We study a discrete model for a non-local diffusion problem with a source, namely (ui) ′ (t) = Nj=−N hJ h(i − j) uj(t) – N j=−N hJ h(i − j) ui(t) + f(ui(t)), with initial datum ui(0) = u0(xi) > 0. We prove the existence and uniqueness of the solution. Moreover, we show that solutions of discrete problem converge to the continuous ones when the mesh parameter goes to zero. We also study the blow-up phenomena of solutions. For some sources f, we obtain the blow-up rate. Finally, we perform some numerical experiments.