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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\MODELING EVICTIONS}



\markboth{Modeling Evictions}{Modeling Evictions}

\author{Mary Vanderschoot\\
Wheaton College\\
Wheaton IL 60187 USA\\
mary.vanderschoot@wheaton.edu
}

\makeStitlePDFLaTex


\begin{abstract}

In this project, students  develop two SIS models to study eviction trends in a population of non-homeowner households using an actual eviction rate. (The Eviction Lab at Princeton University has developed a nationwide database of evictions based on 83 million eviction records \cite{Lab}.)    As the first model is a linear system,  students can readily calculate solutions, sketch the phase portrait, and determine long-term trends without using technology.   Students will design a second model that incorporates the carrying capacity (number of rental units).  
  

\end{abstract}



\section*{SCENARIO DESCRIPTION}

According to the National Law Center on Homelessness and Poverty, unaffordable rents and a lack of legal protections for renters have created a national ``eviction epidemic'' \cite{NLCHP}. Matthew Desmond, author of  \emph{Evicted: Poverty and Profit in the American City} and director of the Eviction Lab at Princeton University, estimates that 2.3 million evictions were filed in the U.S. in 2016 (four evictions per minute).  Desmond writes, ``Eviction is a direct cause of homelessness, but it also is a cause of residential instability, school instability [and] community instability'' \cite{Desmond}.  In this project you will develop and analyze two mathematical models to study eviction trends in a city using an actual eviction rate. 



\section{Part I:  A Linear Model}

Suppose a certain city has 118,000 non-homeowner households and this number remains constant each year.  (For example, if three of these households move to a different city or purchase a home, then three new non-homeowner households move into this city.) Furthermore, suppose that each of these households is either renting an apartment or house or is not renting due to having been evicted.  We can define each of these subpopulations as functions of time, $t$, (years) in the following manner:
\begin{align*}
    R(t) &\text{ is the number of \textit{renting} households at time } t,\\
   E(t) &\text{ is the number of \textit{evicted} households at time } t.
\end{align*}
In order to simplify our calculations, we will consider the fraction of households in each category, that is, if $N$ is the total number of non-homeowner households (in our example $N=118,000$), then 
\begin{align*}
    r(t) =R(t)/N &\text{ is the fraction of \textit{renting} households at time } t,\\
   e(t) =E(t)/N  &\text{ is the fraction of \textit{evicted} households at time } t.
\end{align*}
The Eviction Lab at Princeton University has developed a nationwide database of eviction records, which includes state and city eviction rates \cite{Lab}.  (For example, in 2016 North Charleston, South Carolina, had the highest eviction rate of 16.5\%.)   In our model we will assume that a fixed percent  $\alpha$ from the renting group become evicted each year.  (For North Charleston we would set $\alpha = 0.165$.)  We will also assume that a fixed percent $\beta$ of the evicted group become renters each year.  In this model, the only way a renting household can leave the renting group is by transitioning to the evicted group.  Similarly, the only way an evicted household can leave the evicted group is by transitioning to the renting group.  It is helpful to represent this scenario with a flow diagram in Figure~\ref{fig:evictiondiagram}.

\begin{figure}[htpb]
\begin{center}
\includegraphics{evictiondiagram}

\caption{\label{fig:evictiondiagram} Flow diagram for evictiobn model.}
\end{center}
\end{figure}
\begin{enumerate}

\item Find equations for $\frac{dr}{dt}$ and $\frac{de}{dt}$ that satisfy the above assumptions.  Explain the meanings of $\frac{dr}{dt}$ and $\frac{de}{dt}$ and what each component of your equations represents.


\item Choose a value for $\alpha$ from the 2016 data for city evictions from the Eviction Lab \cite{Lab}.   (State which city's information you chose.)  Set $\beta = \frac{1}{3} \alpha$.  

\item Sketch the phase portrait for this system, including all equilibrium solutions.  (Be sure to sketch only the region relevant  to our situation.) 

\item For our model, we are only concerned with initial conditions on the line $r + e = 1$.  Why?  Sketch this line on your phase portrait.

\item Suppose that, initially, 95\% of non-homeowner households are renters (so 5\% are in the evicted group).  Solve this initial value problem and  plot $r(t)$ and $e(t)$ on one graph.  What does this model predict about the percent of non-homeowner households in each group (renting  and evicted) in the long-run?  Eventually, what will be the (approximate) ratio of renting to evicted non-homeowner households?


\item How might this model be overly simplistic?  What are some additional considerations that should be included in an eviction model? 
\end{enumerate}


\section{PART II:  A Modified Model}

In our first model we assumed that a constant fraction of the renting group, $\alpha r$, transitioned to the evicted group each year regardless of the vacancy rate.  Suppose that the city has exactly $M$ rental units (apartments and houses).  

