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\title{SIR models}
\date{September 1, 2023}

\begin{document}
\ \vspace{-.3in}

\maketitle

\vspace{-.2in}

The SIR model divides a population into 3 groups: the susceptible, the infected, and the recovered/resistant.
We then track those 3 subpopulations using the following variables:
\begin{enumerate}[\ ]
\item $S$ is the number of susceptible individuals
\item $I$ is the number of infected individuals
\item $R$ is the number of recovered/resistant individuals
\end{enumerate}
All 3 variables change over time, so we'll need to remember that they're functions of time.
At all times we assume that the total population is the same: $S(t)+I(t)+R(t)$ is constant.

\picprob{
How do you think $S(t)$, $I(t)$, and $R(t)$ should work?
Sketch a (rough) graph of all 3 functions on a single set of axes.

\vspace{.2in}

\noindent 
\begin{tikzpicture}
%\draw[step=0.6cm,color=lightgray,very thin] (-4,-4) grid (12,4);
\draw[<->, very thick] (-3.6,-4)--(-3.6,4);
\draw[<->, very thick] (-4,-3.6)--(12,-3.6);
\draw (12,-3.6) node[anchor=north]{time (days)};
\draw (-3.6,4) node[anchor=north east]{number};
\end{tikzpicture}

\vspace{.2in}

}

\prob{
Now we'll try to be more sophisticated and mathematical.
Apply pillar 2 and \textbf{track the changes} to fill in the right side of the following equations (we'll do this as a class).
\begin{enumerate}[\ ]
\item $S'=$
\item $I' =$
\item $R'=$
\end{enumerate}

\vspace{1in}

}

\newpage

\prob{
Suppose $S(0) = 100$, $I(0)=1$, and $R(0) = 0$.
Also let $a = 0.01$ and $b=0.125$ with time measured in days.
Apply pillar 5 and go \textbf{one step at a time} to make predictions.
\begin{enumerate}[a) ]
\item Predict $S(1)$, $I(1)$, and $R(1)$

\vfill

\item Predict $S(2)$, $I(2)$, and $R(2)$

\vfill

\item (Optional) Predict $S(3)$, $I(3)$, and $R(3)$
\end{enumerate}
}

\end{document}