Application of Promotion Process Based on Epidemic Models Considering Bidirectionality

: In this paper, on the basis of previous epidemic models, we considered the bidirectionality of a certain process of the model and established the Ignorant, Exposers, Resister of product, Supporter of product, Uninterested (IERSU) model, and applied it to a new field. Firstly, the next generation matrix method was used to obtain the threshold parameter which helps to distinguish the equilibrium point of the model. Secondly, the local and global stability of the equilibrium point of the model were proved by the judgment and properties of the negative definite matrix, Lyapunov 􀆳 s first method and LaSalle 􀆳 s invariance principle. Finally, according to the numerical simulation, the influence of different parameters of the model was analyzed and a series of strategies were dis‐ cussed.


Introduction
Nowadays, with the continuous development of Internet technology, online social networks are rapidly popularized, more and more users have joined online social networks, and companies have also shifted the focus of new product promotion to social media platforms.Therefore, it is necessary to study the process of product promotion in a social network, and it will help companies formulate strategies.
The process of product promotion in social networks is very similar to epidemic diffusion, and the process of product promotion can be seen as a kind of information dissemination in the network.Therefore, the Ignorant, Exposers, Resister of product, Supporter of product, Uninterested (IERSU) model of this paper is based on these epidemic models and the derivative models based on them [1] , such as the classic SIS model [2] , SIR model [3] , and SEIR model [4] .Zhao et al [5] established the SIHR model on the basis of the classic SIR, studied the stability of the model, and discussed the information dissemination threshold.The SIHR model considered the mutual influence of forgetting and memory mechanisms.Wang et al [6] established the SIRaRu model based on the previous information dissemination model, discussed the threshold of information dissemination, and gave a series of strategies for information dissemination.Wang et al [7] analyzed the process of information dissemination in detail, and established the SEINR model.With the help of dataintensive computing theory, Ref. [7] obtained a variety of information dissemination characteristics, improved and perfected the dynamic model, and discussed the basic reproduction number, existence and stability of the equilibrium point of the model.Zhang et al [8] studied the influence of personal behavior on information dissemination, taking it into account the factors that affect personal behaviors, and the delay of information dissemination.Wan et al [9] studied the influence of preferential policies on information dissemination, proposed the SIB model under the scale-free network, and discussed the process of information dissemination in detail using the mean field theory.Kang et al [10] proposed a scale-free network SIS model with delay.The model introduced an average latency, discussed the propagation threshold of the model, and compared the effects of the two strategies on propagation.Zhao et al [11] introduced the concept of propagation force, considered propagation force as a fuzzy variable, and introduced a series of parameter variables with fuzzy properties, established a new model based on SIR, and discussed related strategies.Afassinou [12] considered the influence of education as a factor on information dissemination, proposed a new SEIR model, analyzed and discussed the model, and found that education has a significant influence on information dissemination.
However, there are some differences between the information dissemination model and the infectious disease model [13] .Some scholars have considered other influencing factors and established different information dissemination models to study.Liu et al [14] considered attenuation coefficient and noise coefficient, using them to describe the mutual influence of information transmission between users, and defined the feedback function to discuss the influence of node feedback on information transmission.This model considered the complex interactions between user relationships, social networks, and communities, and drew the conclusion that the intensity of user feedback on information has an significant influence on the process of information dissemination.Wang et al [15] considered the users sentiment towards the event or product, and established an emotion-based information dissemination model to analyze the dissemination process.
In the study of information dissemination, previous studies considered its process as one-way and ignored its bidirectionality.In addition, previous studies ignored the dynamic nature of information dissemination systems and discussed information dissemination in a closed system.Finally, this paper extends the application of information dissemination model to product promotion [16,17] and discusses a series of strategies [18,19] .
From the above analysis, the purpose of this paper is to apply epidemic diffusion model to product promotion, and then discusses a series of strategies.The main work and contributions are as follows: 1) Based on the previous epidemic diffusion model, considering the bidirectionality of the process, the IERSU model is established and applied to product promotion.
2) Through the analysis of the model, the next-generation matrix method is used to obtain the threshold parameter of the model, and we can distinguish the equilibrium point of the model by the threshold parameter.
3) Based on the judgment and properties of the negative definite matrix, Lyapunovs first method and LaSalles invariance principle, the stability of the equilibrium point of the model is proved.
4) Through numerical simulation, the stability of the equilibrium point is verified.The influence of different parameters on the model is analyzed and a series of strategies are discussed.
The rest of this paper is as follows.Section 1 introduces an IERSU information dissemination model and finds the equilibrium point of the model.Section 2 proves the stability of the equilibrium point of the model.Section 3 conducts the numerical simulation to the model, and discusses the influence of each parameter on the model.Finally, Section 4 gives the conclusion of this paper.

