deletions | additions       diff --git a/section_Proofs_subsection_Bacterial_density__.tex b/section_Proofs_subsection_Bacterial_density__.tex  index bc85796..0c07f46 100644  --- a/section_Proofs_subsection_Bacterial_density__.tex  +++ b/section_Proofs_subsection_Bacterial_density__.tex 
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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}{A} \frac{B_u}{L^2}  = \frac{(1-b) B_{\rm tot }}{H (A L^2 A  / c)} c}  = \frac{(1-b) mc}{A L^2} mc}{A}  \end{equation}  \subsection*{Rate of infectivity}  The rate of change of the number ofinfected  host cells with bacteria  (i.e. $\geq$ 1 bacteria have attached) depends on the number of remaining uninfected cells, cells without bacteria attached,  the primary attachment rate and the bacteria bacterial  density. \begin{equation*}  \dot{H}_x \dot{H}_a  = \dot{x}H \dot{a}H  = (N (H  - N_I) H_a)  \Gamma_0 \rho_B = (N (H  - N_I) H_a)  \Gamma_0 \left(\frac{bmc}{a}\right) \left(\frac{bmc}{A L^2}\right)  \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}