Characterisation of the idiotypic immune network through persistent entropy

In the present work we intend to investigate how to detect the behaviour of the immune system reaction to an external stimulus in terms of phase transitions. The immune model considered follows Jerne’s idiotypic network theory. We considered two graph complexity measures—the connectivity entropy and the approximate von Neumann entropy—and one entropy for topological spaces, the so-called persistent entropy. The simplicial complex is obtained enriching the graph structure of the weighted idiotypic network, and it is formally analyzed by persistent homology and persistent entropy. We obtained numerical evidences that approximate von Neumann entropy and persistent entropy detect the activation of the immune system. In addition, persistent entropy allows also to identify the antibodies involved in the immune memory.