Dynamical analysis of spatio-temporal CoVid-19 model

Abstract

This paper focuses on the spatio-temporal CoVid-19 model. The positivity and boundedness are provided based on the standard positivity theorem. The Picard’s iteration is employed to obtain the existence and uniqueness of solution for each state variable. Due to the presence of diffusion terms, the Fourier expansion are employed to provide the locally asymptotical stability for both equilibrium points. Moreover, the appropriate Lyapunov function is a first step before proving the globally asymptotical stability for both equilibrium points. The numerical simulations conclude that the increasing of isolation rate (from $\epsilon =0.03$ to $\epsilon =0.9$) is more effective than the decreasing of social distancing rate (from $\delta =0.4$ to $\delta =0.04$) in reducing the number of infected individuals, where these results are based on the basic reproduction values, i.e., $\mathcal {R}_0=0.8751<1$ for change of isolation rate and $\mathcal {R}_0=0.9344<1$ for change of social distancing rate. The fitting results of our temporal CoVid-19 model with the observed data can be obtained through the least-square technique, neural network, and extended Kalman filter. Based on the model parameters, then we employ neural network (NN) consisting of Levenberg–Marquadt as the training function and Tangent Sigmoid and Purelin as two activation functions to fit the temporal model. The fitting results using neural network of SIQR model are significant based on the mean squared error. Due to the significant results of fitting using extended Kalman filter, we also provide this technique by choosing two tuning parameters of $Q={\text {diag}}([10\;10\;10\;5])$ and $R={\text {diag}}([100\;10\;10\;1])$ for the covariance of process and observation respectively. Moreover, we have the smallest RMSE and MSE of EKF in the date range of $\mathcal {D}_1$ with the computation time 13.002942 s and the fastest computation of EKF in the date range of $\mathcal {D}_5$ with the RMSE 0.014 and the MSE 1.9550e$-$04. The smallest RMSE and MSE of NN are shown in the date range of $\mathcal {D}_1$ with the computation time 11.442112 in unit of seconds and the fastest computation of NN is provided in the date range of $\mathcal {D}_2$ with the RMSE 0.0253 and MSE 6.4100e$-$04.