Proofs
Bacterial density
The bacterial density \(\rho_{B}\) is the number of bacteria (available for attachment) 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}}{HA/c}=\frac{(1-b)mc}{A}\\
\end{equation}
Rate of infectivity
The rate of change of the number of host cells with bacteria (i.e. \(\geq\) 1 bacteria have attached) depends on the number of remaining cells without bacteria attached, the primary attachment rate and the bacterial density.
\begin{equation}
\dot{H}_{a}=\dot{a}H=(H-H_{a})\Gamma_{0}\rho_{B}=(H-H_{a})\Gamma_{0}\left(\frac{bmc}{AL^{2}}\right)\nonumber \\
\end{equation}
In our model, we assume limited invasion (i.e. we impose a maximum number of internalized bacteria per cell). The rate of change of infected cells
\begin{equation}
\dot{H}_{x}=\dot{x}H=(H-H_{x})\\
\end{equation}
\begin{equation}
\dot{x}=(1-x)\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}
Rate of ruffle formation
The rate of change of the number of total ruffles over all cells depends on the number of total host cells, the primary attachment rate, the bacterial density and the probability of ruffle formation.
\begin{equation}
\dot{N}_{R}=N\Gamma_{0}\rho_{B}f=Nf\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}
\begin{equation}
\dot{\tilde{r}}=\frac{\dot{N}_{R}}{N}=f\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}
Rate of ruffling
The rate of change of the number or ruffling cells (i.e. \(\geq\) 1 ruffle) is given by the number of cells without ruffles, the primary attachment rate, the bacterial density and the probability of ruffle formation.
\begin{equation}
\dot{N}_{r}=\dot{r}N=(N-N_{r})\Gamma_{0}\rho_{B}f=(N-N_{r})f\Gamma_{0}\left(\frac{bmc}{a}\right)\nonumber \\
\end{equation}
\begin{equation}
\dot{r}=(1-r)f\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}
Rate of bacteria capture
Bacteria are captured through one of two ways: primary attachment to host cells, or secondary recruitment to host cell ruffles.
\begin{aligned}
\dot{N}_{B}= & \ \dot{b}N_{B}(0)=\dot{b}mN\nonumber \\
= & -N\Gamma_{0}\rho_{B}-N_{R}\Gamma_{1}\rho_{B}=-N\rho_{B}(\Gamma_{0}+\tilde{r}\Gamma_{1})\nonumber \\
\end{aligned}
\begin{equation}
\dot{b}=-\frac{bc}{a}(\Gamma_{0}+\tilde{r}\Gamma_{1})\\
\end{equation}
\begin{equation}
\dot{\tilde{b}}=\frac{-\dot{N}_{B}}{N}=-\dot{b}m\\
\end{equation}
