Modeling activation energy function
Yes, you definitely can, but it's complicated. Because you are a mathematician it might be easier to understand.
Introduction
On your picture is one-dimensional representation of the energy function of multiple variables. Variables are coordinates of all atoms in the molecular system. This function define the potential energy surface or PES. Sometimes you can simplified it and represent as a three-dimensional surface using a linear combination of coordinates. Your picture is the energy profile of the PES. Where minimums are equilibrium states of the system and saddle points (minimax) are transitional states. 
The reaction progress on the picture is the intrinsic reaction coordinate (IRC) which is a one-dimensional representation of the a linear combination of vectors of your molecular system vibration.
Real energy profile
Consider, for example, the reaction $N_2O_2$ with $H_2$. Atoms in both molecules can oscillate in a specific well-defined way which gives you the molecular vibration. You need to solve quantum differential equations for the molecular system to get vibrational frequencies and vibrational vectors. To solve it analytically is incredibly difficult, but you can solve it numerically using quantum chemistry software. It will compute energy over intrinsic reaction coordinate (IRC) as a set of points with a certain displacement or step. Using simple spline function you can get a real energy profile.

Approximation
It is possible just to use a square function. According to the harmonic approximation it is precise at the vicinity of equilibrium states, but useless with transitional states. Sometimes to avoid this problem people use energy profile diagrams. Usually people are not interested in the form of function but the energy differences (activation energies, Gibbs energies and etc.)

If precision is not the case you just need to know energies of initial separated molecules, the transitional state energy (at the saddle point) and the energy of the product (which you can compute with quantum chemistry software) and join square functions at the found energies. You also need to remember that the energy profile is made for the elementary reaction.
Calculation
There are a lot of tutorials how to compute energies of initial systems, transitional states and products with quantum chemistry software. It should be mentioned that you will need to sum equilibrium energy of computed separately molecules to get your initial energy. The most difficult part is to compute the transitional state energy. You need to find a molecular geometry which would be very close to your potential transitional state. You need to calculate Hessian (second derivatives) and start minimax optimization (saddle point optimization).