Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper,we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals,in which only susceptible individuals can travel freely between the patches.The model has multiple equilibria.We determine conditions that lead to the appearance of a backward bifurcation.The results show that the TB model can have exogenous reinfection among the treated individuals and,at the same time,does not exhibit backward bifurcation.Also,conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained.In case without reinfection,the model has four equilibria.In this case,the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations(FDEs).Numerical simulations confirm the validity of the theoretical results.

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

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 Ro 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 Ro>l,we obtain sufficient conditions under which the endernic 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 a 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 a is reduced from l,while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when o limits to 1.

Optimal control of a fractional-order model for the HIV/AIDS epidemic

In this paper,we present a general formulation for a fractional optimal control problem (FOCP),in which the state and co-state equations are given in terms of the left fractional derivatives.We develop the forward-backward sweep method (FBSM)using the Adamstype predictor-corrector method to solve the FOCP.We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS)with treatment and incorporate three control efforts (effective use of condoms,ART treatment and behavioral change control)into the model aimed at controlling the spread of HIV/AIDS epidemic.The necessary conditions for fractional optimal control of the disease are derived and analyzed.The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people.Also,the maximum levels of the controls and the value of objective functional decrease when the derivative order a limits to 1(0.7≤a <1).In addition,the effect of the fractional derivative order a (0.7≤a <1)on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated.The results show that the derivative order a can play the role of using ART treatment in the model.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

In this paper,we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection,in which the treatment is effective for number of infectious individuals and it fails for some other infectious individuals who are being treated.We show that the model exhibits the phenomenon of backward bifurcation,where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity.Also,it is shown that under certain conditions the model cannot exhibit backward bifurcation.Furthermore,it is shown in the absence of re-infection,the backward bifurcation phenomenon does not exist,in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity.The global asymptotic stability of the endemic equilibrium,when the associated reproduction number is greater than unity,is established using the geometric approach.Numerical simulations are presented to illustrate our main results.