# Accurately Identifying Water Contamination Helped by Mathematics

Uncannily similar to the Camelford water pollution incident in Cornwall, England in the late 1980s, in 2009 there was an incident in Crestwood Illinois. Due to human error, drinking water in Crestwood was accidentally contaminated which led to high cancer risk, brain damage, and even death in some instances.

On the heels of these tragedies, Martin Gugat, the author of a paper in the *SIAM Journal on Applied Mathematics *has developed a method for identifying and limiting water contaminations as soon as they occur.

He believes most contaminations are caused by human error and the location of the contamination must be identified immediately to protection the public before major health consequences occur. In his paper, Gugat proposes a water distribution network with a finite number of nodes where contamination can occur in the pipes.

To model the evolution in time of the system and how the pollution spreads, Gugat uses a partial differential equation (PDE). The PDE calculates the solution by using equidistant time grids. This allows researchers to determine values of contamination at all potential contamination locations on the grid. Historical data is then used in the model to increase the accuracy of the prediction.

Gugat employs a least-squares assumption method to estimate travel times. Optimization issues are overcome and can be computed with a method using the tools of numerical linear algebra. This also speeds the identification of the contamination location.

There is a downside. To date, three-dimensional PDEs only work for small networks, thus illuminating the delicate balance between a model’s precision and effectiveness.

Gugat is still working to refine his method and to determine workability in real world scenarios. Once the best method for identification is determined, researchers must devise a flush method. So, too, researchers must find a way to apply this method to large networks in a way that is not cost-prohibitive.