Abstract
Social network information is a measure of the number of infections. Understanding the effect of social network information on disease spread can help improve epidemic forecasting and uncover preventive measures. Many driving factors for the transmission mechanism of infectious diseases remain unclear. Some experts believe that redundant information on social media may increase people’s panic to evade the restrictions or refuse to report their symptoms, which increases the actual infection rate. We analyze the engagement in the COVID-19 topics on the Internet and find that the infection rate is not only related to the total amount of information. In our research, information entropy is introduced into the quantification of the impact of social network information. We find that the amount of information with different distributions has different effects on disease transmission. Furthermore, we build a new dynamic susceptible-exposed-infected-recovered (SEIR) model with information entropy to simulate the epidemic situation in China. Simulation results show that our modified model is effective in predicting the COVID-19 epidemic peaks and sizes.
Keywords: COVID-19, epidemic, information entropy, social media, susceptible-exposed-infected-recovered (SEIR) model
I. Introduction
During an epidemic outbreak, susceptibility to infection can be reduced by raising awareness of prevention from infectious diseases, but the effect of awareness is often unclear, which makes the simulation for disease control and prevention more difficult. Compartmental models proposed by Kermack and McKendrick [1] have been widely generalized and applied to analyze the spreading dynamics of infectious diseases. At the beginning of the 21st century, scholars have made great progress in the study of infectious disease dynamics. A large number of mathematical models are used to analyze various infectious diseases. These models are mainly studied in two aspects. On the one hand, some studies focus on the transmission mechanism of infectious diseases and discuss how direct contact, indirect contact, and vertical transmission have influence on the transmission mechanism [2], [3]. On the other hand, some researchers focus on the estimation of the basic regeneration number. Zhang and Teng [4] have added the saturated incidence rate into the SEIRS epidemic model and calculated the basic reproductive number of the model. By analyzing La Salle’s invariance principle, they have gotten global asymptotic stable conditions of the disease-free equilibrium and endemic equilibrium of the model. Gao et al. [5] have incorporated total population, time delay, and pulse vaccination to the SEIRS model and calculated the basic regeneration number of the model. Most of these models consider the biological characteristics of various diseases [6]–[11], but they often fail to predict the epidemic peaks and sizes. The reason is that these models do not take into considerations of the coupling of social and biological factors [12], [13]. The spread mechanism of epidemics is very complex. The role of social factors on the spread of a disease is often difficult to be represented as a compartment in classical models.
In recent years, scholars have proposed various dynamic models that consider social factors. A number of models [13]–[19] have linked media communication about a disease to protective action. These models postulate that media can suppress the spread of disease. Besides, the human population is also a common social factor that affects the spread of disease. Some models [20]–[23] incorporate population migration data into compartmental models to derive the epidemic curve. They prove that restricting population movement can effectively reduce the number of infected people [23]–[26].
The abovementioned models only consider the positive outcome brought by a single factor. Some negative outcomes are often neglected. Some scholars believe that, in the process of disease transmission, human behaviors are driven by awareness [27]–[30], which motivates us to investigate the problem of “is it possible to use an indicator to measure people’s awareness on the spread of disease?”
Since the outbreak of COVID-19, human beings have been faced with unprecedented serious threats. The World Health Organization (WHO) classified the spread like a pandemic in March 2020. Fig. 1 shows a map of the epidemic in China (as of March 3, 2020). As a pandemic sweeps the globe, most people are demanded to be in quarantine to avoid getting and spreading the contagion, which causes that much of the conversation about epidemics is taking place online. Hence, we suppose that the information on social media can be used to reflect people’s awareness.
Fig. 1.
COVID-19 map in China. The color and size of the circles represent the cumulative number of confirmed cases in the region (as of March 3, 2020).
Numerous studies [15]–[19] have attempted to measure the effect of social network information on the transmission of diseases, but most of them only focus on the positive effect of media data. In the case of a large-scale pandemic, on the one hand, the media coverage may cause a severe societal panic, while, on the other hand, it can certainly reduce the opportunity and probability of contact transmission among the alerted susceptible populations, which, in turn, helps to control and prevent the disease from further spreading [13], [31], [32].
