Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control
Author
Ahmad Naik, Parvaiz
Zu, Jian
Owolabi, Kolade M.
Institutions
Abstract
In this paper, a nonlinear fractional order epidemic model for HIV transmission is proposed and analyzed
by including extra compartment namely exposed class to the basic SIR epidemic model. Also, the infected
class of female sex workers is divided into unaware infectives and the aware infectives. The focus is on
the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The exposed class contains those susceptible males
in the population who have sexual contact with the female sex workers and are exposed to the infection
directly or indirectly. The Caputo type fractional derivative is involved and generalized Adams-BashforthMoulton method is employed to numerically solve the proposed model. Model equilibria are determined
and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. Analysis of the model demonstrates that the population is free from
the disease if R0 < 1 and disease spreads in the population if R0 > 1. Meanwhile, by using Lyapunov
functional approach, the global dynamics of the endemic equilibrium point is discussed. Furthermore,
for the fractional optimal control problem associated with the control strategies such as condom use for
exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model.
The existence of fractional optimal control is analyzed and the Euler-Lagrange necessary conditions for
the optimality of fractional optimal control are obtained. The effectiveness of control strategies is shown
through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the HIV patients. It significantly reduces the number of
HIV/AIDS patients during the whole epidemic.