Abstract
In this paper, the classical Bell replication model is extended to an impulsive analogue incorporating a variable-time impulsive system to analyze the dynamics of antigen–antibody interactions. The model investigates the interaction between antigens and antibody complexes through pulse phenomena to elucidate mechanisms by which antigens can be effectively neutralized via drug action and antibodies reinforced through vaccination. Employing a combination of mathematical modeling, stability theory, and immunological principles, sufficient conditions for the occurrence of pulse phenomena and system stability are established. It is shown that, under appropriate parameter constraints, the solution of the system is asymptotically stable. The analysis further reveals that the inclusion of impulses in the antigen–antibody dynamics enhances antibody efficacy against antigenic invasion. From a theoretical perspective, the results provide a framework for understanding pulse-mediated immune responses and offer potential insights relevant to the design of effective therapeutic and vaccination strategies.