Stability and Bifurcation Analysis for Mathematical Model of Acute Inflammatory Response with Time Delay

In this paper, we present an improved simplified mathematical model for the acute inflammatory response by incorporating two time delays: one for phagocyte migration and another for immune response activation. The model's dynamic behavior is analyzed by studying the existence and local stability of equilibrium points, as well as the emergence of Hopf bifurcation at the positive equilibrium point. Results show that the two delays have the potential to destabilize the equilibrium point associated with septic death and induce oscillation of the two state variables through Hopf bifurcation. Additionally, the normal form of Hopf bifurcation at specific time delays is obtained by using the method of multiple scales. It is discovered that the amplitude of bifurcating periodic solution is sensitive to time delays. Ultimately, numerical simulations of various bifurcation diagrams and attraction basins suggest that proper selection of model parameters can steer the disease toward a healthy state. The correctness of the theoretical analysis is confirmed by using numerical solutions. Overall, this study not only enhances our understanding of the underlying biological mechanisms of acute inflammatory response, but also provides guidance for the clinical treatment of inflammatory response.