Kalman filter based on a fractional discrete-time stochastic augmented CoVid-19 model

Abstract

In this paper, we study the dynamics of the CoVid-19 outbreak in Semarang, Indonesia, using a fractional CoVid-19 model. We first determine the effects of the isolation rate $∊$ and infection rate $\beta$ on the reproduction number $R_0$ and infected number V. We find that $R_0$ is directly proportional to $\beta$ and inversely proportional to $∊$. For V, the effect of physical distancing is not as significant as changing $∊$. As $∊$ increases, V decreases, the number of susceptible individuals increases, the number of quarantined individuals decreases sharply, and the number of recovered individuals decreases. Moreover, the effect of vaccination is also considered. The combination of physical distancing, isolation, and vaccination has a significant impact on reducing the number of infected individuals. Analysis of dynamical systems allows us to understand the characteristics of our model, such as its boundedness and non-negativity, the existence of equilibrium points, the existence and uniqueness of solutions, and the local and global stability. To validate our fractional CoVid-19 model, we introduce the fractional extended Kalman filter (FracEKF) as a prediction method and compare the results against reported CoVid-19 data. FracEKF is a modified version of the basic extended Kalman filter with a time-fractional memory effect. The prediction results illustrate the accuracy of this model in terms of the root mean square error (RMSE), normalized root mean square error (NRMSE), and mean absolute percentage error (MAPE) for each fractional-order. Varying $∊$ reproduces the trends observed in the reported data for the number of infected individuals, i.e., when $∊$ increases, the infected number decreases. Moreover, a higher fractional-order results in higher model accuracy. Furthermore, higher values of the process noise $Q_f$ give smaller errors, whereas higher values of the observation noise $R_f$ produce higher errors. $Q_f$ and the fractional-order $\alpha$ are inversely proportional to $\mathrm{RMSE},\mathrm{NRMSE}$, and MAPE, whereas $R_f$ is directly proportional to $\mathrm{RMSE},\mathrm{NRMSE}$, and MAPE.