A Mathematical Model of COVID-19 Using Piecewise Derivative of Fractional Order

Currently the dynamical systems of infectious disease were studied by using various definitions of fractional calculus. Because the mentioned area has the ability to demonstrate the short and long memory terms involved in the physical dynamics of numerous real world problems. In this work, we consider a seven compartmental model for the transmission dynamics of COVID-19 including susceptible (S), vaccinated (V ), exposed (E), infected (I), quarantined (Q), recovered (R), and death (D) classes. We first revisit the fundamental outcomes related to equilibrium points, basic reproduction numbers, sensitivity analysis, and equilibrium points including disease free (DF) and endemic equilibrium (EE) points. In addition, our main goal is to investigate the considered model under the new aspect of fractional calculus known as piecewise fractional order operators. The mentioned operators have the ability to demonstrate the multi phase behaviours of the dynamical problems. The said characteristics cannot be described by using the traditional operators. We apply the tools of nonlinear analysis to deduce sufficient conditions for the existence of at least one solution and its uniqueness. Additionally, we also investigate the results related to stability analysis of Ulam-Hyers (UH) type. Finally, we extend the concepts of RK2 method to form a sophisticated algorithm to simulate the results graphically. We present the numerical results for various fractional orders.