type
[x] Book Catalogue Framework/Curriculum Framework
key word
* broadcast model (an information source), diffusion model (word-of-mouth), SIR model (considering recovery), susceptible person, infected person, restorer, bath model, hypothesis, probability decomposition, basic reproductive number R0, vaccination threshold, group immunity, critical point depending on environment and direct critical point.
structure
* We use broadcast model, diffusion model and infection model to analyze the spread of information, technology, behavior, beliefs and infectious diseases between people. These models play a central role in the research of communication science, marketing and epidemiology.
* This model relates the microscopic process of thinking and the spread of infectious diseases to the shapes of these adopted curves.
* All models introduced in this chapter assume the existence of relevant population, which is represented by NPOP. Relevant personnel include those who may have infectious diseases, know information or take action.
* At any time, there will always be some people suffering from an infectious disease, knowing specific information or taking certain actions. We call these people infected or insiders (denoted by It), and other members of the relevant population except infected or insiders are susceptible (denoted by St). These susceptible people may be infected with infectious diseases, get information or take action.
* The total number of people involved is equal to the sum of the number of infected people or the number of informed people plus the number of susceptible people: NPOP = IT+ST.
First, the broadcast mode
* In a given period of time, the number of insiders is equal to the number of insiders in the previous period plus the probability of susceptible people hearing information multiplied by the number of susceptible people. The result will be an R-shaped adoption curve.
* It+ 1=It+Pbroad×St
* where Pbroad stands for broadcast probability, and It and St are equal to the number of infected people (insiders) and susceptible people at time T, respectively.
* The initial state is I0=0 and S0=NPOP.
* Broadcasting mode depicts the spread of ideas, rumors, information or technology through television, radio, Internet and other media. The goal of this model is to describe the process of publishing information from information sources, which can be government, enterprises or newspapers. This model is not suitable for infectious diseases or ideas spread from person to person.
* In the broadcast model, everyone in the relevant crowd will eventually know the information. With appropriate data, we can estimate the size of the population concerned.
Second, the diffusion model
* Diffusion model
* Most infectious diseases, as well as information about products, ideas and technological breakthroughs, are spread by word of mouth, and the diffusion model describes these processes. The diffusion model assumes that when a person adopts a certain technology or suffers from an infectious disease, he may pass or infect this technology to the people who come into contact with him.
* In this model, just like in the communication model, in the long run, everyone in the relevant groups will have information. The difference is that the adoption curve of diffusion model is S-shaped.
* In the broadcast model, it is quite simple to estimate the relevant population size according to the data. The number of initial adopters is closely related to the size of the relevant population. On the contrary, it may be very difficult to estimate the size of the relevant population using the data of diffusion model. The increase in product sales may be due to the high diffusion probability in the small related population or the low diffusion probability in the large related population.
* Premise assumption
* In the case of infectious diseases, personal choice will not play any role. The spread of technology is related to the choice of users, so the more useful technology is, the easier it is to be adopted. We didn't explicitly consider this situation in the model.
* The diffusion model assumes random mixing. Random mixing means that any two people in related groups have the same possibility of contact. If it is applied to the urban population, it is problematic. In cities, people are not randomly mixed. A hypothesis doesn't have to be very accurate to be part of a useful model. So we will continue to use this assumption, while keeping an open mind and changing this assumption at any time when it is necessary to change.
* Probability decomposition
* The occurrence of such incidents varies from environment to environment. We can define the diffusion probability as the product of contact probability and sharing probability. We can build the model according to the diffusion probability, but when estimating or applying the model, we must track the contact probability and sharing probability independently.
* Overflow of application software: the first probability is hard to change. In order to increase the second probability, developers can provide some incentives for old users who bring new registered users. Although this can improve the diffusion speed, it will not affect the total sales, at least according to this model. As mentioned above, the total sales volume is equal to the size of the relevant population, regardless of the sharing probability, and increasing the sales speed will not bring long-term impact.
* bathtub model
* Most consumer goods and information are disseminated through broadcasting and communication. The bath model combines these two processes. The difference equation in Bass model is equal to the sum of the difference equations in broadcast model and diffusion model. In bath model, the greater the diffusion probability, the more significant the S-shape of the curve.
Three. Infectious SIR model
* SIR model
* In the model we discussed, once someone adopts a technology, they will never give up. But this does not apply to everything that spreads through diffusion. For example, we will soon recover from infectious diseases, or we can give up after adopting a certain fashion or participating in a certain trend movement.
We call a person who gives up what he has received a healer. The resulting model, namely SIR model (susceptibility, infection and cure), occupies a central position in epidemiology.
* In order to avoid too complicated mathematical calculation, we assume that people who have cured infectious diseases will re-enter the susceptible population, which means that they will not be immune to infectious diseases after curing infectious diseases.
* basic regeneration number R0
* An infectious disease. If R0 is greater than 1, it can spread in the whole population, while infectious diseases with R0 less than 1 tend to disappear.
* The proportion of people who must be vaccinated, that is, the vaccination threshold, can be obtained by the formula Vt≥(R0- 1)/R0. We can deduce this formula from the above model. For infectious diseases with very high R0, such as measles and polio, the government will make efforts to ensure that all people are vaccinated.
* Some people are worried about the side effects of the vaccine and choose not to participate in the vaccination program. If these people are only a small part of the population, then vaccinating others can also prevent these people from contracting this infectious disease, which epidemiologists call group immunity. People who choose not to be vaccinated actually hitchhike with other vaccinated people.
* Super communicator
* If the SIR model is embedded in the network, the importance of degree distribution to the spread of infectious diseases will be observed.
* For hub-and-spoke networks, R0 carries limited information, because if the central node suffers from infectious diseases, infectious diseases will spread. Epidemiologists call people at the highly central node "super diffusers". A high number of nodes can not only spread infectious diseases faster, but also infect infectious diseases faster. The contribution of nodes to the spread of infectious diseases (or ideas) is related to the square of node degree.
* Success and tipping point
* Although the SIR model was originally used to analyze the spread of infectious diseases, we can also apply it to all social phenomena that spread first and then tend to disappear, such as the sale of books, the popularity of songs, the popularity of dance steps, the spread of "hot words", the spread of recipes and fitness methods.
* In these cases, we can also estimate the contact probability, transmission probability and "recovery" probability, as well as the basic regeneration number R0. This model means that as long as these probabilities change slightly, R0 can move to a level higher than zero, thus making a world of difference between success and failure.
Success may depend on a very small difference, and there is only a very small difference between doing one thing well and screwing it up.
* In the SIR model, we derived two key thresholds, namely R0 and vaccination threshold. These two thresholds belong to the critical point where sensitivity depends on the environment, and small changes in the environment (situation) will have a great impact on the results. This critical point is different from the direct tipping point. At the direct critical point, a small action at a specific moment will permanently change the path of the system.
* and at the critical point that depends on the environment, the change of parameters will change the behavior of the system. At the direct critical point, the trajectory of future results has taken a sharp turn.
* Confuse capsizing with sharp rise (fall), leading to the overuse of the term critical point. Critical points mentioned in news media and Internet forums rarely meet the formal definition.
* Model modification
* When applying broadcast model, diffusion model and contagion model to social phenomena, we may find that some assumptions are valid, while others are not. In these cases, we may have to modify the basic model to allow the adoption probability of each contact to increase with the increase of contact times. This modification is usually necessary when expanding the application scope of the model.