© Wuhan University 2022

0 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 data-intensive 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 user's 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, Lyapunov's first method and LaSalle's 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.

1 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.

Fig. 1 The dynamic process of IERSU model

So we have the following equation:

$N(t)=I(t)+E(t)+R(t)+S(t)+U(t)$(1)

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 $\delta$, 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 $\eta$.

The explanation of each parameter in the IERSU model in Fig. 1 is shown in Table 1.

The IERSU model in Fig. 1 can be described by mathematical formulas as follows:

$\{\begin{array}{c}\frac{\mathrm{d}I\left(T\right)}{\mathrm{d}t}=\delta -k_{\mathrm{1}}IR-k_{\mathrm{2}}IS-\eta I\\ \frac{\mathrm{d}E(t)}{\mathrm{d}t}=k_{\mathrm{1}}IR+k_{\mathrm{2}}IS-({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )E\\ \frac{\mathrm{d}R\left(t\right)}{\mathrm{d}t}={\alpha }_{\mathrm{1}}E+{\gamma }_{\mathrm{2}}S-\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)R\\ \frac{\mathrm{d}S(t)}{\mathrm{d}t}={\alpha }_{\mathrm{2}}E+{\gamma }_{\mathrm{1}}R-({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )S\\ \frac{\mathrm{d}U(t)}{\mathrm{d}t}={\alpha }_{\mathrm{3}}E+{\beta }_{\mathrm{1}}R+{\beta }_{\mathrm{2}}S–\eta U\end{array}$(2)

From Eqs. (1) and (2), we can get:

$\frac{\mathrm{d}N(t)}{\mathrm{d}t}=\frac{\mathrm{d}I(t)}{\mathrm{d}t}+\frac{\mathrm{d}E(t)}{\mathrm{d}t}+\frac{\mathrm{d}R(t)}{\mathrm{d}t}+\frac{\mathrm{d}S(t)}{\mathrm{d}t}+\frac{\mathrm{d}U(t)}{\mathrm{d}t}=\delta -\eta N$

Therefore, this paper will discuss IERSU model in the following interval:

$\Omega =\left\{(I,E,R,S,U)\in {\mathbb{R}}_{+}^{\mathrm{5}}:\mathrm{0}\le I+E+R+S+U\le \frac{\delta }{\eta }\right\}$

We define the point where $E,R,S,U$are stable at $\mathrm{0}$ 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 equilibriu ${\omega }_{\mathrm{0}}=(I_{\mathrm{0}},E_{\mathrm{0}},R_{\mathrm{0}},S_{\mathrm{0}},U_{\mathrm{0}})=\left(\frac{\delta }{\eta },\mathrm{0,0},\mathrm{0,0}\right)$. In order to obtain the effective promotion equilibrium of the model ${\omega }_{\mathrm{*}}=(I_{\mathrm{*}},E_{\mathrm{*}},R_{\mathrm{*}},S_{\mathrm{*}},U_{\mathrm{*}})$.

We can set Eq. (2):

$\{\begin{array}{c}\delta -k_{\mathrm{1}}IR-k_{\mathrm{2}}IS-\eta I=\mathrm{0}\\ k_{\mathrm{1}}IR+k_{\mathrm{2}}IS-\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)E=\mathrm{0}\\ {\alpha }_{\mathrm{1}}E+{\gamma }_{\mathrm{2}}S-\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)R=\mathrm{0}\\ {\alpha }_{\mathrm{2}}E+{\gamma }_{\mathrm{1}}R-\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)S=\mathrm{0}\\ {\alpha }_{\mathrm{3}}E+{\beta }_{\mathrm{1}}R+{\beta }_{\mathrm{2}}S–\eta U=\mathrm{0}\end{array}$(3)

The solution of the Eq. (3) can be obtained as follows:

$\{\begin{array}{c}{I}_{*}=\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left[\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\right]}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}\\ E_{*}=\frac{\delta -\eta I_{*}}{{\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta },R_{*}=\frac{{\alpha }_{\mathrm{1}}{E}_{*}+{\gamma }_{\mathrm{2}}{S}_{*}}{{\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta },S_{*}=\frac{{\alpha }_{\mathrm{2}}{E}_{*}+{\gamma }_{\mathrm{1}}{R}_{*}}{{\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta },U_{*}=\frac{{\alpha }_{\mathrm{3}}{E}_{*}+{\beta }_{\mathrm{1}}{R}_{*}+{\beta }_{\mathrm{2}}{S}_{*}}{\eta }\end{array}$(4)

Next, we determine the threshold parameter $R_{\mathrm{0}}$ of the product promotion model. The threshold parameter of the model is calculated by the next generation matrix method ^[20].

