### WHAT ARE MATHEMATICAL MODELS?

Mathematical models are used in many domains to understand and quantify complex system dynamics and test mechanistic and causal hypotheses. In the health sciences, applications of mechanistic models include epidemic forecasting, intra-host models of pathogen and immune response, and modeling cost effectiveness and intervention outcomes.^{[2,3]}

Mechanistic mathematical models differ from statistical models. Statistical models, such as regression methods, generally aim to fit a function or curve to model data. In contrast, mechanistic models are comprised of equations or computation rules that explicitly describe how elements in a system relate to and affect one another.

Mathematical models are comprised of states (or compartments) and the flows between them. Often, mechanistic models employ a system of ordinary or partial differential equations (ODEs or PDEs) to explicitly describe the relationship between states and flow.

Specifically, mathematical models of biological systems and physiology explicitly represent underlying biological processes.^{[4]} Figure 1 shows an example of model parameters and differential equations used in Ki’s Gut and Growth Model. Figure 2 is a simplified representation of the key pathways and clinical endpoints of interest for the Gut and Growth Model.

Mathematical models employ data to accurately specify parameters in the equations, to constrain and calibrate the biological pathways represented, as well as to validate models and model predictions.