Caputo fractional derivative

This article studies a mathematical model of the fractional order of tuberculosis (TB).  It describes the dynamics of the spread of tuberculosis among smokers.  The purpose of this research is to protect vulnerable people against the virus.  According to the survey results, the required model has an equilibrium point: the disease-free equilibrium point $E_f$.  We also analyze the local stability of this equilibrium point of the model, using the basic reproduction number $\mathcal{R}_{0}$ calculated according to the new generation method.  In our model, we include three

Scale insects are parasitic insects that attack many indoor and outdoor plants, including cacti and succulents.  These insects are among the frequent causes of diseases in cacti: for the reason that they are tough, multiply in record time and could be destructive to these plants, although they are considered resistant.  Mealybugs feed on the sap of plants, drying them out and discoloring them.  In this research, we propose and investigate a fractional model for the transmission of the Cochineal.  In the first place, we prove the positivity and boundedness of solutions i