Mean-field Modeling of Social Interactions Using Classical Density Functional Theory

Abstract: Incorporating social interactions is essential to an accurate modeling of epidemic spreading. This work proposes a novel local mean-field density functional theory model by using the sum-of-exponential approximation of convolution kernels for social interactions, which in turn converts the convolution terms into interaction potentials that are governed by the Debye-H\"uckel equation. Thanks to the local formulation of the proposed model, linear stability analysis is able to derive a novel instability condition associated with cross interactions. Global existence of the solution to the proposed model with a simplified self-repulsive interaction potential is established. Extensive numerical simulations are performed to assess the impact of cross social interactions on transmission and isolation, verify the instability conditions obtained from linear stability analysis, and provide theoretical guides for the control of disease spreading.