Set-Based Analysis for Biological Modeling

The understanding of biological systems and processes requires the development of dynamical models characterized by nonlinear laws and often intricate regulation architectures. Differential and difference equations are common formalisms to characterize such systems. Hybrid dynamical systems come in handy when the modeled system combines continuous and discrete evolutions or different evolution modes such as where slow evolution phases are interrupted by fast ones. Biological data with kinetic content are often scarce, thus it can be appropriate to reason in terms of sets of (parametrized) models and sets of trajectories. In doing so, uncertainties and lack of knowledge are explicitly taken into account and more reliable predictions can be made. A crucial problem in Systems Biology is thus to identify regions of parameter space for which model behavior is consistent with experimental observations. In this chapter, we investigate the use of set-based analysis techniques, designed to compute on sets of behaviors, for the validation of biological models under uncertainties and perturbations. In addition, these techniques can be used for the synthesis of model parameter sets, so that the execution of the considered biological model under the influence of the synthesized parameters is guaranteed to satisfy a given constraint or property. The proposed approach is illustrated by several case studies, namely a model of iron homeostasis in mammalian cells and some epidemic models.