The 'hit' phenomenon: a mathematical model of human dynamics interactions as a stochastic process

We start the modeling from the viewpoint of the individual consumer. We define the purchase intention of the individual consumer, labeled i, at time t as I_i(t). We assume that the number of products adopted until time t can be written as

$$\begin{equation}Y(t) = p\int_{t_0 }^t {\mathop \sum \limits_{i = 1}^N I_i (t)\,{\rm{d}}t} ,\end{equation}\noindent \tag{ 7 }$$

where N is the maximum number of adopted persons, p is the price of the product and t₀ is the release date of the product. Thus, our problem is to define the equation of the purchase intention of each consumer I_i(t). We consider the modeling of the effects of advertisements, WOM and rumor for the purchase intention in the following subsections.

The advertisement effect through mass media such as TV, newspapers, magazines, the Web, Facebook and Twitter is modeled as an external force for the equation of the purchase intention of the individual consumer:

$$\begin{equation}\frac{{{\rm{d}}I_i (t)}}{{{\rm{d}}t}} = c_i A(t),\end{equation}\noindent \tag{ 8 }$$

where A(t) is the time distribution of the effective advertisement effect per unit time and the coefficient describes the impression of the advertisement for consumer i. The external force A(t) can be considered as trends in the world or political pressure on the market. In the application to the motion picture business in the Japanese market, we input the real daily advertisement budget used by the largest advertisement office in Japan, Dentsu Inc.

Usually, a film's success spreads through WOM. Such WOM sometimes has a very significant effect on the success of the movie. Thus, the WOM effect should be included in our theory.

The WOM effect can be distinguished into two types: WOM direct from friends and indirect WOM as rumors. We term the WOM effect between friends direct communication, because customers obtain information directly from their friends. In previous marketing theories based on the Bass model [21–24], usually only communications from adopter to non-adopter are taken into account. Here, we include communication between non-adopters. It is very significant for movie entertainment, especially before the opening of the movie. Let us consider that person i hears information from person j. The probability per unit of time for the information to affect the purchase intention of person i can be described as D_ij I_j (t) , where I_j (t) is the purchase intention of person j and D_ij is the coefficient of the direct communication. A schematic image of the direct communication of persons i and j is shown in figure 1. Thus, we can write the effect of the direct communication as follows:

$$\begin{equation}\mathop \sum \limits_{j = 1}^N D_{ij} I_j (t),\end{equation}\noindent \tag{ 9 }$$

where the summation is done without j = i.

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In this paper, the rumor effect is termed as indirect communication.In this form of communication, a person hears a rumor while chatting on the street, overhearing a conversation from the next table in a restaurant or on a train, or finds the rumor in blogs or on Twitter. The situation is illustrated in figure 2(a), where many conversations are conducted on the streets of a city. To construct the theory using mathematics, we focus on one person who listens to a conversation happening around him/her. Let us consider that person I overhears the conversation between person j and person k. The strength of the effect of the conversation can be described as D_jk I_j (t)I_k (t) . The probability per unit time for the conversation to affect the purchase intention of person i is defined as Q_ijk D_jk I_j (t)I_k (t) , where Q_ijk is the coefficient. Thus, the indirect communication coefficient can be defined as P_ijk = Q_ijk D_jk . The situation is shown in figure 2(b).

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Therefore, direct communication is two-body interaction and indirect communication is three-body interaction. Thus, our theory for 'hit' phenomena such as hit movies or music can be described as the equation of the purchase intention of person i with two-body interaction and three-body interaction terms.

A person tends to only watch a particular movie once. This means that the potential number of attendees decreases monotonically after the opening day. In figure 3, we show the typical decline of cinema box office returns for the Japanese movie market. It should be pointed out that DVD sales in Japan for a movie begin several months after the opening day at the cinema. This is the business practice in the Japanese entertainment industry. Thus, on the days shown in figure 3, no one can buy DVDs or online movies in Japan.

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We found in figure 3 that the revenues decrease almost exponentially. This evidence is very natural, because the number of audience members decreases monotonically due to the effect that a person who has watched the movie does not watch the same movie again. In figure 3, only film E shows a sudden decrease. This happened due to a sudden scandal involving the actress in the movie. From the data, we find that the decay factors for most movies in figure 3 seem to be similar. The decay factor of the exponential decay is nearly 0.06 per day. This value is similar to the value reported in [17] for Spider-Man in the US market. It agrees well with the empirical rule of the Japanese movie market that the number of audience members undergoes roughly a 6% decrease.