Here, we introduce an informatics-based model, in which the probability of disease transmission is affected by the total amount and the distribution of information on social media. We assume that the great amount of information on social media prompts people to increase self-protection, which decreases the spread of disease. Meanwhile, we use information entropy [33] to describe the influence of information on people’s awareness. Entropy is sometimes referred to as a measure of the amount of “disorder” in a system. High entropy means lots of disorder, while low entropy means order. Different distributions of information have different values of entropy representing different effects on people’s awareness. If people’s awareness is in a disordered state, the infection rate will increase [34]–[36]. This means that the information entropy and the infection rate are positively correlated.
By incorporating information amounts and distributions of actual media data into disease transmission models, we incorporate social factors in the transmission dynamics and evaluate their significance in the disease transmission process. In this article, we discuss the relationship between the information distributions and infection rates, as the infection rate with information entropy reveals the dynamic characteristics of disease transmission. Furthermore, we incorporate the information entropy into the susceptible-exposed-infected-recovered (SEIR) model and simulate the spreading of COVID-19 in China. Finally, we compare the simulation results of the modified model and other SEIR models by calculating their Sørensen similarity index (SSI).
II. Impact of Social Media on Disease Transmission
A. Data Set
1). Social Media:
This data set contains the engagement in the COVID-19 topics on WeChat, Microblog, QQ, News, and other platforms. When people write or repost a related post with a COVID-19 topic, the topic engagement increases by 1. The data set covers the daily statistics of these platforms from January 1 to March 3.
2). Epidemic Data:
The data set consists of the number of COVID-19 cases in China published by the National Health Council of China (http://www.nhc.gov.cn/). The data include the number of confirmed cases, the number of cured cases, and the number of death cases. Some data examples are shown in Table I.
TABLE I. Some Official Data Released by Hubei Province.
| Date | Number of cumulative infections | Number of cumulative recoveries | Number of cumulative deaths |
|---|---|---|---|
| Jan 10, 2020 | 41 | 2 | 1 |
| Jan 11, 2020 | 41 | 6 | 1 |
| Jan 12, 2020 | 41 | 7 | 1 |
| Jan 13, 2020 | 41 | 7 | 1 |
| Jan 14, 2020 | 41 | 7 | 1 |
| Jan 15, 2020 | 41 | 12 | 2 |
| Jan 16, 2020 | 45 | 15 | 2 |
| Jan 17, 2020 | 62 | 19 | 2 |
| Jan 18, 2020 | 121 | 24 | 3 |
| Jan 19, 2020 | 198 | 25 | 4 |
| Jan 20, 2020 | 270 | 25 | 6 |
B. Statistical Analysis
Social media can be used as a measure of public concern about health-related events [37]–[39]. People’s attention to a public health emergency increases along with the aggravating spread of the pandemic. In other words, the transmission of epidemics can be reflected on the Internet. In order to figure out the correlation between information on the Internet and epidemic trends, we compare social media information with the number of confirmed cases, as shown in Fig. 2. From January 1 to March 3 at 24:00, the discussion (including posts and comments) on the topic of COVID-19 is over 2.5 billion. Among them, more than two billion in WeChat, more than 200 million in QQ, more than 200 million in Microblog, and more than 20 million in reports on news websites (including reprint).
Fig. 2.
Cumulative growth trend of the number of infection and the cumulative growth trend of information on the social media.
In Fig. 2, the trend of the total information and the trend of confirmed cases are roughly similar. We divide the total information into component data sets according to their platforms and then compare them with the trend of confirmed cases. There is a significant correlation between the epidemic transmission of COVID-19 and information from various platforms. In the following, we discuss the quantal relationship between the information and the infection rate.
C. Multicategory Information Entropy
Social media has both positive and negative effects in controlling the pandemic, and how to distinguish this difference is a challenging task. In our study, we use information entropy [33] to quantify the effect of information on epidemics. The calculation formula of information entropy is given as follows:
 |
where
is a random variable with possible outcomes
;
is probability of each
;
is the total number of
(
); and
is the base of the logarithm.
represents multicategory information entropy when
. The maximum value of multicategory information entropy is greater than 1, and it can be easily proved that the maximum value of information entropy occurs when all
’s are equal, i.e.,
. The maximum of entropy
. To avoid calculation errors, we assume that
in the event of
.
