Fractional Optimal Control Problem and Stability Analysis of Rumor Spreading Model with Effective Strategies

This study establishes a fractional-order model (FOM) to describe the rumor spreading process. Members of society in this FOM are classified into three categories that change with time—the population that is ignorant of the rumors and does not know them, the population that is aware of the truth of the rumors but does not believe them, and the spreaders of rumors—taking into consideration awareness programs (APs) through media reports as a subcategory that changes over time where paying attention to these APs makes ignorant individuals avoid believing rumors and become better-informed individuals. We prove the positivity and boundedness of the FOM solutions. The feasible equilibrium points (EPs) and their local asymptotical stability (LAS) are analyzed based on the control reproduction number (CRN). Then, we examine the influence of model parameters that emerge with the CRN through a sensitivity analysis.A fractional optimal control problem (FOCP) is formulated by considering three time-dependent control measures in the suggested FOM to capture the spread of rumors; u1, u2, and u3 represent the contact control between rumor spreaders and ignorant people, control media reports, and control rumor spreaders, respectively. We derive the necessary optimality conditions (NOCs) by applying Pontryagin’s maximum principle (PMP). Different optimal control strategies are proposed to reduce the negative effects of rumor spreading and achieve the maximum social benefit. Numerical simulation is implemented using a forward–backward sweep (FBS) approach based on the predictor–corrector method (PCM) to clarify the efficiency of the proposed strategies in order to decrease the number of rumor spreaders and increase the number of aware populations.