Dynamics of a Generalized Infectious Disease Model with Transmission, Treatment and Vaccination Delays

Abstract

This paper presents a generalized SIR model that accounts for real-world factors such as time delays in disease transmission, treatment, vaccine effectiveness, and waning immunity. The model incorporates a delay representing the time required for individuals to become infectious after exposure, delay in the treatment, as well as a separate delay reflecting the period between vaccination and the development of immunity. In addition, a delay is introduced in the return of recovered individuals to the susceptible class, simulating the gradual loss of immunity over time. The stability of the disease-free and endemic equilibria is examined using both local and global analysis. Local stability conditions are derived through the basic reproduction number and characteristic roots, while global stability is explored using a Lyapunov-based approach. Numerical simulations are carried out to demonstrate how varying the delays influences the disease progression and control, offering practical implications for public health strategies.