A deterministic SAIR model with vaccination and treatment: dynamical behaviors and control strategies

In order to provide valuable information for the timely design and determination of public health measures during epidemic outbreaks, a deterministic SAIR model with vaccination programs and saturated treatment is established in this paper, as they have been demonstrated to be effective ways to change the evolution and prevent the spread of infectious diseases. The dynamical behaviors of the proposed model is analyzed theoretically and numerically. It is found that multiple endemic equilibria occur under certain conditions, suggesting that decreasing the basic reproduction number \(\mathcal {R}_0\) ${\mathcal{R}}_0$ is insufficient for disease eradication and improving the efficiency of vaccination and treatment can have a significant impact on disease progression. Furthermore, sensitivity analysis is conducted to quantify how changes in concerned parameters impact \(\mathcal {R}_0\) ${\mathcal{R}}_0$ , and the proposed model is applied to the pandemic wave of COVID-19. Finally, optimal control problem with time-varying vaccination and treatment is studied and cost-effectiveness analysis is examined under different scenarios.