New crossover lumpy skin disease: Numerical treatments

This work expands on a novel piecewise mathematical model of Lumpy Skin Disease (LSD) in three time intervals by utilizing fractional stochastic derivatives and variable-order differential equations. The LSD model is developed into two crossover hybrid variable-order derivatives. This study's piecewise mathematical model representation of lumpy skin disease has revealed a property that has never been taken into account or seen in previous research employing mathematical models based on classical, different fractional derivatives and variable order fractional derivatives. The Caputo derivative and the Riemann-Liouville integral are merged linearly to produce the hybrid fractional order derivative. The variable-order fractional and hybrid fractional operators are approximated using the Grünwald-Letnikov approximation. We introduce the hybrid variable-order operator combined with the non-standard finite difference method. The stability, boundedness, positivity, and existence of the suggested model are examined. The effectiveness of the techniques and the validity of the theoretical results were verified through a number of numerical experiments.