Numerical methods for stochastic computations: A spectral method approach. Princeton University Press, 2010.


uncertainty quantification, polynomial chaos, infectious disease

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