Mathematical modeling of Buruli ulcer spread
Shimi Chettiparambil Mohanan, Christina Surulescu, Neslihan Nesliye Pelen
TL;DR
This work develops two macroscopic cross-diffusion PDE-ODE frameworks for describing Buruli ulcer spread in tissue, derived from distinct mesoscopic transport descriptions: (i) a kinetic transport equation with velocity jumps and Hilbert expansion, yielding a diffusion-taxis equation for bacterial density $U$ coupled to mycolactone $M$, normal tissue $V$, and necrotic tissue $N$, and (ii) a random-position jump model producing density-dependent diffusion and chemotaxis terms, leading to a comparable PDE-ODE system. The authors prove local and global existence for the resulting systems after nondimensionalization, using fixed-point arguments and a priori estimates, including $L^p$ bounds and Moser iteration to bound $U$. They perform extensive numerical simulations with an IMEX finite-difference scheme to compare the two modeling approaches under multiple scenarios, uncovering how chemotaxis strength and initial tissue availability shape infection dynamics, and they demonstrate that the first, simpler model can capture key behaviors with fewer ad-hoc assumptions while the nonlinear model provides a more conservative propagation estimate. The study offers theoretical guarantees of well-posedness and practical insights into BU progression, informing potential therapeutic strategies for early intervention, resection planning, and tissue preservation.
Abstract
We propose two approaches to a model deduction for Buruli ulcer spread in a tissue, prove global existence of solutions to the obtained macroscopic cross-diffusion PDE-ODE systems, and perform numerical simulations to illustrate the behavior of solutions under various scenarios and compare the outcome of the two approaches.