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\section{}\label{SI2}

{\small{\bf CD8$^+$ T cell depletion model parameters.} Parameters, SSR, and AIC$_\mathrm{c}$ obtained from fitting the CD8$^+$ T cell model (Equations~\eqref{eqn:tc_T}--\eqref{eqn:tc_EM}) to viral titers and CD8$^+$ T cells from mice infected with $75\ \mathrm{TCID}_{50}$ PR8 and with CD8$^+$ T cells depleted at -2, 0, 3, and 7 d pi. The total number of CD8$^+$ T cells is $\hat{E}= E + E_M + \hat{E_0}$, where $\hat{E_0}=5.6e3$ cells, and all other parameters are those in Table~\ref{tab:tab1}. The best model is bolded.}

\begin{table}[h]
\label{tab:tabS1}
\small
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c||c|c|}
\hline 
$T_0$ & $p$ & $\delta$ & $c$ & $\beta$ & $\delta_E$&  $\xi$ & $\eta$& $d_E$ & $K_\delta$ & $\tau_E$ &\textbf{SSR} &\textbf{AIC$_\mathrm{c}$}\\ \hline\hline
3.17e6	&&&&&&&&&&&										342.34 &	60.86\\ \hline
&	0.39		&&&&&&&&&&								320.36 &	60.19\\ \hline
&&		1.09	&&&&&&&&&								177.95 &	54.31\\ \hline
&&&			34.95	&&&&&&&&							295.88&	59.40\\ \hline
&&&&				1.29e-4	&&&&&&&						62.93	&43.92\\ \hline
&&&&&					2.63	&&&&&&					52.72&	42.15\\ \hline
&&&&&&						3.81e4 &&&&&					63.13&	43.95\\ \hline
&&&&&&&							2.81e-7	&&&&			52.24&	42.06\\ \hline
&&&&&&&&								0.84	&&&		53.51 &	42.30\\ \hline
&&&&&&&&&									3.26e-8	&&	346.68 &	60.98\\ \hline
&&&&&&&&&&										3.20 &	60.74 &	43.57\\ \hline
3.57e6 &&&&&&&							9.91e-7 &&&&				5.39	&22.57\\ \hline
&&&&				2.24e-5 &	6.04 &&&&&&						18.40&	34.84\\ \hline
&&	0.45	&&&&&					3.58e-7	&&&&			20.11&	35.72\\ \hline
&&	0.42	&&&			5.19	&	&	&	&	&	&	20.53&	35.93\\ \hline
	
	&	&	&	&	&	&4.29e3	&4.14e-7	&	&	&	&25.72&	38.19\\ \hline
7.97e6	&	&	&	&	&4.48	&	&	&	&	&	&48.50&	44.53\\ \hline
	&	&0.22	&	&	&2.63	&	&	&	&	&	&52.54&	45.33\\ \hline
1.00e7	&	&	&	&	&	&3.28e4	&	&	&	&	&61.30&	46.87\\ \hline
\bf 3.98e6	&	&	&	&	&	&\bf1.33e4	&\bf1.01e-6	&	&	&	&\bf2.44&	\bf18.92\\ \hline
	&	&	&27.88	&	&	&6.20e3	&4.21e-7	&	&	&	&7.33&	29.92\\ \hline
	&0.51	&	&	&	&	&1.04e4	&4.15e-7	&	&	&	&15.22&	37.23\\ \hline
	&0.40	&0.14	&	&	&5.55	&	&	&	&	&	&16.46&	38.01\\ \hline
	&	&0.34	&	&	&	&1.85e3	&5.00e-7	&	&	&	&17.39&	38.56\\ \hline
	&	&	&	&2.21e-05&	7.98&	1.95e4	&	&	&	&	&17.45&	38.59\\ \hline
	&0.47	&0.28	&	&	&	&	&3.55e-7	&	&	&	&18.20&	39.02\\ \hline
	&	&0.23	&	&2.24e-5	&6.08	&	&	&	&	&	&18.38&	39.11\\ \hline
	&0.44	&0.26	&	&	&	&8.10e4	&	&	&	&	&44.31&	47.91\\ \hline
	&0.48	&0.37	&	&	&	&4.57e3	&5.18e-7	&	&	&	&2.40&	24.75\\ \hline
	&	&0.44	&	&2.54e-5	&	&5.92e3	&5.44e-7	&	&	&	&4.35&	30.70\\ \hline
3.62e6	&	&0.24	&	&	&1.63	&	&1.01e-6	&	&	&	&4.80&	31.69\\ \hline
\end{tabular}
\end{center}
\end{table}




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