Numerical simulation of spatiotemporal dynamics of a two-dimensional reaction–diffusion leprosy transmission model

Amir Khan; Shafiullah; Awatif J. Alqarni; Amir Khan; Shafiullah; Awatif J. Alqarni

doi:10.3934/math.20261284

AIMS Mathematics

2026, Volume 11, Issue 10: 32714-32750. doi: 10.3934/math.20261284

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Research article

Numerical simulation of spatiotemporal dynamics of a two-dimensional reaction–diffusion leprosy transmission model

1.
Department of Mathematics and Statistics, University of Swat, KP, Pakistan
2.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha, 61922, Saudi Arabia

Received: 27 June 2026 Revised: 30 August 2026 Accepted: 23 September 2026 Published: 10 October 2026
MSC : 35K57, 65M06, 92D30

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.
- leprosy model,
- reaction–diffusion system,
- partial differential equations,
- spatiotemporal dynamics,
- forward Euler method,
- Crank–Nicolson operator splitting
Citation: Amir Khan, Shafiullah, Awatif J. Alqarni. Numerical simulation of spatiotemporal dynamics of a two-dimensional reaction–diffusion leprosy transmission model[J]. AIMS Mathematics, 2026, 11(10): 32714-32750. doi: 10.3934/math.20261284

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Abstract

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.

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