Global stability and optimal control of a two-patch tuberculosis epidemic model using fractional-order derivatives

Abstract

In this paper, we propose a fractional-order and two-patch model of tuberculosis (TB) epidemic, in which susceptible, slow latent, fast latent and infectious individuals can travel freely between the patches, but not under treatment infected individuals, due to medical reasons. We obtain the basic reproduction number R0 for the model and extend the classical LaSalle's invariance principle for fractional differential equations. We show that if R0 < 1, the disease-free equilibrium (DFE) is locally and globally asymptotically stable. If R0 > 1, we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We extend the model by inclusion the time-dependent controls (effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches), and formulate a fractional optimal control problem to reduce the spread of the disease. The numerical results show that the use of all controls has the most impact on disease control, and decreases the size of all infected compartments, but increases the size of susceptible compartment in both patches. We, also, investigate the impact of the fractional derivative order α on the values of the controls (0.7 ≤ α ≤ 1). The results show that the maximum levels of effective treatment controls in both patches increase when α is reduced from 1, while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when α limits to 1.