Uniform asymptotic stability of a fractional tuberculosis model

Weronika Wojtak,Delfim F.M Torres,Cristiana J Silva,Abdon Atangana,Zakia Hammouch,Gisèle Mophou,Jordan Hristov

doi:10.1051/mmnp/2018015

Weronika Wojtak, Delfim F.M Torres + Show 5 more

Open Access

https://doi.org/10.1051/mmnp/2018015

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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 anyα∈ (0, 1). Numerical simulations for the stability of the endemic equilibrium are provided.

Highlights

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]
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]
In this work we proposed a Caputo fractional-order tuberculosis (TB) model

Summary

Introduction

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]. 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. Existence and uniqueness of solutions are proved, and stability of equilibrium points studied Numerical solutions of such models were obtained [2]. We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB), based on the nonlinear differential system studied in [43], and investigate its stability.

Preliminaries on the Caputo fractional calculus

Uniform asymptotic stability of the endemic equilibrium

Numerical simulations

Conclusions

Methods