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\section{Theoretical Framework}  The wave speed of plasmid spread in a bacterial population is a complex and non-lineal process and as such is hard to grasp some intuitive idea about the main factors controlling the global behavior of the system as a whole. In order to understand the processes some simplifications are required to be made. Let us assume a simple and yet idealized bacterial population of $N$ individuals with a growth rate $\mu = 0$. The population will be sessile, which implies that the topology of network will be static and individuals cells will be distributed side by side conforming an interaction structure which could be abstracted by a linear graph as can be seen in Figure ~\ref{fig:forwarding-delay}.   \begin{figure*}  \centering  \includegraphics[scale=0.6]{figures/paper-20(LinearGraph-0)v2.pdf}  \caption[Forwarding delay]{\label{fig:forwarding-delay} {\bf The forwarding delay scheme}. The $B^r$ and $B^t$ are respectively the recipient and transconjugant bacterial nodes. The figure shows the meaning of forwarding delay which is the time elapsed since a bacteria $B^r_i$ is infected becoming a transconjugant $B^t_i$ and infect the next recipient cell $B^r_{i+1}$ in the linear graph shown.}  \end{figure*}  The Figure ~\ref{fig:interaction-graph} shows the temporal evolution of the linear graph which denotes the bacterial population being infected. In this graph every donor cell, depicted by $B^t_i$ only have a reachable neighbor which can be infected in a conjugative event. In other words the plasmid bearing cell $B^t_i$ can only interact and conjugate with the bacterial agent $B^r_{i+1}$. Thus, in the simplest case where the forwarding delay is fixed, the infection speed can be approximated by a straight line in a form of $y = \alpha + \beta x$ being $\alpha$ the intercept and $\beta$ the slope. The value of slope for the most general case is given by the expression shown in Equation ~\eqref{eq:slope}  \  \begin{equation}  \label{eq:slope}  \beta = \frac{|B^t|t_i - |B^t|t_{i-1}}{\Delta_t}, \forall i > 0  \end{equation}  \  where $|B^t|t_i$ and $|B^t|t_{i-1}$ are respectively the counting of $B^t$ cells at $t_i$ and $t_{i-1}$ and $\Delta_t = t_i - t_{i-1}$. The The intercept $\alpha$ is zero since at $t_0$ the number of $|B^t|$ cells are also zero. The choice of $|B^t|$ as the indicator of network flooding is just for the sake of generality but the same principles will also hold if instead of it the value of $T/(T+R)$ were used.  \begin{figure*}  \centering  \includegraphics[scale=0.61]{figures/paper-20(LinearGraph-1)v2.pdf}  \caption[Interaction graph]{\label{fig:interaction-graph} {\bf The simplified interaction graph}. This figure shows from left to right the snapshots for the temporal evolution of the linear graph representing our bacterial population. The cell types are $B^r$, $B^d$ and $B^t$ standing respectively for recipients, donors and transconjugants.}   \end{figure*}  The Equation ~\eqref{eq:slope}, in the case of unitary increments in the infected nodes which is exactly what is happening in the linear graph show in Figure ~\ref{fig:interaction-graph} , can be simplified to $\beta = 1/\Delta_t$. Withal it is easy to realize that the slope have the following $\lim_{\Delta_t \to \infty} 1/\Delta_t = 0$ $\lim_{\Delta_t \to 0} 1/\Delta_t = \infty$ lower and upper limits.   \begin{figure*}  \centering  \includegraphics[scale=0.8]{figures/forwarding-delay.pdf}  \caption[Forwarding delay effect]{\label{fig:forwarding-delay} {\bf The effect of forwarding delay in infection speed.} This graphic shows the effect of increasing delays in the plasmid forwarding network. As can be observed the slope is inversely proportional to the forwarding delay. }   \end{figure*}