Mathematical modelling of the endemic phase of Covid-19 infection considering vaccination breakthrough, revaccination and reinfection

Resum

The transition of COVID-19 from a pandemic to an endemic phase requires a better understanding of its long-term dynamics, especially regarding vaccination breakthroughs, reinfection, and revaccination. This study develops and analyzes a deterministic mathematical model to quantify the endemic dynamics of COVID-19, using New York, USA, as a case study. The model takes into account vaccination, vaccine breakthrough infections, revaccination of previously immunized individuals, and reinfection. The well-posedness of the model is established by demonstrating the non-negativity of solutions and identifying an invariant region. The disease-free equilibrium (DFE) is derived, and its basic reproduction number (R0) is computed using the next-generation matrix. Local stability of the DFE is confirmed through the Jacobian matrix, while global stability of both the DFE and the endemic equilibrium is proven using suitably constructed Lyapunov functions. A sensitivity analysis of (R0) indicates that the contact rate among non-vaccinated individuals (βc) and the asymptomatic transmission modification factor (ηA) are the most significant parameters, with positive sensitivity indices of 0.987 and 0.518, respectively. In contrast, the recovery rate for symptomatic individuals (γ2) and the vaccination rate (θ) showed substantial negative effects. Numerical simulations, conducted in Python, quantified the endemic equilibrium, revealing stable patterns across population compartments and a notably lower force of infection among vaccinated individuals. Scenarios that examined varying contact rates (βc) demonstrated a direct, linear relationship with the force of infection. Additionally, a three-dimensional analysis of vaccine coverage and efficacy revealed a compensatory relationship, illustrating the herd immunity threshold. Importantly, the revaccination rate emerged as a critical control parameter, with increases from 0.1 to 0.3 day−1 correlating with a 40-60% reduction in peak infections and a 50-70% decrease in hospitalizations. The findings indicate that the endemic persistence of COVID-19 is highly sensitive to contact rates within the unvaccinated population and the infectiousness of asymptomatic cases. The results support the need for targeted interventions, such as ongoing vaccination and revaccination campaigns, rather than broad, uniform measures, to keep the epidemic below critical thresholds and effectively manage the burden on healthcare systems. This study provides a quantitative framework for public health policy in the endemic phase of COVID-19.