In this section you will construct a model for $\frac{dR}{dt}$ and $\frac{dE}{dt}$  that satisfies the following assumptions:
\begin{itemize}
\item The flow rate from the renting group to the evicted group increases as the number of vacancies decrease (i.e. when $R$ is close to $M$).   
\item The flow rate from the evicted group to the renting group decreases as the number of vacancies decrease. 
\end{itemize}

\begin{enumerate}

\item Suppose your roommate suggests using $-\alpha \frac{R^2}{M}$ for the flow rate from the renting group to the evicted group.   Does your roommate's suggested flow rate satisfy the first assumption?  If so, explain why it does; if not, suggest a different formula and show that it satisfies the first assumption.

\item Find a formula that can be used for the flow rate from the evicted group to the renting group.  Explain why your formula satisfies the second assumption. 

\item Suppose that $M = 118,500$.  Using the same values of $\alpha$ and $\beta$ from Part I, use technology to sketch the phase plane and the solution curves that satisfy the initial conditions $R(0)=112,100$ and $E(0)=5900$ (so initially, 95\% of the non-homeowners are renting). What does this model predict about the long term percentages of non-homeowner households who are renting and who are evicted?  How does this compare to your results in the first model?  



\end{enumerate}


\begin{thebibliography}{98}

\bibitem{Desmond}Desmond, M. 2016. {\it Evicted: Poverty and Profit in the American City}. New York: Crown.


\bibitem{Lab} Desmond M., A.~Gromis, L.~Edmonds, J.~Hendrickson, K.~Krywokulski, L.~Leung, and A.~Porton A.  2018. {\it Eviction Lab National Database: Version 1.0\/}. Princeton: Princeton University. Accessed  from \href{https://www.evictionlab.org}{https://www.evictionlab.org} on 12 August 2018.

\bibitem{Nester} Nester, Darryl. 2018. Phase Planes. \href{https://bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html}{https://www.nlchp.org}. Accessed 12 August 2018.

\bibitem{NLCHP}National Law Center on Homelessness and Poverty.  2018.  {\it Protect Tenants, Prevent Homelessness}. Accessed from \href{https://www.nlchp.org}{https://www.nlchp.org} on 12 August 2018.



\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}
This scenario can be used as an in-class group project during the systems of differential equations unit. Following are comments on each problem:

\noindent{\bf Part I}
\begin{enumerate}
\item Students should be able to provide a verbal and written explanation of how the following model follows from the assumptions:  
\begin{eqnarray}
    \frac{dr}{dt}& = &-\alpha r + \beta e\\ \nonumber
    \frac{de}{dt}& =& \alpha r - \beta e \nonumber
\end{eqnarray}
They should explain that $\frac{dr}{dt}$ is the yearly rate of growth/decline of the renting population and that this is found by subtracting the fraction of the renting group who transition to the  evicted group ($\alpha r$),  and adding the fraction of evicted households  ($\beta e$) that transition to the renting group.  They should provide a similar explanation for $\frac{de}{dt}$.
\item The Eviction Lab \cite{Lab} is a helpful resource for information about factors that impact evictions.
\item This system has a line of equilibrium points ($e=3r$.) All other trajectories approach this line on parallel lines.  Their phase portraits should only include $0 \leq r \leq 1$, $0 \leq e \leq 1$.
\item Students can solve this initial value problem with technology or by hand.  They should find that $(r(t),e(t)) \to (0.25, 0.75)$.
\item There are many factors that could be included.  Households from either group could move to a different city or become homeowners.  Our model did not examine factors that impact evictions such as the poverty rate, migration to the city, the number of rental units available, etc.  
\end{enumerate}
\noindent{\bf Part II}
\begin{enumerate}
\item One way to argue that the suggested rate satisfies the assumption is as follows: If $R = \frac{M}{2}$ (many vacancies) then the flow rate from the renting group to the evicted group is $-\alpha \frac{R^2}{M} = -\frac{\alpha}{2} R$.  But if $R \approx M$ (few vacancies) then  $-\alpha \frac{R^2}{M} \approx - \alpha R$, the flow rate in our first model.  Students may have other ideas for modeling the flow rate.
\item One idea is to use $\beta E \left(1-\frac{R}{M}\right)$.  If there are many vacancies, i.e. $R << M$, then $(1-\frac{R}{M}) \approx 1$ so $\beta E \left(1-\frac{R}{M}\right) \approx \beta E$.  But if there are few vacancies, i.e.  $R \approx M$, then $(1-\frac{R}{M}) \approx 0$ so  $\beta E \left(1-\frac{R}{M}\right)$ is very small.  
\item Here is one model
\begin{eqnarray}\label{vacancy}
    \frac{dR}{dt}& = &-\alpha \frac{R^2}{M} + \beta E \left(1-\frac{R}{M}\right)\\ \nonumber
     \frac{dE}{dt}& =&\alpha \frac{R^2}{M} - \beta E \left(1-\frac{R}{M}\right) \nonumber
\end{eqnarray}
\item Students can use a technology like {\it pplane\/} \cite{Nester} to sketch the phase portrait.  The model given by the system of equations in (\ref{vacancy}) predicts there will be 37\% in the renting group and 63\% in the evicted group.
\end{enumerate}}






\end{document}
