“PEOPLE” MEET “MARKOVIANS” — INDIVIDUAL-BASED MODELING WITH HYBRID STOCHASTIC SYSTEMS

Abstract

Individual-based models (IBMs) enable modelers to avoid far-reaching abstractions and strong simplifications by allowing for a state-based representation of individuals. The fact that IBMs are not represented using a standardized mathematical framework such as differential equations makes it harder to reproduce IBMs and introduces difficulties in the analysis of IBMs. We propose a model architecture based on representing individuals via Markov models. Individuals are coupled to populations — for which individuals are not explicitly represented — that are modeled by differential equations. The resulting models consisting of continuous-time finite-state Markov models coupled to systems of differential equations are examples of piecewise-deterministic Markov processes (PDMPs). We will demonstrate that PDMPs, also known as hybrid stochastic systems, allow us to design detailed state-based representations of individuals which, at the same time, can be systematically analyzed by taking advantage of the theory of PDMPs. We will illustrate design and analysis of IBMs using PDMPs via the example of a predator that intermittently feeds on a logistically growing prey by stochastically switching between a resting and a feeding state. This simple model shows a surprisingly rich dynamics which, nevertheless, can be comprehensively analyzed using the theory of PDMPs.