What is Mathematical Modeling
Mathematical modeling is a type of theoretical framework used extensively in the behavioral and applied behavioral sciences. Its essential function is to formulate problems in precise mathematical terms. Effectively, it involves creating a mathematical representation of a system or a phenomenon to understand, predict, and increase control over its behavior.
The process of mathematical modeling involves identifying critical variables in a situation, formulating mathematical expressions that approximate the behavior of those variables, and using those expressions to predict or control the situation. This approach integrates various branches of mathematics such as algebra, calculus, and statistics, among others.
Types of Mathematical Models
There are several types of mathematical models employed in behavioral sciences sharing a common goal—capturing the essence of a system or a phenomenon for purposes of prediction and control. Their efficacy depends on the match between the assumptions underlying a particular model and the realworld situation.

Deterministic Models
These models assume that the same initial conditions will invariably produce the same output. The deterministic model does not factor in the possibility of randomness or chance contributing to the outcome. Examples include mechanics and many physical processes.

Probabilistic or Stochastic Models
Contrary to deterministic models, stochastic models consider the probabilities of different outcomes. They are valuable in situations where random variables are involved. Examples include insurance and finance.

Static Models
A static mathematical model represents a situation at a particular instant in time. It does not factor in the evolution of the situation over time. Static models are used when time does not affect the phenomenon or when time is not a key consideration in the process the model represents.

Dynamic Models
Unlike static models, dynamic models consider the time factor. They show how a situation changes over time. These models are used where time significantly influences the outcome.
The Role of Mathematical Modeling in Behavioral Sciences
The central role of mathematical modeling in behavioral and applied behavioral sciences is to provide a precise language for framing hypotheses, interpreting experimental outcomes, and predicting phenomena. It offers a more thorough understanding of complex systems and facilitates the verification of proposed theories.

Understanding Complex Systems
Mathematical models facilitate a deeper understanding of complex systems by simplifying them into their core principles. By representing the system or phenomena mathematically, researchers can isolate the different variables at play, understand their relationships, and predict how changes to these variables affect the system.

Predicting and Control
Modeling plays a vital role in the prediction and control of outcomes in behavioral sciences. Given the complexity and number of variables that often influence behaviors, mathematical models provide a structured framework through which predictions can be made. Once the model is optimized for prediction, it can be used as a control tool to guide interventions in certain scenarios.

Verification of Theories
Mathematical modeling is a powerful tool for verifying theoretical concepts. By representing a theory as a mathematical model, researchers can experimentally evaluate the model’s predictions. If the outcomes match the predictions, then the theory is validated. If they do not, the theory may need to be revised or discarded.
Application of Mathematical Modeling in Behavioral Design
In the realm of behavioral design, mathematical modeling helps designers understand how different design components influence human behavior and engage that understanding to optimally influence outcomes. The understanding of behavioral and system dynamics—something mathematical models excel in—can be incredibly effective for designing interventions with desired effects.
Critical applications often include the design of incentives, clues, and other behaviorshaping variables. Moreover, these models assist in the quantification of the effects of design elements, thus providing measurable objectives and milestones. Ultimately, mathematical models serve as a strong foundation for evidenceinformed design and for optimizing intervention strategies aimed at influencing behavior towards desirable outcomes.