Uniform asymptotic stability of a fractional tuberculosis model

Abstract

We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB). Uniform asymptotic stability of the unique endemic equilibrium of the fractional-order TB model is proved, for any α ∈ (0, 1). Numerical simulations for the stability of the endemic equilibrium are provided.Uniform asymptotic stability of a fractional tuberculosis model Communicable diseases have always been an important part of human history [6]. Fractional-order differential system models for infectious disease dynamics have been introduced in recent years [1,14,17,28,30,44]. In [30], the authors propose a fractional-order model and show, through numerical simulations, that the fractional models fit better the first dengue epidemic recorded in the Cape Verde islands off the coast of west Africa, in 2009, when compared with the standard differential model. The authors of [17] show that a nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A (H1N1) at the population level. In [28], a fractional model for malaria transmission is considered and numerical simulations are done for the variation of the values of the fractional derivative and of the parameter that models personal protection. In [44], fractionalorder derivatives are introduced into an HIV infection model and local asymptotic stability is proved. The authors of [14] introduce fractional-order derivatives into a model of HIV infection of CD4 + T-cells and analyze the local asymptotic stability of the equilibrium points. In [41], the uniform asymptotic stability is proved extending the Volterra-type Lyapunov functions to fractional-order epidemic systems. Fractional-order predator-prey models are investigated in [2]. In particular, existence and uniqueness of solutions are proved, and stability of equilibrium points studied. Numerical solutions of such models were obtained [2]. Here we are interested to investigate fractional calculus with respect to tuberculosis.

Optimal Control and Sensitivity Analysis of a Fractional Order TB Model

“…Using the values of the parameters as proposed in [24], we obtain results very different from those that the authors present there. We noticed that the value of parameter µ presented in [24] (µ ′ = 0.143), the most sensitive 4 OPTIMAL CONTROL AND SENSITIVITY ANALYSIS OF A FRACTIONAL ORDER TB MODEL Table 2. Sensitivity of R 0 evaluated for the parameter values given in Table 1.…”

“…A sensitivity analysis was carried out for the basic reproduction number of the TB model [24], proving that the most important parameter to have into account is the natural death rate µ. The application of optimal control to the fractional TB model shows that treatment is effective on the reduction of infected individuals.…”

“…In this section, we consider a Caputo fractional-order tuberculosis (TB) model due to [24]. The model describes the dynamics of a population that is susceptible to infection by the Mycobacterium tuberculosis with incomplete treatment.…”

“…In [21], it is proposed a multi-strain TB model with variable-order fractional derivatives and a numerical scheme to approximate the endemic solution, numerically, is developed. More recently, in 2018, a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis was proposed, based on the nonlinear differential system studied in [25], and its stability is investigated [24]. Such model is here subject to fractional optimal control theory [4,7], and its usefulness to tackle a TB epidemic scenario discussed.…”

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

“…Similarly as classical dynamical systems, stability analysis is a central task in the study of fractional dynamical systems, which has attracted increasing interest of many researchers [9,21]. For finite dimensional systems, the stability concept for fractional differential systems equipped with the Caputo derivative is investigated in many works [22].…”