Bayesian Estimation of Spatiotemporal Immune-Viral Dynamics in COVID-19 Using Partial Differential Equations

This study proposes a Bayesian framework for estimating parameters in partial differential equation (PDE) models of viral dynamics. We develop a computational methodology combining Markov Chain Monte Carlo (MCMC) sampling with B-spline basis expansions to address inverse problems in COVID-19 immunology. Applied to clinical data from 30 patients, the model quantifies lymphocyte recruitment kinetics and infection rates during SARS-CoV-2 pathogenesis. Key results demonstrate: (1) mean daily lymphocyte recruitment rate λ̂ = 11.87/day (range: 6.55–14.66), and (2) mean infection rate of pulmonary/lymphoid cells β̂ = 3,556 cells/mL (range: 2,290–5,699). The Bayesian estimator achieved 93.2% posterior coverage probability, confirming its efficacy in characterizing immune response dynamics. These findings provide clinically actionable parameters for optimizing antiviral therapies through precise quantification of host-pathogen interactions. Notably, λ reflects the immune system's capacity to mobilize lymphocytes, with elevated values predicting rapid viral clearance and recovery. In contrast, β serves as a biomarker of viral infectivity severity, where higher values signal increased tissue-level viral load and a greater risk of adverse clinical outcomes.