Abstract
The main purpose of this paper is to provide new vaccinated models of COVID-19 in the sense of Caputo-Fabrizio and new generalized Caputo-type fractional derivatives. The formulation of the given models is presented including an exhaustive study of the model dynamics such as positivity, boundedness of the solutions and local stability analysis. Furthermore, the unique solution existence for the proposed fractional order models is discussed via fixed point theory.
Numerical solutions are also derived by using two-steps Adams-Bashforth algorithm for Caputo-Fabrizio operator, and modified Predictor-Corrector method for generalised Caputo fractional derivative. Our analysis allow to show that the given fractional-order models exemplify the dynamics of COVID-19 much better than the classical ones. Also, the analysis on the convergence and stability for the proposed methods are performed. By this study, we see that how the vaccine availability plays an important role in the control of COVID-19 infection.
Keywords: Fractional mathematical model, Numerical methods, Caputo-Fabrizio and new generalized Caputo fractional-derivatives
1. Introduction
Throughout this pandemic known as COVID-19, we have experimented a great expansion of cases throughout the world. This situation converts into solid actions that affect the population:
social isolation, use of masks, etc. Mathematical models play a key role for describing infectious diseases such as COVID-19 expansion. The development and investigation of this type of models provide us tools for describing and characterizing its transmission, and thus, we are able to propose successful techniques to foresee, prevent, and control infections, also to ensure that population is well-being. Till present time, numerous mathematical models see [1, 2, 3, 4, 5] have been considered and analyzed to ponder the spreading of infections.
COVID-19, has affected nearly 90% countries across the globe with the infection rate rising rapidly at almost 5% per day. However, the COVID-19 infection behavior is different from nation-to-nation, and is depended on numerous factors. In South Africa, with no exception, almost half a million positive cases have been reported already and is currently one of the five most affected countries globally. To date, various mathematical models have been applied to predict infection rates based on only time-series modes [6, 7]. Very few studies attempted to include other related factors to enhancing the modeling process such as the influence of climatic factors for the disease rapid spread. In the last year, numerical models for COVID-19 plague have been taken into consideration by many scientists with respect to the different nature and its behavior by applying different controls to avoid the spread of this pandemic see [8, 9, 10, 11] and references therein.