This exponential decrease can be explained easily using a simple mathematical model. First, we denote the number of potential audience members as N₀ and the number of integrated audience members at time t as N(t). Thus, the number of people who are interested in the movie is N₀ − N(t). Assuming that the probability of watching the movie per day is a, we obtain the equation to describe the number of audience members as

$$\begin{equation}\frac{{{\rm{d}}N(t)}}{{{\rm{d}}t}} = a(N_0 -N(t)).\end{equation} \tag{ 10 }$$

The solution of the equation with N₀ = 0 at t = 0 is

$$\begin{equation}N(t) = N_0 (1-{\rm{e}}^{-at} ).\end{equation} \tag{ 11 }$$

It is clear that the result can explain roughly the exponential decay of the audience shown in figure 3. The purchase intention of person i, I_i(t), can also be considered to decay in a similar manner.

According to the above consideration, we write down the equation of purchase intention at the individual level as

$$\begin{equation}\frac{{{\rm{d}}I_i (t)}}{{{\rm{d}}t}} = -aI_i (t)+\sum\limits_j {d_{ij} I_j (t)} +\mathop \sum \limits_j \mathop \sum \limits_k h_{ijk} I_j (t)I_k (t)+f_i (t),\end{equation} \tag{ 12 }$$

where d_ij, h_ijk and f_i(t) are the coefficient of the direct communication, the coefficient of the indirect communication and the random effect for person i, respectively. We consider the above equation for every consumer so that i = 1,...,N_p.

Taking into account the effect of direct communication, indirect communication and the decline of audience, we obtain the above equation for the mathematical model for the hit phenomenon. The advertisement and publicity effect for each person can be described as the random effect f_i(t).

Equation (12) is the equation for all individual persons, but it is not convenient for analysis. Thus, we consider the ensemble average of the purchase intention of individual persons as follows:

$$\begin{equation}I\left( t \right) = \frac{1}{N}\mathop \sum \limits_i I_i \left( t \right).\end{equation} \tag{ 13 }$$

Taking the ensemble average of equation (12), we obtain for the left-hand side

$$\begin{equation}\frac{{{\rm{d}}I_i (t)}}{{{\rm{d}}t}} = \frac{1}{N}\mathop \sum \limits_i \frac{{{\rm{d}}I_i (t)}}{{{\rm{d}}t}} = \frac{{\rm{d}}}{{{\rm{d}}t}}\left( {\frac{1}{N}\mathop \sum \limits_i I_i (t)} \right) = \frac{{{\rm{d}}I}}{{{\rm{d}}t}}.\end{equation} \tag{ 14 }$$

For the right-hand side, the ensemble average of the first, second and third terms is as follows:

$$\begin{equation}-aI_i = -a\frac{1}{N}\mathop \sum \limits_i I_i (t) = -aI(t),\end{equation} \tag{ 15 }$$

$$\begin{eqnarray}&&\mathop \sum \limits_j {\rm{d}}_{ij} I_j (t) = \mathop \sum \limits_j {\rm{d}}I_j (t) = \frac{1}{N}\mathop \sum \limits_i \mathop \sum \limits_j {\rm{d}}I_j (t) = \mathop \sum \limits_i {\rm{d}}\frac{1}{N}\mathop \sum \limits_j I_j (t) = N{\rm{d}}I(t),\nonumber\\ \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \mathop \sum \limits_j \mathop \sum \limits_k p_{ijk} I_j (t)I_k (t) &= & p\mathop \sum \limits_j \mathop \sum \limits_k I_j (t)I_k (t) \nonumber\\ & = & \frac{1}{N}\mathop \sum \limits_i p\mathop \sum \limits_j \mathop \sum \limits_k I_j (t)I_k (t) \nonumber\\ & = & \mathop \sum \limits_i p\frac{1}{N}\mathop \sum \limits_j \mathop \sum \limits_k I_j (t)I_k (t) \nonumber\\ & = & Np\mathop \sum \limits_i \frac{1}{N}\mathop \sum \limits_j I_j (t)\frac{1}{N}\mathop \sum \limits_k I_k (t) \nonumber\\ & = & N^2 pI(t)^2 , \end{eqnarray} \tag{ 17 }$$

where we assume that the coefficient of direct and indirect communications can be approximated to be

$$\begin{eqnarray*}&&d_{ij} \cong d, \end{eqnarray*}$$

$$\begin{eqnarray*}&&h_{ijk} d_{jk} = p_{ijk} \cong p\end{eqnarray*}$$

under the ensemble average.

