Definition. A model is a simplified representation of a phenomenon of interest.

Examples:

• Physics model. An elastic solid is a simplified representation of the behavior of the atoms in an aluminum bar

• Anatomy model. A mouse lung is a simplified representation of a human lung

• Biological model. Time variation of ${\mathrm{Na}}^{+},{\text{K}}^{+},{\mathrm{Cl}}^{-}$ ions represent synapse formation in a neuron

• Social model. A network is a representation of social structure

Definition. A mathematical model is a quantitative statement of a model

Is a quantitative statement of the above model examples possible?

• Physics model. Yes, $F=kx$, Hooke's law relating force $F$ pulling on bar to elongation $x$.

  Note: In physics, a model with experimental support is called a law.

• Anatomy model. No.

• Biological model. Yes, Hodgkin-Huxley model.

• Social model. Yes, Six degrees of separation model.

Remark. A common scenario is that some states of a system are known along with a rate of change.

• If the system response is continuous, the system can be described through a differential model.

  • System responds to a single varying parameter: ordinary differential equation model

    ${\displaystyle \dot{𝒙}=𝒇(t,𝒙)(\mathrm{explicit}\mathrm{form}),𝑭(t,𝒙,\dot{𝒙})=0(\mathrm{implicit}\mathrm{form})}$

  • System responds to multiple varying parameters: partial differential equation model ${\partial }_{t}u+c{\partial }_{x}u=0$.

• If the system response is discontinuous, the system can be described through a difference model.