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\title{{\huge Supplementary material}\\
Age-specific mixing generates transient outbreak risk following critical level vaccination}
\author[1]{Samit Bhattacharyya\thanks{Corresponding author: Tel: +91 120 3819100 (Extn. 136), Email: szb16@psu.edu, samit.b@snu.edu.in}  }
\author[2]{Matthew J. Ferrari\thanks{mjf283@psu.edu}}
\affil[1]{Dept. of Mathematics, School of Natural Sciences, Shiv Nadar University, India}
\affil[1,2]{Center for Infectious Disease Dynamics, Pennsylvania State University, USA}
\date{}

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\section{Computation of effective reproductive rate $R_E$}
We use next generation method to calculate the effective reproductive rate $R_E$. 
We compute gains and losses from all age-specific (9 age compartments) compartments $E$ and $I$. The gain to $E$ compartments is mainly due to new infection ($\beta SI$), while gain to $I$ compartments is zero. The losses from $E$ compartments are due to death ($\mu$) and latency ($\sigma$), and losses from $I$ compartments also consists recovery ($\gamma$) along with these. The gain and loss term also contains maturation rate $a$. We should note that all parameters such as $\beta$, $\mu$, $\sigma$, $\gamma$ are vectors of length 9. Following we have computation of gain matrix $F$ and loss matrix $V$ at time point $t$:\\
Suppose susceptible density at time $t$ is given by $S_t = (s^1, s^2, ..., s^9)^T$, and infected $I_t = (i^1, i^2, ..., i^9)^T$. Then gain matrix $F_{18\times 18}$ is given by 
\[F (1:2:18, 2:2:18) = \beta (1:9,1:9). (S_t, S_t, ..., S_t).(I_t, I_t, ..., I_t),\]   where both $(S_t, S_t, ..., S_t)$, and $(I_t, I_t, ..., I_t)$ are $9\times 9$ matrices. `.' denotes the element-wise multiplication of matrices. \\

\noindent The elements of loss matrix $V_{18\times 18}$ is given as follows:\\
We introduce $\Gamma (1:9) = (\gamma, \gamma, ... \gamma)$, $\Sigma (1:9) = (\sigma, \sigma, ...\sigma)$;\\
$V(2:2:18,2:2:18)= diag(-(\mu+\Gamma+a))$,\\
$V(1:2:18,1:2:18)=diag(-(\mu+\Sigma+a))$,\\
$V(2:(2*(18+1)):18\times 18) = \sigma$,\\
$V(3:(2*(18+1)):(18-3)*18) = a(1:8)$,\\
$V((18+4):(2*(18+1)):(18-2)*18) = a(1:8)$.\\

\noindent The largest eigenvalue of the product matrix $FV^{-1}$ is the required effective reproductive rate $R_E$ at time $t$. 


\newpage


\begin{figure*}[ht]
\begin{center}
\noindent\makebox[\textwidth]{\includegraphics[width=1.2\paperwidth]{CmbndOtherMat.jpg}}
\caption{\textbf{Ref.: Figure 1 in main text: }Illustration of transient risk for different contact matrix from Fumanelli (UK) \cite{Fum2012}, and other POLYMOD matrices such as Germany, Great Britain and Italy \cite{Mossong08}. Age-class in the right panel indicates the corresponding age group in the contact matrices. Descriptions and interpretations are same as in Figure 1 in main text.}
\label{RiskAreaDis}
\end{center}
\end{figure*}

















\begin{figure*}[ht]
\begin{center}
\noindent\makebox[\textwidth]{\includegraphics[width=0.8\paperwidth]{cmbndAreaDistRisk.eps}}
\caption{\textbf{Ref.: Figure 2 in main text: Plot of actual Risk data for different values of intermediate vaccination coverage:} (a) Area and (b) duration, and (c) epidemic size while $R_E$ remains above 1 for different values of intermediate vaccination coverage. }
\label{RiskAreaDis}
\end{center}
\end{figure*}


\begin{figure*}[ht]
\begin{center}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{cmbndErrAreaYrsHBLB.eps}}
\caption{\textbf{Area and duration of transient risk in (a) growing and (b) declining population.} The vertical axis of the right figure in the panel (b) shows there are data points with more than 50 years of duration. In this analysis, we only observe the dynamics till 50 years. Compare Figure 2 in main text.}
\label{SingleCIP}
\end{center}
\end{figure*}

\begin{figure*}[ht]
\begin{center}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{cmbndCurveTypeAllVac.eps}}
\caption{\textbf{Characterization types of risk for all different values of intermediate vaccination coverage:} (a) 75\%, (b)80\%, (c) 85\%, and (d) 90\%.  Figure description is same as Figure 3 in the main text. For details discussion, see the main text.}
\label{SingleCIP}
\end{center}
\end{figure*}



\bibliographystyle{vancouver}
\begin{thebibliography}{1}

\bibitem{Fum2012}
Fumanelli, L., Ajelli, M., Manfredi, P., Vespignani, A. and Merler, S., 2012. {\em Inferring the structure of social contacts from demographic data in the analysis of infectious diseases spread.} PLoS Comput Biol, 8(9), p.e1002673.

\bibitem{Mossong08}
Mossong, J., Hens, N., Jit, M., Beutels, P., et. al. 2008 Social contacts and mixing patterns relevant to the spread of infectious diseases. {\em PLoS medicine}, 5(3), e74.

\end{thebibliography}

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