For the fourth term, the random effect term, we consider that the random effect can be divided into two parts: the collective effect and the individual effect:

$$\begin{equation}f_i (t){\rm{ }} = {\rm{ }}f(t)+\Delta f_i (t),\end{equation} \tag{ 18 }$$

$$\begin{equation}f_i (t) = \frac{1}{N}\mathop \sum \limits_i f_i (t) = f(t),\end{equation} \tag{ 19 }$$

where Δ f_i (t) means the deviation of the individual external effects from the collective effect, f(t) . Thus, we consider here that the collective external effect term f(t) corresponds to advertisements and publicity to persons in society. The deviation term Δ f_i (t) corresponds to the deviation effect from the collective advertisement and publicity effect for individual persons, which we can assume to be

$$\begin{equation}{\rm{\Delta }}f_i (t) = \frac{1}{N}\mathop \sum \limits_i {\rm{\Delta }}f_i (t) = 0.\end{equation} \tag{ 20 }$$

Therefore, we obtain the equation for the ensemble-averaged purchase intention in the following manner:

$$\begin{equation}\frac{{{\rm{d}}I(t)}}{{{\rm{d}}t}} = -aI(t)+DI(t)+PI(t)^2 +f(t),\end{equation} \tag{ 21 }$$

where

$$\begin{eqnarray*}&&Nd = D, \end{eqnarray*}$$

$$\begin{equation}N^2 p = P.\end{equation} \tag{ 22 }$$

Equation (21) can be applied to the purchase intention in the real market. Equation (21) is the equation we assumed in our previous works without derivation [25–30]. In this paper, we apply this equation to the motion picture business.

In much of the previous marketing science research [1–15, 20–24], advertising costs are considered only in total. However, the time distribution of the daily advertising cost is very important. In figure 4, we show the advertising cost and related gross rating point (GRP) for The Da Vinci Code in the Japanese market. GRP is a term used in advertising to measure the size of the audience reached by a specific media vehicle or schedule [31]. It is just the product of the percentage of the target audience reached by an advertisement times the frequency at which they see it in a given movie campaign. Both the daily advertising costs and the GRP for each movie in the Japanese market are obtained from Dentsu Inc., the largest advertising agency in Japan. In Japan, Dentsu handles most advertisements appearing on every TV station except NHK (the national station), so all campaigns for all movies in all Japanese prefectures are organized entirely by Dentsu; thus, the advertisement cost data from Dentsu are exactly equal to the advertisement costs of the movies in the Japanese market. The value of the daily advertising costs does not include the discount for major movie production agencies in Japan, but the discount rate is the same during the campaign of each movie.

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It was clearly seen in figure 4 that the advertising costs and GRP are distributed for two months after the opening of a movie. This feature can be found for every movie campaign in the Japanese market. Thus, not only the total advertising cost but also the time distribution of the costs is very important for the success of the advertising campaign for a motion picture.

The number of daily audience members for each movie is very important for the mathematical study of a motion picture campaign. In Japan, sales of DVDs or online access to movies is forbidden until several months after the opening of the movie in movie theaters. Thus, the number of audience members is the exact number of people who watch the movie. During this time, no one can watch the movie without going to the movie theater. We obtain the data for daily audience numbers from Box Office Japan (Kogyo Tsushinsha, Tokyo).

Daily blog postings for a movie are a very important signal to measure the movement of purchase intention among persons in the society. We measure the daily data of the number of posts for movies using the site Kizasi, which is a service for observing blog postings in Japan. We measure the number of blog posts for 25 movies in Japan in order to compare this information with the box office gross income for each movie in Japan. We show several results in figures 5–9. In these figures, the curve of daily blog posts is multiplied by a certain constant to normalize the value to be similar to the revenue value.

