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On the basis of nonlinear dynamic model, we research the propagation of rumors after emergencies, describe the impact of media coverage and emergency strategies by government on the transmission dynamics of information, and then obtain the basic reproduction number of rumor spreading. In order to overcome the limit of traditional methods of the static decision problem, the dynamic optimal control model for the rumor spreading is proposed based on the theorem of the optimal control. An optimal objective based on the maximum social utility is established and the optimal solution is acquired by using the Pontryagin Maximum Principle. Finally, numerical simulations show that the dynamic optimal control method has obvious superiority in modeling compared with the method without control. By means of the dynamic optimal control of media coverage and emergency strategies for government, the final rumor scale and the peak of spreaders can be effectively reduced.

When emergencies occur, people's thirst for information is different from normal. If the propagation of information is not transparent, detailed, and controlled by the government and the media at this time, it may lead to people’s psychological and emotional tension, unnecessary public panic, and economic loss [

In recent years, many scholars have discussed the rumor spreading from different perspectives through qualitative analysis and theoretical modeling. Rapoport [

With the development of dynamic theories, rumor spreading based on dynamics has attracted more and more attention. Dickinson [

Media reports are known as “subcutaneous injection” and “magic bullet” theory, and it is believed that media coverage has a great influence on public’s understanding and emergency management [

When an emergency occurs, it will cause or may cause serious social harm. In order to avoid losses, some measures must be taken as soon as possible in response to the sudden incidents, such as natural disasters, accident disasters, public health events, and social security incidents. However, the main effect of media coverage is to popularize science education, improve the identification of the ignorant, reduce the probability of the ignorant believing rumors, and thus reduce the spreading rate of rumor dissemination, rather than prevent the spread of rumors. On the contrary, when rumors break out, relevant emergency strategies can not only prevent individuals from believing in rumors, but also reduce the spreader's enthusiasm for dissemination. Huo et al. [

The spread of rumor not only caused the people to panic, but also brought huge economic losses. Therefore, many scholars have made great contributions to the immune control of rumors [

From the previous researches, the scholars mainly focus on the state of the final spreading, but ignore the influence factors and the control problems during the process. The individual’s spreading behavior will be determined by the amount of information obtained by an individual preference; meanwhile, the media and government measures will affect the rumor spreading. The facts indicate that the popularization of scientific knowledge, emergency strategies, authoritative information published by the media, expansion of information coverage, and guiding public opinion are the key to rumor governance in emergency management.

Considering the positive impact of media coverage and the emergency strategies of government, we extended the classical DK model. On the basis of the optimal control theory, the optimal control strategy of rumor spreading is discussed, and a feasible management and control scheme is proposed for the control of the rumor in emergency management and obtainment of an emergency management strategy with maximization of social utility.

The organization of this paper is as follows: the rumor spreading model is formulated in the next section. Section

The spread of rumors will be influenced by media coverage. The media coverage improves personal discrimination ability through daily popular science information, thereby reducing the spreading rate of rumors. On the other hand, in order to reduce the loss caused by emergencies, it is necessary to take timely and effective emergency strategies in dealing with rumor spreading. Such measures include holding a press conference and releasing relevant facts for emergencies, which can effectively prevent the spread of rumors.

Based on the study of the classic D-K model, this paper adds two factors, media coverage and emergency strategies by government, to the spread of rumors, analyzes the spread mechanism of the rumor under the influence of these two factors, and lays a theoretical foundation for further discussion of the control strategy of the rumors.

According to the basic assumptions of the model, an open and mixed population in a certain area can be divided into 3 categories: the ignorant (nodes who have never heard rumors), the spreaders (nodes who know and spread rumors), and the stiflers (nodes who understand the truth of rumors and stop spreading rumors). Similar to the traditional models, at any time each and every node in propagation network is assumed to be in one of three possible categories. Let

Schematic diagram of rumor spreading under the influence of media coverage and emergency strategies.

Based on the idea of differential equation modeling, the D-K model is further extended to establish a nonlinear dynamic model considering the impact of media coverage and emergency strategies by government. According to the above assumptions, one has the following system of

According to the previous studies,

This paper analyzes the mathematical model based on the stability related theory of differential equations, and due to the fact that the model depicts the dynamic propagation process of rumor in the crowd, the parameters involved are assumed to be nonnegative.

