Atangana-Baleanu fractional dynamics of dengue fever with optimal control strategies

Dengue fever, a vector-borne disease, has affected the whole world in general and the Indian subcontinent in particular for the last three decades. Dengue fever has a significant economic and health impact worldwide; it is essential to develop new mathematical models to study not only the dynamics of the disease but also to suggest cost-effective mechanisms to control disease. In this paper, we design modified facts about the dynamics of this disease more realistically by formulating a new basic $S_h{E}_h{I}_h{R}_h$ host population and $S_v{I}_v$ vector population integer order model, later converting it into a fractional-order model with the help of the well-known Atangana-Baleanu derivative. In this design, we introduce two more compartments, such as the treatment compartment $T_h$, and the protected traveler compartment $P_h$ in the host population to produce $S_h{E}_h{I}_h{T}_h{R}_h{P}_h$. We present some observational results by investigating the model for the existence of a unique solution as well as by proving the positivity and boundedness of the solution. We compute reproduction number ${\mathcal{R}}_0$ by using a next-generation matrix method to estimate the contagious behavior of the infected humans by the disease. In addition, we prove that disease free and endemic equilibrium points are locally and globally stable with restriction to reproduction number ${\mathcal{R}}_0$. The second goal of this article is to formulate an optimal control problem to study the effect of the control strategy. We implement the Toufik-Atangana scheme for the first time to solve both of the state and adjoint fractional differential equations with the ABC derivative operator. The numerical results show that the fractional order and the different constant treatment rates affect the dynamics of the disease. With an increase in the fractional order and the treatment rate, exposed and infected humans, as well as the infected mosquitoes, decrease. However, the optimal control analysis reveals that the implemented optimal control strategy is very effective for disease control.