Uniform asymptotic stability of a fractional tuberculosis model

When defining a differential operator, one often employs, besides ordinary derivatives, generalized derivatives, which appear in a natural way when considering extensions of differential operators defined on differentiable functions, and weak derivatives, related to the transition to the adjoint operator [9]. Derivatives of fractional and negative orders appear when the differentiation is defined by means of an integral transform, applicable to the domain of definition and range of such generalized differential operator [18]. This is often done in order to obtain the simplest possible representation of the corresponding differential operator of a function and to attain a reasonable generality in the formulation of problems and satisfactory properties of the objects considered [24]. Problems in the theory of differential equations, e.g., problems of existence, uniqueness, regularity, continuous dependence of the solutions on the initial data or on the right-hand side, or the explicit form of a solution of a differential equation defined by a given differential expression, are readily interpreted in the theory of operators as problems on the corresponding differential operator defined on suitable function spaces [5]. One advantage of fractional order differential equations is that they provide a powerful instrument to incorporate memory and hereditary properties into the systems, as opposed to the integer order models, where such effects are neglected or difficult to incorporate [17]. Moreover, in order to precisely reproduce the nonlocal, frequencyand history-dependent properties of power law phenomena, some different modeling tools, based on fractional operators, have to be introduced: see, e.g., [3] and references therein.