















































International Journal of Social and Educational Innovation (IJSEIro) 

Volume 10/ Issue 20/ 2023 

 

 

53 
 

 

CONSTRUCTION AND ANALYSIS 

OF THE ADVERTISING DISTRIBUTION MODEL 

 

T.R. NIKITINA  

Tavria State Agrotechnological University, Melitopol, Ukraine 

 

Abstract: The study investigates the construction characteristics and evaluates the advertising 

distribution model by utilizing the case of beauty products as an example. 

 

Keywords: advertising, distribution model, promoting campaign, customer analytics;  

 

1. Research context 

It is widely acknowledged that advertising plays a crucial role in the success of trade and often 

proves decisive in promoting a product. The advertising industry attracts highly talented 

individuals, including top-notch professionals who create their own unique styles. Advertising 

has become an integral part of our society's culture, influencing cinema, literature, and theater. 

Additionally, advertising shapes the mentality of the country, ultimately leading to changes in 

people's characters, desires, and ways of thinking. This, in turn, results in an accelerated pace 

of life for society as a whole [2]. 

As advertising is a form of information, we can create a mathematical model based on the 

classic model of information dissemination [] within a given society. The intense competition 

between manufacturers compels them to seek novel and innovative approaches to promote their 

products. Therefore, building such a model and analyzing its behavior based on varying 

parameters (such as advertising intensity and a person's propensity to change their mindset due 

to advertising) can undoubtedly prove useful for both a theoretical understanding and practical 

application. 

 

2. Literature Review 

The mathematical models studied in this paper are based on differential equations, including 

stochastic ones. Many well-known scientists, including A. V. Skorokhod, Y. I. Gikhman, M. 

M. Bogolyubov, devoted their work to the study of evolutionary systems in the form of 

stochastic equations. An extensive bibliography on this issue can be found in the book by V.S. 



International Journal of Social and Educational Innovation (IJSEIro) 

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54 
 

Monographs of Korolyuk. In particular, in [7], the approaches used in this article to the study 

of the asymptotic behavior of an evolutionary system with a diffusion perturbation were 

presented for the first time. The question of the asymptotic behavior of limit generators is of 

paramount importance. Similar problems were also solved earlier using qualitatively different 

methods (see, for example, [17]). The methods presented in our study allow us to explore a 

model consisting of Markov transitions that fit a random environment. In addition, these 

methods make it possible to reveal an additional diffusion component and significant jumps in 

the perturbation process in the boundary equation. The authors examine a social community of 

N0, potentially exposed to several types of advertising through two different channels. N1(t), 

N2(t), … denote the number of "followers" who have adopted the new ad product over time. 

In the presented article, a more natural generalization of the model is presented and the 

generator of limit processes is explicitly formulated, and an interpretation of the model is also 

proposed. 

 

3. Purpose and aims of the study 

The purpose of this article is to present the advancements in the asymptotic theory for stochastic 

evolutions, which are solutions of stochastic differential equations. The focus is specifically on 

those influenced by impulse disturbances and non-classical approximation schemes. 

Furthermore, the research aims to develop and scrutinize a model for advertising distribution. 

The effectiveness of this model will be evaluated using a small business as an example. 

 

4. The study 

In this study, we will utilize the classic model of information dissemination [1] to examine the 

efficacy of advertising the company's new beauty product to consumers. 

Let us revisit the fundamental assumptions underlying the model. Consider a community of 

consumers comprising 𝑁0 individuals who are targeted with advertising for our product. In 

other words, this community is receptive to advertising, which implies that the probability of 

modifying their attitude towards the product can be increased through the dissemination of 

relevant information. At a specific time point 𝑡 = 𝑡0, the advertising distribution source 

initiates its operations, leading to the proliferation of the product's advertisement within the 

community. 

The primary measure of the extent of advertising distribution is the quantity 𝑁(𝑡) which varies 

with time t and represents the number of individuals who have embraced the new product. For 



International Journal of Social and Educational Innovation (IJSEIro) 

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55 
 

ease of analysis, we will assume that at the onset 𝑡 = 𝑡0, 𝑁(𝑡0) = 𝑁(0) = 0, indicating that 

there are no initial supporters of the new product. 