For the sake of simplicity, we set

With no loss of generality, we assume that

Obviously, system (

Furthermore, the existence of rumor spreading equilibrium point

Following the recipe of van den Driessche and Watmough [

where the rate of appearance of new spreaders is

the Jacobian matrices of

According to the concept of next generation matrix and reproduction number given in [

The basic reproductive number

Based on the analysis of the previous section, we find that it is not enough for rumor management to discuss the dynamic process of rumor, and the optimal control techniques are of great use in optimal strategies to control various kinds of rumors. Based on the optimal control theory and numerical simulation method, we further expand the system (

In real life, when rumors break out, some control measures will be taken by officials including precontrol measures (such as popularization of science education, Microblog hot tweens, press conference, and other media coverage measures) and postcontrol measures (by means of isolating and compulsive punishments of spreaders to transform them into stiflers).

On one hand, when rumors prevail, the media should increase the broadcast to suppress them; on the other hand, when the rumor is controlled, the amount of media coverage should be reduced accordingly. So, we assume media coverage is a dynamically adjusted process with the density of the spreaders in the current environment (the range of rumor spreading). Meanwhile, when there are mass spreaders in the environment, a series of emergency strategies by government must be taken to reduce the harm of rumors, such as forced isolation, enhanced punishment, and cut off sources of rumors spreading. But taking into account the control costs and other factors, the emergency strategies by government can be slackened a little when the rumor situation slows down. From this perspective, emergency measures are also a dynamic adjustment process.

In the actual model, we select the Lebesgue square-integrable control functions

The objective of optimal control is to minimize the negative effects of rumor dissemination, in the meantime, and maximize the positive social utility. This paper selects the following aspects to consider the issue of social utility of rumor spreading:

Maximize the number of ignorants and stiflers.

Try to minimize the number of rumor spreaders.

Minimize the controlling costs of variable control in the process.

Now, we choose an objective function to be

where

The optimal control variable

where

According to the conclusion of the existence of the optimal control solution by Fleming and Rishel [

Under the limit of (

In order to prove the conclusion of Theorem

The control set and corresponding state variable are not empty.

The control set is a closed set of convexity by definition.

The upper bound of the solution of system (

The integral of the system (

According to Theorem 9.2.1 of Lukes [

Then system (

Theorem

Pontryagin Maximum Principle requires the use of auxiliary functions to explore the optimal value problem. The differential equation system (

Based on the above analysis, the Hamiltonian for the control problem is constructed as

where

Next, we will derive the necessary conditions for the optimal control strategy by means of the Pontryagin Maximum Principle [

Let

To determine the adjoint equations and the transversality conditions, we use the Hamiltonian (5.6). By using the necessary condition for optimal control problems and differentiating the Hamiltonian (5.6) with

Hamiltonian minimizing condition: at the interior points, we have

The transversality conditions are obvious, namely,

Therefore, according to the analysis above, we can easily obtain the following optimality system:

To complement the mathematical analysis carried out in the previous section, a series of numerical simulations were performed to support and extend our theoretical results on BA scale-free networks (

From Figure

The relationship between the basic reproduction number

First, the dynamic propagation process of the rumor without the optimal control is discussed. Figure

The impact of different values of

The impact of different values of

The influence of different values

In this part, based on the theory of optimal control, combined with Pontryagin Maximum Principle, we respectively simulated the dynamic changes of four groups of people in different situations, such as Figures

Relevant parameter values.

Parameters | | | | | | | | | | | | | |

| |||||||||||||

Values | 0.0001 | 0.5 | 0.133 | 0.133 | 0.001 | 0.000005 | 0.5 | 5 | 0.05 | 2 | 100 | 100 | 200 |

Evolution of densities of spreaders and stiflers with control or without control.

Time evolution of control variables ((a) for

Figure

Figure

Based on the classical SIR model, an improved SIRM rumor spreading model is proposed. On the one hand, this paper introduces the media coverage into the rumor spreading model, and with the propaganda function of the media, the rumor can recognize the reality and thus manage the rumor from the source. On the other hand, the emergency strategies by government are also used to block the diffusion of spreaders. Then the dynamic equations are obtained and the threshold of rumor spreading is calculated. At the same time, for the purpose of maximizing the social utility and minimizing the cost of rumor governance, an optimal control problem has been formulated. The existence of an optimal control has been shown, and the corresponding optimality system has been derived. The proposed optimal control scheme can achieve a low level of spreaders at a low cost. Finally, the theoretical analysis is verified by numerical simulation.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The work was partially supported by the National Natural Science Foundation of China (71774111).