ODE and PDE models for COVID-19, with reinfection and vaccination process for Cameroon and Germany

We develop and analyze a reaction-diffusion model for the transmission dynamics of the coronavirus (COVID-19) that accounts for reinfection and vaccination, and compare it to the ODE model. For the latter, we calculate the control reproduction number $ \mathcal{R}_c $ and demonstrate the global stability of the DFE for $ \mathcal{R}_c1 $, the DFE becomes unstable and co-exists with at least one endemic equilibrium point. We use data from Germany and Cameroon to calibrate our model. We find $ \mathcal{R}_c\approx 1.13 $ for Germany and $ \mathcal R_c \approx 1.2554 $ for Cameroon, indicating a persistence of the disease. We use sensitivity analysis (SA) to identify model factors that contribute to the disease dynamics. According to this analysis, the most crucial factors are the transmission rate, the recovered rate of asymptomatic humans, and the recovered rate of symptomatic humans. Then, a modification to a reaction-diffusion PDE model, accounting for spatial mobility, is studied; solutions exist and are non-negative and unique. The DFE is locally stable, and globally stable when $ \mathcal{R}_c1 $, the DFE is unstable and coexists with at least one endemic equilibrium point. Numerical simulations validate our theoretical predictions. We then compare the ODE and PDE models.