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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 21
    (Friday)}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical
    Modeling $|$ Spring 2022 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

    % Allman/Rhodes Exercise 7.1.6 (modified)
\item Suppose a disease is modeled by the SIR model, but many people
  who get the disease self-quarantine. The net result is that a
  fraction $q\geq 0$ of the infectives are prevented from having contacts
  with the susceptibles.
  \begin{enumerate}
  \item Modify the equations of the standard SIR model to reflect this.
  \item Quarantining can be viewed as a way of modifying the
      transmission coefficient. Suppose an SIR model has transmission
      coefficient $\beta$, and a fraction $q$ of the infectives are
      successfully quarantined. Then the model with quarantining is
      identical to a standard SIR model with some other transmission
      coefficient $\beta'$, the \emph{effective transmission
        coefficient}. Give a formula for $\beta'$ in terms of
      $\beta$ and $q$.
    \item Repeat the previous parts using the SIR model with births
      and deaths. Use $\mu$ for the constant birth rate and uniform
      death rate.
    \item Use MATLAB (e.g., Program 2.1 of Keeling/Rohani) to
      investigate the behavior of your quarantine models for some fixed
      $\beta$, and $\gamma$, and vary $q$ from $0$ to $1$. Explain the
      qualitative behavior you see.
    \item Pick two common diseases that might be well-suited for the
      SIR model, with quite different values of $\mathcal{R}_0$. For each
      one, experiment with various values of $q$ (at least 5) and
      report back how much of a difference this makes in the final
      epidemic size (classic model), and in the endemic equilibrium
      (model with births and deaths).
    \item In the previous part, can you find a value of $q$ that
      prevents an epidemic from occurring, regardless of $I_0$? Does this depend on whether there are births and deaths? Justify your answer mathematically. \\
  \end{enumerate}

  
\item Since vampires have circulated endemically in the human population for thousands of years, they are well-suited for the following \emph{SVA} model:
  \begin{align*}
    \frac{dS}{dt}&=\mu-\beta SV-\delta_SS \\
    \frac{dV}{dt}&=\beta SV-\delta_V V \\
    \frac{dA}{dt}&=\alpha\delta_VV-\delta_AA,
  \end{align*}
  where $\alpha\in[0,1]$. The ``A'' class represents the \emph{abstinent} vampires that are not sucking blood, presumably because they are no longer willing or able to hunt. Of course, this will eventually lead to their demise. 
  \begin{enumerate}
  \item Carefully justify this model. Make sure you explain what the
    rate constants represent, and why the terms are what they
    are. Include a diagram showing each compartment and the
    transitions.
  \item Find the basic reproductive number $\mathcal{R}_0$, for which a vampire epidemic will spread if $\mathcal{R}_0>1$ but die out if $\mathcal{R}_0<1$. 
  \item What other disease(s) might be well-modeled by these
    equations? Fully justify your answer.
  \end{enumerate}

  
\item Whereas vampires circulate among humans almost unnoticed in an endemic manner, other members of the living dead, such as zombies, occasionally pop up in brief but virulent outbreaks. The possibility of deceased individuals resurrecting back into zombies requires a different approach, such as the following \emph{SZR} model:
  \begin{align*}
    \frac{dS}{dt}&=\mu-\beta SZ-\delta S \\
    \frac{dZ}{dt}&=\beta SZ+\zeta R-\alpha SZ \\
    \frac{dR}{dt}&=\delta S+\alpha SZ-\zeta R
  \end{align*}

    \begin{enumerate}
  \item Carefully justify this model. Make sure you explain what the
    rate constants represent, and why the terms are what they are. Include a diagram showing each compartment and the transitions.
  \item Compute the Jacobian at the zombie-free equilibrium $(S,Z,R)=(1,0,0)$ and determine its stability.
  \item Compute the Jacobian at the doomsday equilibrium $(S,Z,R)=(0,Z^*,0)$ and determine its stability. 
  \item Discuss the implications of the previous parts of this problem
    on humanity. Is there any way to save us? 
    \end{enumerate}
    

\end{enumerate}

\end{document}
