Mathematical structure of mosaic disease using microbial biostimulants via Caputo and Atangana–Baleanu derivatives

In this research collection, we analysed two different fractional non-linear mathematical models of a well-known mosaic epidemic of plants, which is underlying by begomoviruses and is distributed to plants by whitefly. We included the role of natural microbial biostimulants which are used to increase plant performance and protects them against mosaic infection. Cause of the big expansion of the mosaic epidemic in various geographical areas, and its large privative economic and societal impacts, it is of major consequence to define dominant optimal control means of this disease. In this paper, we used Caputo (singular type kernel) and Atangana–Baleanu (Mittag-Leffler type kernel) fractional derivatives to define the structure of the proposed mosaic model. We performed some important existence and uniqueness analyses for both models by the applications of fixed point theory and the Picard–Lindelof technique. We derived the numerical solution of the Caputo fractional model by the application of the fourth-order Runge–Kutta method and the Atangana–Baleanu model by the Predictor–Corrector algorithm. A long-term discussion on the graphical interpretations of both models with different infection transmission rate and application proportion rate of MBs (microbial biostimulants) at different fractional-order values have established. We exemplified that under the case of the Mittag-Leffler kernel, the effects of different fractional-order values are much clear as compared to the singular type kernel. The main contribution of this paper is to study the dynamics of mosaic disease at different transmission rates and MBs application rates in the sense of two different kernel types.