Global Dynamics Analysis of Homogeneous New Products Diffusion Model

Abstract

A mathematical model with stage structures is presented that incorporates the awareness stage and the decision-making stage; individuals exchange product information by two channels: mass media and interpersonal communication. When the persuasive advertisement is neglected in the decision-making stage, we find a threshold value about whether new products diffusion is successful or not. When the persuasive advertisement is considered, there must exist a positive equilibrium under some parameter condition; moreover, it must be globally asymptotically stable as long as it exists. Results show that the persuasive advertisement in the decision-making stage does not influence new products' successful diffusion, but the critical value that new products diffuse successfully decreases when the persuasive advertisement is considered. Some numerical simulations confirm our theoretical analysis and demonstrate the influence of parameters on the system. Corresponding to the analysis results, some diffusion strategies are provided.

1. Introduction

New products diffusion is the process by which some new products information is communicated through certain channels over time among the members of a social system. It is a special type of communication in which the messages are concerned with new ideas. Previously we defined communication as the process by which participants create and share information with one another in order to reach a mutual understanding. The essence of the new products diffusion process is the product information exchanged by which one individual communicates a new product to one or several others. At its most elementary form, the process involves (1) a propagation about some new product by mass media, (2) an individual or other unit of adoption that is aware of, or experienced with, using some new product, (3) another individual or other unit that is not yet aware of this new product, and (4) a communication channel connecting the two units. The heart of new products diffusion process is that more people know and adopt new products; therefore, on one hand, mass media channels are all those means of transmitting new products information that involve a mass medium, such as radio, television, newspapers, and so on, which enable a source of one or a few individuals to reach an audience of many; on the other hand, most people depend mainly upon a subjective evaluation of an innovation that is conveyed to them from other individuals like themselves who have previously adopted the innovation. So, interpersonal channels are more effective in persuading an individual to adopt new products, especially if the interpersonal channel links two or more individuals who are near-peers.

The Bass model has become an important exemplar in marketing science. For over three decades, this model has been the main impetus underlying diffusion research and has been widely used to understand the spread of new products [1–3]. On the base of the Bass model, many dynamics models are used to study new products diffusion [4–7]. In fact, the consumer's aware formation of new products and decision making process on the new product adoption involves a complex interaction of various external and internal factors like mass media, advertising, word-of-mouth, personal preferences, and experience. Interpersonal communications, without a doubt, constitute an important communication media especially for social groups that are hardly reachable by mass media advertising [8–10]. In this paper, for the simplicity of mathematical analysis, we consider two stages in the diffusion process: the stage of awareness of information and the stage of decision making. Specifically, in the first stage, enterprises transfer new production information to consumers through the informative advertisement, or individuals transfer new product information to each other. So, individuals become aware of information and they are interested in the product and evaluate its value to adopt it or not to adopt it. In the decision-making stage, enterprises change consumers’ preferences through the persuasive advertisement, or a friend's opinion or advice also influences individuals’ preference, so individuals try or adopt the product in the second stage. In view of these, assuming that the product information is spread due to word-of-mouth among people’s direct contacts and mass media channels, we propose a homogeneous model with two stages of new products diffusion.

The organization of this paper is as follows. In the next section, we set up a homogeneous model about new products diffusion and discuss its equilibriums and their stability when the persuasive advertisement is neglected or considered in the decision-making stage and, respectively, obtain conditions that the diffusion is successful; further, by comparing two critical values, we analyse the effect of the persuasive advertisement in the decision-making stage. In Section 3, theoretical results are confirmed by numerical simulations. In Section 4, we give sensitive analysis of parameters and the influence of parameters on the adopters by simulations when diffusion is successful. Finally, we give a brief conclusion and discussion in Section 5.

