On some recent advances in fractional order modeling in engineering and science

In this chapter, we discuss and illustrate recent developments in the areas of the application of fractional calculus. We adopt the basic knowledge of the fractional operator, especially the Caputo fractional operator, to represent these models. We first demonstrate a fractional model simulating the spread of COVID-19, which has an impact on people and spreads quickly among individuals. We use the base collocation method of the dynamic Vieta–Lucas polynomial type to shed light on the dynamics of such model. The basis for the fractional COVID-19 model is given along with the properties of the proposed technique. The existence and uniqueness of the solution for the suggested fractional COVID-19 model are proved, and we establish that the fractional COVID-19 model dissipates, is symmetric, and is invariant. The precision an validation of the mathematical model are shown, and 2D plots are presented to demonstrate the effectiveness and success of the strategy. As a second model, we simulate the spread of Zika virus through a fractional order model using the Sumudu transform method. The Zika virus is considered as one of the most dangerous viruses which can rapidly spreads and is transmitted through either mosquitoes or sexual contact and has a dangerous effect especially on pregnant women. This fractional model succeeds in providing effective simulations for different compartments. The symmetry, invariance, and dissipation of the nonlinear fractional model are examined and another version of the model with optimal control is discussed. We tested the method for two cases of two different sets of initial conditions, which in both cases provide accurate approximate solutions.