Let

$\mathit{F}=\left(\begin{array}{c}{k}_{\mathrm{1}}IR+k_{\mathrm{2}}IS\\ \mathrm{0}\\ \mathrm{0}\end{array}\right),\mathit{P}=\left(\begin{array}{c}\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)E\\ -{\alpha }_{\mathrm{1}}E-{\gamma }_{\mathrm{2}}S+{\beta }_{\mathrm{1}}R+{\gamma }_{\mathrm{1}}R+\eta R\\ -{\alpha }_{\mathrm{2}}E-{\gamma }_{\mathrm{1}}R+{\beta }_{\mathrm{2}}S+{\gamma }_{\mathrm{2}}S+\eta S\end{array}\right)$

Then, taking the derivative of F and P, we have:

$\mathit{H}=\left(\begin{array}{ccc}\mathrm{0}& k_{\mathrm{1}}{I}_{\mathrm{0}}& k_{\mathrm{2}}{I}_{\mathrm{0}}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right),\mathit{W}=\left(\begin{array}{ccc}{\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta & \mathrm{0}& \mathrm{0}\\ -{\alpha }_{\mathrm{1}}& {\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta & \mathrm{0}\\ -{\alpha }_{\mathrm{2}}& -{\gamma }_{\mathrm{1}}& {\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \end{array}\right),\mathit{H}\mathit{W}{}^{-\mathrm{1}}=\left(\begin{array}{ccc}A& B& C\\ \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right)$

$A=\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta ({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )[({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]}$

$B=\frac{\delta k_{\mathrm{1}}({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )+\delta k_{\mathrm{2}}{\gamma }_{\mathrm{1}}}{\eta [({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]},C=\frac{\delta k_{\mathrm{2}}({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )+\delta k_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta [({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]}$

Thus, the threshold parameter of the model can be obtained as:

$R_{\mathrm{0}}=\rho \left(\mathit{H}{\mathit{W}}^{-\mathrm{1}}\right)=\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta ({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )[({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]}=\frac{\delta }{\eta I_{*}}$

2 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 $R_{\mathrm{0}}<\mathrm{1}$, invalid promotion equilibrium ${\omega }_{\mathrm{0}}$ is locally asymptotically stable. When $R_{\mathrm{0}}>\mathrm{1}$, effective promotion equilibrium ${\omega }_{\mathrm{*}}$ is locally asymptotically stable.

Proof   The Jacobian matrix of the model at ${\omega }_{\mathrm{0}}$ is:

$\mathit{J}\left({\omega }_{\mathrm{0}}\right)=\left(\begin{array}{ccccc}-\eta & \mathrm{0}& -\frac{k_{\mathrm{1}}\delta }{\eta }& -\frac{k_{\mathrm{2}}\delta }{\eta }& \mathrm{0}\\ \mathrm{0}& -{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}-{\alpha }_{\mathrm{3}}-\eta & \frac{k_{\mathrm{1}}\delta }{\eta }& \frac{k_{\mathrm{2}}\delta }{\eta }& \mathrm{0}\\ \mathrm{0}& {\alpha }_{\mathrm{1}}& -{\beta }_{\mathrm{1}}-{\gamma }_{\mathrm{1}}-\eta & {\gamma }_{\mathrm{2}}& \mathrm{0}\\ \mathrm{0}& {\alpha }_{\mathrm{2}}& {\gamma }_{\mathrm{1}}& -{\beta }_{\mathrm{2}}-{\gamma }_{\mathrm{2}}-\eta & \mathrm{0}\\ \mathrm{0}& {\alpha }_{\mathrm{3}}& {\beta }_{\mathrm{1}}& {\beta }_{\mathrm{2}}& -\eta \end{array}\right)$

The characteristic polynomial of matrix $\mathit{J}({\mathrm{\omega }}_{\mathrm{0}})$ is:

$\left|\lambda E-\mathit{J}\left({\omega }_{\mathrm{0}}\right)\right|=\left(\lambda +\eta \right)\left(\lambda +\eta \right)\left|\begin{array}{ccc}\lambda +{\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta & -\frac{k_{\mathrm{1}}\delta }{\eta }& -\frac{k_{\mathrm{2}}\delta }{\eta }\\ -{\alpha }_{\mathrm{1}}& \lambda +{\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta & -{\gamma }_{\mathrm{2}}\\ -{\alpha }_{\mathrm{2}}& -{\gamma }_{\mathrm{1}}& \lambda +{\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \end{array}\right|=\mathrm{0}$