The IERSU Model
When a new product enters the market, it will be supported or resisted in the promotion process.In order to gain more supporters, the company needs a series of strategies to increase supporters and reduce resisters.Based on this fact, we assume that the population is divided into five classes in an open social network system, and different classes have a certain conversion relationship.As shown in Fig. 1, the five classes of people are: Ignorant (I) who has never heard of relevant product information, Exposers (E) who have heard relevant product information, Resister of product (R), Supporter of product (S) , and the Uninterested (U) who are not interested in product.Let N(t) represent the total population of the system at the initial time t.I(T), E(t), R(t), S(t), U(t) respectively represent the number of I, E, R, S, U in the system at time t.
So we have the following equation: The process of IERSU model is shown in Fig. 1.We establish models under a dynamically changing system.The population of system increases at the immigration rate δ, the initial state of the new person entering the system is I, and the population of five classes in the system decreases at the emigration rate η.
The explanation of each parameter in the IERSU model in Fig. 1 is shown in Table 1.
The IERSU model in Fig.  ( ) From Eqs. ( 1) and (2), we can get: We define the point where E R S U are stable at 0 as the invalid equilibrium point and the point where it is stable at non-zero as the valid equilibrium point.It is easy to see from Eq. ( 2) that the model has an invalid promotion ) .In order to obtain the effective promotion equilibrium of the We can set Eq. ( 2): The solution of the Eq. ( 3) can be obtained as follows: ) ) Next, we determine the threshold parameter R 0 of the product promotion model.The threshold parameter of the model is calculated by the next generation matrix method [20] . Let ) Then, taking the derivative of F and P, we have: Thus, the threshold parameter of the model can be obtained as:

Stability Analysis of IERSU Model
The purpose of this section is to analyze the stability of the equilibrium point of the IERSU model.Theorem 1 When R 0 < 1, invalid promotion equilibrium ω 0 is locally asymptotically stable.When R 0 > 1, effective promotion equilibrium ω * is locally asymptotically stable.
Proof The Jacobian matrix of the model at ω 0 is: The characteristic polynomial of matrix J(ω 0 ) is: It is easy to know that the two characteristic roots of matrix J(ω 0 ) are as follows: λ 1 = -η < 0, λ 2 = -η < 0, the other three eigenvalues of matrix J(ω 0 ) can be regarded as the eigenvalues of matrix J(ω 0 ) 1 : The first-order sequential principal minor of the matrix The second-order sequential principal minor of the matrix J(ω 0 ) 1 is ( Due to R 0 < 1, that is : And then we have: ) After dividing both sides by (β 2 + γ 2 + η) at the same time, we can get: ) So the second order sequential principal minor of the matrix J(ω 0 ) 1 is: The third-order sequential principal minor of the matrix J(ω 0 ) 1 is: ) ) ) Simplifying the above formula, we can get: ) So the third-order sequential principal minor of the matrix J(ω 0 ) 1 is: ) ) Based on the above results, we can know that when R 0 < 1, matrix J(ω 0 ) 1 is a negative definite matrix, and its three eigenvalues are all less than 0, that is λ 1  λ 2  λ 3 are less than 0, when R 0 > 1, matrix J(ω 0 ) has non-negative eigenvalues.According to Lyapunovs first method [21] , it can be obtained: when R 0 < 1, the invalid promotion equilibrium point is locally asymptotically stable, and when R 0 > 1, the invalid promotion equilibrium point is unstable.The local asymptotic stability of the invalid promotion equilibrium point has been proved.
Next, we will prove the local asymptotic stability of the effective promotion equilibrium point.Since the first four equations in Eq. ( 2) are not affected by the last equation, it can be simplified when calculating the Jacobian matrix of J(ω * ).
The Jacobian matrix of the model at J(ω * ) is: The first-order sequential principal minor of the matrix J(ω * ) is -( The second-order sequential principal minor of the matrix J(ω * ) is The third-order sequential principal minor of the matrix J(ω * ) is: -( ) ) ) ) ) ) ) From Eq. ( 5), we can get: And due to R 0 > 1, we have: > η, so we have the following inequality: δ ) ) ) ) The fourth-order sequential principal minor of the matrix J(ω * ) is: From Eq.( 4), we have: Therefore, the fourth-order sequential principal minor can be simplified to: Based on the above results, we can know that the matrix J(ω * ) is a negative definite matrix, and its eigenvalues are all less than 0. According to the first method of Lyapunov, when R 0 > 1, effective promotion equilibrium point ω * is locally asymptotically stable.
Based on the above results, we can know that: When R 0 < 1, ω 0 is the unique equilibrium point.When R 0 > 1, ω * is the unique equilibrium point.

