Kinetic models of opinion-driven epidemic dynamics modulated by graphons

We introduce kinetic models to simulate epidemic spread while accounting for individuals’ opinions on protective behaviors. Opinion exchanges occur on a social network represented by a graphon, leading to scenarios with or without opinion leaders. We prove convergence to equilibrium in the strong $L^1$ norm via relative entropy methods and in homogeneous Sobolev spaces ${\dot{H}}^{-s}$, $s\in (\frac{1}{2},1)$, using Fourier-based techniques. We then design a structure-preserving scheme for the coupled opinion-epidemiological system, highlighting graphon effects: opinion leaders supporting protective behaviors limit disease spread, whereas influenceable individuals may shift toward opposing views, worsening epidemics. Finally, we introduce a time-dependent quantity, analogous to the reproduction number, whose oscillations can generate epidemic waves without explicit external forcing. The MATLAB code implementing our algorithms is made publicly available.