Abstract

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Text and results for this section, as per the individual journal’s instructions for authors. (Koonin 1996, Orengo 1999, Kharitonov 2014, Zvaifler 1999, Jones 1996, Schnepf 1993, Ponder 1998, Smith 1996, Margulis 1970, Hunninghake 1995, authors 1999, Kohavi 1995, Smith 2014)

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Text for this sub-sub-heading …Text for this sub-sub-sub-heading …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 \(\alpha=0\))

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

\begin{aligned} E\bigl{[}Z_{1}(vT_{x})\bigr{]} & = & \frac{\mu}{r}\log x\int_{0}^{v\wedge 1}x^{1-u}x^{({\lambda_{1}}/{r})(v-u)}\,du \\ & = & \frac{\mu}{r}x^{1-{\lambda_{1}}/{\lambda_{0}}v}\log x\int_{0}^{v\wedge 1}x^{-u(1+{\lambda_{1}}/{r})}\,du\nonumber \\ & = & \frac{\mu}{\lambda_{1}-\lambda_{0}}x^{1+{\lambda_{1}}/{r}v}\biggl{(}1-\exp\biggl{[}-(v\wedge 1)\biggl{(}1+\frac{\lambda_{1}}{r}\biggr{)}\log x\biggr{]}\biggr{)}.\nonumber \\ \end{aligned}Thus we observe that this expected value is finite for all \(v>0\) (also see (Koonin 1996, Zvaifler 1999, Jones 1996, Margulis 1970)).

\begin{aligned}
E\bigl{[}Z_{1}(vT_{x})\bigr{]}=E\biggl{[}\mu T_{x}\int_{0}^{v\wedge 1}Z_{0}(uT_{x})\exp\bigl{(}\lambda_{1}T_{x}(v-u)\bigr{)}\,du\biggr{]}.\\
\end{aligned}