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\def\Rzero{\mathcal{R}_0}
\def\Rvac{\mathcal{R}_{vac}}
\def\Rc{\mathcal{R}_c}
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\title{\blue The basic reproduction number $\Rzero$}
\subtitle{Two examples}
\author{Julien Arino}
\institution{McMaster University}
\email{arino@math.mcmaster.ca}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
%%%%%%%%%%
\begin{slide}{The basic reproduction number $\Rzero$}
\begin{itemize}
\item Average number of new infectives produced, in a totally
susceptible population, by the introduction of an infective
individual. 

\item Characterizes the capacity of a disease to ultimately invade
a population.

\item Using such a definition, difficult quantity to estimate.
\end{itemize}
\end{slide}
%%%%%%%%%%%%
\overlays{2}{
\begin{slide}{$\Rzero$ is a threshold}
\begin{itemize}
\item In terms of stability, $\Rzero$ is a threshold.
\item Suppose that there exists a disease free equilibrium (DFE),
  \emph{i.e.}, one where there are no individuals infected by the
  disease, then 
  \begin{itemize}
  \item If $\Rzero<1$, the DFE is l.a.s.
  \item If $\Rzero>1$, the DFE is unstable.
  \end{itemize}
\end{itemize}

\FromSlide{2}
\begin{itemize}
\item This gives a method for computing $\Rzero$:
\begin{itemize}
\item Find the value of the DFE.
\item Study the local stability at the DFE.
\item Compute $\Rzero$ such that the DFE loses stability as $\Rzero$
  crosses $\Rzero=1$ from left to right.
\end{itemize}
\end{itemize}
\end{slide}}

%%%%%%%%%%%%
\begin{slide}{A simple example}
Consider a population. Individuals can be either susceptible to a
disease, or infected by it (and infectives, in that they propagate
it). Infectives recover.
\begin{subeqnarray*}
\frac{dS}{dt}&=& d(1-S)+\gamma I-\beta SI \\
\frac{dI}{dt}&=& \beta SI -(\gamma+d)I 
\end{subeqnarray*}
\end{slide}

\begin{slide}{\mbox{}}
From $I'$ equation, $I=0$ is an EP, and then $S_{DFE}=1$ from $S'$ equation. 
\[
J_{DFE}=
\left[
\begin{array}{cc}
-d & \gamma-\beta \\
0 & \beta-(\gamma+d)
\end{array}
\right]
\]
Thus l.a.s. depends on $\sgn\left(\beta-(\gamma+d)\right)$. So let
\[
\Rzero=\frac{\beta}{\gamma+d}
\]
(average time spent in infective class times probability of entering
it).
\end{slide}


%%%%%%%%%%
\begin{slide}{Limitations of this approach}
\point $\Rzero$ should have some sort of biological
interpretation. Here, it is okay, but any multiple of $\Rzero$ would
be okay, as well as any monotone function.

\imply How can we be sure that we have ``the right $\Rzero$''?\\[1cm]

\point When the number of equations in the system is large, it can be
difficult to do a local stability analysis.
\end{slide}

%%%%%%%%%%%%%%
\begin{slide}{The next generation operator approach}
See, \emph{e.g.}, \cite{DiekmannHeesterbeek1999}, in a very general
setting. In the case of ODE's, a constructive approach was given by
\cite{VdDWatmough2002}. 

Consider a system 
\[
x'=f(x)
\]
Note it 
\[
x'=\mathcal{F}-\mathcal{V}
\]
where $\mathcal{F}$ is the inflow of new infectives because of the
infection process, and $\mathcal{V}$ are all the other flows.
\end{slide}

\begin{slide}{\mbox{}}
Keep only classes where individuals are infected (infective, exposed,
...). Compute $F=D\mathcal{F}$ and $V=D\mathcal{V}$, evaluate at the
DFE. Then
\[
\Rzero=\rho(FV^{-1})
\]
where $\rho$ is the spectral radius.
\end{slide}


%%%%%%%%%%%%%
\begin{slide}{Example: the basic SIS system}
Order the variables as $I,S$, and consider only $I$. Here
$\mathcal{F}=\beta SI$ and $\mathcal{V}=(\gamma+d)I$. So $F=\beta S$
and $V=\gamma+d$. Thus, at the DFE,
\[
\Rzero=\rho(FV^{-1})=FV^{-1}=\frac{\beta}{\gamma+d}
\]
\end{slide}

%%%%%%%%%%%%%%
\begin{slide}{Two more detailed examples}
\begin{itemize}
\item {\red A multicity epidemic model} A system that has high
  dimensionality: computation of $\Rzero$ is hard; however, the
  behavior is then simple. 
\item {\red A model with vaccination} A low dimension system (4
  equations), but with more complicated behavior. 
\end{itemize}
\end{slide}

