<p>A weighted networked susceptible–asymptomatic infected–symptomatic infected–recovered (SAIR) epidemic model has been established to investigate the influence of heterogeneous population movement on the dynamics of infectious diseases by using a graph Laplacian diffusion. Both endemic and disease-free equilibria exist in our model. By determining eigenvalues of the graph Laplacian, we prove that the disease-free equilibrium point is locally asymptotically stable when the basic reproduction number is below unity, and the endemic equilibrium point is locally asymptotically stable when the basic reproduction number is above unity. Moreover, we show that this local stability implies global stability for our model with the help of a suitable Lyapunov functional and Green formula. A control problem is subsequently formulated to manage the spread of the disease within the network, incorporating vaccination strategies. The implementation of optimal control not only significantly mitigates the disease but also minimizes the overall cost of vaccination. To support theoretical findings, numerical simulations have been carried out by considering a Watts–Strogatz small-world network. Furthermore, we analyze the effects of perturbations on transmission and mobility parameters. Finally, the determination of risk factors is demonstrated using a community detection method.</p>