The propagation dynamics during phase 1 for one piece of posted information is modeled based on the traditional susceptible-forwarding-immune [35] model, with a novel stratification of the immune population. Namely, there will be two classes of immune populations (as far as information 1 is concerned): those who have forwarded the posted information but are no longer in the active period of forwarding this posted information (I₁₊) and those who have been exposed to the information but are not interested in forwarding it (I_1–). This distinction of immunity is important, as individuals in these two distinct compartments will have different levels of interest in other relevant information that will be posted later. This will allow us to introduce different measures of attractiveness to new relevant information.

So, in our model, we stratify the population (N₁) into four states: the susceptible state of the posted information (S₁), the forwarding state of the posted information (F₁), the inactive immune state (I₁₊), and the direct immune state (I_1–). A susceptible user can be exposed to the posted information with an average exposure rate β₁ and will forward the information with the forwarding probability p₁. The forwarding users can become inactive immune users with an average rate α₁. So a user may have a unique state, with S₁(t), F₁(t), I₁₊(t), and I_1–(t) denoting the number of users in the susceptible, forwarding, inactive, and direct immune state, respectively. We obtain the following delay in transmission susceptible-forwarding-immune (DT-SFI) dynamics model in phase 1:

The state transition of different populations in phase 1 can be interpreted as follows: an active forwarding user will contact an average number of β₁N₁ users per unit time, and the probability of a user being a susceptible user of the posted information is S₁(t)/N₁, so an active forwarding user will contact β₁S₁(t) susceptible users. There are F₁(t) active forwarding users of information 1 in total at time t, so p₁β₁S₁(t)F₁(t) susceptible users will choose to forward the information and become new forwarding users, and (1 – p₁) β₁S₁(t)F₁(t) will not. As time goes by, α₁F₁(t) will go to the immune state from the forwarding period when they do not influence other users as far as information 1 is concerned.

For data fitting purposes, we noted that the Sina Microblog provides important data about any piece of information relevant to COVID-19, the number of cumulative forwarding quantity, given by:

Now we consider that a piece of new information is posted at time t_τ when the posted information is already in the quasi–steady state. We introduce the following indexes:

Accordingly, we introduce three states of the population (N₂) for the newly posted information: the susceptible state (S₂), the forwarding state (F₂), and the immune state (I₂). We summarize the notations in Textbox 1.

Each user may have a unique state, with I₁₊(t), I_1–(t), S₂(t), F₂(t), and I₂(t) denoting the number of users in the susceptible, forwarding, and immune state, respectively. We obtain the following LTI DT-SFI dynamics model in phase 2:

The mass action in phase 2 can be interpreted as follows: an active forwarding user will contact an average number of β₂₁N₂ inactive immune users of the posted information per unit time, and the probability of a user being an inactive immune user is I₁₊(t)/N₂, so an active forwarding user will contact β₂₁I₁₊(t) inactive immune users, among which m₂₁p₂β₂₁I₁₊(t)F₂(t) will choose to forward the new information and (1 – m₂₁p₂)β₂₁I₁₊(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t; an active forwarding user will contact an average number of β₂₂N₂ direct immune users of the posted information per unit time, and the probability of a user being a direct immune user is I_1–(t)/N₂, so an active forwarding user will contact β₂₂I_1–(t) direct immune users, among which m₂₂p₂β₂₂I_1–(t)F₂(t) will choose to forward the new information and (1 – m₂₂p₂)β₂₂I_1–(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t; an active forwarding user will contact an average number of β₂₃N₂ susceptible users of the newly posted information per unit time, and the probability of a user being a susceptible user is S₂(t)/N₂, so an active forwarding user will contact β₂₃S₂(t) susceptible users, among which m₂₃p₂β₂₃S₂(t)F₂(t) will choose to forward the new information and (1 – m₂₃p₂)β₂₃S₂(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t.

