2.1. The Model of Information Disclosing
The information dissemination system resp. behavioral response system is embedded in the information network resp. physical network. Both networks are given as follows.
Information network: the network has (N+1) nodes, first N are individual nodes representing N individuals denoted as i,i=1,2,⋯N, and one government information node denoted as j. The degree of an individual node i is denoted as yi, which obeys a power-law distribution, that is, Fyi∝yi−v, where F(·) is the CDF and yi satisfies ϵ≤1∕yi≤1, where ϵ is a small constant to avoid the degree to blow up. Degree and degree distribution are concepts used in graph theory and network theory. A graph (or network) consists of a number of vertices (nodes) and the edges (links) that connect them. The number of edges (links) connected to each vertex (node) is the degree of the vertex (node). The degree distribution is a general description of the number of degrees of vertices (nodes) in a graph (or network), and, for random graphs, the degree distribution is the probability distribution of the number of degrees of vertices in the graph, which usually assumes a power-law distribution. Throughout the following analysis, we take v=−1 and ϵ=0.01. The government node j (representing real-world government) discloses information to every individual node and can only obtain information from n1 (n1≪N) (The notation “≪” means that the number n1 must be far less than the number N.) random nodes. The neighborhood of an individual node i is the set of all other nodes (including j) it connects with, denoted as Oi.
Physical network: the physical network has M nodes, including n2 “special” nodes defined as the “gathering spots”, which predisposes these nodes to this epidemic. Mt denotes the distribution of locations of all N individuals during period t, and M0 is the initial distribution that can be viewed as the “home” for every individual (node), thus at the beginning of each period t the individuals move from M0 to Mt and return back to M0 at the end of period t. Home coordinates M0 and gathering spots are randomly assigned and different from each other, so we have N+n2<M. Suppose there are n3 random nodes, each with identical initial information ξ, who disseminate information at the outbreak of disease; n4 random nodes are initially affected by the public crisis, representing the “patient zero”.
Without loss of generality, we unitize the information between 0 and 1. The rules for information dissemination in each period are as follows.
Stage i. Individual nodes send information to neighbors. Each node that has information at the beginning of each period sends its information to all its neighbors, so all (N+1) nodes might receive information from others. As information is spontaneously [19], rapidly, and extensively [22] misrepresented during transmission, and most people do not send more accurate information than they receive [17,18], we assume that information gets distorted and misrepresented during each transmission. Thus, the actual amount of information received is δxi due to information decay, where δ∼U(0,1), and we assume xi∈0,1 without loss of generality.
Stage ii. Individual nodes receive information from neighbors. Each node might have multiple information sources, and it merges the information from all its neighbors weighted by their degrees (and including itself). Each individual updates its information based on Equation (1) at each period before the government intervenes:(1) xi,t+1=∑k∈Oiδxk,tyk+xi,tyi∑k∈Oiyk+yi.
Stage iii. The government node censors and screens the information. The government has a threshold XD once it receives information from individuals (otherwise, the government would not act in this stage), the government will screen out all individuals with above-threshold information at the beginning of current period, among whom the government pinpoints the nearest ones and takes the maximum amount of information they carry denoted as xd.
Stage iv. Government node discloses information. The government is not able to intervene until it censors and screens the information; thus, there is a lag between receiving information and disclosing, which as we can see in Figure 4e, increases with XD. After the lag (otherwise, the government would not act in this stage), the government shall disclose xd to all nodes in each period with a weight of λ, where λ∈0,1. The higher the λ, the more credible the information.
Stage v. Individual nodes update information again. The government intervention switches the updating rule to (2) xi,t+1=λxd+(1−λ)∑k∈Oiδxk,tyk+xi,tyi∑k∈Oiyk+yi, which is also the final amount of the information after government intervention. In addition, we assume that the amount of information of initial information holders (those who have information in period 0) is constant, i.e., they do not apply for Equations (1) and (2).
In short, in the first period, only a few people disseminate information, which will be randomly decayed in each subsequent period, this process simulates the misrepresentation of information.
Twitter data show that there was a significant heterogeneity in the behavioral response to the COVID-19 epidemic [52]. Some people, once informed about the epidemic, wear a mask and practice social distance to not expose themselves to the virus—while others panicked, herded, and behaved irrationally because of bad news, exemplified by flocking to churches for psychological comfort [53], to supermarkets for daily supplies [54], and taking radical actions like repeated hospital visits [55]. Thus, in this paper, we group the population by susceptibleness to irrational behavior caused by information described by an exogenous parameter—individual threshold XI that distinguishes whether an individual is panic-prone or non panic-prone by comparing it with the amount of information the individual has. An above-threshold (under-threshold) information denotes a (non) panic-prone individual. For a panic-prone node, we assume its probability of going to gathering spots instead of maintaining the original trajectory is 1−x·,a, where x·,a is the amount of information it has. For a non panic-prone node, we assume that its probability of not moving is r·,N=x·,a. Thus, the behavioral routine is as follows (see Figure 1 for a simplified example): a node moves along with its path with a maximum radius d1, and the actual distance it moves obeys a uniform distribution in (0,d1); this node will randomly choose one of the gathering spots if intending to go to one in this period; every individual node follows this routine, then we have an evolving geographical distribution Mt of the population moving in period t. The uninfected will contact everyone within the maximum infection radius d2 and there is a probability μ of being infected for each contact.
Throughout the simulation analysis, we focus on the impact of three key parameters, the initial information (ϵ), individual threshold (XI), and disclosing threshold (XD), which are the most important quantities to measure the impact of government intervention on the coupled information-disease dynamics. The initial information is the source of all information, which denotes the medical awareness of the virus; the individual threshold is a parameter to distinguish the population by groups set above, the smaller it is, the higher the level of public health awareness. Disclosing the threshold, chosen by the government, measures its relative priority to speed and accuracy in information dissemination. One of the objectives of our experiment is to ascertain the optimal disclosing threshold. Government prioritizes speed more as its threshold is lower, “0“ means that government discloses the information immediately upon receipt; “1” means that government only discloses completely accurate information.