Information with different distributions has different values of entropy. We use the efficiency of entropy [40] as an indicator to quantify the differences. The efficiency is defined as
 |
where
is a random variable with possible outcomes
;
is probability of each
;
is the total number of
; and
is the base of the logarithm. Equation (2) can be simplified to
 |
Equation (3) shows that the constant
cannot influence the efficiency. The efficiency is only related to the distribution of information and the total number of
. The efficiency reaches the maximum when
. It means that information with uniform distribution has the largest information entropy, and at this time, information brings people lots of disorder.
III. SEIR Model With Information Entropy
We invent a model of disease transmission that incorporates social media into disease spread dynamics. In the model, we explicitly quantify the influence of social media from different sources, which leads to a variation in the per-contact probability of disease transmission.
We modify the original SEIR model to implicitly incorporate a dynamic infection rate by utilizing the information entropy. Conceptually, the flowchart of our model is shown in Fig. 3, which describes the dynamic process of the whole system.
Fig. 3.
Flowchart of the SEIR model with information entropy.
The dynamic model shown in Fig. 3 can be expressed as follows:
 |
and
 |
where
is the total number of people in the transmission system;
is the susceptible;
is the exposed people who are in the incubation period;
is the number of infected people;
is the number of removals, including recoveries and deaths;
is the average number of people in contact;
is the dynamic infection rate;
is the information entropy;
is the information amount (the total engagement in the COVID-19 topics on social media);
is the rate of the exposed being infected, and it is generally the reciprocal of the average incubation period; and
is the composite coefficient representing recovery or mortality.
A. Dynamic Infection Rate
The infection rate is the probability of an infection in a population. It is used to measure the frequency of occurrence of new instances of infection within a population during a specific time period. In mathematical modeling, the infection rate is expressed as
, which is usually a constant. In the research of infectious diseases, the infection rate is a very important parameter that can influence the public health department’s judgment on the cycle of infection. At the early stage of the epidemic, there are few case data. If the infection rate calculated from these data is used to judge the epidemic situation, it is not conducive to prevention and control. Because sufficient data are not available at the beginning of the epidemic outbreak. Especially for infectious diseases with a latent period, there are fewer early relevant data. The incubation period makes many characteristics of the disease not shown in the early stage of the outbreak, which leads to the misunderstanding of the disease.
People have different estimates of infection rates at different times. This is not because their method is incorrect. They use different data sets and have different understandings of the disease. We assume the infection rate as a variable and use the SEIR model to recall the actual infection rate [or infection impact rate (IIR)] of COVID-19. The differential equations of the original SEIR model are as follows:
 |
where
represents the susceptible;
represents the exposed;
represents the infected;
represents the removal;
is the total number of people in the transmission system;
is the average number of people in contact;
is the infection rate; and
reflects the intensity of disease transmission, i.e., the IIR. For discrete data,
, and
. Combining the above differential equations, we have the IIR of the SEIR model as follows:
 |
Equation (7) tells us that the IIR may be a dynamic value. Knowing real epidemic data, we can estimate the dynamic IIR
.
B. Infection Impact Rate
Different kinds of information drive people to take different measures, including effective and invalid measures, which changes the infection rate. To accurately calculate the infection rate, we consider quantifying the impact of information on the infection rate by using information entropy
.
We model the IIR as
 |
where
represents the total amount of information;
is a random variable with possible outcomes
;
is probability of each
;
is the total number of
(
);
is the base of the logarithm; and
and
are constant coefficients.
Let
(natural logarithm); then, we have
 |
where
representing the IIR is a function about information amounts
and distributions
. We transform (8) into a linear function. Hence, the least-squares method can be used for fitting parameters. With the actual outbreak data, we can calculate the IIR
, as shown in Fig. 4.
Fig. 4.
Dynamic IIR
with
.
It is depicted in Fig. 4 that the IIR
is a variable rather than a constant. The IIRs have been oscillating with large amplitudes until February, which implies the changes in people’s awareness and their behaviors. Based on these actual changes, we can estimate the parameters
and
in (8) by the least-squares method. Using the real data of social media and epidemics in China, we can get
, and
.
