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

TL;DR

An efficient positivity-preserving, implicit-explicit scheme with second-order accuracy is constructed that allows to study the nucleation, spread, and dissipation of crime hotspots, as well as validate that clustering of population density may accelerate virus transmission in epidemic dynamics and potentially aggravate the second infectious wave.

Abstract

The system of chemotaxis PDEs with singular sensitivity was originally proposed by Short et al. [Math. Mod. Meth. Appl. Sci., 18:1249-1267, 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. 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. Typical numerical examples are conducted to validate our theoretical results and to demonstrate the necessity of correction strategy. The proposed scheme allows us to study the nucleation, spread, and dissipation of crime hotspots, as well as validate that clustering of population density may accelerate virus transmission in epidemic dynamics and potentially aggravate the second infectious wave.

Figures (3)

• Figure 1: Population density at $T = 800$ day in PDE-based and particle-based crime modelings: A similar pattern is observed in both particle system and PDE system. From the horizontal view, the hotspots become more concentrated as the diffusion coefficient $D$ increases. When the value of $D$ is small, the density of the crime population shows a local clustering pattern such as "points" or "rings". In contrast, when the value of $D$ is large, the density distribution exhibits several large spikes. From the vertical view, when the diffusion coefficient $D$ is fixed, the parameter $\eta$ has a qualitative impact on the pattern of distribution of crime. When $\eta$ decreases (from $\eta = 0.2$ to $\eta = 0.03$), the clustered 'points' become more regular, and the phase diagram transits from diffusion-dominated to aggregation-dominated.
• Figure 2: 2-D spatial distribution of mobile agents ($S+E+P+A+I^-+I^++R$) and the time evolution of virus carriers (i.e., the renormalized total population of $E$, $P$, $A$, $I^+$ and $I^-$) under different combinations of $D$ and $\eta$, with the corresponding ODE results used as references. As $\eta$ or $D$ increases, the phase diagram clearly reveals the transition among three phases: the aggregating phase (I), intermediate phase (II), and dissipative phase (III). Panels (A)–(D) show that aggregation has little impact on the first epidemic peak. In panels (E)–(H), high-density aggregation significantly aggravates the second epidemic peak.
• Figure 3: Numerical simulations of an infectious-disease model before and after applying the Lagrange-multiplier correction are compared. In panel (a), the left subplot displays results from three schemes: the uncorrected method produces nonphysical negative values and becomes unstable, whereas the corrected method closely matches the reference ODE solution. The middle and right subplots show the state of the system at $T=80$ without or with corrections, respectively, demonstrating the instability caused by the uncorrected non-physical values. Panel (b) presents only corrected results: the left plot confirms the strict non-negativity of $\psi_i~(i=1,\dots,8)$; the middle plot depicts the number of iterations required by the semi-smooth Newton solver, illustrating its efficiency and robustness; and the right plot verifies discrete mass conservation of $\bm{\Psi}$.

Theorems & Definitions (21)

• Remark 1
• Remark 2
• Theorem 1
• Lemma 2
• Proof 1
• Lemma 3: Lemma 4.3 in TongFenghua2024Positivity
• Proof 2
• Lemma 4
• Proof 3
• Proof 4: Proof of Theorem \ref{['main_theorem']}