An Algebraic Framework for Stock & Flow Diagrams and Dynamical Systems Using Category Theory

Mathematical modeling of infectious disease at scale is important, but challenging. Some of these difficulties can be alleviated by an approach that takes diagrams seriously as mathematical formalisms in their own right. Stock & flow diagrams are widely used as broadly accessible building blocks for infectious disease modeling. In this chapter, rather than focusing on the underlying mathematics, we informally use communicable disease examples created by the implemented software of StockFlow.jl to explain the basics, characteristics, and benefits of the categorical framework. We first characterize categorical stock & flow diagrams, and note the clear separation between the syntax of stock & flow diagrams and their semantics, demonstrating three examples of semantics already implemented in the software: ODEs, causal loop diagrams, and system structure diagrams. We then establish composition and stratification frameworks and examples for stock & flow diagrams. Applying category theory, these frameworks can build large diagrams from smaller ones in a modular fashion. Finally, we introduce the open-source ModelCollab software for diagram-centric real-time collaborative modeling. Using the graphical user interface, this web-based software allows the user to undertake the types of categorically-rooted operations discussed above, but without any knowledge of their categorical foundations.