%Author: Erin C. McKiernan
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\title{Mitigating effects of influenza vaccination given constraints
  in supply and administration capacity}

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\author{Erin C. McKiernan$^{1*}$, Maytee Cruz-Aponte$^{3,4}$, Marco Arieli Herrera-Valdez$^{1,2}$}
 
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\institute{$^1$ Institute of Interdisciplinary Research, University
  of Puerto Rico at Cayey \\
  $^2$ Department of Mathematics and Physics, University of Puerto Rico at Cayey\\
  $^3$ Mathematical, Computational, and Modeling Sciences Center, Arizona State University \\
  $^4$ School of Human Evolution and Social Change, Arizona State University\\
} \email{$*$erin.mckiernan@upr.edu }

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%\conference{FORO RCM 2012}

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\section{Overview}
Influenza viruses are a major cause of morbidity and mortality
worldwide. Vaccination is a powerful tool for preventing or mitigating
influenza outbreaks. Yet, vaccine supplies and daily administration
capacities are limited, even in developed countries. Understanding how
such constraints can alter the mitigating effects of vaccination is a
crucial part of influenza preparedness plans. We present a
mathematical model that explicitly takes into account vaccine supply
and the number of vaccines administered per day and places
data-informed limits on these parameters. We use the model to test
several vaccination scenarios. \textbf{The model can be used by
government and medical officials to create customized pandemic
preparedness plans based on the supply and administration constraints
of specific communities.}

\section{Epidemiological Model}
We developed a SIR-like epidemiological model to study the spread of
influenza \citep{KermackMcKendrick1927, Coburn2009}.

\vspace{1cm}


% -------------------------------------------
% Figure 1
% -------------------------------------------
\begin{figure}
\begin{minipage}{0.6\columnwidth}
\centering
  \includegraphics[width=\textwidth]{Figures/Fig1_modelschematic.png}
\end{minipage}
\hspace{2.5cm}
\begin{minipage}{0.4\columnwidth}
S = Susceptible \\ 
V$_S$ = Vaccinated susceptibles \\
I$_C$ = Infected confirmed \\
I$_U$ = Infected unconfirmed \\
R$_C$ = Recovered confirmed \\
R$_U$ = Recovered unconfirmed \\
V$_C$ = Vaccinated confirmed \\
V*$_U$ = Vaccinated unconfirmed \\
D = Deceased due to infection\\
\\
\\
*still infected/infectious
\end{minipage}
\end{figure}


The dynamics of the model are defined by: \\
\begin{minipage}{0.6\columnwidth}
\begin{flushleft}
 \begin{eqnarray*}
   &\dot{S}&= - \lambda(S,I,t) - v_S (t) \\ 
   &\dot{I}_C& = p \lambda (S,I,t) - (c+\delta) I_C \\
   &\dot{I}_U&= (1-p) \lambda (S,I,t)- (c+\delta)I_U - v_U(t) \\
   &\dot{V}_U&= {v_U}(t) - (c+\delta)V_U 
 \end{eqnarray*}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.4\columnwidth}
\begin{flushleft}
 \begin{eqnarray*}
   &\dot{R_C}&= cI_C- v_{R} (t) \\
   &\dot{R_U}&= c(I_U + V_U) \\
   &\dot{V}_{SC}&= v_S(t)  + v_R(t) \\
   &\dot{D}&= \delta (I_C + I_U+V_U) 
 \end{eqnarray*}
\end{flushleft}
\end{minipage}

\vspace{2cm}

New infections per unit time are:
\begin{equation*}
\lambda (S,{I},t) =b\frac{S }{N}\left[ {I_C} + \alpha { 
  \left(I_U+V_U \right)} \right] + \phi(t), \label{eq:forceinfection}
\end{equation*}
where $p$ is probability of being infected and confirmed, $c$ is rate
of recovery, $\delta$ is death rate due to infection, $b$ is
probability of infection per contact, $\alpha$ is infectiousness of
unconfirmed cases, and $\phi(t)$ is a small pulse used to initiate
an outbreak.

\columnbreak

\section{Modeling Vaccination}
Modling vaccination according to \textit{proportion} of
population may over- or underestimate number of people that can
realistically be vaccinated per day. We propose a
\textit{non-proportional} scheme with a daily vaccine limit.

