Leprosy remains one of the biggest public health problems in endemic countries, even with the availability of effective treatments. To better understand how the disease spreads over both time and space, this study extends a standard six-compartment mathematical model into a two-dimensional spatial model using partial differential equations. By including spatial movement (Fickian diffusion), we can simulate how different groups of people interact and move across a given geographical area. We construct two numerical schemes in order to solve this mathematical system: the forward Euler scheme and the Crank–Nicolson operator-splitting method. We analyze the stability properties of both schemes: the forward Euler method is conditionally stable subject to a Courant–Friedrichs–Lewy (CFL) condition, and the Crank–Nicolson operator-splitting method is conditionally stable due to the explicit treatment of the reaction terms, with its diffusion stages being individually unconditionally stable. Moreover, we simulate our mode using MATLAB the long-term stability of the disease. The results of the simulation clearly demonstrate the temporal evolution and geographical spread of leprosy over time, which is a useful tool for public health planning and disease control interventions.
BT - AIMS Mathematics DA - 10/2026 DO - 10.3934/math.20261284 IS - 10 LA - ENG M3 - Article N2 -Leprosy remains one of the biggest public health problems in endemic countries, even with the availability of effective treatments. To better understand how the disease spreads over both time and space, this study extends a standard six-compartment mathematical model into a two-dimensional spatial model using partial differential equations. By including spatial movement (Fickian diffusion), we can simulate how different groups of people interact and move across a given geographical area. We construct two numerical schemes in order to solve this mathematical system: the forward Euler scheme and the Crank–Nicolson operator-splitting method. We analyze the stability properties of both schemes: the forward Euler method is conditionally stable subject to a Courant–Friedrichs–Lewy (CFL) condition, and the Crank–Nicolson operator-splitting method is conditionally stable due to the explicit treatment of the reaction terms, with its diffusion stages being individually unconditionally stable. Moreover, we simulate our mode using MATLAB the long-term stability of the disease. The results of the simulation clearly demonstrate the temporal evolution and geographical spread of leprosy over time, which is a useful tool for public health planning and disease control interventions.
PB - American Institute of Mathematical Sciences (AIMS) PY - 2026 SP - 32714 EP - 32750 T2 - AIMS Mathematics TI - Numerical simulation of spatiotemporal dynamics of a two-dimensional reaction–diffusion leprosy transmission model UR - https://www.aimspress.com/article/doi/10.3934/math.20261284 VL - 11 SN - 2473-6988 ER -