%%%%%%%%%%%%%
\overlays{2}{
\begin{slide}{A multicity epidemic model}
\onlySlide*{1}{
\begin{itemize}
\item Diseases spread in space.
\end{itemize}}
\onlySlide*{2}{
\begin{itemize}
\item Diseases spread in space.
\item Spatial models exist. Most use PDE's.
\begin{itemize}
\item Assumes a relatively ``dense'' support for the disease.
\item For tractability, assumes spatial homogeneity of the disease
  characteristics.
\end{itemize}
\end{itemize}}
\end{slide}}
%%%%%%%%%%%%%%
%\overlays{7}{
%\begin{slide}{Diffusion by contact}
%\onlySlide*{1}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_1}}
%\onlySlide*{2}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_2}}
%\onlySlide*{3}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_3}}
%\onlySlide*{4}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_4}}
%\onlySlide*{5}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_5}}
%\onlySlide*{6}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_6}}
%\onlySlide*{7}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion1_7}}
%\end{slide}}
%%%%%%%%%%%%%%
%\overlays{6}{
%\begin{slide}{Diffusion by migration}{%
%\onlySlide*{1}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_1}}%
%\onlySlide*{2}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_2}}%
%\onlySlide*{3}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_3}}%
%\onlySlide*{4}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_4}}%
%\onlySlide*{5}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_5}}%
%\onlySlide*{6}{\includegraphics[angle=270,width=0.85\textwidth]{diffusion2_6}}%
%\end{slide}}

%%%%%%%%%%%%%%%
\overlays{2}{
\begin{slide}{Western Canada}
\onlySlide*{1}{
\begin{itemize}
\item BC area: 947 800 km$^2$.
\item 2001 population estimates.
\begin{itemize}
\item Vancouver 2,229,434
\item Victoria 313,530
\item Rest of BC: less than 1 million (995,377), 50\% of whom live in
  9 cities of more than 20,000 inhabitants.
\end{itemize}
\item 154 municipalities in BC, municipalities composing Greater
  Vancouver and Greater Victoria included.
\end{itemize}}
\onlySlide*{2}{
\begin{itemize}
\item BC area: {\red 947 800 km$^2$}. 
\item France (547 030 km$^2$) + Germany (357 021 km$^2$) + Netherlands
  (41 526 km$^2$) = {\red 945 577 km$^2$}.
\item 2001 population estimates. BC $\simeq$ {\blue 3 500 000}.
\item 2002 population estimates. France (59,765,983) + Germany
  (83,251,851) + Netherlands (16,067,754) = {\blue 159 085 588}.
\item {\green 154} municipalities in BC.  
\item {\green 53,323} municipalities in France + Germany + Netherlands.
\end{itemize}}
\end{slide}}
%%%%%%%%%%%%%%%
\begin{slide}{A multicity setting}
\begin{itemize}
\item Directed graph with cities as vertices, and (valued) arcs
  representing the travels between the cities.
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%%
\overlays{3}{%
\begin{slide}{The model}
%1
\onlySlide*{1}{
\begin{subeqnarray*}
\frac{dS_{ii}}{dt} &=& {\red \sum_{k=1}^n r_{ik}S_{ik} -g_iS_{ii}}
-\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p} +
d(N_i^r-S_{ii}) +\gamma I_{ii} \\
\frac{dI_{ii}}{dt} &=& {\red \sum_{k=1}^n r_{ik}I_{ik}-g_iI_{ii}}
+\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p} 
-(\gamma+d)I_{ii}
\end{subeqnarray*}
and, for $j\neq i$, 
\begin{subeqnarray*}
\frac{dS_{ij}}{dt} &=& {\red g_im_{ji}S_{ii}-r_{ij}S_{ij}}
-\sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j} 
-dS_{ij}+\gamma I_{ij} \\
\frac{dI_{ij}}{dt} &=& {\red g_i m_{ji}I_{ii}-r_{ij}I_{ij}} 
+ \sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j}
-(\gamma+d)I_{ij}
\end{subeqnarray*}}
%2
\onlySlide*{2}{
\begin{subeqnarray*}
\frac{dS_{ii}}{dt} &=& \sum_{k=1}^n r_{ik}S_{ik} -g_iS_{ii}
{\red -\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p}} +
d(N_i^r-S_{ii}) +\gamma I_{ii} \\
\frac{dI_{ii}}{dt} &=& \sum_{k=1}^n r_{ik}I_{ik}-g_iI_{ii}
{\red +\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p}} 
-(\gamma+d)I_{ii}
\end{subeqnarray*}
and, for $j\neq i$, 
\begin{subeqnarray*}
\frac{dS_{ij}}{dt} &=& g_im_{ji}S_{ii}-r_{ij}S_{ij}
{\red -\sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j}} 
-dS_{ij}+\gamma I_{ij} \\
\frac{dI_{ij}}{dt} &=& g_i m_{ji}I_{ii}-r_{ij}I_{ij} 
{\red + \sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j}}
-(\gamma+d)I_{ij}
\end{subeqnarray*}}
%3
\onlySlide*{3}{
\begin{subeqnarray*}
\frac{dS_{ii}}{dt} &=& \sum_{k=1}^n r_{ik}S_{ik} -g_iS_{ii}
-\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p}
{\red +d(N_i^r-S_{ii}) +\gamma I_{ii}} \\
\frac{dI_{ii}}{dt} &=& \sum_{k=1}^n r_{ik}I_{ik}-g_iI_{ii}
+\sum_{k=1}^n\kappa_i\beta_{iki}\frac{S_{ii}I_{ki}}{N_i^p} 
{\red -(\gamma+d)I_{ii}}
\end{subeqnarray*}
and, for $j\neq i$, 
\begin{subeqnarray*}
\frac{dS_{ij}}{dt} &=& g_im_{ji}S_{ii}-r_{ij}S_{ij}
-\sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j} 
{\red -dS_{ij}+\gamma I_{ij}} \\
\frac{dI_{ij}}{dt} &=& g_i m_{ji}I_{ii}-r_{ij}I_{ij} 
+ \sum_{k=1}^n\kappa_j\beta_{ikj}\frac{S_{ij}I_{kj}}{N^p_j}
{\red -(\gamma+d)I_{ij}}
\end{subeqnarray*}}
\end{slide}}

