deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index 031bee7..a01d408 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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\dot{h}_a = \frac{\dot{H}_a}{H_{\rm tot}} = \frac{H}{H_{\rm tot}} \Gamma_a \rho_B = h \Gamma_a b m c  \end{equation}  The change in infectivity $h_v$ $h_x$  is controlled by invasion rate $\Gamma_v$, rates $\Gamma_v$ and $\Gamma_c$,  and is limited by the fraction of uninfected cells and attached bacteria per cell $\tilde{b}_a$. \begin{equation}  \dot{h}_v \dot{h}_x  = \frac{\dot{H_v}}{H_{\rm \frac{\dot{H_x}}{H_{\rm  tot}} = \frac{H_a - H_v}{H_{\rm H_x}{H_{\rm  tot}} \Gamma_v (\Gamma_v + \Gamma_c)  \tilde{b}_a = (h_a - h_v) \Gamma_v h_x) (\Gamma_v + \Gamma_c)  \frac{b_a m}{h_a} = \left(1 - \frac{h_v}{h_a}\right) \Gamma_v \frac{h_x}{h_a}\right) (\Gamma_v + \Gamma_c)  b_a m \end{equation}  Once infected by vacuolar bacteria, Similar equations can be derived for  host cells gain cytosolic bacteria at escape rate $\Gamma_c$, governed by the number of with  vacuolar bacteria per infected cell $\tilde{b}_v$. and host cells with cytosolic:  \begin{equation}  \dot{h}_v = \frac{\dot{H_v}}{H_{\rm tot}} = \frac{H_a - H_v}{H_{\rm tot}} \Gamma_v \tilde{b}_a = (h_a - h_v) \Gamma_v \frac{b_a m}{h_a} = \left(1 - \frac{h_v}{h_a}\right) \Gamma_v b_a m  \end{equation}  \begin{equation}  \dot{h}_c = \frac{\dot{H_c}}{H_{\rm tot}} = \frac{H_v \frac{H_a  - H_c}{H_{\rm tot}} \Gamma_c \tilde{b}_v \tilde{b}_a  = (h_v (h_a  - h_c) \Gamma_c \frac{b_v m}{h_v} \frac{b_a m}{h_a}  = \left(1 - \frac{h_c}{h_v}\right) \frac{h_c}{h_a}\right)  \Gamma_c b_v b_a  m \end{equation}  %Once infected by vacuolar bacteria, host cells gain cytosolic bacteria at escape rate $\Gamma_c$, governed by the number of vacuolar bacteria per infected cell $\tilde{b}_v$.  %\begin{equation}  % \dot{h}_c = \frac{\dot{H_c}}{H_{\rm tot}} = \frac{H_v - H_c}{H_{\rm tot}} \Gamma_c \tilde{b}_v = (h_v - h_c) \Gamma_c \frac{b_v m}{h_v} = \left(1 - \frac{h_c}{h_v}\right) \Gamma_c b_v m  %\end{equation}  Host cell ruffles form at rate $\Gamma_r$, depending on the number of attached bacteria.  \begin{equation} 
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\subsection{Bacteria dynamic equations}  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 ruffle recruitment with rate $\Gamma_b$, and the third is a loss of attached bacteria as they invade with limited invasion rate $\Gamma_v^*$. $\Gamma_x^*$.  \begin{equation}  \begin{align}  \dot{b}_a = \frac{\dot{B_a}}{B_{\mathrm{tot}}} &= \frac{H_{\rm tot} \Gamma_a \rho_B}{B_{\mathrm{tot}}} + \frac{R \Gamma_b \rho_B}{B_{\mathrm{tot}}} - \frac{B_a \Gamma_v^*}{B_{\mathrm{tot}}} \Gamma_x^*}{B_{\mathrm{tot}}}  \\ &= \Gamma_a b c + \tilde{r} h_r h_a  \Gamma_b b c - b_a \Gamma_v \Gamma_x  \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right) \end{align}  \end{equation}  Bacterial internalization by SCV has two means of change: increasing by Bacteria will either internalize and remain vacuolar at  rate $\Gamma_v^*$ and decreasing by or  escape early and become cytosolic at  rate $\Gamma_c$. $\Gamma_c^*$:  \begin{equation}  \dot{b}_v = \frac{\dot{B_v}}{B_{\rm tot}} = \frac{B_a}{B_{\rm tot}} \Gamma_v^*- \frac{B_v}{B_{\rm tot}} \Gamma_c  = b_a \Gamma_v \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right)- b_v \Gamma_c  \end{equation}  And finally the change in the fraction of cytosolic bacteria is simply the inverse of the second term in the above equation.  \begin{equation}  \dot{b}_c = \frac{\dot{B_c}}{B_{\rm tot}} = \frac{B_v}{B_{\rm \frac{B_a}{B_{\rm  tot}} \Gamma_c \Gamma_c^*  = b_v b_a  \Gamma_c \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right)  \end{equation}  \subsection{Ruffle dynamic equation}