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Optimization Algorithm of New Media Hot Event Push Based on Nonlinear Differential Equation

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 22 Feb 2022
Accepted: 19 Apr 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

New media hotspot mass emergencies refer to emergencies caused by contradictions among the people or caused by improper handling. Actively preventing, resolving, and properly handling various mass emergencies has become a hot issue that governments at all levels must focus on solving.

Differential game theory is a new way of thinking for solving coordinated control problems. This theory originated from the research on the pursuit and escape of the two sides in the military confrontation carried out by the US Air Force in the 1950s. It is a combination of optimal control and game theory [1]. The algorithm has been used in more and more fields such as economics, management, and environmental science. Most of the research on differential games is concentrated on advertising. Among them, academia first proposed the dynamic advertising competition model. Some scholars have proposed three kinds of equilibrium under the linear objective function. Some scholars have modified the original model by changing the dynamic equation of the system. Some scholars have proposed a differential game approach to consider IT constraints in the e-retail industry.

Differential game is a dynamic game in which players participate in uninterrupted time. The time difference for each stage is narrowed down to the minimum limit. This is in line with the characteristics of repetition, continuity, and a long cycle of mass emergencies. Therefore, this paper constructs a differential game model for an emergency response to mass incidents based on Kumar's research on the e-commerce enterprise advertising optimization model and other related literature [2]. We strive to achieve the goal of minimizing the total social loss through economic subsidies, taking into account the government's use of police force and the degree of social legality.

Differential game model of mass emergencies

The basic assumptions and definitions of this paper are as follows:

At the time t, the proportion of the social gathering population in the total local population is expressed as x(t) ∈ (0, 1). Certain social conflicts cause the gathering of people. At the same time, it is formed temporarily by the aggregation of a specific group of people or an unspecified majority [3]. Gathering crowds may cause certain harm to social order. Assuming that the total local population remains unchanged, it is set to 1. h It represents the unit social loss caused by the unit gathering of people. r is the discount rate.

If an event occurs randomly and independently with fixed intensity x, then the number (number) of the event in unit time can be regarded as obeying the Poisson distribution. The participants in the aggregation follow a non-homogeneous Poisson distribution. At time t the mean is x(t).

Legal thinking and disposal methods have become an important way to resolve social conflicts. According to the requirement of “improving the ability of leading cadres to use the rule of law thinking and the rule of law to resolve conflicts and maintain stability,” the degree of the social legal system at the time t indicates v (t), 0 ≤ v (t) ≤ 1. This paper assumes that the degree of illegality that the gathering crowd may take to protest follows an exponential distribution with mean 1 / v. The deployment intensity of the government police force is μ (t). The social rule of law is greater than police deployment (v > μ). The economic subsidy used by the government to appease the gathering masses at the time t is A (t). The unit utility of the government's economic subsidy to appease the gathered masses to the evacuation of the gathered masses is represented by the parameter β0 here.

The number of imitators depends on the size of the potential group that could become a protest crowd [4]. The parameter represents the imitation ratio β1. The social and economic losses caused by the protesters of the unit are recorded as h. The local government's economic compensation for the protesters’ reasonable economic interests is expressed as A (t). These losses will continue to accumulate as the protest events are delayed. R(t)=vx(t)vx(t)+μ R\left(t \right) = {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}

The above formula is stable if and only when the social rule of law is greater than the aggregated population in the total local population. So the percentage of dropouts is always positive [5]. According to the Vidale-Wolfe model, the dynamic equation of the system is x˙(t)=dx(t)dt=β1(1x(t))β0A(t)x(t)x(0)=x0 \matrix{{\dot x\left(t \right) = {{dx\left(t \right)} \over {dt}} = {\beta _1}\left({1 - x\left(t \right)} \right) - {\beta _0}A\left(t \right)x\left(t \right)} \hfill \cr {x\left(0 \right) = {x_0}} \hfill \cr}

The objective function is the minimum sum of the social losses caused by mass events such as protest marches and the government's economic subsidies to appease the gathering crowd: min{0[h(x(t)vx(t)vx(t)+μ)+A(t)]ertdt} \min \left\{{\int_0^\infty {\left[{h\left({x\left(t \right) - {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}} \right) + A\left(t \right)} \right]{e^{- rt}}dt}} \right\}

Among them, (x(t)vx(t)vx(t)+μ) \left({x\left(t \right) - {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}} \right) represents the gathering people participating in mass events. [h(x(t)vx(t)vx(t)+μ)+A(t)] \left[{h\left({x\left(t \right) - {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}} \right) + A\left(t \right)} \right] represents the social losses caused by mass events and social subsidies provided by the government. ert represents discounting. We rewrite the objective function as: max{0[h(vx(t)vx(t)+μx(t))A(t)]ertdt} \max \left\{{\int_0^\infty {\left[{h\left({{{vx\left(t \right)} \over {v - x\left(t \right) + \mu}} - x\left(t \right)} \right) - A\left(t \right)} \right]{e^{- rt}}dt}} \right\}

