Mathematical Analysis of Agent-Based Models: Discrete and Heuristic Methods

Agent-based models (ABMs) are computer simulations wherein discrete agents interact with their environment. Agents follow certain rules (applied locally) in order to determine how their states change; agents may move, grow, reproduce, die, or represent other common natural phenomena. ABMs are used in a wide range of fields, including social science, biology, bioinformatics, epidemiology, climatology, geology, and computer science. They are used as predictive and analytical tools that can preclude expensive laboratory experimentation and save researchers valuable time.

There currently exists no universally agreed-upon method for rigorous mathematical analysis of such models. While some work has been done in the field of continuous modeling (i.e, translating ABMs into systems of differential equations), analysis from the discrete approach is limited. Conversion of ABMs to math models is critical in order to apply the wealth of mathematical theory to study and analyze such models.

We present scaling methods to improve simulation as well as conversion of ABMs into discrete mathematical models: both as difference equations and as Markov chain models. Case studies include spatially homogeneous and heterogeneous examples. Given that ABMs are designed to help investigate natural systems, the development of optimal control theory is critical for successful analysis. We demonstrate optimal control methods using ABM simulation and using discrete math models.

 ABMs are used in many areas of interdisciplinary research. Application and development of a mathematical framework for such models allows more and better results to be attained, further enhancing the potential for ABMs to inform and guide future scientific study.