2. Model

In this section, incorporating an awareness stage and a decision-making stage into the Bass model, we put forward the model as follows with the standard incidence rate: 𝑑𝑈(𝑡)𝑑𝑡=−𝑝𝑈−𝛽₁𝑈𝐼𝑁−𝛽₂𝑈𝐴𝑁(𝑡)+𝛾𝐼+𝜇𝐴,𝑑𝐼𝑑𝑡=𝑝𝑈+𝛽₁𝑈𝐼𝑁+𝛽₂𝑈𝐴𝑁−𝛾𝐼−𝛼𝐼−𝛽₃𝐼𝐴𝑁,𝑑𝐴(𝑡)𝑑𝑡=𝛼𝐼+𝛽₃𝐼𝐴𝑁−𝜇𝐴. (1) Here, 𝑈(𝑡) denote the number of those individuals who have not been aware of the product at time 𝑡, 𝐼(𝑡) denote the number of those individuals who have been aware of the information about the product but have not yet adopted it at time 𝑡, and 𝐴(𝑡) denote the number of those individuals who have adopted the product at time 𝑡. Consider that 𝑁=𝑈(𝑡)+𝐼(𝑡)+𝐴(𝑡), the total number of people, is a constant. All parameters are positive constants and summarized in Table 1. Note that who have been aware of the information about the product but have not yet adopted it also spread the information of the product, so we incorporate the item 𝛽₁𝑈(𝐼/𝑁) into our model which extends the earlier model [11]. When 𝛽₁=0, system (1) becomes the model in [11].

Table 1 

Parameters of the model.

Mass media channels are considered a more effective way to generate awareness of the new product [12]; for this reason, we further assume that 𝑝>𝛽₁,𝑝>𝛽₂. Denote the relative densities of 𝑈(𝑡), 𝐼(𝑡), and 𝐴(𝑡) at time 𝑡 by 𝑢(𝑡), 𝑖(𝑡), and 𝑎(𝑡), respectively; then, system (1) can be rewritten as 𝑑𝑢(𝑡)𝑑𝑡=−𝑝𝑢−𝛽₁𝑢𝑖−𝛽₂(𝑡)𝑢𝑎+𝛾𝑖+𝜇𝑎,𝑑𝑖𝑑𝑡=𝑝𝑢+𝛽₁𝑢𝑖+𝛽₂𝑢𝑎−𝛾𝑖−𝛼𝑖−𝛽₃(𝑡)𝑖𝑎,𝑑𝑎𝑑𝑡=𝛼𝑖+𝛽₃𝑖𝑎−𝜇𝑎. (2) Noting that 𝑢(𝑡)+𝑖(𝑡)+𝑎(𝑡)=1, system (2) is reduced to 𝑑𝑖(𝑡)=𝑑𝑡𝑝+𝛽₁𝑖+𝛽₂𝑎(1−𝑖−𝑎)−𝛾𝑖−𝛼𝑖−𝛽₃(𝑡)𝑖𝑎,𝑑𝑎𝑑𝑡=𝛼𝑖+𝛽₃𝑖𝑎−𝜇𝑎. (3)

Theorem 1. System (3) has no cycle.

Proof. Leting 𝐺=1/(𝛼𝑖+𝛽₃𝑖𝑎), we denote the right sides of system (3) by 𝑓₁ and 𝑓₂, respectively, and it follows that 𝜕𝐺𝑓₁+𝜕𝜕𝑖𝐺𝑓₂=1𝜕𝑎𝛼𝑖+𝛽₃𝑎𝑖²−𝛽₁𝑖²+𝑝+𝛽₂𝑎(×1−𝑎)𝛼+𝛽₃𝑎−𝜇𝛼𝑖<0. (4) So, Dulac’s criteria imply that system (3) does not have a limit cycle [13].

As we all know, interpersonal communication is considered more effective in influencing an individual's decision to adopt [12]. For investigating the persuasive advertisement's effect in the decision-making stage, we will analyze the dynamics of system (3) in two kinds of parameter assumptions: in the decision-making process, I: 𝛼=0, the persuasive advertisement's effect is neglected, and II: 𝛼≠0, the persuasive advertisement's effect is considered. We first discuss system (3) under the assumption Case I.

Case I (𝛼=0). When 𝛼=0, we can derive that system (3) has the equilibrium 𝑃⁰(𝑖⁰,0), where 𝑖⁰=(1/2𝛽₁)(𝛽₁−𝑝−𝛾+(𝑝+𝛾−𝛽₁)²+4𝑝𝛽₁)<1. For considering its local stability, we first give that the jacobian matrix of system (3) at the generic equilibrium is

𝛽𝐽=¹(1−𝑖−𝑎)−𝑝+𝛽₁𝑖+𝛽₂𝑎−𝛾−𝛽₃𝑎𝛽₂(1−𝑖−𝑎)−𝑝+𝛽₁𝑖+𝛽₂𝑎−𝛽₃𝑖𝛽₃𝑎𝛽₃𝑖−𝜇. (5)

Furthermore, 𝐽∣_𝑃⁰=𝛽₁1−𝑖⁰−𝑝+𝛽₁𝑖⁰+𝛾𝛽₂1−𝑖⁰−𝑝+𝛽₁𝑖⁰+𝛽₃𝑖⁰0𝛽₃𝑖⁰−𝜇, (6)

and we assume that its character roots are𝜆₁ and 𝜆₂; then 𝜆₁=𝛽₁(1−𝑖⁰)−(𝑝+𝛽₁𝑖⁰+𝛾)=−𝑝/𝑖⁰−𝛽₁𝑖⁰<0, 𝜆₂=𝛽₃𝑖⁰−𝜇. Define the threshold 𝑅≜𝛽₃𝑖⁰/𝜇; the result as follows is obtained.

