Efficient structure-preserving scheme for chemotaxis PDEs with singular sensitivity in crime and epidemic modeling

The chemotaxis PDE system with singular sensitivity was originally proposed by Short et al. (Math. Mod. Meth. Appl. Sci., 2008) as the continuum limit of a biased random walk model to account for the formation of crime hotspots and environmental feedback successfully. Recently, this idea has also been applied to epidemiology to model the impact of human social behaviors on disease transmission. In order to characterize the phase transition, pattern formation and statistical properties in the long-term dynamics, a stable and accurate numerical scheme is urgently demanded, which still remains challenging due to the positivity constraint on the singular sensitivity and the absence of an energy functional. In particular, the loss of positivity may produce nonphysical states and even cause spurious blow-up. To address these numerical challenges, this paper constructs an efficient positivity-preserving, implicit-explicit scheme with second-order accuracy. A rigorous error estimation is provided with the Lagrange multiplier correction to deal with the singular sensitivity. The whole framework is extended to a multi-agent epidemic model with degenerate diffusion, in which both positivity and mass conservation are achieved. Numerical experiments are performed to validate the theoretical results and demonstrate the necessity of the correction strategy. Our simulations reveal rich dynamical behaviors, including the phase transition between aggregation-dominated and dissipative regimes, as well as the nucleation, spread, and dissipation of crime hotspots. For the epidemic model, the results further show that spatial clustering of population density may accelerate virus transmission and significantly amplify the infectious wave.