It is easy to know that the two characteristic roots of matrix $\mathit{J}({\omega }_{\mathrm{0}})$ are as follows: ${\lambda }_{\mathrm{1}}=-\eta <\mathrm{0}$, ${\lambda }_{\mathrm{2}}=-\eta <\mathrm{0}$, the other three eigenvalues of matrix $\mathit{J}({\omega }_{\mathrm{0}})$ can be regarded as the eigenvalues of matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$:

$\mathit{J}{\left({\omega }_{\mathrm{0}}\right)}_{\mathrm{1}}=\left(\begin{array}{ccc}-{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}-{\alpha }_{\mathrm{3}}-\eta & \frac{k_{\mathrm{1}}\delta }{\eta }& \frac{k_{\mathrm{2}}\delta }{\eta }\\ {\alpha }_{\mathrm{1}}& -{\beta }_{\mathrm{1}}-{\gamma }_{\mathrm{1}}-\eta & {\gamma }_{\mathrm{2}}\\ {\alpha }_{\mathrm{2}}& {\gamma }_{\mathrm{1}}& -{\beta }_{\mathrm{2}}-{\gamma }_{\mathrm{2}}-\eta \end{array}\right)$

The first-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$is $-{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}-{\alpha }_{\mathrm{3}}-\eta <\mathrm{0}$.

The second-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$ is $\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)-\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}}{\eta }$.

Due to$R_{\mathrm{0}}<\mathrm{1}$, that is :

$\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta ({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )[({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]}<\mathrm{1}$

And then we have:

$\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta }<({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )[({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]$

After dividing both sides by $({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )$ at the same time, we can get:

$\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}}{\eta }+\frac{\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta \left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}<({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )-\frac{({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta ){\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}$

So the second order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$ is:

$\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)-\frac{\delta \alpha k_{\mathrm{1}}}{\eta }>\mathrm{0}$

The third-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$ is:

$\left(-{\beta }_{\mathrm{2}}-{\gamma }_{\mathrm{2}}-\eta \right)\left|\begin{array}{cc}-{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}-{\alpha }_{\mathrm{3}}-\eta & \frac{k_{\mathrm{1}}\delta }{\eta }\\ {\alpha }_{\mathrm{1}}& -{\beta }_{\mathrm{1}}-{\gamma }_{\mathrm{1}}-\eta \end{array}\right|-{\gamma }_{\mathrm{2}}\left|\begin{array}{cc}-{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}-{\alpha }_{\mathrm{3}}-\eta & \frac{k_{\mathrm{1}}\delta }{\eta }\\ {\alpha }_{\mathrm{2}}& {\gamma }_{\mathrm{1}}\end{array}\right|+\frac{k_{\mathrm{2}}\delta }{\eta }\left|\begin{array}{cc}{\alpha }_{\mathrm{1}}& -{\beta }_{\mathrm{1}}-{\gamma }_{\mathrm{1}}-\eta \\ {\alpha }_{\mathrm{2}}& {\gamma }_{\mathrm{1}}\end{array}\right|$

$\begin{array}{l}=\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right){\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\eta +\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}{\eta }\\ -\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\end{array}$

Due to $R_{\mathrm{0}}<\mathrm{1}$, that is:

$\frac{\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{\eta }<({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )[({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}]$

Simplifying the above formula, we can get:

$\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right){\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\eta +\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}{\eta }$

$<\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)$

So the third-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$ is:

$\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right){\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\eta +\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+\delta {\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+\delta {\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}{\eta }$

$-\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)<\mathrm{0}$

Based on the above results, we can know that when $R_{\mathrm{0}}<\mathrm{1}$, matrix $\mathit{J}({\omega }_{\mathrm{0}}{)}_{\mathrm{1}}$ is a negative definite matrix, and its three eigenvalues are all less than 0, that is ${\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},{\lambda }_{\mathrm{3}}$are less than 0, when$R_{\mathrm{0}}>\mathrm{1}$, matrix $\mathit{J}({\omega }_{\mathrm{0}})$ has non-negative eigenvalues. According to Lyapunov's first method ^[21], it can be obtained: when $R_{\mathrm{0}}<\mathrm{1}$, the invalid promotion equilibrium point is locally asymptotically stable, and when $R_{\mathrm{0}}>\mathrm{1}$, 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 $\mathit{J}({\omega }_{\mathrm{*}})$.

