\documentclass[11pt,letterpaper]{article}
\usepackage{amsthm}

\usepackage[hmargin=2cm,vmargin=2.5cm]{geometry}
\newtheorem{theorem}{Theorem}
\newtheorem{col}{Corollary}
\newtheorem{lem}{Lemma}
\usepackage[utf8x]{inputenc}
\newtheorem{ass}{Assumption}
\usepackage{amsmath}
\usepackage{verbatim}
\usepackage[round]{natbib}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
  colorlinks,
  citecolor=black,
  filecolor=black,
  linkcolor=black,
  urlcolor=black
}

\bibliographystyle{plainnat}


\author{Pierre Chauss\'e}
\title{The Susceptible, Exposed, Infectious, Removed (SEIR) Model}
\begin{document}

\maketitle

\newcommand{\E}{\mathrm{E}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\Prob}{\mathrm{Pr}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Vect}{\mathrm{Vec}}
\newcommand{\Cov}{\mathrm{Cov}}
\newcommand{\conP}{\overset{p}{\to}}
\newcommand{\conD}{\overset{d}{\to}}
\newcommand\R{ \mathbb{R} }
\newcommand\N{ \mathbb{N} }
\newcommand\C{ \mathbb{C} }
\newcommand\rv{{\cal R}}
\newcommand\Q{\mathbb{Q}}
\newcommand\PR{{\cal R}}
\newcommand\T{{\cal T}}
\newcommand\Hi{{\cal H}}
\newcommand\La{{\cal L}}
\newcommand\plim{plim}
\renewcommand{\epsilon}{\varepsilon}

\abstract{This vignette presents how to use the SEIR package}
%\VignetteIndexEntry{SEIR Model}
%\VignetteKeywords{virus, spread}
%\VignettePackage{SEIR}
%\VignetteEngine{knitr::knitr}
<<echo=FALSE>>=
library(knitr)
opts_chunk$set(size='footnotesize')
@ 

\section{The SEIR Model}\label{sec:single}

This package solves the SEIR model for the spread of a virus. In the
example, the parameters of \cite{wang-all20} are used for the case of
Wuhan, China. The model is available in Matlab on Peter Forsyth
webpage
(\href{https://cs.uwaterloo.ca/~paforsyt/SEIR.html}{https://cs.uwaterloo.ca/~paforsyt/SEIR.html}). In
fact, the R package is based on the work of Peter. You are invited to
read his page for more details about the model. The system of
differencial equations is:
\begin{eqnarray*}
\dot{S}(t) &=& -\beta(t) \frac{S(t)I(t)}{N}\\
\dot{E}(t) &=& \beta(t) \frac{S(t)I(t)}{N} - \sigma E(t)\\
\dot{I}(t) &=& \sigma E(t) - \gamma I(t) +c\frac{R(t)I(t)}{N}\\
\dot{R}(t) &=& \gamma I(t)- c \frac{R(t)I(t)}{N}
\end{eqnarray*}
where $\beta(t)=\gamma RZero(t)$ and $RZero(t)$ is the number of
individuals infected by an infectious person. It is a function of time
because the transmission if function of the social distancing policy
in place.

\section{The Package}

The main function is solveSEIR. It creates an object of class ``seir''. The main arguments are (See Peter's Matlab code for the source):
\begin{itemize}
\item h: The time interval used in the Runge-Kutta method.
\item T: The number of days to solve.
\item c: Mutation parameter (0 when no mutation is present)  
\item sigma: Set to 1/5.2 for the Wuhan case.
\item gamma: Set to 1/18 for the Wuhan case.
\item y0: The initial value  $\{S(0), E(0), I(0), R(0)\}$.
\item r0: The matrix of breaks for RZero(t).  See below.
\item type: Type of RZero(t). See below.
\end{itemize}

In this document, we use the default parameter used by Peter
Forsyth. The important factor is $RZero(t)$, which is determined by
the arguments tyee an r0. The default type is ``LIN'' and the default r0 is:

<<>>=
matrix(c(0, 20, 70, 
    84, 90, 3, 2.6, 1.9, 1, 0.5), ncol = 2)
@ 

It means that the value of RZero(t) changes at t=20, 70, 84 and 90
(days). The type argument determines how it changes. By default it
looks like the following:
\begin{center}
  \begin{minipage}{.7\textwidth}
<<echo=FALSE, fig.height=4.5>>=
library(SEIR) # we first load the package
s <- solveSEIR()
plot(s, "rzero")
@     
  \end{minipage}
\end{center}
Therefore, $RZero(t)$ is a linear interpolation between the different
values. The other option is to have a constant $RZero(t)$ between
periods as in the following graph.
\begin{center}
  \begin{minipage}{.7\textwidth}
<<echo=FALSE, fig.height=4.5>>=
s <- solveSEIR( type="C")
plot(s, "rzero")
@     
\end{minipage}
\end{center}
The first $RZero(t)$ is the default option. If we run the function
with all default values we obtain (the print method summarizes the result):

<<>>=
Sol <- solveSEIR()
Sol
@ 

The object contains the following elements:

<<>>=
names(Sol)
@ 

The element y is an $nT\times4$ matrix, where $nT$ is the number of
grid points (the closest integer of T/h), with the $i^{th}$ row being
the solution $\{S(t_i), E(t_i), I(t_i), R(t_i)\}$. The element t is
the time grid and $RZero$ is an $nT$ vector with the $i^{th}$ element
being $RZero(t_i)$. The "["  method can be used to get the solution
  at different points in time:
<<>>=
Sol[c(10,50,100,150)]  
@   

It is possible to use those elements to plot the solution, but the
easiest ways is to use the plot method. The plot method gives you
options to plot any of the curve:

\begin{center}
  \begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "acase")
@     
\end{minipage}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "tcase")
@     
\end{minipage}
\end{center}

\begin{center}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "both")
@     
\end{minipage}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "rzero")
@     
\end{minipage}
\end{center}

You can also plot them one by one:

\begin{center}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "E")
@     
\end{minipage}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "I")
@     
\end{minipage}
\end{center}

As a last example, we can reproduce the second wave case produced by
Peter Forsyth on his website. Suppose RZero stays at 0.5 for 30 days
and then the government start relaxing the social distancing
restrictions. The following is what he proposes:

\begin{center}
\begin{minipage}{.7\textwidth}
<<fig.height=5>>=
zbreaks <- matrix(c(0,  20,  70,  84,  90, 120, 240, 360,
                    3, 2.6,  1.9, 1,  0.5, 0.5, 1.75, 0.5), ncol = 2)
Sol <- solveSEIR(T=450, r0=zbreaks)
plot(Sol, "rzero")
@ 
\end{minipage}
\end{center}

The new solution is:
\begin{center}
  \begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "acase")
@     
\end{minipage}
\begin{minipage}{.4\textwidth}
<<fig.height=5>>=
plot(Sol, "both")
@     
\end{minipage}
\end{center}






\bibliography{seir}
\end{document}