Mathematical Modeling

Mathematical or equation-based modelling is an approach to describe real-world systems with the tools of calculus, typically in terms of systems of differential or difference equations. They usually allow only for the description on one (macro, aggregate) level. The master equation approach can be used to describe interactions between a micro (individual) and a macro (aggregate) level, converting assumptions about the stochastic behaviour of (relatively simple) micro units into statements about distributions of attributes of the macro unit or units.

Only few of these mathematical models have closed solutions, thus necessitating numerical treatment, and this is kind of simulation, such that more complex systems of more complex elements profit much from agent-based models whose structural validity is often better that the structural validity of mathematical models of social and economic systems. 

Whereas in physics mathematical model are often sufficient and sometimes the best way of describing the interaction between fields and particles, this is only very rarely the case for social systems.

Literature:

Troitzsch, Klaus G. (2009): Perspectives and Challenges of Agent-Based Simulation as a Tool for Economics and Other Social Sciences. In: Proc. of the 8th Int. Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 2009). S. 35-42.

Troitzsch, Klaus G. (1998): Multilevel Process Modeling in the Social Sciences: Mathematical Analysis and Computer Simulation. In: Liebrand, Wim B.G.; Nowak, Andrzej; Hegselmann, Rainer: Computer Modeling of Social Processes. London: Sage. S. 20--36.