% -------------------------------------------
% Figure 2
% -------------------------------------------
\begin{figure}
\begin{minipage}{0.4\columnwidth}
\flushright
  \includegraphics[width=\textwidth]{Figures/Fig2_decay_propvsnonprop.png}
\end{minipage}
\hspace{1.7cm}
\begin{minipage}{0.45\columnwidth}
  Comparison of decay in vaccinable population over time for
  proportional (dashed line) and non-proportional (solid line) models
  of vaccination.  Proportional decay is given by $x(t)=x_0e^{-kt}$,
  time constant of decay is $k$=0.1.  Non-proportional decay is given
  by $x(t) = x_0 - \bar{v}_D t$, where $\bar{v}_D$=$kx_0$ and
  represents max total number of vaccines per day.
\end{minipage}
\end{figure}

We calculate weights of each subpopulation (epidemiological class):\\
\begin{eqnarray*}
  &&  w_S(t) = \frac{S(t)}{M(t)},
  \quad {w_U(t)}= \frac{{I_U(t)}}{M(t)},
  \quad w_R(t) =\frac{{R_C(t)}}{M(t)}, \label{eq:homweights}
\end{eqnarray*}

\vspace{1cm}

where $M(t) = S(t) + {I_U (t) + R_C(t)}$ is total eligible for
vaccination at time $t$.\\

\vspace{0.5cm}

Max number of vaccines per day each subpopulation can receive is then:\\
\begin{equation*}
\label{vDbarnus}
  \bar{v}_S(t)= \bar{v}_D w_S(t), 
\quad  {\bar{v}_U(t)} = \bar{v}_D {w_U(t)}, 
\quad   \bar{v}_R(t)= \bar{v}_D w_R(t). 
\end{equation*}

\vspace{0.5cm}

For more details on the model, see \cite{cruz2011mitigating}

\section{Results}

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% Figure 3
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\begin{figure}
 \begin{minipage}{0.65\columnwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig3_campaignstarts.png} 
 \label{fig:vacc30d_vday1M}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.3\columnwidth}
  Comparison of proportional (dashed) and non-proportional (solid)
  models for different vaccination campaign starts. Initial outbreak
  occurs on day 10 ($t_0$=10; solid vertical line). Vaccination
  campaign initiated on days 20 \textbf{(a, b)}, 50 \textbf{(c, d)},
  or 80 \textbf{(e, f)}, and lasts for 28 days, at rate of 1\% of
  population per day (proportional; $k$=0.01), or maximum of $10^6$
  vaccines per day (non-proportional, $\bar{v}_D=10^6$). 
\end{minipage}
\end{figure}

\textbf{$\blacksquare$ Proportional model predicts epidemics that peak
  sooner with larger peak, but are often shorter, than
  non-proportional model epidemics.}

\columnbreak

\section{Results}

% -------------------------------------------
% Figure 4
% -------------------------------------------
\begin{figure}
 \begin{minipage}{0.55\columnwidth}
\centering
  \includegraphics[width=\textwidth]{Figures/Fig4_epidemicmeasures.png} 
 \label{fig:outbreak_measures}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.4\columnwidth}
  Quantification of final size, peak size, peak time, and epidemic
  duration for proportional (open circles) and non-proportional
  (filled circles) models. Measures plotted as function of difference
  between vaccination start time ($t_a$) and initial outbreak ($t_0$;
  solid gray line). Vaccination scenarios were: (1) 60 day campaign
  with k=0.001 (proportional) or $\bar{\nu}_D=10^5$ (non-proportional)
  \textbf{(a1-a4)}, (2) 30 day campaign with k=0.01 or
  $\bar{\nu}_D=10^6$ \textbf{(b1-b4)}, and (3) {5 day campaign} with
  k=0.1 or $\bar{\nu}_D=10^7$ \textbf{(c1-c4)}.
\end{minipage}
 \end{figure}

 \textbf{$\blacksquare$ In moderate/aggressive regime, final and
   peak sizes are smaller, while peak times and epidemic durations are
   larger, in non-prop. vs. proportional model.}

% -------------------------------------------
% Figure 5
% -------------------------------------------
 \begin{figure}
 \begin{minipage}{0.7\columnwidth}
\centering
  \includegraphics[width=\textwidth]{Figures/Fig5_vaccinecoverage.png}
\end{minipage}
\hspace{0.7cm}
\begin{minipage}{0.25\columnwidth}
  Effects of vaccine coverage, start times, and unconfirmed cases in
  non-prop. model ($\bar{v}_D$=$10^6$).  Outbreak at $t_0$=10.
  Vaccination starts $t_a$=20 \textbf{(a)}-\textbf{(d)}, 50
  \textbf{(e)}-\textbf{(h)}, or 80 \textbf{(i)}-\textbf{(l)}.
  Confirmed case probablity, $p$, 0.20 (thick gray) or 0.65 (thin
  black).
\end{minipage}
\end{figure}

\textbf{$\blacksquare$ 20\% coverage does not mitigate outbreak.
  40-60\% coverage effective in mitigating outbreak, the earlier the
  better. Beyond 60\% coverage, no additional benefits, vaccines are
  wasted.  Confirmed case probablity has little effect.}


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