Transmission Dynamics of an Infectious Disease With Vaccination and Multiple Infection Classes

In this paper, we propose an improved SEIR epidemic model that incorporates waning vaccine-induced immunity and divides the infected population into three distinct classes. The model captures more realistic disease dynamics by accounting for waning vaccine-induced immunity among vaccinated individuals and heterogeneous infectious stages. Fundamental properties of the model, including the existence, uniqueness, positivity and boundedness of solutions, are established to ensure its mathematical and epidemiological validity.The disease-free equilibrium is derived and the vaccination reproduction number ($R_v$) is computed using the next-generation matrix approach. Conditions for the existence of an endemic equilibrium are established, and numerical simulations indicate that, for the parameter regimes considered, the disease-free equilibrium is locally asymptotically stable when $R_v<1$, while the endemic equilibrium is locally stable when $R_v>1$. Additional simulations are carried out to illustrate the effects of key epidemiological parameters.Furthermore, sensitivity analysis is performed to assess the influence of model parameters on $R_v$ and the state variables. The results highlight the dominant roles of the effective contact/transmission rates and the vaccine waning rate in shaping $R_v$. Overall, the proposed model provides a useful framework for evaluating vaccination strategies and informing public health interventions aimed at controlling infectious disease transmission.