Rich dynamics and data analysis of immune decline against SARS-CoV-2

The global pandemic of SARS-CoV-2 has constituted a serious threat to public health. Although natural infection or vaccination can induce immune protection in individuals, such immunity typically wanes nonlinearly over time. Therefore, to characterize the dynamic process of immunity, we construct a model that includes a two-stage immune waning mechanism. Theoretical analysis reveals under certain conditions there are forward and backward bifurcations in the system, and the coexistence of multiple endemic equilibria can also occur. Numerical simulations further show that under specific parameter conditions, the endemic equilibrium undergoes a saddle-node bifurcation along the extended branch of the forward bifurcation and exhibits bistability, allowing the epidemic to persist even when the basic reproduction number $\mathcal{R}_{0}<1$. Subsequently, using COVID-19 epidemiological data from the United States, we employed the Markov chain Monte Carlo (MCMC) algorithm for parameter estimation, obtaining optimal values for key model parameters. Finally, the sensitivity analysis of key parameters reveals that increasing vaccination rate, prolonging the duration of secondary waning immunity, and reducing relative susceptibility can collectively help reduce the risk of infection and delay the peak of the epidemic. These findings highlight the importance of optimizing vaccination strategies and developing more durable and effective vaccines for epidemic control.