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GAM(M) for longitudinal measurements from paired control treatment samples in R

I have a test dataset looking like;

ID Sample Group Week
X1 Sample1 Healthy 0
X1 Sample2 Disease 0
X1 Sample3 Healthy 2
X1 Sample4 Disease 2
X1 Sample5 Healthy 5
X1 Sample6 Healthy 16
X1 Sample7 Disease 16
X2 Sample8 Healthy 0
X2 Sample9 Disease 0
X2 Sample10 Healthy 2
X2 Sample11 Disease 2
X2 Sample12 Healthy 5
X2 Sample13 Disease 5
X2 Sample14 Healthy 16
X2 Sample15 Disease 16
X3 Sample16 Healthy 0
X3 Sample17 Disease 0
X3 Sample18 Healthy 2
X3 Sample19 Disease 2
X3 Sample20 Healthy 5
X3 Sample21 Disease 5
X3 Sample22 Healthy 16
X3 Sample23 Disease 16
X4 Sample24 Disease 0
X4 Sample25 Healthy 2
X4 Sample26 Disease 2
X4 Sample27 Healthy 5
X4 Sample28 Disease 5
X4 Sample29 Healthy 16
X4 Sample30 Disease 16

The ID columns define each subject that paired samples collected from. There are total of four time points; 0, 2, 5, and 16.

As some part of the data were exampled above, I have 200 different measurements taken from each paired sites at each time point, but not all subjects have complete sample sets. Assuming the measurements are not following linear trend, can you please suggest (to someone who is pretty new to the GAMs/GAMMs) how can I perform testing whether measurements have significantly different longitudinal trend between healthy and disease groups by taking care of the paired sampling strategy using GAM or GAMMs in R?

PS: I tried some models such as

gam(Measurement1 ~ Group + Week +
            s(ID, bs = 're'),
          data = data, method = 'REML')

but I need some other families to test rather than Gaussian.