A reaction-diffusion model for the transmission dynamics of the Coronavirus pandemic with reinfection and vaccination process
Abstract
This work aims at deriving and analysing a reaction-diffusion model for the transmission dynamics of the Coronavirus (COVID-19) which takes into account the reinfection and the vaccination process, and to compare it with the ODE model. After formulating the time-dependent ODE model, we compute the control reproduction number $\mathcal{R}_c$ and prove the global stability of the disease-free equilibrium whenever $\mathcal{R}_c1$, the disease-free equilibrium becomes unstable and coexists with at least one endemic equilibrium point. We then use data from Germany to calibrate our model and estimate some model parameters. We find that $\mathcal{R}_c\approx 1.13$ expressing that the disease persists in the population. To determine key parameters which influence the model dynamics, we perform global sensitivity analysis by computing partial rank correlation coefficients between model parameters and the control reproduction number (respectively model state variables). After that, we include in the previous model the mobility in space by transforming it into a reaction-diffusion PDE model. For this last initial value boundary problem (IVBP), we prove the non-negativity, existence, and uniqueness of solutions. We also prove the local and global stability of the disease-free equilibrium whenever $\mathcal{R}_c1$ implies the instability of the DFE and its coexistence with at least one endemic equilibrium point. To validate our theoretical results, we perform numerous numerical…
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