Mathematical analysis of the spreading of a rumor among different subgroups of spreaders

📝 Abstract

This paper presents a system of differential equations that describes the spreading of a rumor when it is propagated by different subgroups of spreaders. The system that is developed is a generalization of the model proposed by Daley and Kendall. Finally, the system is applied to the exchange rate of the parallel dollar in Venezuela, where the source data that was used were obtained from Google Trends.

💡 Analysis

This paper presents a system of differential equations that describes the spreading of a rumor when it is propagated by different subgroups of spreaders. The system that is developed is a generalization of the model proposed by Daley and Kendall. Finally, the system is applied to the exchange rate of the parallel dollar in Venezuela, where the source data that was used were obtained from Google Trends.

📄 Content

Mathematical analysis of the spreading of a rumor among different subgroups of spreaders Raúl Isea1* and Rafael Mayo-García2 1Fundación Instituto de Estudios Avanzados, IDEA. Hoyo de la Puerta, Baruta, Venezuela. 2CIEMAT. Avda. Complutense, 40, 28040 Madrid. Spain (*) Corresponding author: Raúl Isea. Email: risea@idea.gob.ve CITATION: R. Isea and R. Mayo-García. Mathematical analysis of the spreading of a rumor among different subgroups of spreaders. Pure and Applied Mathematics Letters (2015), Vol. 2015, pp 50-54

Abstract This paper presents a system of differential equations that describes the spreading of a rumor when it is propagated by different subgroups of spreaders. The system that is developed is a generalization of the model proposed by Daley and Kendall. Finally, the system is applied to the exchange rate of the parallel dollar in Venezuela, where the source data that was used were obtained from Google Trends. Keywords: Rumor, Daley and Kendall, parallel dollar, Google Trends. Introduction The following discussion is based on a social action in which we express our point of view. The interest of them is to transmit some information and the need to listen it by others. In fact, many times our communications depend upon the interests of the people who generate this information as well as upon those who are willing to receive it. Usually, we make a decision based on such information in order to verify the validity of it, and at that moment, a rumor could be created. According to the Dictionary of the Royal Spanish Academy (Spanish: Real Academia Española, RAE), a rumor is defined as “the voice which runs between the public". Thanks to Information and Communications Technologies, it is no longer required to have direct conversations between people. It is now possible to hold them through social networks in order to spread the idea, employing, for example, twitter, Facebook, blogs, etc. On the other hand, the aforementioned act of the dissemination of ideas is very similar to the spreading of an epidemic. Thus, from the initial information, there could be a widespread dissemination of these ideas among different its groups of people who endure different interactions. In this sense, Daley and Kendall established in their published work of 1965, the mathematical basis for modeling the spreading of a rumor and the disease in an epidemic by just emphasizing that similarity [1]. This work proposes a mathematical model to understand the transmission of a rumor among diverse subgroups of spreaders. In this sense, it will explain the behavior of the exchange rate of the parallel dollar in Venezuela under the theory of rumors that have been disseminated in the community. To accomplish this task, the scope of the model proposed by Daley-Kendall [1] will be generalized in order to explain the spreading of a rumor among different groups, i.e. rumor spreaders.

Daley and Kendall’s model Daley and Kendall proposed in 1965 [1], a mathematical model to simulate the process of the spreading a rumor, the so-called DK model. This model classifies the population into three different groups:

The ignorant population U which starts a rumor.

The spreading population V which spreads the rumor.

The stifler population W. which hears of the rumor and decides not to spread it. This model, which is depicted in figure 1, assumes that the rumor spreads according to the interaction between the ignorant and the spreader populations with a probability defined by /N. We defined the degree of acceptance of the rumor with . When a spreader interacts with a stifler, the rumor stops spreading and the likelihood of that happening is

(see the corresponding details in equations 1-3).

Figure 1. Scheme proposed by the Daley and Kendall’s model to describe the spreading of rumor (details of the notation are described in the text).

As it might be expected, the rumor loses its value over time [2-5]. Such a probability is defined by the factor. This fact is equivalent to considering that the rumor does not remain a novelty, either because it is already known or because it lacks the value to be broadcasted. From all this, the equations describing the model above are given by:

(1)

(2)

(3) The solution of the system of equations (1-3) is:

(4)

(5)

(6) Defining and , equations (4-6) can be written as:

(7)

(8)

(9) In the next section, the DK model will be generalized to a case where a source of the rumor is spread by subgroups of spreaders. The proposed model A model to describe the diffusion process of a single r

This content is AI-processed based on ArXiv data.