Cognitive epidemic models for non-local vaccination behavior driven by infection risk perception

This paper considers a class of nonlocal vaccination epidemic models, where vaccination willingness decreases with increasing distance from the outbreak epicenter due to diminishing perception of the risk of contracting diseases. By establishing SVIR reaction–diffusion epidemic models with cognition, we continue to explore the application of Fick’s law and Fokker–Planck’s law in the diffusion of cognition. Meanwhile, we investigate the impact of different diffusion strategies adopted by vaccinated individuals on the final scale of nonlocal vaccination in spatially heterogeneous environment. Firstly, we conduct well-posedness analysis for both the random diffusion model and the symmetric diffusion model. We calculate the basic reproduction numbers of these models and conduct threshold dynamics analysis. Then, we obtain the corresponding degenerate model and prove the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium using the Lyapunov function. Finally, the results of numerical simulations demonstrate that different diffusion strategies and vaccination radii are associated with distinct spatial segregation phenomena. In the random diffusion model, if the diffusion strategy of vaccinated individuals is the same as that of susceptible individuals, the steady-state dispersion of vaccinated individuals closely resembles that of susceptible individuals. Conversely, if the diffusion strategy of vaccinated individuals is the same as that of infected individuals, they also demonstrate comparable equilibrium distributions. Intriguingly, this particular phenomenon failed to manifest in the remaining two model systems. Therefore, we speculate that Fokker–Planck’s law may better describe the human transmission patterns in infectious disease models.