Within the framework of a random network, where the network architecture specifies the directional flow between nodes, this study delves into the SIR (Susceptible-Infected-Recovered) model. We incorporate two distinct time delays into the model: one representing the recovery period of infected individuals and the other the incubation time for susceptibles to become infectious. The model is developed using continuous-time and discrete-space equations that account for dual delays within the diffusion network. By employing the delay as a bifurcation parameter, we derive the essential conditions for the onset of Hopf bifurcation. The stability of this Hopf bifurcation is then scrutinized utilizing the central manifold theory. Our numerical simulations confirm that the stability features associated with the Hopf bifurcation can trigger disease outbreaks. Furthermore, our research reveals that the connection probability within the network significantly alters the spatio-temporal dynamics of the epidemic spread. This insight underscores the importance of network connectivity in influencing epidemic patterns and has implications for public health interventions and disease control strategies.