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From figures 5–9, we find that the curve of the daily blog posts for each movie is very similar to the curve of the daily revenue we reported in our previous works [27, 28]. We observed 25 movies in the Japanese market and found a similar feature for all 25 movies. Since the selected 25 movies were a random sampling of hit movies in Japan, we can consider that the similarity can be applied to the most popular motion pictures in the Japanese market. This observation means that the ratio of blog postings for each person is almost constant during the duration of each movie so that the daily number of blog postings is very similar to the daily revenue.

Blog postings for each film can be distinguished into positive, negative and neutral opinions. A positive opinion means that the blogger wants to watch the film or judges the watched film in a positive way. In figure 10, we show that more than half of the blogs show a positive opinion for several movies. Moreover, we find that the ratio of positive, negative and neutral opinions is almost constant during the duration of the movie opening. Thus, the observed blog posting counts can be considered to be proportional to the counts of positive blog posts.

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According to this observation, we propose to use the daily number of blog posts as the daily 'quasi-revenue.'. Quasi-revenue is very useful for analysis, because it can be defined even before the opening of the movie. We can observe the increase in anticipation of a movie.

For the calculation of each movie, we should derive the actual formulation for the calculation where the adopter and the non-adopter are distinguished. We can write the direct communication term of the non-adopter-to-non-adopter interaction in equation (21) as follows:

$$\begin{eqnarray} DI(t) &= & \displaystyle\frac{1}{N}\mathop \sum \limits_i \mathop \sum \limits_j {\rm{d}}I_j (t) \nonumber\\[-4pt] &= & \displaystyle\frac{1}{N}d\mathop \sum \limits_i N\frac{1}{N}\mathop \sum \limits_j I_j (t) \nonumber\\[-4pt] &= & \displaystyle\frac{{N_p -N(t)}}{{N_p }}(N_p -N(t))\frac{1}{{N_p -N\left( t \right)}}\mathop \sum \limits_j {\rm{d}}^{nn} I_j (t) \nonumber\\[-4pt] &= & \displaystyle\frac{{N_p -N(t)}}{{N_p }}(N_p -N(t)){\rm{d}}^{nn} I(t), \end{eqnarray}\noindent \tag{ 23 }$$

where

$$\begin{equation}N(t) = N_p \int_0^t {I(\tau )\,{\rm{d}}\tau } .\end{equation} \tag{ 24 }$$

The direct communication term for the adopter to the non-adopter is

$$\begin{equation}DI(t)\mathop \Rightarrow \limits_{} \frac{{N(t)}}{{N_p }}(N_p -N(t)){\rm{d}}^{ny} I.\end{equation} \tag{ 25 }$$

Similarly, we obtain the indirect communication term due to the communication between non-adopters at time t:

$$\begin{eqnarray} PI(t)^2 &= & p\displaystyle\frac{1}{N}\mathop \sum \limits_i \mathop \sum \limits_j \mathop \sum \limits_k I_j (t)I_k (t) \nonumber\\ & = & p\displaystyle\frac{1}{N}\mathop \sum \limits_i N\frac{1}{N}\mathop \sum \limits_j N\frac{1}{N}\mathop \sum \limits_k I_j (t)I_k (t) \nonumber\\ &= & \left( \displaystyle{\frac{{N_p -N(t)}}{{N_p }}} \right)^3 N_p^2 p^{nn} I^2, \nonumber\\ && \frac{{\left( {N_p -N(t)} \right)^3 }}{{N_p }}p^{nn} I^2 , \end{eqnarray} \tag{ 26 }$$

where pⁿⁿ is the factor of the indirect communication between non-adopters at time t. For the indirect communication, we obtain two more terms corresponding to the indirect communication due to the communication between adopters and that between an adopter and a non-adopter as follows:

$$\begin{equation}\frac{{\left( {N(t)} \right)^2 \left( {N_p -N(t)} \right)}}{{N_p }}p^{yy} I^2 +\frac{{N(t)\left( {N_p -N(t)} \right)^2 }}{{N_p }}p^{ny} I^2 ,\end{equation} \tag{ 27 }$$

where p^yy is the factor of the indirect communication between adopters and p^ny is the factor of the indirect communication due to the communication between adopters and non-adopters at time t.

