Mathematical modeling of COVID-19 and chronic kidney disease co-infection with vaccination and optimal control: a bifurcation and sensitivity analysis approach

Chronic kidney disease (CKD), affecting approximately 843 million individuals globally, substantially increases susceptibility to severe COVID-19 outcomes, while SARS-CoV-2 infection independently accelerates renal deterioration through cytokine storms and microvascular injury. Despite this clinically significant bidirectional interaction, rigorous mathematical characterization of their co-dynamics within a unified framework integrating stability theory, bifurcation analysis, and optimal control remains limited. We formulated a seven-compartment deterministic model incorporating CKD'S irreversibility, the enhanced blue susceptibility of CKD patients, and vaccine imperfection, calibrated against Indian epidemiological data. The basic reproduction number ${\mathcal{R}}_0$ was derived using the next-generation matrix method, and stability and bifurcation analysis were performed. Sensitivity analysis identified transmission and immunity waning as dominant drivers, while optimal control strategies significantly reduced co-infections and hospitalizations, demonstrating the effectiveness of coordinated intervention policies.