A Novel SIR Model-Based Public Opinion Propagation and Optimal Control

A strategy for optimal control is introduced to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$SI_{1}I_{2}R$</tex> public opinion propagation model. This incorporates the enhanced epidemic model along with real-world public opinion propagation characteristics. This paper primarily focuses on identifying the right method to steer public opinion trends. Initially, The basic reproduction number of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$SI_{1}I_{2}R$</tex> model is calculated using the next-generation matrix method, which can determine whether rumors will prevail. Subsequently, to construct a harmonious network environment, control variables are introduced to formulate an optimal control problem. The controllability of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$SI_{1}I_{2}R$</tex> model with control terms is analyzed, and the minimum value principle is employed to develop an optimal control strategy. Finally, we deduce that false information can be controlled by executing effective control measures. Moreover, the control cost can be minimized. The validity and efficiency of our approaches are demonstrated through comprehensive simulation examples.