Fractional-Order COVID-19 Model in Indonesia with Comorbidity and Immunization: PID Control, Ulam-Hyers Stability, and Biosecurity Implications - Muhammad Farman; Cicik Alfiniyah; Fatmawati Fatmawati; Muhammad Abdurrahman Rois; Khadija Jamil (Department of Mathematics, Universitas Negeri Gorontalo, 2025)
In this paper, we developed a fractal fractional model for Covid-19 dynamics in Indonesia with comorbidity and various immunization stages doses is presented and examined. The system is analysed disease-free according to reproductive number. We conducted both qualitative and quantitative research on the COVID-19 model using the Atangana-Baleanu fractal-fractional operator. We demonstrated the existence and uniqueness of the model with the Atangana-Baleanue fractal-fractional operator as continuous and compact integral components, by means of Krasnoselskii fixed point theorem. We ensure that our proposed model has a unique fixed-point solution by including the properties of both the Schauder and Krasnoselskii theorems into the contraction mapping. We conduct a thorough examination of the suggested model’s stability using the Ulam-Hyers stability concept. We discuss how the Proportional Integral Derivative (PID) impact in a fractional COVID-19 model improves stability. Since these control methods have a great potential to improve overall treatment outcomes, minimise side effects, and correctly regulate these treatments to achieve this goal, their use will stabilise the dynamics behaviour while accurately regulating the administration, leading to better vaccination outcomes with fewer adverse effects inferred from this. A numerical approach based on Lagrange interpolation is presented. The dynamics of disease transmission throughout a range of fractional-order ϖ and fractal dimensions ϑ are then visually represented by the numerical results that have been obtained. The findings demonstrate the deep impact of fractional dynamics and fractal dimensions on the processes of vaccination, recovery, and propagation, exposing intricate, time-dependent epidemic characteristics.