A Stochastic Galerkin Method for the Fokker–Planck–Landau Equation with Random Uncertainties

The ability to accurately predict the course of an epidemic is extremely important. This article looks at an influenza outbreak that spread through a small boarding school. Predictions are made on multiple days throughout the epidemic using the stochastic Galerkin method to consider a range of plausible values for the parameters. These predictions are then compared to known data points. Predictions made before the peak of the epidemic had much larger variances compared to predictions made after the peak of the epidemic. References 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. Applied Mathematics and Computation, 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. Mathematical and Computer Modelling, 52:1004 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 individualised parameter distributions. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C160C176, August 2016. doi:10.21914/anziamj.v57i0.10394 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with uncertainty in the parameters. Mathematical Biosciences, 277:25 37, 2016. doi:10.1016/j.mbs.2016.03.012 D. B. Harman and P. R. Johnston. Boarding house: find border. 2019. doi:10.6084/m9.figshare.7699844.v1 D. B. Harman and P. R. Johnston. SIR uniform equations. 2 2019. doi:10.6084/m9.figshare.7692392.v1 H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599653, 2000. doi:10.1137/S0036144500371907 R.I. Hickson and M.G. Roberts. How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. Journal of Theoretical Biology, 350(0):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. Proceedings of the Royal Society of London. Series A, 115(772):700721, August 1927. doi:10.1098/rspa.1927.0118 M. G. Roberts. A two-strain epidemic model with uncertainty in the interaction. The ANZIAM Journal, 54:108115, 10 2012. doi:10.1017/S1446181112000326 M. G. Roberts. Epidemic models with uncertainty in the reproduction number. Journal of Mathematical Biology, 66(7):14631474, 2013. doi:10.1007/s00285-012-0540-y F. Santonja and B. Chen-Charpentier. Uncertainty quantification in simulations of epidemics using polynomial chaos. Computational and Mathematical Methods in Medicine, 2012:742086, 2012. doi:10.1155/2012/742086 Communicable Disease Surveillance Centre (Public Health Laboratory Service) and Communicable Diseases (Scotland) Unit. Influenza in a boarding school. BMJ, 1(6112):587, 1978. doi:10.1136/bmj.1.6112.586 G. Strang. Linear Algebra and Its Applications. Thomson, Brooks/Cole, 2006. D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010.

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