On the Solvability of Time-Fractional Spatio-Temporal COVID-19 Model with Non-linear Diffusion

COVID-19 (Coronavirus Disease-2019) is a highly contagious disease that began spreading towards the end of 2019 and quickly became a global pandemic. Despite extensive efforts taken by public health authorities and policymakers, the disease continues to persist globally. To overcome this scenario, we construct a spatio-temporal time-fractional COVID-19 model that incorporates non-linear density dependent diffusion coefficients as well as the Caputo time-fractional derivative, which captures the disease dynamics more relevantly. The main objective of this article is to explore the existence and uniqueness of global weak solutions for the proposed model. We first construct weak solutions as sequences in a finite-dimensional space. We then ensure the convergence of these sequences through energy estimates and compactness results, thereby establishing the existence of weak solutions for the proposed model using the Faedo–Galerkin method. We then prove the uniqueness of weak solutions. Finally, we study stability of the model in the Mittag–Leffler sense.