# Julien Vincent (ESR11)

Modelling the mathematical way

As a PhD student in mathematics coming from a different field, I chose to write this blog post about mathematical modelling, a subject that at the beginning of my PhD I did not suspect to like that much but for which I ended up developing great interest.In simple words, mathematical modelling is the expression of physical, biological and chemical events into differential equations. These equations can depend on space or time (Ordinary Differential Equations), or both (Partial Differential Equations). The resolution of these equations or system of equations can predict the behaviour of the systems analysed.

When systems of equations are too complicated to solve analytically, i.e. using mathematical methods, “numerics” come into play and the systems can be solved using computers. To that end, the time and/or space is separated in a mesh or grid of points (called discretization), and by knowing what happens at the boundary of the systems and at the initial time, we can deduce, point by point, the solution for all the points on the grid by using numerical methods.

For someone that is used to laboratory work, mathematical modelling is really fun! Small constraints related to the practical aspects of working in a lab such as missing reagents or equipment failure are not present, and this is refreshing. However, real data is necessary to calibrate the models developed, making mathematical modelling and laboratory work very intertwined.

The main limitations of mathematical modelling are getting unknown parameters that can be very cumbersome, and modelling an event for which equations are still unknown or known but too difficult to solve. The range of validity of equations can also be a limitation to the applicability of the models. In addition, mathematical modelling requires multidisciplinary collaboration which can turn out to be difficult since biologists, physicists and mathematicians do not describe the same scientific problems in the same way. Similarly, different fields approach the same scientific question with a different angle and focus.

In my research, mathematical modelling is used to understand and/or predict the behaviour of biofilms, complex bacterial communities. That saves experimental time (some reactors can take months to develop a stable community) and costs, but a developed model can also be helpful because of the complexity of analysing biofilms without degrading them. 