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How do engineers predict the exact trajectory of a rocket through changing atmosphere, or epidemiologists pinpoint the peak of an outbreak? They translate a complex physical situation into a mathematical modelβ€”a set of equations capturing how key quantities behave and interact.
The Mathematical Modelling Cycle begins by abstracting an unstructured real-world context into a tractable problem. This requires careful problem specification, defining key state variables, and setting explicit simplifying assumptions.

The Mathematical Modelling Cycle

The Mathematical Modelling Cycle begins by abstracting an unstructured real-world context into a mathematically tractable problem through specification, variable definition, and explicit simplifying assumptions.
Before writing an equation, our first job is to isolate the quantities that change from those that stay constant.

Variables vs. Parameters

A variable is any measurable quantity that changes value during the scenario, such as time tt, population PP, or temperature TT. A parameter is a fixed constant specific to the particular setting, such as a growth rate coefficient kk or environmental temperature TenvT_{\text{env}}.
Once we identify our variables, how do we translate everyday descriptive language into strict mathematical relations?

Translating English to Rates of Change

Beyond descriptive words, differential equations arise directly from fundamental physical laws. For example, moving bodies obey Newton's Second Law (βˆ‘F=mdvdt\sum F = m\frac{dv}{dt}), while flowing substances follow rate balance laws (dQdt=Rateinβˆ’Rateout\frac{dQ}{dt} = \text{Rate}_{\text{in}} - \text{Rate}_{\text{out}}).
Words like "rate of growth" or "speed of decay" signify a derivative with respect to time, dydt\frac{dy}{dt}. When a quantity changes proportionally to another value zz, we multiply by a parameter kk, writing dydt=kz\frac{dy}{dt} = kz.