Fig 1.
Flow diagram of disease transmissions.
Table 1.
Sensitivity index for μ2 has a higher impact on the
.
The δ and Ψ are the most sensitive parameters but both (birth and natural death rate) cannot be controlled. The force of interaction μ2 and treatment rate of the infectious ρ2 are the most positive sensitive parameters to .
Fig 2.
Sensitivity of the parameters against the selective model parameters.
The increasing graphs show positive sensitivity index and the decreasing curves show the negative sensitivity index of the parameter to .
Fig 3.
Flow diagram of the updated model: This figure is the updated form of the Fig 1 with controls and
which represent the use of condoms and change in sexual habits, respectively.
Table 2.
Parametric values.
Fig 4.
Cost functional and optimal control (use of a condom) versus time.
It is evident that the functional reaches its lowest under the optimal control in the 32th, 30th, and 28th iterations for α = 0.6, α = 0.8, and α = 0.95, respectively. The optimal control rates are at maximum for the whole tenure and identify that to control the further spread of HIV/AIDS, the infectious population should always use condoms during sex.
Fig 5.
State variables versus time before (doted) and after (solid) optimization for .
Susceptible individuals increase under the application of optimal control with each fractional-order derivative. A slight difference can be observed for different fractional orders. In addition, the number of both identified and unidentified exposed individuals decreased after optimal control.
Fig 6.
Continuations of Fig 5, showing that the number of infectious individuals in the population declines rapidly after optimal control .
Furthermore, the number of people with full-blown AIDS goes to zero under the use of condoms as optimal control. Thus, a decrease in exposed and infectious population is the target of this optimal control strategy.
Fig 7.
The objective functional and the behavior of the optimal control variable (change in sexual habits) are given here.
A change in sexual habits is also an effective strategy to control the spread of HIV/AIDS. It is also clear that a greater fractional order derivative is more suitable to minimize the objective functional and the cost of the control.
Fig 8.
For Case-II, the system’s behavior is simulated with optimal control for , the rate at which susceptible individuals have changed and maintained their sexual habits and recovered.
The number of susceptible people recovers without entering an infectious period more rapidly after optimization. Thus, the number of exposed individuals gradually decreases to zero.
Fig 9.
Continuous of Fig 8 Case-II.
Infectious and later stages of the infection, i.e., fully developed AIDS (A(t)) and treatment (T(t)) are also decreased. However, the number of people who recovered grows gradually.
Fig 10.
The objective functional is minimized under the influence of both optimal controls .
The graph of the optimal controls shows that when a maximum number of susceptible individuals have changed their sexual habits, the burden of using condoms also decreases. Notably, an increase in fractional order increased the number of iterations but minimized the objective functional and the cost of the controls.
Fig 11.
State variables before and after optimization of (Case-III).
The dynamics of the state variables decrease rapidly after optimization of change in sexual habits and the use of condoms
.
Fig 12.
State variables before and after optimization for Case-III.
The curves for state variables I, A, and T decrease gradually due to the change in sexual habits and the use of condoms. The increase in recovered individuals is largely due to the implementation of changing sexual habits.