C. Identification of Parameters
In order to use mathematical models to simulate the actual epidemic situation, the first problem to be solved is to determine the parameters, including the IIR
and the removal rate
. We use results from clinical studies [41]–[43] as a reference.
In our research, we presume that the virus infectivity of these potential persons (
) is as same as that of the infected ones (
). According to [42], the first patient is inferred to have onset in the middle of December 2019. We set the initial time for our study as about December 15. The initial value of the infected population (
) is set as 1, so the initial value of the susceptible (
) is
-1. The initial value of the exposed (
) and the initial value of the removal (
) are 0. Guan et al.
[43] tell us that the incubation period of COVID-19 is two to seven days, and the median incubation is about four days. Therefore, the probability of the incubation period developing into a patient is
. The man who suffers from COVID-19 has an average recovery period of about ten days, so we set the removal rate
to 0.1.
Next, we mainly discuss how to identify the initial infection rate. At the beginning of the epidemic outbreak, people did not know the incubation period of COVID-19. Only confirmed patients are treated by the hospital and included in the epidemic data. Therefore, we can use the SIR model [6] to estimate the early infection rate. The differential equations of the SIR model are
 |
where
represents the susceptible;
represents the infected;
represents the removal;
is the total number of people in the transmission system;
is the average number of people in contact; and
is the initial infection rate at the beginning of the epidemic.
In the beginning,
. We have
 |
The general solution of the differential equation (11) is
 |
where
is constant. If
, we can get
. Then
 |
Now, we can construct the following identification problems.
Decision Variables:
.
Objective Function:
, where
is the actual number of patients and
is the time set containing all dates of the training set. We set
; then, the original problem can be equivalent to the following problem.
Decision Variables:
.
Objective Function:
.
In short, we transform the fitting function into a linear function. Hence, the least-squares method can be used for fitting parameters. Since January 22, Wuhan has decided to adopt “lockdown” as a method of reducing the epidemic spread. In order to avoid counting error, we use the data of confirmed cases from January 10 to January 21 for fitting the initial IIR
. By calculation, we get
. Substituting 0.1 for
, we have
.
D. Analysis of Equilibrium
In order to find out the stable state of the modified model, we discuss the equilibrium point of the model. We know that the IIR is a dynamic variable that changes with time. We set
, and let the equations in (4) all equal to 0; then, we have
 |
Thus, we can calculate the equilibrium point as follows:
 |
From the above equations, it is easy to know whether the equilibrium point of the model exists depends on the IIR. In other words, the amount of information determines the equilibrium point. Given (8), we have
 |
When time
,
. At this time, it can be considered that
. Therefore, we can know that the equilibrium point of the model exists and
 |
To sum up, the final stable state of the model is that there are no longer infected and exposed persons in the transmission system.
IV. Case Study
In this section, we use the modified model to simulate: 1) the number of infected cases in China from mid-December 2020 to March 3, 2020; 2) the number of infected cases in Hubei province before March 3, 2020; and 3) the number of infected cases in other provinces before March 3, 2020. Furthermore, we compare our modified model with the original SEIR model in the quality of simulation results.
Due to the inconsistency of detection standards, we divide the data into three groups, i.e., China, Hubei, and Others, and compare them with the simulation results. The data for China only contains data after February 12. The data for Hubei include data after January 10. The data for other provinces contain data before March 3. The data of Hubei need to be preprocessed because of the inclusion of clinical diagnosis data on February 12. First, we need to select the window function for sampling. In this article, we use a Kaiser window function [44]. The function interval is from the beginning of the data to the peak of the data. Next, we calculate the convolution of the original data and the window for further smoothing. Finally, we combine the preprocessed data with the second half of the real data. The experimental results are shown in Fig. 5. The black line represents the simulation result of the exposed (
); the blue line represents the simulation result of the infected (
); and the dotted line represents the real data of the infected people (
). It can be seen that the curve of the exposed (
) simulated by our model is not the same smooth curve as the original SEIR model. This difference results from incorporating information entropy into models. By adding information factors that can reflect people’s behaviors, the IIR changes dynamically, and then, the simulation result of our model is closer to the real data.
Fig. 5.