Usually, the current Hamiltonian value is calculated first: H=h(x(t)vx(t)vx(t)+μ)A(t)+λ[β1(1x(t))β0A(t)x(t)] H = h\left({x\left(t \right) - {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}} \right) - A\left(t \right) + \lambda \left[{{\beta _1}\left({1 - x\left(t \right)} \right) - {\beta _0}A\left(t \right)x\left(t \right)} \right]

λ(t) is the shadow price of the state variable x(t). It is the change value of the objective function caused by each unit change of x(t) at time t. We rewrite the Hamiltonian value as H=h(x(t)vx(t)vx(t)+μ)+λβ1(1x(t))HAA H = h\left({x\left(t \right) - {{vx\left(t \right)} \over {v - x\left(t \right) + \mu}}} \right) + \lambda {\beta _1}\left({1 - x\left(t \right)} \right) - {H_A}A

Among them HA = λβ0x(t) + 1. It can be known from formula (6): (a) if HA > 0, A = 0; (b) if HA = 0, A is to be determined; (c) if HA < 0, A is infinite.

This paper mainly studies the case of time HA = 0 solution. Because this situation can keep the system in a stable state, we can obtain a unique analytical solution [6]. In a steady state, the above relationship must be satisfied simultaneously: HA=0 {H_A} = 0 H˙A=dHAdt=0 {\dot H_A} = {{d{H_A}} \over {dt}} = 0

From the necessary conditions of Pontrayagin, it can be seen that the following formula holds dλdt=rλHx=rλ[h(v(v+μ)(vx+μ)2)λβ1λβ0A] {{d\lambda} \over {dt}} = r\lambda - {{\partial H} \over {\partial x}} = r\lambda - \left[{h\left({{{v\left({v + \mu} \right)} \over {{{\left({v - x + \mu} \right)}^2}}}} \right) - \lambda {\beta _1} - \lambda {\beta _0}A} \right]

From formula (6) and formula (7), it can be known that: λ=1/β0x \lambda = - 1/{\beta _0}x

We bring Equation (9) into Equation (7) to get: (rλ+h[1v(v+μ)(vx+μ)2]+λβ1+λβ0A)β0xλβ0[β0Axβ1(1x)]=0 \left({r\lambda + h\left[{1 - {{v\left({v + \mu} \right)} \over {{{\left({v - x + \mu} \right)}^2}}}} \right] + \lambda {\beta _1} + \lambda {\beta _0}A} \right){\beta _0}x - \lambda {\beta _0}\left[{{\beta _0}Ax - {\beta _1}\left({1 - x} \right)} \right] = 0

We can obtain from equations (10) and (11): h[1v(v+μ)(vx+μ)2]=β1+rxβ0x2 h\left[{1 - {{v\left({v + \mu} \right)} \over {{{\left({v - x + \mu} \right)}^2}}}} \right] = {{{\beta _1} + rx} \over {{\beta _0}{x^2}}}

The proportion x of the aggregated population in the total local population must satisfy the above formula (12). Since the proportion x of the aggregated population in the total local population remains unchanged in a steady-state, Equation (2) is equal to 0. At this point we can get A=β1(1x)/β0x A = {\beta _1}\left({1 - x} \right)/{\beta _0}x

Theorem 1

If x<v/((2hv2β0/β1)13+1) x < v/\left({{{\left({2h{v^2}{\beta _0}/{\beta _1}} \right)}^{{1 \over 3}}} + 1} \right) , then dxdu<0 {{dx} \over {du}} < 0 and dAdu>0 {{dA} \over {du}} > 0 .

Proof: Rewrite formula (11) into the following formula F(x,μ)=h[1v(v+μ)(vx+μ)2]=β1+rxβ0x2=0 F\left({x,\mu} \right) = h\left[{1 - {{v\left({v + \mu} \right)} \over {{{\left({v - x + \mu} \right)}^2}}}} \right] = {{{\beta _1} + rx} \over {{\beta _0}{x^2}}} = 0 , then F'μ=dFdμ=hv[v+μ+x(vx+μ)3] {F^{'}}_\mu = {{dF} \over {d\mu}} = hv\left[{{{v + \mu + x} \over {{{\left({v - x + \mu} \right)}^3}}}} \right] , F'x=dFdx=rx+2β1β0x32hv(v+μ)(vx+u)3 {F^{'}}_x = {{dF} \over {dx}} = {{rx + 2{\beta _1}} \over {{\beta _0}{x^3}}} - {{2hv\left({v + \mu} \right)} \over {{{\left({v - x + u} \right)}^3}}} .

