Global Stability and Optimal Control in a Dengue Model With Fractional-Order Transmission and Recovery Process

Abstract

The current manuscript introduces a single-strain dengue model developed from stochastic processes incorporating fractional-order transmission and recovery. The fractional derivative has been introduced within the context of transmission and recovery process, displaying characteristics similar to tempered fractional ( $TF$) derivatives. It has been established that under certain condition, a function's $TF$ derivatives are proportional to the function itself. Applying the following observation, we examined the stability of several steady-state solutions, such as disease-free and endemic states, in light of this newly formulated model, using the reproduction number ( $R_0$). In addition, the precise range of epidemiological parameters for the fractional-order model was determined by calibrating weekly registered dengue incidence in the San Juan municipality of Puerto Rico, from April 9, 2010, to April 2, 2011. We performed a global sensitivity analysis method to measure the influence of key model parameters (along with the fractional-order coefficient) on total dengue cases and the basic reproduction number ( $R_0$) using a Monte Carlo-based partial rank correlation coefficient (PRCC). Moreover, we formulated a fractional-order model with fractional control to assess the effectiveness of different interventions, such as reducing the recruitment rate of mosquito breeding, controlling adult vectors, and providing individual protection. Also, we established the existence of a solution for the fractional-order optimal control problem. Finally, the numerical experiment illustrates that policymakers should place importance on the fractional-order transmission and recovery parameters that capture the underlying mechanisms of disease, along with reducing the spread of dengue cases, carried out through the implementation of adult vector control.