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% noteref={n1}, % id's of article notes, if any
email={jane.e.doe@cambridge.co.uk} % email address
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\author[
addressref={aff1,aff2},
email={john.RS.Smith@cambridge.co.uk}
]{\inits{J.R.S.}\fnm{John R.S.} \snm{Smith}}
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In this section we examine the growth rate of the mean of $Z_0$, $Z_1$ and $Z_2$. In
addition, we examine a common modeling assumption and note the
importance of considering the tails of the extinction time $T_x$ in
studies of escape dynamics.
We will first consider the expected resistant population at $vT_x$ for
some $v>0$, (and temporarily assume $\alpha=0$)
%
\[
E \bigl[Z_1(vT_x) \bigr]=
\int_0^{v\wedge
1}Z_0(uT_x)
\exp (\lambda_1)\,du .
\]
%
If we assume that sensitive cells follow a deterministic decay
$Z_0(t)=xe^{\lambda_0 t}$ and approximate their extinction time as
$T_x\approx-\frac{1}{\lambda_0}\log x$, then we can heuristically
estimate the expected value as
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\begin{equation}\label{eqexpmuts}
\begin{aligned}[b]
& E\bigl[Z_1(vT_x)\bigr]\\
&\quad = \frac{\mu}{r}\log x
\int_0^{v\wedge1}x^{1-u}x^{({\lambda_1}/{r})(v-u)}\,du .
\end{aligned}
\end{equation}
%
Thus we observe that this expected value is finite for all $v>0$ (also see \cite{koon,xjon,marg,schn,koha,issnic}).
\section*{Appendix}
Text for this section\ldots
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\section*{Acknowledgements}%% if any
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\section*{Competing interests}
The authors declare that they have no competing interests.
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\section*{Authors' contributions}
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\section*{Authors' information}%% if any
Text for this section\ldots
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