Mathematical Models Connected to the Respiratory System

Abstract

The respiratory system is intricately connected to various other systems in the human body. When abnormalities arise, whether their cause is known or unknown, mathematical models are crucial for identifying underlying issues that may not be detectable with current medical tools. This work focuses on mathematical models of normal and pathological respiratory conditions, including Cheyne-Stokes Respiration (CSR), a form of Central Sleep Apnea (CSA). These models, often described by delayed differential equations (DDEs), account for feedback control loops and time delays, as seen in the work of Mackey and Glass and Landa and Rosenblum. Our primary contribution is the modification of the Landa and Rosenblum model. We introduce a refined model with two delayed differential equations and perform numerical simulations analysis, conducted using MATLAB, aiming to pinpoint a new bifurcation parameter and keep the qualitative behavior of the original model that accounts for the cumulative description of CSR even at a longer time. Additionally, we address ambiguities in the parameterization presented by Dong and Langford (2008) and apply Hopf Bifurcation analysis to explore alternative causes of CSR beyond time delay.