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\section{Proofs}
The bacterial density $\rho_B$ is the number of unattached, uninvaded bacteria per unit area, but is more helpful in terms of MOI and $b$.
\begin{equation}
\rho_B = \frac{B_u}{L^2} = \frac{(1-b) B_{\rm tot }}{H A / c} = \frac{(1-b) mc}{A}
\end{equation}
\subsection{Bacterial attachment}
The change in the fraction of host cells with attached bacteria $a$ will only depend on the primary attachment rate $\Gamma_a$ and not secondary attachment $\Gamma_b$, because for bacteria to be recruited by ruffles, a bacterium has already attached to form that ruffle. Of course, this rate of change will be limited by the bacterial density, $\rho_B$.
\begin{equation}
\dot{a} = \frac{\dot{H}_a}{H} = \frac{H - H_a}{H} \Gamma_a \rho_B = (1-a) \Gamma_a (1-b) \frac{mc}{A}
\end{equation}
It is important to note that bacterial invasion/entry is not a consideration in the above equation. Consequently, if all the bacteria attached to a host cell have invaded, we still consider that host cell to have attached bacteria, even though it has no extracellular bacteria left.
A more complicated quantity is the fraction of bacteria which are attached to host cells, which has three means of change. The first is by regular primary attachment with rate $\Gamma_a$, the second is secondary ruffle recruitment with rate $\Gamma_b$, and the third is a loss of attached bacteria as they invade with rate $\Gamma_x$.
\begin{equation}
\dot{b_a} = \frac{\dot{B_a}}{B_{\mathrm{tot}}} = \frac{H \Gamma_a \rho_B}{B_{\mathrm{tot}}} + \frac{R \Gamma_b \rho_B}{B_{\mathrm{tot}}} - \frac{B_a \Gamma_x}{B_{\mathrm{tot}}}
\end{equation}