The forwarding quantity of the newly posted information is given by:

Since the newly posted Weibo starts at different times and develops differently under the influence of prior information, we defined the information reproduction ratio as:

This is the total number of information 2 users generated by introducing a typical information 2 user, at time τ after information 1 was posted, during their entire period of active forwarding. The initial population for information 2 has been stratified by the exposure of the entire population to information 1 during the time interval [0, τ]. The relative size of to the unity determines if information 2 can generate an information outbreak; since (0) = (m₂₁p₂β₂₁I_{1 + ,τ} + m₂₂p₂β₂₂I_{1 – ,τ} + m₂₃p₂β₂₃S₂₀ – α₂) F₂ (0), we concluded that (0)>0 if >1.

The model is consistent with that of phase 1 of the LTI DT-SFI model.

Considering that the new information is posted during the outbreak period of the old information, we developed our phase 2 model to describe the concurrent dynamic process of the two related pieces of information. We considered the difference between the population in active forwarding state or the immune state out of the active period and the population in the direct immune state of the first (old) piece of information. Here, we set the following indexes:

Assuming that the number of users (N₃) who can contact the information in the process of information propagation on Sina Microblog remains unchanged, we introduced three states of the population of the newly posted information: the susceptible state (S₁) that includes the users who can be exposed to the old information and the new information, the forwarding state of the new information (F₂), and the immune state (I₂). The parameters are shown in Textbox 2.

Each user may have a unique state, with S₁(t), F₁(t), I₁₊(t), I_1–(t), F₂(t), and I₂(t) denoting the number of users in the susceptible, forwarding, and immune state when t>0, respectively. We obtain the following STI DT-SFI dynamics model in phase 2:

The mass action in phase 2 can be interpreted as follows: an active forwarding user will contact an average number of β₂₁N₃ inactive immune users and forwarding users of posted information per unit time, and the probability of a user being an inactive immune user and a forwarding user are I₁₊(t)/N₃ and F₁(t)/N₃, respectively, so an active forwarding user will contact β₂₁I₁₊(t) inactive immune users and β₂₁F₁(t) forwarding users, among which m₂₁p₂β₂₁I₁₊(t)F₂(t) and m₂₁p₂β₂₁F₁(t)F₂(t) will choose to forward the new information, however, (1 – m₂₁p₂)β₂I₁₊(t)F₂(t) and (1 – m₂₁p₂)β₂₁I₁(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t. Here, the individuals in the forwarding state and the individuals in the immune state have the same familiarity with the topic content who have forwarded information 1; an active forwarding user will contact an average number of β₂₂N₃ direct immune users of posted information per unit time, and the probability of a user being a direct immune user is I_1–(t)/N₃, so an active forwarding user will contact β₂₂I_1–(t) direct immune users, among which m₂₂p₂β₂₂I_1–(t)F₂(t) will choose to forward the new information and (1 – m₂₂p₂)β₂₂I_1–(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t; an active forwarding user will contact an average number of β₂₃N₃ susceptible users, and the probability of a user being a susceptible user is S₁(t)/N₃, so an active forwarding user will contact β₂₃S₁(t) susceptible users, among which m₂₃p₂β₂₃S₁(t)F₂(t) will choose to forward the new information and (1 – m₂₃p₂)β₂₃S₁(t)F₂(t) will not, where F₂(t) is the number of new active forwarding users at time t.

The forwarding quantity of the newly posted information is given by:

Considering the initial condition in phase 2, we can obtain the following public opinion reproduction ratio . The new information entered at time τ during an outbreak period of the posted information, and (0) = (m₂₁p₂β₂₁F_1τ + m₂₁p₂β₂₁I_{1 + ,τ} + m₂₂p₂β₂₂I_{1 – ,τ} + m₂₃p₂β₂₃S_1τ – α₂)F₂(0). The population will never take off if (0) = (m₂₁p₂β₂₁F_1τ + m₂₁p₂β₂₁I_{1 + ,τ} + m₂₂p₂β₂₂I_{1 – ,τ} + m₂₃p₂β₂₃S_1τ – α₂)F₂(0)<0 due to the decrease of S_10τ. It is therefore natural to introduce the following as the STI DT-SFI reproduction ratio:

In the same way, the of the STI DT-SFI model denotes the comprehensive public opinion generated by the newly posted Weibo starting at the outbreak period of the posted information. When the reproduction ratio <1, it means that the new public opinion will decline. When the reproduction ratio >1, it indicates that the new public opinion will initially grow exponentially.

