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In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown‐like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS‐CoV‐2 virus into a system of equations to examine the existence and uniqueness results with the Atangana–Baleanu derivative by using a fixed‐point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India.

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.

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This paper describes the required and adequate conditions for controllability and optimal controls of Atangana–Baleanu–Caputo (ABC) neutral fractional integrodifferential equations (NFIE) with noninstantaneous impulses. Measure of noncompactness, semigroup theory, fractional calculus and fixed point techniques act as the main tools in deriving the results. An illustration is offered to enhance our theoretical outcomes.