Fitting ICC Curves

Theoretically Fitting of ICC Curves

Without diving into too many math details (see disseration for those). We can think of the ICC curve as dependent on 4 primary parameters:

1. beta - the transmissibility of the disease

2. N - the population size

3. gamma - the recovery rate

4. C_0 - the initial number of infectious individuals

These four variables can be combined to find the theoretical ICC curve. In practice though, data will be messy, and we need a method to identify what the optimal parameters are for a given epidemic trajectory.

Adding Noise

To best test the plausibility of this method, we want to first generate data. We do this by simulating SIR infections and tracking the number of total cases and the change in number of cases (C, dC/dt) to generate the ICC Curve. A collection of ICC curves with beta = 0.2, gamma = 0.1, N = 10000, C_0 = 10 is shown below.

Using the statistics described in the ICC Introduction, we can see that the theoretical distribution matches the system well. However, what happens if we only have one curve.

When one single epidemic trajectory is present, we use our algorithm to find the optimal parameters to represent the epidemic data as an ICC curve. Below, we show an example with both the parameters used to generate the ICC curve and those found by our solver.

We can see that the trajectory falls nicely within the original parameters distribution, but that those parameters might not fully capture the behavior of the curve. On the right-hand side, we see the discovered parameters give an ICC curve that falls nicely along the full data.

This example illustrates our ability to fit ICC curves using our parameter optimization method. 

Recall the Arizona data (shown below) appears to be made up of multiple ICC curves. This now begs the questions: can we use this with real data? What if the parameters change? If they do change, how do we detect the change?