Since the COVID-19 outbreak in China, intensive information that were clearly relevant to each other have been frequently posted. Figure 4 shows the total forwarding quantities of Weibos during each 1-hour time frame on January 25, 2020, and January 26, 2020 (data acquisition up to February 19, 2020), for the top 10 opinion leaders of this outbreak event in the Chinese Sina Microblog. As illustrated, those pieces of information with high influence were posted frequently by each opinion leader. At the same time, there was a strong correlation among a series of Weibos posted by these opinion leaders during certain periods. Of all the data shown in Figure 4, People’s Daily issued five Weibos in 1 hour from 10 PM to 11 PM on January 25, and they were forwarded by a total of more than 300,000 users. Therefore, the frequent release of relevant information by original post owners was a common phenomenon in the COVID-19 information propagation, and understanding its effectiveness is important.

Figure 5 shows the trend of cumulative forwarding users for the three pieces of information in Tables 1-3. It shows that when information A broke out, information B was posted almost immediately. Compared with the information, the outbreak period of information B was shorter and the trend was flatter. Information C was released during the quasi–steady-state period of information B. In comparison with this information, the outbreak period of information C lasted longer; meanwhile, the cumulative forwarding quantity was also larger.

Sequentially releasing two related pieces of information by the same original post owners within the same COVID-19 theme was a common phenomenon. Importantly, different entering times of new information during the spreading process of an old (previously posted) information exhibited different promoting effects on the cross-propagation and cross-promotion of relevant public opinions. Here we focus on users who have been exposed to one posted information that may have a special interest in, and hence are susceptible to, new and relevant information. This represents a remarkable difference from the spread of rumors and other traditional public hot events. Our information cross-propagation DT-SFI models, including the STI DT-SFI dynamics model and the LTI DT-SFI dynamics model, were developed to take into consideration the situations when the relevant information is posted during the outbreak period or during the quasi–steady-state period of the previously posted (old) information.

To fit our model with real data from the Sina Microblog, we used the least squares (LS) method to estimate the LTI DT-SFI model parameters and the initial data. In phase 1, the parameter vector is set as Θ₁ = (p₁, β₁, α₁, S₁₀), and the corresponding numerical calculation based on the parameter vector for C₁(t) is denoted by (k, Θ₁). The follow LS error function was used in our calculation:

where C_1k denotes the actual cumulative forwarding populations of the posted information. Similarly, in phase 2, the vector is set as Θ₂ = (β₂₁, β₂₂, β₂₃, m₂₁, m₂₂, m₂₃, p₂, α₂, S₂₀), and the corresponding numerical calculation based on the parameter vector for C₂(t) is denoted by (k, Θ₂). The following LS error function was used in our calculation:

where C_2k denotes the actual cumulative forwarding populations of the newly posted information. Here, n=1, 2 ... represents the different phases, and k=0, 1, 2, ... is the sampling time n=1, 2, 3. We estimated the parameters of our LTI DT-SFI model with the data of information B and information C.

Figure 6 reports our data fitting results for information B and information C on the real data given in Tables 2 and 3, where the blue star denotes the actual cumulative number of forwarding users of information B; the red star denotes the actual cumulative number of forwarding users of information C; and the green line and the black line denote the estimated cumulative number of forwarding users of information B and information C, respectively.