Finally, we obtain the equation of purchase intention for the actual calculation as follows:

$$\begin{eqnarray} \frac{{{\rm{d}}I(t)}}{{{\rm{d}}t}} &= & -aI(t)+f(t) \nonumber\\ [3pt] && +\frac{{N_p -N(t)}}{{N_p }}\left( {N_p -N(t)} \right)d^{nn} I+\frac{{N(t)}}{{N_p }}\left( {N_p -N(t)} \right){\rm{d}}^{ny} I \nonumber\\ && +\frac{{\left( {N_p -N(t)} \right)^3 }}{{N_p }}p^{nn} I^2 +\frac{{\left( {N(t)} \right)^2 \left( {N_p -N(t)} \right)}}{{N_p }}p^{yy} I^2 \nonumber\\ [3pt] && +\frac{{N(t)\left( {N_p -N(t)} \right)^2 }}{{N_p }}p^{ny} I^2 . \end{eqnarray} \tag{ 28 }$$

For the period before the opening of the movie, there are no adopters. Furthermore, the total number of adopters is zero. Thus, for the period before opening, equation (28) is reduced to the following form:

$$\begin{equation}\frac{{{\rm{d}}I(t)}}{{{\rm{d}}t}} = -aI(t)+f(t)+N_p {\rm{d}}^{nn} I(t)+N_p^2 p^{nn} I(t)^2 .\end{equation}\noindent \tag{ 29 }$$

Equation (28) with (24) is the nonlinear integro-differential equation. However, since the data are handled daily, the time difference is 1 day. We can solve the equation numerically as a difference equation.

For the purpose of reliability, we introduce here the so-called 'R-factor' (reliable factor), well known in the field of low-energy electron diffraction (LEED) experiments [32]. In LEED experiments, the experimentally observed curve of current versus voltage is compared with the corresponding theoretical curve using the R-factor.

For our purpose, we define the R-factor as follows:

$$\begin{equation}R = \frac{{\mathop \sum \nolimits_i \left( {f(i)-g(i)} \right)^2 }}{{\mathop \sum \nolimits_i \left( {f(i)^2 +g(i)^2 } \right)}},\end{equation}\noindent \tag{ 30 }$$

where the functions f(i) and g(i) are defined in figure 11. The smaller the R-factor, the better the match in functions f and g. We use this R-factor as a guide to obtain the best adjustment of our parameters for each movie.

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We perform calculations using equation (29) for many movies in the Japanese market. We show several results in figures 12–16. The solid line shows our calculation with the real daily advertisement costs as input data of 〈f(t)〉 of equation (29). In the calculation, we use trial and error to decide the parameters to minimize the R-factor. The durations we calculate the R-factors are equal to those of each corresponding figure. We also perform Metropolis-like random number sampling for some parameters to minimize the R-factor, but not for all the parameters, because it would require too much calculation time. The R-factor for each calculation is shown in the figure captions of figures 12–16. For D_nn and P_nn , the coefficients for direct and indirect communication between non-adopters, we decide the values before and after the opening day separately, because the audience response can change from the impression of the movie itself.

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The results are compared with the number of blog posts as the quasi-revenue shown as a red histogram in the figures. We found that the agreement of the calculation with the quasi-revenue (blog posts) is very good.

The parameters we use in the above calculations are shown in table 1. We found that the value is similar for these movies, although not equal. The value for Avatar will be discussed later.

In figures 12–16, we find that our calculations agree well with the observed daily number of blog posts for each movie. Since we find that the daily number of blog posts for a movie is proportional to the revenue for the movie, this agreement means that our calculation can reproduce the revenue for each movie. For the Japanese motion picture market, new movies are watched in cinemas, because online downloads and DVDs are available only several months after the opening day of the movie. Moreover, the cinema entrance fees in the Japanese market are similar for almost all cinemas. Thus, the agreement of our calculation with the daily blog post numbers means that our calculation can describe the movement of people in society very well, at least for the Japanese motion picture market.

The agreement here can be considered to mean that our theory can describe the collective motion of people in society, at least for the entertainment market in Japan. Since our theory is considered to be general, we expect that it can be applied to other entertainment markets in the world.

As we saw in section 3.1, the advertisement cost has a distribution in time. The calculated result using equations (28) and (29) of our theory depends strongly on the time distribution of the advertisement cost. This means that, even for the same total cost, the total revenue can change depending on the time distribution of the advertisement cost. Thus, using our equations (28) and (29), we can consider how to optimize the time distribution of the advertisement cost to obtain the maximum value of the total revenue at the condition of the fixed total advertisement cost.