dxdμ=F'μF'x=v[v+μ+x(vx+μ)3]rx+2β1β0x32hv(v+μ)(vx+u)3 {{dx} \over {d\mu}} = - {{{F^{'}}_\mu} \over {{F^{'}}_x}} = {{v\left[{{{v + \mu + x} \over {{{\left({v - x + \mu} \right)}^3}}}} \right]} \over {{{rx + 2{\beta _1}} \over {{\beta _0}{x^3}}} - {{2hv\left({v + \mu} \right)} \over {{{\left({v - x + u} \right)}^3}}}}} can be known from the implicit function derivation theorem. The numerator part is always greater than 0, so if the derivative is greater than 0, the denominator part is less than 0. If the derivative is less than 0, the denominator part is greater than 0.

(rx + 2β1) (vx + u)3 − 2hv(v + μ) β0x3 > 0 if the denominator part is greater than 0. We use scaling to get x<v/((2hv2β0/β1)13+1) x < v/\left({{{\left({2h{v^2}\,{\beta _0}/{\beta _1}} \right)}^{{1 \over 3}}} + 1} \right) .

Therefore, when the above formula holds, dxdμ<0 {{dx} \over {d\mu}} < 0 . We can get dAdx<0 {{dA} \over {dx}} < 0 according to formula (12), so dAdμ>0 {{dA} \over {d\mu}} > 0 at this time.

Theorem 1 shows that when x<v/((2hv2β0/β1)13+1) x < v/\left({{{\left({2h{v^2}\,{\beta _0}/{\beta _1}} \right)}^{{1 \over 3}}} + 1} \right) , the proportion of the gathering crowd to the total local population x decreases with police deployment μ. At this time, the economic subsidy A increases with the increase of the police force deployment μ.

Therefore, in the process of emergency response to mass incidents, it is necessary to intervene as soon as possible in the budding state when the protest crowd is small. At this stage, the police force is deployed to intervene with greater intensity. At the same time, the amount of economic compensation for reasonable economic interests can effectively resolve mass incidents.

Theorem 2

If x>[4v(2β1+r)hβ0]13 x > {\left[{{{4v\left({2{\beta _1} + r} \right)} \over {h{\beta _0}}}} \right]^{{1 \over 3}}} , then dxdμ>0 {{dx} \over {d\mu}} > 0 and dAdu<0 {{dA} \over {du}} < 0 .

Proof: Similarly, if the denominator part is less than 0, then (rx + 2β1) (vx + u)3 − 2hv(v + μ) β0x3 < 0. We can obtain x>(4v(2β1+r)hβ0)13 x > {\left({{{4v\left({2{\beta _1} + r} \right)} \over {h{\beta _0}}}} \right)^{{1 \over 3}}} using the scaling method. Therefore, when the above formula holds, dxdμ>0 {{dx} \over {d\mu}} > 0 . We can obtain dAdx<0 {{dA} \over {dx}} < 0 from equation (12), so dAdμ<0 {{dA} \over {d\mu}} < 0 .

Theorem 2 shows that when x>(2v(2β1+r)hβ0)13 x > {\left({{{2v\left({2{\beta _1} + r} \right)} \over {h{\beta _0}}}} \right)^{{1 \over 3}}} , the proportion of aggregated people in the total local population, x, increases with the increase of police deployment μ. At this time, the economic subsidy A decreases as the police force deployment μ increases.

When mass incidents are in the outbreak or spread period, if the local government simply takes measures to increase police deployment, it will easily lead to more extreme emotions among the protesters. This could easily lead to further deterioration of the situation. Since the protesters have put forward too many interest demands, it is not helpful for the local government to provide economic subsidies to ease the development of the situation. Therefore, the local government should take legal measures to punish a very small number of extremist lawbreakers [7]. At the same time, the police need to deploy the police force scientifically to control the expansion of the situation. At the same time, comprehensive governance methods such as dividing and ruling for protesters with different demands are also needed.

Numerical Analysis

Because mass emergencies are socially sensitive, unreproducible, and emergent, the statistics on such events are lacking [8]. We visualize the changes in the relevant parameters in the above theorem. Parameters include the appeasement rate, imitation efficiency, and the impact of changes in social losses per unit time caused by protesters on the proportion of gathering crowds in the total local population. When we choose h = 1000, v = 0.8, β0 = 0.1, β1 = 0.1, r = 0.1, observe the relationship between μ and x.

Figure 1 shows the relationship between the proportion x of the aggregated crowd in the total local population and the local government police deployment intensity μ when the proportion x is at a low level. In this case x decreases as μ increases. If we improve the social rule of law, we will increase the proportion of aggregated people in the total local population x. If we increase the social loss per unit, we will reduce the proportion of aggregated people in the total local population.

