\newcommand{\theversion}{0.13.0}

A \py{TimeFrame} is almost identical to a \py{DataFrame}, which we used in Section~\ref{worldpopdata}, with just a few changes I made to make them easier to use for our purposes. For more details, you can read the \py{modsim} library source code.

The first line creates an empty \py{TimeFrame} with one column for each state variables. Then, before the loop starts, we store the initial conditions in the \py{TimeFrame} at \py{t0}. Based on the way we've been using \py{TimeSeries} objects, it is tempting to write:

But when you use the bracket operator with a \py{TimeFrame} or \py{DataFrame}, it selects a column, not a row. For example, to select a column, we could write

When we plot a time series, we get a view of everything that happened when the model ran, but often we want to boil it down to a few numbers, ideally one, that summarize the outcome. These summary statistics are called {\bf metrics}.

One option is to treat immunization as a short cut from \py{S} to \py{R}, that is, from susceptible to removed (or resistant). We can implement this feature like this:

where $T$, the temperature of the object, is a function of time, $t$; $T_{env}$ is the temperature of the environment, and $r$ is a constant that characterizes how quickly heats moves across the boundary between the system and the environment.

\caption{Final temperature as a function of when we add the milk.}

The initial state of the model is $X(0) = I_b$ and $G(0) = G_0$, where $G_0$ is a constant that represents the concentration of blood sugar immediately after the injection. In theory we could estimate $G_0$ based on measurements, but in practice it takes time for the injected glucose to spread through the blood volume. Since $G_0$ is not measurable, it is treated as a {\bf free parameter} of the model, which means that we are free to chose it to fit the data.

To develop and test the model, I'll use data from Pacini and Bergman\footnote{"MINMOD: a computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous glucose tolerance test", {\em Computer Methods and Programs in Biomedicine} 23: 113-122, 1986.}. The dataset is in a file in the repository for this book, which we can read into a \py{DataFrame}:

where $H$ is another function and $y'$ is shorthand for $dy/dx$. In particular, we will work with one of the most famous, useful, and possibly the first second order ODE, Newton's second law of motion:

where $F$ is a force or the total of a set of forces, $m$ is the mass of a moving object, and $a$ is acceleration.

But for medium to large things with constant mass, moving at speeds that are medium to slow, Newton's model is a phenomenally useful. If we can quantify the forces that act on such an object, we can figure out how it will move.

Given that the Empire State Building is \SI{381}{\meter} high, and assuming that the penny is dropped with 0 velocity, the initial conditions are:

Next we need a \py{System} object to contains the system parameters:

The result is a sequence containing \SI{0}{\meter\per\second} for velocity and \SI{-9.8}{\meter\per\second\squared} for acceleration. Now we can run \py{odeint} like this:

As the number of system parameters increases, and as we need to do more work to compute them, we will find it useful to define a \py{Condition} object to contain the quantities we need to make a \py{System} object.

The way we create a \py{Condition} object is the same as for \py{System} and \py{State} objects. In fact, these three object types have the same capabilities; I have given them different names just to identify their different roles. Here's the \py{Condition} object for the falling penny:

Then it uses \py{loc} to extract only the points after \py{t_peak}. The index here is a {\bf slice}. In general, you can use a slice to select a range of values from a \py{Series}, specifying the start and end indices. In this example we only specify the beginning, so we get everything from \py{t_peak} the end of the \py{Series}.

Finally, it uses \py{interpolate} to make \py{X}, which is an interpolation function that maps from time to $x$ component. So \py{X(T_land)} is the distance from home plate when the ball lands.

And when you applying a twisting force to a rotating object, the effect is often contrary to intuition. For an example, see this video on gyroscopic precession \url{https://www.youtube.com/watch?v=ty9QSiVC2g0}.

In the most general case, a 3D object rotating around an arbitrary axis, moment of inertia is a tensor quantity, which is like a function that takes a vector as a parameter and returns a vector as a result.