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Result for the “BFGS” method with 1992-1993 epidemic data (7 parameters, different alpha values):

For the guess of parameters beta1 = 0.3 alpha1 = 0.2 Ne = 1434 I1 = 1 beta2 = 0.3 alpha2 = 0.2 I2 = 1, we obtained

beta1 = 1.744934e+00 alpha1 = 1.155532e+00 Ne = 1.433791e+03 I1 = 7.364403e-03 beta2 = 5.099094e+00 alpha2 = 4.375838e+00 I2 = 2.700597e-01

So, this implies that f = 1433.791/27653146 = 0.000051849

The covariance matrix was

            [,1]          [,2]          [,3]          [,4]          [,5]

[1,] 1.808317e-01 1.878818e-01 138.899440 2.323948e-05 5.393679e-01 [2,] 1.878818e-01 1.954287e-01 144.767963 4.960058e-05 5.629935e-01 [3,] 1.388994e+02 1.447680e+02 110205.711377 8.517400e-02 4.330268e+02 [4,] 2.323948e-05 4.960058e-05 0.085174 3.283574e-06 5.012389e-04 [5,] 5.393679e-01 5.629935e-01 433.026845 5.012389e-04 1.929288e+00 [6,] 5.397251e-01 5.633125e-01 432.906798 4.891948e-04 1.911758e+00 [7,] -2.928666e-02 -3.063522e-02 -24.024505 -4.253060e-05 -1.349541e-01

            [,6]          [,7]

[1,] 5.397251e-01 -2.928666e-02 [2,] 5.633125e-01 -3.063522e-02 [3,] 4.329068e+02 -2.402451e+01 [4,] 4.891948e-04 -4.253060e-05 [5,] 1.911758e+00 -1.349541e-01 [6,] 1.896301e+00 -1.309859e-01 [7,] -1.309859e-01 1.419956e-02

The MLE was 3782.433

Here’s a plot of the model with the data (red is the data, blue is the sum of the two models, green is the lamb2 model, and purple is the lamb1 model)

two_sir_bfgs_7_param_1992-1993_epi.pdf

Weston: Once more, it seems to be a good fit.