Stability analysis and optimal control of rumor spreading model under media coverage considering time delay and pulse vaccination

Abstract

Information asymmetry often interferes with the public's judgment, creates unnecessary suspicion, and leads to the wanton spread of rumors. Therefore, it is necessary to carry out regular science education to improve personal cultural literacy and restrain the spread of rumors. In this paper, regular popular science education is introduced into the rumor spreading model as periodic pulse vaccination. On this basis, an improved rumor spreading model under media coverage is constructed in combination with time delay. Two thresholds are calculated by using the comparison theorem of impulse differential equation. The conditions of global attractively and persistence of rumor spreading are analyzed. Then, the optimal control problem under pulse vaccination is designed to minimize the rumor spreading scale and control cost, then the optimality conditions are obtained by using the Pontryagin's minimum principle. Finally, the theoretical results are verified by numerical simulations. It shown that increasing the proportion of pulse vaccination or shortening the pulse period or prolonging time delay contribute to the restraint of rumors. Increasing the pulse period or decreasing the delay time both aggravate the rumor spreading. Meanwhile, the introduction of the optimal pulse control can shorten the spreading time of rumors and reduce the losses caused by rumors.