Figure 1

The impact of police deployment on the proportion of gathering people in the total local population when the proportion of low gathering people in the total local population

Figure 2 shows the relationship between the proportion x of the aggregated population in the total local population and the local government's police force deployment intensity μ. In this case x increases as μ increases [9]. If we improve the social rule of law, we will reduce the proportion of aggregated people in the total local population. The increase of social loss per unit will increase the proportion of aggregated people in the local population.

Figure 2

The impact of police deployment on the proportion of gathering people in the total local population

Figure 3 shows the relationship between the proportion of aggregated people in the local population x and the optimal economic subsidy A, at a lower level of local government police deployment intensity [10]. In this case A increases as μ increases. Improving the social rule of law will reduce the economic subsidy A, and increasing the social loss per unit will increase the economic subsidy.

Figure 3

The impact of police deployment on economic subsidies when low-aggregation groups account for the proportion of the total local population

Figure 4 shows the relationship between the local government's police force deployment intensity μ and the optimal economic subsidy A at a relatively high level of the proportion of aggregated people in the total local population x. In this case A decreases as μ increases. Improving the social rule of law will increase the economic subsidy A, while increasing social loss per unit will reduce the economic subsidy.

Figure 4

The impact of police deployment on economic subsidies when high gatherings account for the proportion of the total local population

Discussion

When the proportion of the gathering crowd in the total local population is high, the higher the local government's police force deployment, the higher the proportion of the gathering crowd in the total population. At this time, the optimal economic subsidy used by the government to appease the crowd is less. When the number of people participating in mass incidents is relatively large, if the government deploys fewer police forces in advance, the mass incidents will not intensify. At this time, the government paid a high amount of economic subsidies to “let things go.” If the government arranges a higher level of police deployment in advance, it will intensify the conflict. This will lead to an increase in the number of crowds gathered. The government can’t provide high economic subsidies under this circumstance [11]. In areas with a high degree of the social rule of law, once a problem that may lead to mass incidents occurs, many prioritize solving it through legal means. Under the same conditions, the proportion of aggregated people in the total population will be reduced. In economically developed areas, mass incidents cause larger unit losses, and more people gather to participate in mass incidents to seek solutions to the incident. When the proportion of the gathering crowd in the total local population is low, the higher the local government's police force deployment, the lower the proportion of the gathering crowd in the total population. At this time, the more optimal economic subsidies the government uses to appease the gathering crowd. When the number of people participating in mass events is small, if the government arranges a higher degree of police force deployment in advance, it will have a greater deterrent effect on the participating people. The government can solve mass incidents without providing higher economic compensation. The government's pre-arranged police deployment is low. Although the number of gatherings is small, it will still have a greater impact on the government and cause greater social problems. To avoid the emergence of larger social problems, the government should provide higher economic compensation for the gathering of the masses. Through this method, the economic demands of the gathering masses are met. Public events in areas with a high degree of the social rule of law will lead to dissatisfaction among more people. Social losses per unit caused by mass incidents are relatively small in economically impoverished areas. The local government will not pay attention to the incidents of a small number of gathering people. The gathering people will spontaneously give up their participation in the incident when they find that they have not attracted the government's attention. Therefore, when making decisions, the local government should comprehensively consider the degree of the local social legal system, the follow-up rate of mass incidents, and the government's appeasement rate. Governments make different decisions according to different situations.

Conclusion

Properly handling mass emergencies is important for the government to strengthen and innovate social management. This paper considers that the government will adopt two methods of police force deployment and economic subsidies to achieve the smallest social loss. We characterize events as time-continuous games by constructing a differential game model. Economic subsidies can alleviate the dissatisfaction of the gathering crowd to a certain extent. Based on the social rule of law and the deployment of the local government police force, this paper explores how the government can efficiently handle mass incidents. This paper studies two government means providing economic subsidies and deploying police force. In future research, we can enrich the government's means of dealing with mass incidents. In addition, the research in this paper applies to mass emergencies and has reference significance for related issues such as environmental governance and credit risk.

Figure 1

The impact of police deployment on the proportion of gathering people in the total local population when the proportion of low gathering people in the total local population
The impact of police deployment on the proportion of gathering people in the total local population when the proportion of low gathering people in the total local population

Figure 2

The impact of police deployment on the proportion of gathering people in the total local population
The impact of police deployment on the proportion of gathering people in the total local population

Figure 3

The impact of police deployment on economic subsidies when low-aggregation groups account for the proportion of the total local population
The impact of police deployment on economic subsidies when low-aggregation groups account for the proportion of the total local population

Figure 4

The impact of police deployment on economic subsidies when high gatherings account for the proportion of the total local population
The impact of police deployment on economic subsidies when high gatherings account for the proportion of the total local population

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