On nonstandard finite difference schemes in biosciences

Area of Science:

• Computational Biosciences
• Mathematical Biology
• Numerical Analysis

Background:

• Differential equations are crucial for modeling biological systems.
• Existing numerical methods may not preserve essential properties of biological models.
• Reliable numerical schemes are needed for accurate simulations in biosciences.

Purpose of the Study:

• To design, analyze, and implement novel nonstandard finite difference (NSFD) schemes.
• To ensure these NSFD schemes are reliable for various bioscience differential models.
• To validate the schemes' performance in terms of dynamic consistency, stability, and solution property preservation.

Main Methods:

• Development and analysis of nonstandard finite difference (NSFD) schemes.
• Application of NSFD schemes to one-dimensional models, MSEIR epidemiological models, and advection-reaction/reaction-diffusion equations.
• Verification of topological dynamical consistency and global asymptotic stability properties.

Main Results:

• NSFD schemes demonstrate topological dynamical consistency for one-dimensional models.
• The schemes successfully replicate the global asymptotic stability of the disease-free equilibrium for the MSEIR model when the basic reproduction number is less than 1.
• Positivity and boundedness properties of solutions are preserved for advection-reaction and reaction-diffusion equations.

Conclusions:

• The developed NSFD schemes offer a reliable numerical approach for bioscience differential models.
• These schemes accurately capture essential qualitative properties of biological systems.
• NSFD methods provide a robust framework for advancing computational biosciences.