Temporal dynamics of immunity: modeling susceptibility delay in antibody-shielded populations

This study presents a mathematical model that incorporates multiple time delays and a distinct compartment for antibody-protected immune individuals to analyze the transmission dynamics of infectious diseases. We ensure through analytical results that our model produced positive and bounded solutions, which is essential for realistic predictions. Parameter estimation is performed using real-time data to accurately determine the time delays associated with the system. In the absence of time delays, the analysis demonstrates that the disease transmission rate ($\beta$) plays a critical role in determining the system's behavior. When $\beta$ exceeds a threshold value ($\beta_c$), a forward bifurcation occurs. The study further investigates the impact of time delays on the stability of disease-free and endemic equilibria and identifies conditions under which the system undergoes a Hopf bifurcation, resulting in periodic oscillations. Numerical simulations are conducted to validate the theoretical findings, providing insights into the influence of immunity delays on disease persistence and intervention strategies.