<p>Given that the network topology can accurately describe the intricate connections between many individuals, this paper studies the dynamic behavior of the Susceptible-Infected-Recovery (<i>SIR</i>) model with network structure, and obtains the necessary conditions for the system to have Turing bifurcation in homogeneous networks and heterogeneous networks. Based on the optimal control theory, the loss function is constructed to identify the parameters of the system in the heterogeneous network. The numerical simulation shows that the network topology in which susceptible population and infected population exist affects the shape of Turing patterns, while changes in the network structure of recovered population have almost no effect on the patterns. When the network structure is fixed, the density of infected population will increase with the increase of the propagation rate. Then, the projection gradient (<i>PG</i>) algorithm, the Barzilar–Borwein (<i>BB</i>) algorithm and the Broyden–Fletcher–Goldfarb–Shanno (<i>BFGS</i>) algorithm are used to realize the effective identification of two parameters and three parameters in heterogeneous networks. In addition, we apply the theoretical method to practice, realize the effective identification of spatio-temporal heterogeneous parameters in the generative network, and simulate the epidemic diffusion process to a certain extent. Finally, based on the continuous space, we use the identification of spatial heterogeneous parameters to reproduce the mechanism of biological pattern formation.</p>