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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 Fig.  ~\ref{fig:forwarding-delay}. \begin{figure*} \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*} \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}  \