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\title{Some (simple) examples of differential equations}
\date{January 20, 2021}

\begin{document}\ 

\vspace{-.25in}

\maketitle

\vspace{-.15in}

\eg{
The SIR model of disease spread looks at the interaction of 3 quantities at time $t$: $S(t)$ is the number of susceptible individuals, $I(t)$ is the number of infected individuals, and $R(t)$ is the number of recovered (or removed) individuals.
The model starts with assumptions about how the disease works:
\begin{enumerate}[a)]
\item The population remains constant at $p_0$
\item A constant proportion $k$ of infected individuals recover in each unit of time
\item Each susceptible individual has probability $b$ of a contact that could transmit the disease (if the other person is infected)
\item The number of susceptible individuals decreases as they are infected
\item The number of infected increases as susceptibles are infected and decreases as the infected recover
\end{enumerate}
These assumptions translate into mathematics:
\begin{align*}
\frac{dR}{dt} &= k I \\
\frac{dS}{dt} &= -b\left(\frac{I}{p_0}\right)S \\
\frac{dI}{dt} & = b\left( \frac{I}{p_0}\right)S -k I
\end{align*}
The full model is a system of differential equations (and it's a bit too complex for us), so we'll just look at a simple model for $I(t)$ near time $t=0$ when $S \approx p_0$. 
This gives a simplified picture that should be reasonably accurate for initial spread of the disease. 
}

\newpage

\prob{
Population tend to grow in proportion to their size: $P'(t) = rP(t)$ where $r$ is a constant growth rate parameter.
This means $\displaystyle P(t) =$?
\begin{enumerate}[a)]
\item Human population is currently about $7.8$ billion and the estimated growth rate is about 1\% per year.
This means that $P(0) =7.8$ billion and $P(1) = 1.01p_0$.
Use these two facts to solve for $r$ and any other unknown constants.
\item Predict the population in 10 years and in 100 years.
\item What does the model predict in the long-term? Is this realistic?
\item $r = b-d$ where $b$ is a birth rate parameter and $d$ is a death rate parameter.
What has to be true of $b$ and $d$ if the population isn't growing exponentially?
\end{enumerate}
}

\vfill

\chal{
The exponential model for population growth $P' = rP$ predicts unbounded population sizes.
The logistic model resolves this problem: $P' = aP(1- b P)$ where $a$ and $b$ are positive constants.
If it helps you to solve the problem, you may use $a = 1$ and $b=1/2$.
\begin{enumerate}[a)]
\item Solve the integral equation for $P(t)$ (you may need to use the method of partial fractions): 
\[ \int \frac{1}{P(1-bP)}\ dP = \int a\ dt \]

\item Calculate $\lim_{t \to \infty}P(t)$. Does the logistic model predict unbounded growth?
\end{enumerate}
}

\newpage

Newton's law of cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the temperature of its environment.
As a differential equation: $T' = -k(T-T_m)$ where $k$ is a positive constant of proportionality and $T_m$ is the (constant) temperature of the environment.
%This means that $\displaystyle T = T_m + T_0 e^{-kt}$ where $T_0 = T(0)-T_m$.

\prob{
Solve the differential equation for $T$.

\vspace{1 in}

}

\prob{[CSI Gonzaga]
A cooling cup of coffee is found outside on a $5^{\circ}$C day.
At 12:15 its temperature is $35^{\circ}$C and at 12:45 its temperature is $25^{\circ}$C.
\begin{enumerate}[a)]
\item Use the two points to find a formula for $T(t)$, the temperature of the coffee $t$ hours after 12:15 (so $t=0$ is 12:15).
\item Coffee is usually brewed at about $95^{\circ}$C. Use this to estimate how long ago the coffee was brewed.
\end{enumerate}
}

\end{document}