The Jacobian matrix of the model at $\mathit{J}({\omega }_{\mathrm{*}})$ is:

$\mathit{J}\left({\omega }_{*}\right)=\left(\begin{array}{cccc}-\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta \right)& \mathrm{0}& -k_{\mathrm{1}}{I}_{*}& -k_{\mathrm{2}}{I}_{*}\\ \left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}\right)& -({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta )& k_{\mathrm{1}}{I}_{*}& k_{\mathrm{2}}{I}_{*}\\ \mathrm{0}& {\alpha }_{\mathrm{1}}& -({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta )& {\gamma }_{\mathrm{2}}\\ \mathrm{0}& {\alpha }_{\mathrm{2}}& {\gamma }_{\mathrm{1}}& -({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta )\end{array}\right)$

The first-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{*}})$ is $-\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta \right)<\mathrm{0}$

The second-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{*}})$is $\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta \right)\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)>\mathrm{0}$

The third-order sequential principal minor of the matrix $\mathit{J}({\omega }_{\mathrm{*}})$ is:

$-\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta \right)\left[\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)-{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}{I}_{*}\right]-\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}\right){\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}{I}_{*}$

$=-\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta \right)\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}$

From Eq. (4), we can get:

$\{\begin{array}{c}{R}_{*}=\frac{{\alpha }_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{\gamma }_{\mathrm{2}}}{\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{E}_{*}\\ S_{*}=\frac{{\alpha }_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{\gamma }_{\mathrm{1}}}{\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{E}_{*}\\ =\frac{{\alpha }_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{\gamma }_{\mathrm{1}}}{{\alpha }_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{\gamma }_{\mathrm{2}}}R\\ k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}=\frac{{\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta }{I_{*}}{E}_{*}=\frac{\delta -\eta I_{*}}{I_{*}}\\ \end{array}$(5)

From Eq. (5), we can get:

$-\left(k_{\mathrm{1}}R+k_{\mathrm{2}}S+\eta \right)\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}=-\frac{\delta }{I_{*}}\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}$

And due to $R_{\mathrm{0}}>\mathrm{1}$, we have: $\frac{\delta }{\eta I_{\mathrm{*}}}>\mathrm{1}\Rightarrow \frac{\delta }{I_{\mathrm{*}}}>\eta$, so we have the following inequality:

$-\frac{\delta }{I_{*}}\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}<{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}-\eta \left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right),$

$\begin{array}{l}{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta I_{*}-\eta \left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)=\frac{\mathrm{1}}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta }\left(I_{*}-\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}}\right)\\ =\frac{\mathrm{1}}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\eta }\left(\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left[\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\right]}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}}-\frac{\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}{{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)}\right)<\mathrm{0}\end{array}$

The fourth-order sequential principal minor of the matrix $\mathit{J}({\mathrm{\omega }}_{\mathrm{*}})$ is:

$\begin{array}{l}(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta )\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left[\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\right]-(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta )[{\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}{I}_{*}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}I{}_{*}]\\ -(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}+\eta )[{\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}I{}_{*}+{\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}I{}_{*}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)]+k_{\mathrm{1}}I{}_{*}(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*})[{\alpha }_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{\gamma }_{\mathrm{2}}]\\ +k_{\mathrm{2}}I{}_{*}(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*})[{\alpha }_{\mathrm{1}}{\gamma }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)]\end{array}$

From Eq.(4), we have:

$\left({\alpha }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}+{\alpha }_{\mathrm{3}}+\eta \right)\left[\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)-{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\right]={\alpha }_{\mathrm{1}}{k}_{\mathrm{1}}I{}_{*}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{k}_{\mathrm{2}}I{}_{*}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)+{\alpha }_{\mathrm{1}}{k}_{\mathrm{2}}{\gamma }_{\mathrm{1}}I{}_{*}+{\alpha }_{\mathrm{2}}{k}_{\mathrm{1}}{\gamma }_{\mathrm{2}}I{}_{*}$

Therefore, the fourth-order sequential principal minor can be simplified to:

$k_{\mathrm{1}}{I}_{*}\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}\right)\left[{\alpha }_{\mathrm{1}}\left({\beta }_{\mathrm{2}}+{\gamma }_{\mathrm{2}}+\eta \right)+{\alpha }_{\mathrm{2}}{\gamma }_{\mathrm{2}}\right]+k_{\mathrm{2}}{I}_{*}\left(k_{\mathrm{1}}{R}_{*}+k_{\mathrm{2}}{S}_{*}\right)\left[{\alpha }_{\mathrm{1}}{\gamma }_{\mathrm{1}}+{\alpha }_{\mathrm{2}}\left({\beta }_{\mathrm{1}}+{\gamma }_{\mathrm{1}}+\eta \right)\right]>\mathrm{0}$