Theorem 2. System (3) always has a trivial equilibrium𝑃⁰(𝑖⁰,0); if 𝑅<1, 𝑃⁰ is locally asymptotically stable; furthermore, it is globally asymptotically stable; if 𝑅>1, 𝑃⁰ is unstable.

Remark 3. In the early stage of new products diffusion, there are some people who have been aware of the information about the product through mass media in the awareness stage but have not yet adopted it, and at this moment, the fraction of population 𝑈 is 1−𝑖⁰, the fraction of population 𝐼 is 𝑖⁰, and then 𝑅 is the average number (related to the number of population 𝐼 at the initial time) of new adopters caused by a single adopter introduced into population exclusive adopters in the early stage of the diffusion.

Next, we consider the existence and stability of the positive equilibrium. Letting the right side of equations in system (3) equal to zero, we obtain 𝑖^∗=𝜇/𝛽₃, and 𝐹₁(𝑎)≜𝛽₂𝑎²+𝑝−𝛽₂+𝛽₁+𝛽₂+𝛽₃𝜇𝛽₃𝑎+𝛽₁𝜇𝛽₃²+𝑝+𝛾−𝛽₁𝜇𝛽₃−𝑝=0. (7) When 𝑅>1,𝑖^∗=𝜇/𝛽₃<𝑖⁰<1; notice that 𝛽₁(𝑖⁰)²+(𝑝−𝛽₁+𝛾)𝑖⁰−𝑝=0, so 𝛽₁(𝜇/𝛽₃)²+(𝑝+𝛾−𝛽₁)(𝜇/𝛽₃)−𝑝<0, so (7) has a unique solution 𝑎^∗ in(0,1); when 𝑅<1, (7) has no positive solution.

Theorem 4. When 𝑅>1, system (3) has the equilibrium 𝑃⁰ and 𝑃^∗(𝜇/𝛽₃,𝑎^∗), 𝑃⁰ is unstable, and 𝑃^∗ is globally asymptotically stable.

Remark 5. When 𝛽₃>𝜇/𝑖⁰>𝜇, new products can still diffuse successfully even if the influence of mass media is neglected in the decision-making stage.

Case II (𝛼≠0). When 𝛼≠0, let 𝑝+𝛽₁𝑖+𝛽₂𝑎(1−𝑖−𝑎)−𝛾𝑖−𝛼𝑖−𝛽₃𝑖𝑎=0,𝛼𝑖+𝛽₃𝑖𝑎−𝜇𝑎=0. (8) From the second equation of the formula (8), we derive 𝑖̂=𝜇𝑎/(𝛼+𝛽₃𝑎)<𝜇/𝛽₃, so, we assume 𝜇<𝛽₃<1 to guarantee the positive equilibrium's existence. Substituting 𝑖̂ into the first equation of the formula (8), we obtain 𝐹₂(𝑎)≜𝑞₀𝑎⁴+𝑞₁𝑎³+𝑞₂𝑎²+𝑞₃𝑎+𝑞₄=0. (9) Here 𝑞₀=𝛽₂𝛽²₃>0,

𝑞₁=2𝛼𝛽₂𝛽₃+(𝑝−𝛽₂)𝛽²₃+𝜇(𝛽₁+𝛽₂+𝛽₃)𝛽₃>0,

𝑞₂=𝛼²𝛽₂+2𝛼(𝑝−𝛽₂)𝛽₃+𝛼𝜇(𝛽₁+𝛽₂+𝛽₃)+𝜇²𝛽₁+𝜇(𝑝+𝛼+𝛾−𝛽₁)𝛽₃−𝑝𝛽²₃,

𝑞₃=(𝑝−𝛽₂)𝛼²+𝜇(𝑝+𝛼+𝛾−𝛽₁)𝛼−2𝑝𝛼𝛽₃,

𝑞₄=−𝑝𝛼²<0.

