On the Numerical Solution of the Multiscale Models Characterising Noroviral Infectious Disease Systems Using the Multistage Spectral the Relaxation Method and the Nonstandard Finite Difference Method

Multiscale models for infectious disease systems are highly nonlinear. This presents substantial difficulties for both analysis and computation. Consequently, there is a continuous demand for the development of efficient numerical methods that provide reliable solutions. Over time, various numerical techniques have been developed for single-scale models. However, multiscale models have increasingly relied on built-in solvers and, more recently, the nonstandard finite difference method. The primary aim of this study is to offer an in-depth examination of the multistage spectral relaxation method (MSRM) applied to the multiscale model characterising norovirus infection. This study also provides a comprehensive comparison with the nonstandard finite difference method (NSFDM) in terms of convergence and accuracy, as well as their handling of highly nonlinear systems of ordinary differential equations. Both schemes were validated against an adaptive ODE45 scheme in MATLAB2025a, which served as the reference solver. The numerical outcomes indicate that the MSRM achieves a better accuracy level, with relative L2 errors ranging from 10−9 to 10−7 for Δτ≤0.10 and Nch=8. In contrast, the NSFDM provides solid first-order accuracy, yielding L2 errors around 10−4, while ensuring strict unconditional positivity and stability across all tested step sizes from h=0.01 to 2.0. This study features a parameter sensitivity analysis that varies βH,δH,αh,μV,μH, as well as semi-log error graphs, convergence rate visuals, and CPU performance benchmarks. The MSRM costs about 24 times more per run, but it offers better accuracy for between-host variables. This makes it the preferred method for high-fidelity applications. In contrast, the NSFDM is the best option for quick parameter sweeps and long simulations that require guaranteed positivity.