SIR (susceptible-infective-recovery) model is a widely investigated model to explain the time evolution of infectious diseases. Outbreak of infectious diseases is affected by diffusion of infected, which is true especially in COVID-19 outbreak. Therefore, it is imperative to construct a diffusion network in the model for spatial consideration; However, the inclusion of a diffusion network is seldom considered for the studies. In this work, we first modified the SIR model for COVID-19 and then performed its stability and bifurcation analysis in qualitative research. Based on our analysis, we propose some of the advice to mitigate the spread of COVID-19. Then, a random diffusion network is constructed, which shows its vital role in the Turing instability and bifurcation. We noticed that the stability of network-organized SIR could be determined by the maximum of eigenvalues of the network matrix. The maximum of eigenvalues of the network matrix is proportional to network connection rate and infection rate of the network. Therefore, these two rates play a critical role in Turing instability. We perform the numerical simulations to verify the analytical results. We try to explain the spread mechanism of infectious diseases and provide some feasible strategies based on our analysis of these two models. Also, the reduced system method for a network-organized system is proposed, which is a novel approach to investigate the complex system.