Numerical Simulations and Discussion
This paper establishes an IERSU model to describe the process of product promotion.The stability of the IERSU has been proven in the previous section.From the proof process in the previous section, we can use the threshold parameter R 0 to distinguish the steady state of the model.When R 0 < 1, invalid promotion equilibrium ω 0 is globally asymptotically stable.When R 0 > 1, effective promotion equilibrium ω * is globally asymptotically stable.
In this section, the paper conducts numerical simulation to verify the above analysis results, and discusses the influence of each parameters on the number of S under different values.According to the results of the discussion, a series of marketing strategies helping product promotion are proposed for different parameters.We set N = 10 000 I = 9 998 E = 0 R = 1 S = 1 U = 0 at t 0 .The results are shown in Fig. 2 and Fig. 3.
Figure 2 describes the number of five classes over time t when R 0 < 1.From Fig. 2, we can know that the number of I continues to decrease until reaching 0. The number of E first increases, reaches a peak, and then begins to decrease to 0. And S, R, U have roughly the same trend, the numbers of S, R, U first increase, reach a peak, and then decrease to 0. But it is different for the time of S, R, U to reach the peak.R reaches the peak first, S is slightly behind, and U reaches at the last.This shows that when the system reaches stability, the system is at a point of invalid promotion equilibrium.
Figure 3 describes the number of five classes over time t when R 0 > 1.From Fig. 3, we can know that the number of I continues to decrease until reaching 0. The number of E first increases, reaches a peak, then begins to decrease, and finally stabilizes at a non-zero value.The numbers of S, R first increase, reaching the peak point, and then begin to decrease, finally stabilize at a non-zero value, with roughly the same trend.R peaks before S.However, U increases slowly and then reaches a non-zero stable value.Compared with Fig. 2, the numbers of E, R, S, and U finally stabilize at a non-zero value.This shows that when the system reaches stability, the system is at a point of effective promotion equilibrium.Based on the above analysis, we can distinguish the equilibrium point of the model by R 0 , and when R 0 > 1, it is meaningful to discuss the product promotion model.Next, we use the number of S as a dependent variable and discuss the influence of different parameters on it.The following discussion is based on the parameter values in Fig. 3 Figure 4(a) describes the number of S over time t under different α 1 .When α 1 decreases, the number of S increases.When α 1 decreases from 0.08 to 0.008, the number of S increases little.When α 1 decreases from 0.008 to 0.000 8, the number of S increases significantly.α 1 continues to decrease and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of E turning into R decreases, so the number of E turning into R decreases, and more E can turn into S compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to increase the conversion of E to S, the company can try to build a good social image to gain more consumers psychological recognition when promoting new products, and a good social image will drive more consumers to support its new product.This strategy reflected in the model will increase α 2 while reducing α 1 and α 3 .
Figure 4(b) describes the number of S over time t under different α 2 .When α 2 increases, the number of S increases.When α 2 increases from 0.000 8 to 0.008, the number of S increases significantly, while α 2 continues to increase, and the number of S increases slightly.α 2 continues to increase and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of E turning into S increases, so the number of E turning into S increases compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to increase the conversion of E to S, the company can push a large number of advertisements on the target group when promoting new products, so as to make as many people as possible to leave an impression on the product.This strategy reflected in the model will increase α 2 while reducing α 3 .
Figure 4(c) describes the number of S over time t under different α 3 .When α 3 decreases, the number of S increases.When α 3 decreases from 0.01 to 0.000 1, the number of S increases significantly, and when α 3 decreases from 0.000 1 to 0.000 01, the number of S increases little.α 3 continues to decrease and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of E turning into U decreases, so the number of E turning into U decreases, and more E can turn into S compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to increase the conversion of E to S, the company can invite celebrities to help promote the product in the whole process of product promotion, using the influence of celebrities to improve the visibility of product and promote the product to more potential customers.This strategy reflected in the model will increase α 2 while reducing α 1 .
