Optimal social distancing through cross-diffusion control for a disease outbreak PDE model

We consider a human population residing in a two-dimensional domain at the onset of an ongoing disease outbreak. The population dynamics are governed by two coupled partial differential equations, featuring a quadratic force of infection. Our aim is to indirectly control the dynamics of the infected population by mathematically incorporating social distancing measures for susceptible individuals. Traditional control approaches used in previous literature are based on the introduction of control variables within the reaction terms or the boundary conditions of the reaction–diffusion system governing the dynamics. However, the complex nature of social distancing makes these approaches less practical since they do not fully capture the induced behavioral changes, resulting from such a control measure, within the state equations. Hence, the main novelty of our work is the development of a new practical approach by relying instead on the mathematical concept of cross-diffusion. To this end, we introduce a cross-diffusion term in the susceptible equation, effectively using spatio-temporal cross-diffusivity as a control variable. By employing truncation procedures, suitable interpolation spaces and fixed-point theory, we establish the existence of a unique, essentially bounded weak solution to the state system. Moreover, by demonstrating the Gateaux differentiability of the control-to-state mapping and based on a suitable adjoint problem, we provide a characterization of optimal social distancing measures. To validate our theoretical results, we perform numerical investigations that illustrate the effect of adopting social distancing as an optimal control measure on the disease spread.