Tables 4 and 5 give estimated values of important parameters for information B and information C, respectively. We can see in phase 2, when information C was posted during the quasi–steady-state period of information B, the average exposure rate β₂₁ was the largest, indicating that an inactive user of the posted information B was more susceptible to the newly posted information C; the average exposure rate β₂₃ was small, indicating that a susceptible user of the newly posted information C contacted the information at a lower rate. In addition, among the three attractiveness indexes, the index m₂₂ is the largest, which indicates that information C had the strongest appeal to a direct immune user of information B and has the least attractiveness to an inactive user of the posted information B.

By comparison, there is a difference between the initial time of a new piece of information at the outbreak phase and at the quasi–steady-state phase of the posted information. When the initial time of new information is in the outbreak phase of the posted information, the value of the average immune rate α₂ is generally higher than the value in the quasi–steady-state phase, which is due to the rapid outbreak of information (a large amount of information updates and iterations). The average active duration 1/α₂ of forwarding users of the new piece of information where users can influence other users to contact information is shorter. Similarly, the average forwarding probability p₂ of the initial time in the outbreak period is also higher than the value in the quasi–steady-state phase, which conforms to the fact that people are more willing to participate in the discussion successively when exposed to relevant information in the short term. In comparison, the value of average contact rate β₂₁ in the outbreak phase is lower than that in the quasi–steady-state phase, indicating that the population who has forwarded information will be larger based on a relatively larger contact population. β₂₂ and β₂₃ of the initial time in the outbreak period are higher than those of the initial time in the quasi–steady-state phase, indicating that continuous exposure to relevant information in the short term (when the initial time of the new piece of information is in the outbreak phase of a posted information) would attract people who had not participated in the new transmission to forward and spread the information. All average attractiveness indexes of the initial time in the quasi–steady-state phase are larger, which indicates that information that is re-exposed to users after a period of time will inspire their freshness and make them pay more attention to the information itself.

To use our model to explore some distinctions of the qualitative behaviors for prediction, we used the LS method to estimate the STI DT-SFI model parameters and the initial data of our model. The vector is set as Θ₃ = (p₁, β₁, α₁, p₂, β₂₁, β₂₂, β₂₁, m₂₁, m₂₂, m₂₃, α₂, S₁₀), and the corresponding numerical calculation based on the parameter vectors for C₁(t) and C₂(t) are denoted by (k, Θ₃) and (k, Θ₃), respectively. The following LS error function was used in our calculation:

where C_1k and C_2k denote the actual cumulative forwarding populations of the posted information and the newly posted information; here, n=1, 2, 3 represents the different phases, and k=0, 1, 2, ... is the sampling time n=1, 2, 3. We estimated the parameters of our STI DT-SFI model with the data of information A and information B.

In the data fitting of the STI DT-SFI model, we used the same method as for the LTI DT-SFI model to fit the data of information A and information B. As shown in Figure 7, we performed data fitting of information A and information B on the real data in Tables 1 and 2, where the pink star denotes the actual cumulative number of forwarding users of information A; the red star denotes the actual cumulative number of forwarding users of information B; the green line and the blue line denote the estimated cumulative number of forwarding users in the early and later period of information A, respectively; and the black line denotes the estimated cumulative number of forwarding users of information B. It can be seen that our STI DT-SFI model achieves accurate estimation.

Table 6 gives some important values of parameter (relevant to the early period of the outbreak) estimation for information A, and Table 7 gives some important values of parameter estimation for the later period data of information A and all data of information B. We can see in phase 2, when information B was posted during the outbreak period of information A, the average exposure rate β₂₁ and β₂₂ are much larger than β₁ and β₂₃, which indicates that users who have been exposed to information A will contact information B at a greater rate than new susceptible users. In addition, the unexposure attractiveness index m₂₃ is the largest among the three attractiveness indexes since the time interval between two information posts is small, and people who have not been exposed to relevant information may have a greater interest in new information; the outbreak of information B has the strongest appeal to susceptible users.