Applied Mathematics and Computational Sciences
Bringing an old proof to modern problems
A physically consistent and self-adaptive numerical modelling approach opens the door to more accurate and reliable computational simulations of clinical and biological processes.

A mathematical proof established more than 140 years ago provided the key for a KAUST-led team to develop a computational method for accurately simulating complex biophysical processes, such as the spread of disease and the growth of tumors[1].
Reaction-diffusion equations are widely used to model the dynamics of complex systems by providing a macroscopic mathematical description of the interplay between local interactions and random motion.
“Reaction-diffusion equations play a crucial role in many clinical applications, including computational epidemiology — an interdisciplinary field that combines mathematics and computational science — to enhance our understanding and control of the spatiotemporal spread of diseases in real time,” says Rasha Al Jahdali from the KAUST research team.
“This discipline is vital for informing health policy decisions worldwide. Since studying epidemiological phenomena is often complex and sometimes unfeasible, it is important to develop efficient, robust, and predictive algorithms for reaction-diffusion processes.”
These equations capture real physical mechanisms in the form of continuous ‘partial differential’ equations that describe rates of change over space and time. While mathematically ideal, such equations can be very difficult to solve for the purposes of simulating complex biophysical processes because computers require numerical methods involving discrete calculations of real numbers.
“Discretizing continuous reaction-diffusion equations using numerical schemes allows us to use computers to solve them numerically,” says Al Jahdali. “This involves approximating continuous variables and their derivatives or rates of change at specific points in space and time. The problem is that existing discretization methods often lack the ability to maintain stability and accuracy when simulating complex, nonlinear interactions like those inherent in biological systems.”
That means existing numerical schemes often produce ‘unphysical’ results or break down, particularly with spatially varying reaction-diffusion equations, which limits their utility for making reliable predictions in a clinical context.
To solve this problem, Al Jahdali and her colleagues turned to an old mathematical proof, called the Lyapunov direct method, that tests for the existence of a stable solution of time-varying system without needing to solve the underlying partial differential equations.
“Our computational framework leverages the Lyapunov’s direct method to develop fully discrete and ‘smart’ self-adapting schemes of arbitrary accuracy in space and time,” says Al Jahdali. “This new computational framework provides robust and accurate solutions suitable for applications in complex environments, capturing correctly the dynamics of phenomena, which is crucial for accurately modeling real-world scenarios in biological and clinical applications.”
The researchers applied their method to a commonly used spatially varying ‘susceptible-infected’ model for predicting the endemic spread of disease, showing that their numerical approach stayed consistent with the original physically accurate reaction-diffusion solution. They also used their approach to model the treatment of a tumor in the brain using virotherapy, involving complex interactions in space and time based on known biochemical processes.
“Our approach demonstrates superior performance compared to traditional numerical methods for solving reaction-diffusion partial differential equations, which will enable more reliable and physically consistent results,” says Al Jahdali. “This represents a significant step in developing computational methods that are not only theoretically sound, but practically useful for addressing pressing global problems.”
Reference
- Al Jahdali, R., Del Rey Fernández, D.C., Dalcin, L. et al. Fully-discrete Lyapunov consistent discretizations for parabolic reaction-diffusion equations with r species. Communications in Applied Mathematics and Computation (2024). | article.
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