%%%%%%%%%%%%%%
\begin{slide}{\mbox{}}
\begin{itemize}
\item Extended to the SEIRS case without too much trouble.
\item Same type of approach applied to a multispecies spatial SEIRS
  model (simpler migration, more complicated disease dynamics,
  \emph{e.g.}, plague).
\item Applied by Wang \& Zhao to SEIRS with nonlinear migration.
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%
\begin{slide}{More complicated behavior?}
\begin{itemize}
\item Previous example: as $\Rzero$ crosses 1 from left to right, a
  nontrivial (endemic) equilibrium is born, through a pitchfork
  bifurcation.
\item Behavior is sometimes more complicated. In particular, there can
  be backward bifurcation: saddle node followed by transcritical.
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%
\overlays{2}{
\begin{slide}{An SIRS model with vaccination}
\begin{itemize}
\item An SIR model describes a disease with
\begin{itemize}
\item Susceptibles
\item Infectives
\item Recovered individuals (who have immunity from the disease).
\end{itemize}
\item As $S\to I\to R\to S$, immunity is only temporary.
\end{itemize}

\FromSlide{2}
\begin{itemize}
\item Suppose susceptibles are vaccinated
\begin{itemize}
\item Vaccination at rate $\phi$
\item Vaccine is not 100\% effective: $1-\sigma$ is the efficacy
\item Vaccine wanes at rate $\theta$
\end{itemize}
\item Fraction $\alpha$ of newborns is vaccinated.
\end{itemize}
\end{slide}}

%%%%%%%%%%%%%
\begin{slide}{Flow diagram}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{SIRV_newborns}
\end{center}
\end{figure}
\end{slide}
%%%%%%%%%%%%%%
\begin{slide}{\mbox{}}
Constant total population so normalize ($N=1$), and the model is
\begin{subeqnarray*}
\frac{dS}{dt} &=& (1-\alpha)d-dS-\beta{SI}-\phi S+\theta
V+\nu R \\
\frac{dI}{dt} &=& \beta{SI}
+\sigma\beta{VI} -(d+\gamma)I \\
\frac{dR}{dt} &=& \gamma I-(d+\nu)R \\
\frac{dV}{dt} &=& \alpha d+\phi S-(d+\theta)V
-\sigma\beta{VI}
\end{subeqnarray*}
\end{slide}

%%%%%%%%%%%%%%
\begin{slide}{Global stability}
\begin{itemize}
\item It can be shown using the compound matrix approach to global
  stability that under mild () conditions, in a backward bifurcation
  region every orbit limits to an equilibrium.
\item Thus, except for a set of measure 0 consisting of the
  separatrix, we have global stability, with solutions going to the
  stable EEP or to the DFE depending on initial conditions.
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%
\begin{slide}{Conclusion}
\myitem{1}{\includegraphics[width=.4cm]{green-bullet-on-white.ps}}
The basic reproduction $\Rzero$ is an elementary tool that can give
some information about the \emph{asymptotic} behavior of an epidemic
model.
\begin{itemize}
\item Pros:
  \begin{itemize}
    \item Gives condensed summary of the behavior of the system.
    \end{itemize}
\item Cons:
  \begin{itemize}
  \item Only provides \emph{local} information.
  \item More analysis can be required.
  \end{itemize}
\end{itemize}   
There have been uses in mathematical ecology, \emph{e.g.}, as a
condition for invasion.
\end{slide}

%%%%%%%%%%%%%%
\begin{slide}{References}
\bibliographystyle{apalike}
\bibliography{epidemio}
\end{slide}


\end{document}