1. Advertising will be disseminated within the community through two distinct 

information channels.  

a) The first channel is external to the community and may involve an advertising campaign 

conducted via social media, as well as seminars and meetings. The intensity of this channel, 

measured by the number of equivalent information actions per unit of time, is denoted by the 

parameter 𝛼11 > 0. 

b) The second channel is the internal channel, which involves interpersonal 

communication among members of the community. The intensity of this channel, which is 

measured by the parameter 𝛼12 > 0, is determined by the number of equivalent informational 

contacts. During communication, consumers who have been exposed to advertising (whose 

number is 𝑁(𝑡)), influence other consumers who have not yet been influenced (whose number 

is 𝑁0 − 𝑁(𝑡), thereby making an additional "personal" contribution to the process of generating 

interest in the new product. We will assume that any consumer who has not yet been exposed 

to advertising will always have the chance to receive the information that is distributed through 

external channels, with a certain probability of perceiving it. As a result, the rate at which 

consumers are recruited through external channels is determined not only by the factor 𝑁0 −

𝑁(𝑡) and the value of 𝛼11, but also by the value of 𝛼12, which represents the probability or 

tendency to perceive the information. This probability may depend on factors such as the level 

of trust in the information. It is important to note that although the internal channel is local and 

operates on a person-to-person basis, unlike the external channel, the speed of recruitment is 

still directly proportional to the number of consumers who have not been exposed to advertising 

yet. This value is denoted as 𝑁0 − 𝑁(𝑡), and is also determined by the intensity of contacts 

𝛼21, and the tendency of consumers to perceive information through the second channel, which 

is represented by the value 𝛼22. 

 

2. The change rate in the number of followers 𝑁(𝑡), which refers to the number of new 

members joining per unit of time, is determined by two factors.  

a. Firstly, the speed of external recruitment, which is directly proportional to the product 

of the recruitment intensity 𝛼11, the recruitment probability 𝛼12, and the number of potential 

members who have not yet been recruited ((𝑁0 − 𝑁(𝑡)).  



International Journal of Social and Educational Innovation (IJSEIro) 

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56 
 

Mathematically, this can be expressed as  

𝛼11𝛼12(𝑁0 − 𝑁(𝑡)) = 𝛼1(𝑁0 − 𝑁(𝑡)), where 𝛼1 = 𝛼11𝛼12 

b. Secondly, the speed of internal recruitment, which is directly proportional to the product 

of the recruitment intensity 𝛼21, the recruitment probability 𝛼22, the number of active followers 

𝑁(𝑡) and the number of potential members who have not yet been recruited 𝑁0 − 𝑁(𝑡)). This 

can be mathematically expressed as 𝛼21𝛼22𝑁(𝑡)(𝑁0 − 𝑁(𝑡)) = 𝛼2𝑁(𝑡)(𝑁0 − 𝑁(𝑡)), where 

𝛼2 = 𝛼21𝛼22. 

 

By summarizing assumptions 1 and 2, we can derive the differential equation that constitutes 

the advertising distribution model (1). 

𝑑𝑁(𝑡)

𝑑𝑡
= (𝛼1 + 𝛼2𝑁(𝑡))(𝑁0 − 𝑁(𝑡)), 𝑁(0) = 0, 𝑡 > 0 (1.1) 

The simplest macro model under consideration does not account for various factors that could 

be significant for the researched process, such as the heterogeneity of the social community, 

the effects of information "forgetting," the potential interdependence of α1and α2, values, the 

possibility of deliberate opposition to the advertising campaign, among others. Nonetheless, 

under these assumptions, the model permits an analytical solution. Specifically, the solution 

𝑁(𝑡) of problem (1.1) 𝑡 ≥ 𝑡0  =  0, with the initial condition 𝑁(0) = 0, takes the following 

form: 

 

𝑁(𝑡) =
𝑁0𝑒𝑥𝑝{𝛼1 + 𝛼2𝑁0}𝑡 − 𝛼1

(𝛼2 −
𝛼1
𝑁0

)𝑒𝑥𝑝{𝛼1 + 𝛼2𝑁0}𝑡
 

An analysis reveals that the solution, excluding the time t=0, is invariably positive, 

monotonically increasing, and gradually converges to the value of 𝑁0 over time. This model 

characterizes the diffusion of advertising when, ultimately (after a sufficiently long time, 

specifically t → ∞), all individuals within the group have been "recruited." To facilitate 

understanding, we will explore the discrete-time equivalent of model (1). By applying the 

definition of the derivative, we obtain: 

𝑑𝑁𝑖
𝑑𝑡

=
𝛥𝑁

𝛥𝑡
=

𝑁
𝑖
(𝑛+1)

− 𝑁
𝑖
(𝑛)

1
, 

where 𝑛 𝜖 (0, ∞) – time. 

