Mathematical exploration of a diffusive infection's dynamics and simulation
Rassim Darazirar1,2, Ahmed A Mohsen3,4, Aziz Khan5
1Department of Mathematics, Faculty of Exact Sciences and Computer Science, Hassiba Benbouali University, Chlef, Algeria.
Scientific Reports
|August 26, 2025
Summary
This study uses Lyapunov functionals to analyze biological reaction-diffusion models. The research demonstrates global stability for disease-free equilibria when the basic reproduction number is less than one, and for endemic equilibria when it exceeds one.
Area of Science:
- Mathematical Biology
- Dynamical Systems Theory
- Epidemiology
Background:
- Reaction-diffusion systems are crucial for modeling spatially explicit biological processes.
- Understanding equilibrium stability in these systems is vital for predicting disease dynamics.
- Lyapunov functionals offer a powerful tool for stability analysis.
Purpose of the Study:
- To extend Lyapunov functional methods for analyzing global stability in reaction-diffusion systems.
- To investigate the stability of disease-free and endemic equilibria using these functionals.
- To apply the methodology to epidemiological models with spatial diffusion.
Main Methods:
- Construction and application of Lyapunov functionals.
- Extension of ordinary differential equation (ODE) based functionals to partial differential equation (PDE) systems.
- Analysis of equilibrium stability in relation to the basic reproduction number (R0).
Main Results:
- Global asymptotic stability of the disease-free equilibrium is proven for R0 < 1.
- Global stability of the endemic equilibrium is established for R0 > 1 under specific conditions.
- The methods are demonstrated with examples from epidemiology and public health.
Conclusions:
- Lyapunov functionals provide a robust framework for analyzing the global stability of reaction-diffusion models.
- The basic reproduction number serves as a critical threshold for determining disease persistence or eradication.
- The study offers theoretical insights and numerical validation for epidemiological models with spatial spread.
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