Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment

A multi-patch HIV/AIDS epidemic model with fractional order derivative is formulated to investigate the effect of human movement on the spread of HIV/AIDS epidemic among patches. We derive the basic reproduction number R0 and prove that if R0<1, the disease-free equilibrium (DFE) is locally and globally asymptotically stable. In the case of R0>1, we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We also formulate a fractional optimal control problem (FOCP), in which the state and co-state equations are given in terms of the left fractional derivatives. We incorporate into the model time dependent controls aimed at controlling the spread of HIV/AIDS epidemic. The necessary conditions for fractional optimal control of the disease are obtained. The simulation of the model is done with two patches. The numerical results show that implementing all the control efforts decreases significantly the number of HIV-infected and AIDS people in both patches. In addition, the value of Rc, the control reproduction number, for a long time is at its minimum level, and the value of objective functional J(u) increases when the fractional derivative order α is reduced from 1 (0.8≤α≤1).