  We propose a new model that describes the dynamics of epidemic spreading on
connected graphs. Our model consists in a PDE-ODE system where at each vertex
of the graph we have a standard SIR model and connexions between vertices are
given by heat equations on the edges supplemented with Robin like boundary
conditions at the vertices modeling exchanges between incident edges and the
associated vertex. We describe the main properties of the system, and also
derive the final total population of infected individuals. We present a
semi-implicit in time numerical scheme based on finite differences in space
which preserves the main properties of the continuous model such as the
uniqueness and positivity of solutions and the conservation of the total
population. We also illustrate our results with a selection of numerical
simulations for a selection of connected graphs.
