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\section{}\label{SI1}

{\small{\bf Statistical comparison of alternate models.} Comparison of the Akaike Information Criteria (AIC) of the CD8$^+$ T cell model in Equations~\eqref{eqn:tc_T}--\eqref{eqn:tc_EM}, the alternate model in Appendix~\ref{A1}-Equations~\eqref{eqn:ta_T}--\eqref{eqn:ta_EM}, and the Baral model in  Appendix~\ref{A1}-Equations~\eqref{eqn:ta2_I2}--\eqref{eqn:ta2_E}. The fit of these models is shown in Appendix~\ref{A1} Figure~\ref{fig:A1}.}

\begin{table}[h]
\small
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline 
&\multicolumn{2}{l|}{\textbf{Total CD8}} &\multicolumn{2}{l|}{ \textbf{Lung CD8} } \\ \hline 
&\textbf{All data} & \textbf{Excl. d3-5} & \textbf{All data} & \textbf{Excl. d3-5}  \\ \hline 
CD8$^+$ T cell Model (Eqns~\eqref{eqn:tc_T}--\eqref{eqn:tc_EM}) & 32.1 &26.0 & 77.9 &  63.2 \\ \hline
Alternate Model (Appendix~\ref{A1}-Eqns~\eqref{eqn:ta_T}--\eqref{eqn:ta_EM}) & 69.5 &64.9 & 84.7 & 72.5 \\ \hline
Baral Model (Appendix~\ref{A1}-Eqns~\eqref{eqn:ta2_I2}--\eqref{eqn:ta2_E}) & 151.9 & 135.2 & 118.2 & 103.2 \\ \hline
\end{tabular}
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\end{table}



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