Our initial models will be SIR models -- Susceptible, Infected, Recovered. There is no death, so we start with a certain population, and we'll end with that population -- everyone is always in one of the three groups.
Now let's get started....
Once again we'll need to talk about differential equations:
The simplest looks like this:
\[ \frac{dS}{dt} = -\alpha S I\\ \frac{dI}{dt} = \alpha S I - \beta I\\ \frac{dR}{dt} = \beta I \]
This simple SIR model generally assumes that
But for most of this lab, you'll be taking $\gamma=0$.
These equations give the rates of change of the three populations. But, since the total never changes (no death), we just reshuffle the populations each time; and we must have a constant total population $N(t)=S(t)+I(t)+R(t)$.
Write down everything you can about the derivative $\frac{dN}{dt}$.
Similarly, the $- \gamma R$ is like a death of the recovered, into a new life (birth) of a susceptible person. We think of it as a loss of immunity. Some vaccines wear off, and you need to be re-vaccinated: this is because you gradually lose your immunity.
The other type of term -- lost from the $S(t)$ population, but gained by the $I(t)$ population -- is an example of what we call "Mass-action dynamics": $\alpha S I$ represents a term that comes from the interaction of the $S$ and $I$ population. They have to collide -- come into contact -- to pass the infection from the $I$ to the $S$.
Check your initial settings:
$S(0)=S_0$ | $60$ |
$I(0)=I_0$ | $20$ |
$R(0)=R_0$ | $20$ |
$\alpha$ | $0.09$ |
$\beta$ | $1.3$ |
$\gamma$ | $0.0$ |
Notice that I've written the initial populations so that their sum, the total population, is 100. So we can interpret the populations as the percentage of the total population.
Notice also that you can set your rates with the sliders on the right, under the Insight information.
Set your time setting for months, 0 to 4; RK-4, with step-size of .1.
So here's the story:
Reset your initial populations to their original values. But in "Settings", change the final time to 0 to 1 -- that should show things off better (adjust as needed!).
Now using the Insightmaker model with the values shown in the tables below, estimate the values of $S$ at the time $t^*$ when the number of infected is a maximum (don't submit plots):
a) $\alpha=0.125, \beta=3$: \[ \begin{array}{|c|c|c|} {S(0)} & {I(0)} & {S(t^*)}\cr \hline {60}&{40}&{}\cr \hline {60}&{22}&{}\cr \hline {60}&{10}&{}\cr \hline {40}&{60}&{}\cr \hline \end{array} \] | b) $\alpha=0.167, \beta=3$: \[ \begin{array}{|c|c|c|} {S(0)} & {I(0)} & {S(t^*)}\cr \hline {60}&{40}&{}\cr \hline {60}&{22}&{}\cr \hline {60}&{10}&{}\cr \hline {40}&{60}&{}\cr \hline \end{array} \] |
(Include graphs to help support your claims about these statements.)
Set the time to 144 months (twelve years), simulate and include your graphs, and answer these questions:
Links: