Mathematical Modelling

The modelling is an activity, a cognitive activity in which we thing about and make models to describe how devices or objects of interest behave. We usually use words, drawings or sketches, physical models, computer programs, or mathematical formulas (or more generally said different languages). If we use language of mathematics to make models we are providing mathematical modelling.

According to Dym and Ivey (2004) mathematical modelling is a principled activity that has principles behind it and methods that can be successfully applied. As he mentions the principles are over-arching or meta-principles phrased as questions about the intentions and purposes of mathematical modelling.

Usually mathematical modelling is the use of mathematics to describe real or conceptual world phenomena, to investigate important questions about the observed world, to explain world phenomena, to test ideas, and to make predictions about the world around or inside us. Mathematical (or equation-based modelling) is an approach to describe a system with the tools of calculus, typically in terms of systems of differential or difference equations. As mentioned in Toitzsch (1998) they usually allow only for the description on one macro level. The master equation approach can be used to describe interactions between a micro and a macro level, converting assumptions about the stochastic behaviour of micro units into statements about distributions of attributes of the macro unit or units.

According to Troitzsch (2009) 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 than the structural validity of mathematical models of social and economic systems. Whereas in physics mathematical models 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.

Related terms: Mathematical model

References:

Dym, C. L., Ivey, E. S. (2004). Principles of Mathematical Modeling (Computer Science and Applied Mathematics). Academic Press.

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). p. 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. p. 20--36.