A Non-linear Fractional Model for Analyzing the Impact of Vaccination on the Dynamics of COVID-19 in Indonesia

COVID-19, caused by the novel coronavirus SARS-CoV-2, remains a global public health challenge. This study proposes and analyzes a mathematical model to monitor the progression of COVID-19 and assess the impact of immunization efforts. The model incorporates key epidemiological factors and is calibrated using publicly available data on cumulative daily cases of COVID-19 in Indonesia, spanning from July 1, 2021, to July 21, 2022. The basic reproduction number, R₀, is derived and the equilibrium states are established. Bifurcation analysis is conducted using the Center Manifold Theorem to understand the potential transition dynamics of the disease. A local sensitivity analysis reveals that the effective transmission rate (β), the natural mortality rate (µ), the vaccination rate (τ) and the treatment rate for symptomatic individuals (θ) are the most influential parameters. Model simulations suggest that reducing transmission, improving treatment, and increasing vaccine uptake significantly reduce disease burden. To better capture the memory effect inherent in disease transmission, the model is extended to a fractional-order Caputo derivative framework. The existence, uniqueness, and stability of the fractional model are established via fixed-point theory. Numerical results demonstrate that the decrease in fractional order slightly shifts the dynamics, suggesting behavioural changes in response to past outbreaks. These findings provide valuable information on disease control strategies and highlight the importance of sustained public health measures.