In this article, a fractional model of the Capsicum annuum (C. annuum) affected by the yellow virus through whiteflies (Bemisia tabaci) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal control methods are discussed to decrease the infectious B. tabaci and C. annuum by applying the Verticillium lecanii (V. lecanii) with the Atangana–Baleanu derivative. Numerical results described the population of plants and comparison values of using V. lecanni. The results show that using 60% of V. lecanni will control the spread of the yellow virus in infected B. tabaci and C. annuum in 10 days, which helps farmers to afford the costs of cultivating chili plants.

We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section.