Mathematical Modeling of the Cardiovascular and Respiratory Systems With Delay Effects: Case of a 30-Year-Old Woman in Resting, Walking, Jogging, and Running

The human cardiovascular and respiratory systems are critical and appealing study areas, as cardiovascular and respiratory disorders become more prevalent and burdensome for families. This paper is aimed at developing a mathematical model with delays for determining blood pressure’s response to cardiac and respiratory parameters. Two nonlinear coupled ordinary differential equations are created with the intention of looking for the behavior of the model in stable, unstable, and Hopf bifurcation solutions for mathematical models of arterial and venous pressures during physical activity (rest, walking, jogging, and running cases). A comprehensive mathematical model integrating cardiovascular and respiratory systems, with a focus on delay effects, is necessary to account for physiological response lags between oxygen demand, heart rate, and ventilation rate. A mathematical model representing the mass balance between the systemic arterial and systemic venous compartments has been developed. We also carried out a stability analysis with different delays. The findings reveal that when a delay exceeds a threshold value, we have Hopf bifurcation, and as delays increase, the time lag between changes in systemic arterial and systemic venous pressures increases, resulting in a loss of stability. The results show that the model can accurately predict cardiovascular and respiratory responses and provide insights into system stability and adaptability across exercise intensities. This work advances our understanding of cardiorespiratory dynamics and establishes a framework for individualized health monitoring and activity planning.