Pattern dynamics in the epidemic model with diffusion network

It is well known that the outbreak of infectious diseases is affected by the diffusion of the infected. However, the diffusion network is seldom considered in the network-organized SIR model. In this work, we investigate the effect of the maximum eigenvalue on Turing instability and show the role of network parameters (the network connection rate, the network's infection, etc.) on the outbreak of infectious diseases. Meanwhile, stability of network-organized SIR is given by using the maximum eigenvalue of the network matrix which is proportional to the network connection rate and the networks infection rate. The bridge between the two rates and Turing instability was also revealed which can explain the spread mechanism of infectious diseases. In the end, some measures to mitigate the spread of infectious diseases are proposed and the feasible strategies for prevention and control can be provided in our paper, the data from COVID-19 validated the above results.