Containing the outbreak

To determine which points in the list make up the vertices of the containment polygon, I first put all the points (x,y) into a 2 dimensional array and sorted with respect to the x value. I then start with the first element of the array (the leftmost point), and determine the slope between it and each of the other points.

The largest slope will be with the next point along the top half of the polygon, and the smallest slope will be the next point along the bottom. I then add those to a list. Then I take the next point along the top, and search all the points after it in the array (to the right graphically), and find the next largest slope, and add it as being the next point along the top. I repeat this with the bottom, and perform this step over and over until I reach the last element in the array for both the top and bottom halves of the polygon.

After I have the points that make up the top and bottom halves of the polygon, I find the integral of the shape formed by the top half of the polygon from the range of the first element in the array to the last (still talking x values here), and subtract from that the integral of the bottom half of the polygon.

I found the integral of the halves using trapezoidal sums. Here's the formula for the integral between point 1 and point 2:

$integral = \frac{(x_{2}-x_{1}) \times (y_{2}+y_{1})}{2}$

To solve this we want to build a convex hull around the points and then calculate the area of the hull.

I made a visualizer for a convex hull algorithm a while back and with some tweaking it was able solve this particular problem.

Khan Academy - Convex Hull - Brilliant: Containing the outbreak

Feel free to play around with the code!

Mathematica has a built in function for computing the convex hull of a set of points

ch = ConvexHullMesh[plst];

Show[ch, Graphics[Point[plst]]]

Area[ch]