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Section1.2Introduction to Modeling

Subsection1.2.1What is a Model?

Models are simplified abstract representations of something of interest. Airplane and automobile manufacturers create scale models to test aerodynamics in wind tunnels. Architects build models of future projects, whether a physical mock-up or a computerized 3-d representation, to see how plan will fit together and give clients a vision of what they will get. The models do not need to include every detail of the actual object of interest, just those details that are relevant to the purpose of the model.

Scientists also regularly use models. Physicists use high energy collisions of extremely fast particles to create conditions that they expect are comparable to the moments immediately after the big bang. A biologist may use mice from a well-controlled population as a model to study cancer, considering its biology to mimic that of humans at some level of approximation. A climatologist might use a computational model where a computer program tracks changes in the makeup of the air, pollutant levels and air and ocean temperatures according to known and assumed interactions.

Subsection1.2.2Mathematical Models

A mathematical model is an abstract representation of measurable phenomena that is characterized through mathematical equations. We will use the word system to mean the physical collection of objects and environment involved in the phenomenon. The state of the system refers to the collection of all measurable quantities related to the system at a given instant in time and under particular environmental conditions. A state variable refers to a particular measurable quantity, whether or not we can construct a physical way to make the measurement. When the phenomenon is considered under a single environmental condition, there may be measurements of those environmental conditions. Since they are controlled to have no variability, these measurements are called state parameters, although mathematically they may also be correctly called variables.

To help think about this, consider the physical example of a sink being filled with water. At each instant in time, the sink is in a different state. The state can be characterized by measurements at that time, which might include the height of water in the sink, the volume of water in the sink, the rate of water flowing into the sink from the spigot and the rate of water flowing out of the sink through the drain. We might also measure things like water temperature or the dilution of some substance or chemical in the water.

In our example, the variables correspond to the measurements. For simplicity of communication, we often associate a single symbol (usually a letter) to represent that variable. We might use \(t\) to represent our measurement associated with the time of observation, \(V\) to represent the volume of water, \(h\) to represent the height of water in the sink, \(F_{\hbox{in}}\) to represent the rate of water flowing in and \(F_{\hbox{out}}\) to represent the rate of water flowing out. We might consider a situation in which the spigot has water flowing at a constant rate. In this case, the variable \(F_{\hbox{in}}\) is never changing; we could call this a state parameter instead of a state variable. A diagram can often help summarize this information.

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Figure1.2.1Diagram illustrating the variables associated with the water in a sink.

Mathematical models attempt to find and interpret relationships between state variables. Static models characterize relationships between variables that come from the state of a system at the same time. Dynamic models characterize relationships between variables over different states of the system, usually as the state changes in time. It is possible (and often reasonable) to think of time as a state variable, and so models that find relationships between state variables and time can be both static and dynamic. We will call such a model dynamic if the model describes how state variables change in time and static if the model establishes an equation directly relating the state variables and time.

Thinking back to our example of a sink, to create mathematical models would be to create mathematical relationships that capture (exactly or approximately) the relationships between our observed variables. For example, the rate of water flowing out through the drain \(F_{\hbox{out}}\) might change depending on how much water there is in the sink, which could be measured by either the volume \(V\) or the height \(h\) of water in the sink.

For example, we might test a model that the flow out is proportional to the volume of water, \begin{equation*}F_{\hbox{out}} = \alpha V,\end{equation*} or that it is proportional to the height of water, \begin{equation*}F_{\hbox{out}} = \beta h,\end{equation*} where \(\alpha\) and \(\beta\) are model parameters that would be chosen to best fit the data. These equations are examples of static models.

A dynamic model could be built using one of these static models. Suppose we decided that the model \(F_{\hbox{out}} = \alpha V\) was valid and the flow of water coming in was constant \(F_{\hbox{in}}\). Then the net rate of change of volume of water in the sink is found by the difference \(F_{\hbox{in}} - F_{\hbox{out}}\), with a negative value being interpreted as the total volume decreasing. Calculus will teach us that the rate of change of volume is called a derivative and has a symbol \(\frac{dV}{dt}\) (that fraction is a single but complex symbol). So our dynamic model would be \begin{equation*} \frac{dV}{dt} = F_{\hbox{in}} - \alpha V, \end{equation*} which is called a differential equation.