One of the original ideas of the present theory is the effect of indirect communication, as explained in section 2.3. To demonstrate the effect of indirect communication, we perform two calculations: one including indirect communication and one without indirect calculation. The example we show here is the calculation for Avatar, which was a very big hit in 2010. Both calculations are optimized as much as possible using the Metropolis-like method of parameter optimization with random numbers to minimize the R-factor. The result is shown in figure 17. We found that the indirect calculation is better. The parameters for the best fit are shown in table 1. As we can see in table 1, the parameters of indirect communication are larger for Avatar. The result shows us that indirect communication played a very significant role in causing the success of Avatar. Since this calculation can be applied to many other big hit movies, we can conclude that indirect communication as included in our theory is very important in describing human interactions in real society.

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The Bass model [21–22] is well known to be the model of the spread of WOM. In this subsection, we verify that the Bass model is derived from the equation of the mathematical model of the hit phenomenon. For sales of the production, the equation of our mathematical model for the hit phenomenon is written as follows:

$$\begin{equation} \frac{{{\rm{d}}I_i (t)}}{{{\rm{d}}t}} = A_i +\mathop \sum \limits_j D_{ij} I_j (t)+\mathop \sum \limits_j \mathop \sum \limits_k P_{ijk} I_j (t)I_k (t).\end{equation} \tag{ 31 }$$

The total number of sales for the product, N, can be defined using the number of people, m, and the purchase intention, as follows:

$$\begin{equation}\mathop \sum \limits_i I_i (t) = mI(t) = N(t).\end{equation} \tag{ 32 }$$

In the Bass model, advertisement is spread from adopters to non-adopters. Thus,

$$\begin{equation}\mathop \sum \limits_i A_i = a\left( {m-N(t)} \right) = \mathop \sum \limits_i a,\end{equation} \tag{ 33 }$$

where the sum is only the persons who do not buy the product.

For the direct communication term in equation (31), $\sum\nolimits_i {\sum\nolimits_j {D_{ij} I_i I_j } }$, we assume here that the coefficient D_ij is not zero only for the pair of persons, adopter and non-adopter. The purchase intention for the adopter is considered to be I_j = 0 . Thus, we can express the direct communication term as follows:

$$\begin{equation}\mathop \sum \limits_i \mathop \sum \limits_j D_{ij} I_j = \left( {m-N(t)} \right)N(t)\,{\rm{d}}I = \left( {m-N(t)} \right)N(t)\,b.\end{equation} \tag{ 34 }$$

Thus, summing up for i, we obtain from equation (21)

$$\begin{equation}\frac{{{\rm{d}}\mathop \sum \nolimits_i I_i (t)}}{{{\rm{d}}t}} = \mathop \sum \limits_i \left[ {A_i +\mathop \sum \limits_j D_{ij} I_j (t)+\mathop \sum \limits_j \mathop \sum \limits_k P_{ijk} I_j (t)I_k (t)} \right].\end{equation} \tag{ 35 }$$

Therefore, substituting (34) into (28), we obtain

$$\begin{equation}\frac{{{\rm{d}}N(t)}}{{{\rm{d}}t}} = a\left( {m-N(t)} \right)+\left( {m-N(t)} \right)N\left( t \right)b+\mathop \sum \limits_i \mathop \sum \limits_j \mathop \sum \limits_k P_{ijk} I_j (t)I_k (t).\end{equation} \tag{ 36 }$$

Here, if we neglect the indirect communication term as P_ijk = 0, we obtain the well-known Bass model equation

$$\begin{equation}\frac{{{\rm{d}}N(t)}}{{{\rm{d}}t}} = a\left( {m-N(t)} \right)+\left( {m-N(t)} \right)N(t)b.\end{equation} \tag{ 37 }$$

Therefore, we find that our mathematical model for the hit phenomenon includes the Bass model and addresses the previously neglected indirect communication. Thus, the indirect communication in equation (21) is the new WOM effect.

In this paper, we calculate the time variation of the averaged action of humans in real society using our theory. Equation (21) or equations (28) and (29) seem to describe the collective action of humans for entertainment well. As in the usual physics approach, we calculate the prediction and compare it with observation. Thus, if we obtain a large quantity of observation data from real society, the usual method of statistical physics or many-body physics is available to investigate the human interaction in real society. For this purpose, of course, the human interaction for each individual problem should be investigated carefully with a social scientist specialising in communication.