Figure 5(a) describes the number of S over time t under different β 1 .When β 1 decreases, the number of S increases.When β 1 decreases from 0.01 to 0.000 1, the number of S increases significantly.And β 1 decreases from 0.000 1 to 0.000 01, the number of S increases slightly.β 1 continues to decrease and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of R becoming U decreases, so the number of Based on the above analysis, in order to increase the conversion of R to S, the company can randomly give consumers rewards such as free orders or discounts when promoting new products.Such random rewards will attract R, prompting some of R to turn into S, and may make more E turn into S.This strategy reflected in the model will reduce β 1 while increasing α 2 .
Figure 5(b) describes the number of S over time t under different β 2 .When β 2 decreases, the number of S increases.When β 2 decreases from 0.005 to 0.000 05, the number of S increases significantly.And β 2 decreases from 0.000 05 to 0.000 005, the number of S increases slightly.β 2 continues to decrease and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of S becoming U decreases, so the number of S turning into U decreases compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to decrease the conversion of S to U, the company can develop S into members, by turning customers into members and maintaining them regularly, the company can keep their members active and reduce the loss of supporters.This strategy reflected in the model will reduce β 2 .
Figure 6(a) describes the number of S over time t under different γ 1 .When γ 1 increases, the number of S increases.When γ 1 increases from 0.000 05 to 0.005, the number of S increases significantly.And γ 1 increases from 0.005 to 0.05, the number of S increases slightly.γ 1 continues to increase and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of R turning into S increases, so the number of R turning into S increases compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to increase the conversion of R to S, the company can display positive reviews of products during product promotion, and product reviews may give R the idea to try the product, thus some of R will turn into S.This strategy reflected in the model will increase γ 1 .
Figure 6(b) describes the number of S over time t under different γ 2 .When γ 2 decreases, the number of S increases.When γ 2 decreases from 0.01 to 0.0001, the number of S increased significantly.And γ 2 decreases from 0.000 1 to 0.000 01, the number of S increases slightly.γ 2 continues to decrease and the degree of increase of S becomes smaller and smaller.The reason for the above phenomenon is that the probability of S turning into R decreases, so the number of S turning into R decreases compared with the original probability, thus increasing the number of S.
Based on the above analysis, in order to increase the conversion of R to S, the company can focus on product quality and product after-sales service, good products and excellent after-sales service can enhance customer experience, which can help company gain more loyal customers.This strategy reflected in the model will reduce γ 2 .
Figure 7 describes the number of S over time t under different k 1 and k 2 .When k 1 and k 2 increase or decrease, the number of S remains unchanged on the whole.The reason for the above phenomenon is that k 1 and k 2 influence the number of S by affecting α 1 , α 2 , so the influence of k 1 , k 2 on the number of S can be referred to α 1 , α 2 .
In this section, we discuss the influence of each parameter on the number of S. It can be known that the each parameter influences the number of E turning into S, where the influence of k 1 and k 2 on the number of S can be reflected by α 1 and α 2 .Based on the influence of each parameter on the number of S, a series of strategies are proposed to increase the number of S.

Conclusion
This paper proposes an IERSD model to describe the dynamic process of product promotion and discusses the influence of each parameter on the number of S. Based on Lyapunovs first method and LaSalles invariance principle, the stability of the model is proved.The analysis find that the dynamic characteristics of the model are related to the threshold parameter, and the threshold parameter can be used to distinguish the steady state of the model.When the threshold parameter is less than 1, the invalid promotion equilibrium is globally asymptotically stable.The number of S, R and U in the system finally stabilized at 0. When the threshold parameter is bigger than 1, the effective promotion equilibrium is globally asymptotically stable.The number of S, R and U in the system will stabilize at a non-zero value.Finally, the paper conducts a series of numerical simulations to verify the above analysis and discusses the influence of each parameter on the number of S, and a series of strategies are proposed to increase the number of S.
The IERSU model established in this paper further improves the information dissemination model by considering 1 can be described by mathematical formulas as follows:

Fig. 1
Fig. 1 The dynamic process of IERSU model

Fig. 4
Fig. 4 The number of S over time t under different α

Fig. 5
Fig. 5 The number of S over time t under different β

Fig. 6 Fig. 7
Fig. 6 The number of S over time t under different γ