Fig 1.
The mathematical structure of the particle filtering aggregate model.
Fig 2.
The mathematical structure of the particle filtering age stratified model.
Fig 3.
The monthly reported measles cases in Saskatchewan from 1921 to 1956.
The values given are normalized by the population employed in the model (863,545).
Table 1.
Table showing the value of parameters.
Table 2.
Comparison of the average discrepancy of all seven models by incorporating empirical data across all observation points.
Fig 4.
Box plots of monthly and yearly discrepancy of all models by incorporating empirical data across all observation points.
Each of the five particle filtering models is run 5 times (the random seed generated from the same set). Then the average monthly and yearly discrepancy among these five runs at each time between the particle filtering models and the empirical data are plotted.
Fig 5.
2D histogram prior result of total timeframe of the minimum discrepancy model (monthly).
Fig 6.
2D histogram posterior result of total timeframe of the minimum discrepancy model.
(a) the monthly particle filtering result across all population. (b) the yearly particle filtering result of the child and adult age groups.
Fig 7.
2D histogram results for the S, E, I, R stocks with different age groups of the minimum discrepancy model with splitting the age groups at 15 years by incorporating the empirical data across all timeframe.
The results shown consider both the yearly and monthly empirical data, with monthly discrepancy 90.7, the sum of all age groups discrepancy in Month is 36.9. (a) across all population. (b) the child age group (those within their first 15 years of life). (c) the adult age group (years 15 and up).
Fig 8.
2D histogram of predicting from the first or second time point of an outbreak of the minimum discrepancy model.
(a) predicted from the month 121, with monthly prediction discrepancy 306.0, and the sum of yearly prediction discrepancy of all age groups per month is 246.7. (b) predicted from the month 190, with monthly prediction discrepancy 320.4, and the sum of yearly prediction discrepancy of all age groups per month is 237.2.
Fig 9.
2D histogram of predicting from the peak of an outbreak of the minimum discrepancy model.
(a) predicted from the month 242, with monthly prediction discrepancy 305.7, and the sum of yearly prediction discrepancy of all age groups per month is 205.2. (b) predicted from the month 312, with monthly prediction discrepancy 306.9, and the sum of yearly prediction discrepancy of all age groups per month is 201.6.
Fig 10.
2D histogram of predicting from the end of an outbreak of the minimum discrepancy model.
(a) predicted from the month 138, with monthly prediction discrepancy 302.6, and the sum of yearly prediction discrepancy of all age groups per month is 248.0. (b) predicted from the month 201, with monthly prediction discrepancy 316.7, and the sum of yearly prediction discrepancy of all age groups per month is 217.3.
Fig 11.
2D histogram of predicting before the next outbreak of the minimum discrepancy model.
(a) predicted from the month 52, with monthly prediction discrepancy 324.9, and the sum of yearly prediction discrepancy of all age groups per month is 198.8. (b) predicted from the month 150, with monthly prediction discrepancy 353.0, and the sum of yearly prediction discrepancy of all age groups per month is 268.5. It is notable that the values of two parameters have been selected differently, to get a more certain predict result—the diffusion coefficient of the transmission rate of child age group is 0.12, and the particle number of sampling to plot the 2D histogram is 1000 in this two cases.
Table 3.
The confusion matrix of classifying outbreak occurrence at threshold θk = 0.5.
Fig 12.
The ROC curve of the prediction classification result of the minimum discrepancy model.
The AUC is 0.893.
Fig 13.
Scatter plot and regression result of the empirical data vs. mean and median of the model predicted next month results over all sampled particles with the minimum discrepancy model of measles.
The regression result is: ymean = 0.80x + 84.80, ymedian = 0.78x + 72.31.