For more than two centuries differential equations (DEs) have been the mainstay of modelling in areas of science as disparate as electrical engineering and biology. The ever-increasing demands for better predictive ability of models require constant monitoring of the assumptions on which such models are based. This is particularly true for equations used in the formulation of models acutely needed to guide policy (in bioconservation or public health, for example).

However, the usage of ODEs in population and chemical models relies on the assumption that the numbers of molecules or individuals in the sample/population being analysed is large, an assumption that is routinely violated. This has given rise to hybrid models, in which some species are considered abundant, and their dynamics are modelled by ODEs, and other species are considered scarce and are modelled using variants of the chemical master equation (CME). While such hybrid systems have been used effectively in the context of biochemical control systems, where the division of species into abundant and scarce is fixed in time, there are many cases where species can move between abundant and scarce classes. Examples of such situations arise in the modelling of epidemiological systems and coagulation-fragmentation processes, which are of importance for example, in

planetoid formation, dynamics of aerosols and of schools of fish, and in protein aggregation in Alzheimer's disease. This network is interested in developing sound theoretical frameworks to model such systems (which we call temporally inhomogeneous hybrid systems). Such a framework will be widely applicable, including in the contexts outlined above.

A separate issue, of paramount importance in biology (e.g. in the modelling of efferocytosis or of viral reproduction and cell lysis) is the relaxation of the assumption routinely made in aforementioned coagulation-fragmentation equations (CFEs), of all reactants having zero volume. Relaxing this assumption calls for a multiscale approach to CFEs, i.e. assuming that some species (e.g. cells) are much larger than others (e.g. viruses) and allowing the macro-objects to have non-zero volume. Developing a rigorous mathematical framework for multiscale CFEs is the second principal objective of the network.

This network is initially funded by an INI network support grant (January 2026 to December 2027).