Notice that 𝐹₂(0)<0, 𝐹₂(1)=𝑞₀+𝑞₁+𝑞₂+𝑞₃+𝑞₄=𝜇(𝛼+𝛽₃)(𝛽₂+𝛽₃+𝑝+𝛼+𝛾)+𝜇²𝛽₁>0, so (9) at least has one positive solution in the interval (0,1), and

𝐹^′₂(𝑎)=4𝑞₀𝑎³+3𝑞₁𝑎²+2𝑞₂𝑎+𝑞₃,

𝐹₂^′′(𝑎)=12𝑞₀𝑎²+6𝑞₁𝑎+2𝑞₂,

𝐹₂^′′(1)=5𝛽₂𝛽²₃+6𝛼𝛽₂𝛽₃+2(𝑝−𝛽₂)(𝛽²₃+𝛼𝛽₃)+𝜇(𝛽₁+𝛽₂+𝛽₃)(𝛼+𝛽₃)+𝛼²𝛽₂+𝜇²𝛽₁+𝜇(𝑝+𝛼+𝛾−𝛽₁)𝛽₃>0.

When 𝑞₂≥0, because 𝑞₁>0, 𝐹₂^′′(𝑎)=0 has no positive root on (0,1) and 𝐹₂^′′(1)>0, this means that 𝐹₂(𝑎) is concave on [0,1]; moreover 𝐹₂(𝑎) either increases on [0,1] or decreases first and then increases on [0,1], and it follows that (9) has a unique root on (0,1).

When 𝑞₂<0, on one hand, we can deduce that 𝐹₂^′′(𝑎)=0 has a positive root on (0,1) which means that 𝐹₂(𝑎) has inflection points on (0,1). Combining 𝐹₂^′′(1)>0, it is proved that 𝐹₂(𝑎) is convex first and then concave on (0,1). On the other hand, we can also deduce that 𝐹^′₂(0)<0, which show that (9) has a unique root on (0,1).

In summary, (9) has a unique root 𝑎 on (0,1); therefore, we have the following result.

Theorem 6. When 𝛽₃>𝜇, system (3) has a unique positive equilibrium 𝑃(𝑖,̂𝑎), and it is globally asymptotically stable.

Remark 7. New products diffusion is successful when 𝛽₃>𝜇, and it is obvious that the critical value of successful diffusion decrease when 𝛼≠0, which means new products' successful diffusion is relatively easier than which there is no persuasive advertisement's effect in the decision-making stage.

3. Numerical Simulation

To verify our analysis, we give the simulation results. Figure 1 is time series of system (3) with different initial values; parameters are chosen as 𝑝=0.001, 𝛽₁=0.0001, 𝛽₂=0.0004, 𝛾=0.0001, and 𝜇=0.015. In Figure 1, we fix 𝑢(0)=0.3; by 𝑖(0)+𝑎(0)=1−𝑢(0), we take the initial value (𝑖(0),𝑎(0)) as (0.6,0.1), (0.3,0.4), (0.1,0.6), (0.2,0.5), (0.4,0.3), and (0.5,0.2), respectively. (a) Figure 1 suggests that the equilibrium 𝑃⁰ is globally asymptotic stable when 𝛼=0,𝛽₃=0.01, and 𝑅=0.6107. Namely, when 𝑅<1, although there are many people who are aware of the product, no one buys it, and new products cannot diffuse on the market; (b) suggests the equilibrium 𝑃^∗ is globally asymptotically stable when 𝛼=0,𝛽₃=0.03, and 𝑅=1.8322; it shows that new products can diffuse successfully on the market when 𝑅>1.

(a)

(b)

Figure 1 

(a) and (b) are time series of system (3) when 𝛼=0.

