Stability and Bifurcation Analysis of a Delay Differential Equation Modeling the Human Respiratory System

The human respiratory system takes oxygen and releases carbon dioxide in air with the help of lungs and other respiratory organs. Understanding the human respiratory system is important for many medical conditions. There are many disorders associated with respiratory system that affect a large number of people. Thus studying this system with a good mathematical model has far-reaching implications. We study the two state model which describes the balance equation for carbon dioxide and oxygen. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. So the dynamics of a two state one delay model are investigated. By choosing the delay as a parameter, the stability and Hopf bifurcation conditions are obtained. We notice that as the delay passes through its critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. The stable region of the system with delay against the other parameters and bifurcation diagrams are also plotted. The three dimensional stability chart of the two state model is constructed. We find that the delay parameter has effect on the stability but not on the equilibrium state. The explicit derivation of the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions are determined with the help of normal form theory and center manifold theorem to delay differential equations. We also study the five state model with four delays numerically. We investigated the stability of the system at several delays and stability chart near the measured values are constructed. Finally, some numerical example and simulations are carried out to confirm the analytical findings. The numerical simulations verify the theoretical results.