We have obtained a discrete version of the model of competitive advertising struggle (1): 

𝑁 (𝑛+1) = (𝛼1 + 𝛼2𝑁
(𝑛))(𝑁0 − 𝑁

(𝑛)) + 𝑁(𝑛)  (1) 



International Journal of Social and Educational Innovation (IJSEIro) 

Volume 10/ Issue 20/ 2023 

 

 

57 
 

The primary limitation of the classical model is twofold: first, it assumes a constant intensity 

of information influence, and second, it fails to account for sudden and unpredictable events 

that may significantly impact the consciousness of information consumers. Given the current 

state of the world, where information spreads rapidly and reaches a broad audience, it is clear 

that rare yet highly influential factors must be considered. In this study, we develop and analyze 

a model of competitive advertising struggle in the following form: 

𝑑𝑁𝜀 (𝑡) = 𝐶 (𝑁𝜀 , 𝑥 (
𝑡

𝜀2
)) 𝑑𝑡 + 𝑑𝜂𝜀 (𝑡), 𝑁𝜀 (𝑡) ∈ ℝ. (22) 

where 

𝐶 (𝑁𝜀 , 𝑥 (
𝑡

𝜀2
)) = 

= (
−𝛼1(𝑥) + 𝛽1(𝑥)𝑁0 − 𝛽1(𝑥)𝑁1

𝜀 (𝑡) −𝛼1(𝑥) − 𝛽1(𝑥)𝑁1
𝜀 (𝑡)

−𝛼2(𝑥) − 𝛽2(𝑥)𝑁2
𝜀 (𝑡) −𝛼2(𝑥) + 𝛽2(𝑥)𝑁0 − 𝛽2(𝑥)𝑁2

𝜀 (𝑡)
) ×  

× (
𝑁1

𝜀 (𝑡)

𝑁2
𝜀 (𝑡)

) + (
𝛼1𝑁0
𝛼2𝑁0

) 

 

The proposed model considers both the stochastic impact of the environment on the level of 

information diffusion of an advertising campaign, as well as infrequent random fluctuations 

that cause significant short-term changes in the number of supporters of pertinent ideas. The 

primary finding of this study is that the impact of significant jumps is sustained in the limit 

process. These large jumps, for instance, represent high-impact events that immediately and 

substantially influence individuals' thoughts. While they are rare, their effect is substantial, 

which is not captured in any known models. In our problem formulation, the average model of 

competitive advertising struggle has the following form: 

𝐋𝜑(𝑤) = �̂�(𝑢)𝜑′(𝑤) + Γ𝜑(𝑤), 

where �̂�(𝑢) = ∫ 𝜋(𝑑𝑥)𝐶(𝑢, 𝑥)
𝑋

. 

 

If a switching process with "favorable" characteristics, such as the Ornstein-Uhlenbeck 

process, is known, the above equations can be explicitly computed while considering the type 

of potential operator. 

Consider the example of four vectors and their initial conditions as follows: 𝑁0 = 30000, 𝑁1(0) 

= 140, 𝑁2(0) = 180, 𝑁3(0) = 100, 𝑁4(0) = 129. The coefficients associated with these vectors 



International Journal of Social and Educational Innovation (IJSEIro) 

Volume 10/ Issue 20/ 2023 

 

 

58 
 

are 𝛼1 = 0.000012, 𝛼2 = 0.000015, 𝛼3 = 0.000018, 𝛼4 = 0.000015, 𝛽1 = 0.00000012, 𝛽2 = 

0.00000009, 𝛽3 = 0.0000001, and  𝛽4 = 0.00000012. 

 

 

As depicted in the graph, although the first type of advertising had a larger number of 

supporters at the outset, it ultimately lost to the second type due to the latter's higher growth 

rate, as ensured by its corresponding parameters. 

 

4. Summary and conclusions 

This study proposes a novel model for advertising campaigns that incorporates randomness, 

which may be more suitable for today's context where breaking news can have swift and 

substantial effects on audiences via television and the Internet. Unlike the classical case, the 

behavior of this generalized model cannot be explicitly analyzed for a fixed moment in time. 

As is common practice for stochastic models, it is feasible to derive functional limit theorems 

that capture the behavior of the process over extended time intervals. This enables the 

averaging of the process's limiting characteristics and the construction of explicit solutions. Put 

differently, any functions dependent on the Markov process must be averaged over the 

stationary measure of its transitions. 

 

 

 



International Journal of Social and Educational Innovation (IJSEIro) 

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