4. Sensitivity Analysis and Diffusion Strategies

In real world applications, our main objective is to guarantee that new products can diffuse on the market. So, we consider the influence of the parameter of the system on products diffusion; that is to say, we study the relationship between the basic reproduction number 𝑅 and parameters in the model. By some calculations, it is obvious that 𝜕𝑅=𝛽𝜕𝑝2𝜇𝛽³₁−1+𝑝+𝛾+𝛽₁𝑝+𝛾−𝛽₁²+4𝑝𝛽₁>0,𝜕𝑅𝜕𝛽₁=𝛽₃𝑝+𝛾𝑝+𝛾−𝛽₁²+4𝑝𝛽₁−𝑝+𝛾−𝛽₁𝑝+𝛾+2𝑝𝛽₁×2𝜇𝛽²₁𝑝+𝛾−𝛽₁²+4𝑝𝛽₁⁻¹>0𝜕𝑅𝜕𝛽₃=𝛽₁−𝑝−𝛾+𝑝+𝛾−𝛽₁²+4𝑝𝛽₁2𝜇𝛽₁>0, (10) while 𝜕𝑅𝛽𝜕𝜇=−³𝛽₁−𝑝−𝛾+𝑝+𝛾−𝛽₁²+4𝑝𝛽₁2𝜇²𝛽₁<0,𝜕𝑅=𝛽𝜕𝛾2𝜇𝛽³₁−1+𝑝+𝛾−𝛽₁𝑝+𝛾−𝛽₁²+4𝑝𝛽₁<0. (11) It can be seen that, among these parameters, 𝑅 is an increasing function of 𝑝, 𝛽₁, and 𝛽₃. Opposed to this, 𝜇 and 𝛾 have an inversely proportional relationship with 𝑅. So, in order to guarantee new products can diffuse on the market, namely, to make 𝑅 larger than one, we can take some measures such as enhancing the effect of mass media in the awareness stage and reinforcing the interpersonal persuasion in the decision-making stage and so forth.

It is not enough that new products can diffuse on the market; we are more willing to see that there are more and more adopters. For this purpose, we will show the influences of parameters on the population 𝑎(𝑡) in Figures 2 and 3. It is observed that the peak and the stable state of 𝑎(𝑡) increase while 𝑝, 𝛽₁, 𝛽₂, and 𝛽₃ are increasing, respectively, or while 𝜇 and 𝛾 is declining, respectively, and that the stable state is arrived in advance, while the peak is postponed. The peak and the stable state of 𝑎(𝑡) do not change clearly when 𝛽₁, 𝛽₂, 𝛽₃, or 𝛾 varies respectively. When 𝛽₃ is increasing, the stable state of 𝑎(𝑡) has no obvious variation, but its peak obviously increases.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

When 𝛼=0, evolution of 𝑎(𝑡) for different values of 𝑝, 𝛽1, 𝛽2, 𝛽3, 𝛾, and 𝜇, respectively.

Figure 3 

Time series of 𝑎(𝑡) with changes of 𝛼.

Figure 3 is time series of system (3) when 𝛼=0,𝛼=0.0005,𝛼=0.005,𝛼=0.05, and 𝛽₃=0.03. We find that with the increment of 𝛼, the density of adopters 𝑎 gradually increases, and the speed tending to steady states becomes more faster.

To sum up, enhancing the effect of mass media in two stages, reinforcing the interpersonal persuasion in the decision-making stage, and decreasing the loss of adopters are effective measures to new products diffusion with the number of adopters increasing and the arrival time of the stable state decreasing.

5. Conclusion and Discussion

In this paper, we have proposed a homogeneous model with the awareness stage and the decision-making stage to expound new products diffusion processes. We assume product information spreads due to two routes: word-of-mouth among people's direct contacts and mass media channels. If the influence of mass media is neglected in the decision-making stage, we have shown that the model admits a threshold 𝑅. when 𝑅<1, namely, 𝛽₃<𝜇/𝑖⁰, new products diffusion is unsuccessful, few people being aware of the product and no one buying it; when 𝑅>1, namely, 𝛽₃>𝜇/𝑖⁰>𝜇, new products diffusion is successful, which means new products can diffuse on the market even without the influence of mass media in the decision-making stage. If the influence of mass media is considered in the decision-making stage, when 𝛽₃>𝜇, new products diffusion is successful. It is obvious that the critical value of successful diffusion decreases. All theoretical results are confirmed by numerical simulations, and some diffusion strategies are given by parameters' sensitive analysis.

Simulations imply that persuasive advertisement is very helpful to the prevalence of new products from the perspective of economics, and it can accelerate new products prevalence. In addition, simulations also indicate that enterprises should take different advertisement strategies in different stages: before adopters tend to stable states, enterprises should take the persuasive advertisement strategy to tell consumers why they should choose a particular brand; after adopters tend to stable states, namely, once the brand is popularized, the enterprise should take the remindful advertisement strategy to remind consumers that there is such a product, which is useful to save advertising cost.

Acknowledgments

This work is supported by a Grant from the National Natural Science Foundation of China (11171314, 11201434, and 11301491), TianYuan Special Foundation of the National Natural Science Foundation of China (11226259), Shanxi province Natural Science Foundation (2012011002-1 and 2013021002-1), and Scientific research item for the Returned Overseas Chinese Scholars of Shanxi province (2010-074).