We develop a multi-variant, reaction–diffusion epidemic framework that couples mutation, spatial movement, vaccination, and symptomatic/asymptomatic transmission for SARS-CoV-2. The model tracks \(\{S,E_i,I_{i,1},I_{i,2},R_i,V\}\) over \(\Omega \subseteq \mathbb {R}^d\) with strain-specific infection, recovery, and disease-induced mortality, and with mutation flows between exposed classes. Methodologically, we prove positivity and boundedness, characterize the disease-free equilibrium (DFE) with vaccination, and derive a next-generation reproduction number \(R_0\) that explicitly accounts for mutation and spatial diffusion. Using Lyapunov techniques, we establish global stability of the DFE when \(R_0<1\) , show existence and stability of endemic equilibria when \(R_0>1\) , and provide conditions under which a higher- \(R_0\) mutant becomes dominant. We also conduct sensitivity analysis and incorporate spatially varying vaccination \(\omega (y)\) . Our results yield interpretable thresholds and dominance criteria that help: (i) anticipate strain replacement under given mutation profiles; (ii) quantify how mobility (diffusion) and immunity loss shift peak timing and size; and (iii) design spatially targeted vaccination strategies that lower effective \(R_0\) and suppress mutant establishment. Simulations illustrate how tuning \(\omega (y)\) and limiting opportunities for mutation (via faster case resolution or reduced contact) change epidemic trajectories and reduce severe disease burden. The framework thus provides a transferable tool for prioritizing interventions across space and variant landscapes.
