Biomathematics, from the discrete to the continuous, at the service of modeling living systems

Biomathematics brings together mathematical modeling and simulation techniques for dynamic phenomena observed in nature, and applied to the field of living systems. These modeling techniques can be broken down into two main families:

• discrete biomathematics, illustrated by the theory of cellular automata (deterministic or random) and its applications to the modelling of genetic regulation networks and contagious diseases;

• continuous biomathematics, illustrated by the theory of partial differential equations applied to embryological development and the modeling of the spread of infectious diseases.

These two families of modeling techniques have in common a fairly recently explored field, that of hybrid systems, having a discrete and a continuous part.

Despite the apparent disparity of the fields of application, the spectrum of life sciences being very broad, the constancy in the choice of classical tools, through recent articles in international reference journals in the field, leads us to believe that the originality of biomathematics lies more in the complexity of the systems to which it applies, at the limit of computational possibilities in terms of the dimension of the systems studied and the number of interactions between their components (which obliges the implementation of computational methods optimizing execution times), than in the creation of new theoretical tools. The introduction of multi-scale methods in time and space, hybrid systems and energy approaches such as Hodge decomposition (potential-Hamiltonian) represents an innovative attempt to find specific methodologies, without in itself representing a break with the paradigm of classical modeling, which would introduce totally new mathematical methods required by the specificities of living systems. However, such a development cannot be ruled out in the future, and we will outline a few prospects.