West Nile virus reaction–diffusion dynamics on higher-order directed network

West Nile virus (WNv) has a unique mosquito–bird–mosquito diffusion cycle, so the common compartment model cannot describe its complex spatial spread process. This study generalizes a reaction–diffusion system on continuous space to a complex directed network with a nonlinear diffusion function. By incorporating the Logistic growth of mosquitoes, a three-variable reaction–diffusion system is established. We deduce the basic reproduction number R 0 and Turing instability conditions, and prove that nonlinear diffusion can induce spatial patterns. A periodic diffusion coefficient can also result in patterns, as discussed via Floquet theory. Using optimal control, the optimization problem and parameter identification are derived. Numerical simulations verify the theoretical conclusions, and we further use a physics-informed neural network (PINN) to predict the evolutionary trend based on real data from New Jersey, incorporating temperature into the time-varying transmission rate.