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$\quad b_x \equiv $ fraction of bacteria that have invaded a host cell $ = \frac{B_x}{B_{\rm tot}}$  $r $\tilde{r}  \equiv $ ratio of ruffles to host cells with ruffles $ = \frac{R}{H_r}$ $\tilde{b} \equiv $ ratio of attached bacteria to host cells with bacteria $ = \frac{B_a}{H_a}$  \begin{subsection}  To understand how these quantities evolve in time, we must define the rates which describe the various stochastic processes of bacterial invasion. A swimming bacterium can either attach to a host cell normally, or be recruited to a membrane ruffle.  $\Gamma_a \equiv $ primary attachment rate per bacterial density  $\Gamma_b \equiv $ secondary attachment by ruffle recruitment rate per bacterial density  Once attached, a bacterium can cause ruffling or invade the host cell.  $\Gamma_r \equiv $ ruffle formation rate per attached bacteria  $\Gamma_x \equiv $ invasion rate per attached bacteria  There is evidence that suggests a physical limit to the number of ruffles per cell, and the number of invaded bacteria per cell, so we define the following effective rates.  $\Gamma_r^{*} \equiv $ effective (limited) ruffle formation rate per attached bacteria $ = \Gamma_r (1 - \frac{\tilde{r}}{\tilde{r_{\rm max }}})$  $\Gamma_x \equiv $ invasion rate per attached bacteria