Simulation of the number of confirmed cases by using (a) our model in China, (b) our model in Hubei province, (c) our model in other provinces, (d) original SEIR model in China, (e) original SEIR model in Hubei province, and (f) original SEIR model in other provinces. (g)–(i) Modified SEIR model with migration.
Fig. 5(a)–(c) shows the epidemic trend simulated by our model. Fig. 5(d)–(f) shows the epidemic trend simulated by the original SEIR model. Fig. 5(g)–(i) shows the epidemic trend simulated by a modified SEIR model considering human migration behavior [23]. We get parameters of the original SEIR model using the Monte Carlo simulation. The objective function of optimization is to maximize the SSI of the original SEIR model. In the original SEIR model, the simulation results are underestimated. In this case, the simulated infection cycle is shorter than the actual infection cycle. Parameters of the modified model with migration are obtained in the same way as our proposed model in this article. Comparing Fig. 5(a)–(i), we can know that incorporating information entropy into the SEIR model is very helpful to the simulation of epidemics. The simulation result of our modified model is more consistent with the actual data. It is not as effective as our model to use the migration data directly to improve the model because migration behavior only reflects intercity mobility. Most cases due to cross-infection in one city cannot be taken into considerations. The parameters of each model are shown in Table II.
TABLE II. Parameters for the Transmission Model.
| Parameter | Description | A | B | C | D | E | F |
|---|---|---|---|---|---|---|---|
 |
Total number of people in the communication system | 180000 | 150000 | 24000 | 146000 | 128000 | 20000 |
 |
Initial infection impact rate | 0.65 | 0.625 | 1.1 | |||
 |
Probability of latent disease (i.e., the reciprocal of the incubation period) | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
 |
Probability of removal | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
 |
Constant coefficient of dynamic infection impact rate | 0.001 | 0.001 | 0.002 | |||
 |
Constant coefficient of dynamic infection impact rate | 0.39 | 0.39 | 0.35 |
The dot plot method used in Fig. 5 cannot allow an explicit comparison to distinguish the performance of the two models. Thus, we exploit the SSI to quantify the degree of similarity with real data to offer a better comparison. SSI is a statistic tool for identifying the similarity between the two samples. A modified version of the index [45], [46] is used to measure whether the real data are correctly reproduced by prediction models, defined as
 |
where
is the value predicted by models and
represents the real data. It is obvious that, if each
is equal to
, the index is 1; if all
’s are far from the real data, the index is close to 0.
The calculation results are shown in Fig. 6. For these three cases, our model outperforms the original SEIR model and the modified SEIR model with migration. Besides, our proposed model exhibits relatively high index values (more than 0.95), indicating that the modified model captures the coupling mechanism of the social media and epidemics.
Fig. 6.
Comparison between the prediction ability of the modified model and other SEIR models in terms of the SSI.
V. Conclusion
Simulating and forecasting the spread of epidemics are challenging and significant research problems in public health. The spread of disease is affected by many important social factors. If these social factors are ignored, the prediction result may not accurate. By statistical analysis, we find that the spreading of information from different platforms and the disease transmission are approximately similar in the growth trend. In this article, we analyze the information entropy of COVID-19 topics on the Internet. Moreover, we use information entropy to measure the impact of social network information on the spread of disease and incorporate information entropy into the SEIR model to derive the epidemic curve. We have also carried out comparative experiments with other existing models. Intuitively, our dynamical model with information entropy is effective in predicting the COVID-19 epidemic peaks and sizes. To quantify the degree of similarity with real data, we introduce SSI to offer a better comparison. The result of the comparison shows the obvious advantage of our model, indicating that our modified model captures the coupling mechanism of social media and the epidemic. The transmission of information has an impact on the spreading of disease. Taking the information entropy of social media into account can effectively increase the accuracy of the epidemic simulation.
Biographies
Qi Nie received the M.Sc. degree in systems science from Beijing Jiaotong University, Beijing, China, in 2018. He is currently pursuing the Ph.D. degree with Wuhan University, Wuhan, China.
His current research interests include data mining, complex networks, machine learning, and human mobility.
Yifeng Liu received the Ph.D. degree in electronic engineering from Wuhan University, Wuhan, China, in 2016.
He is currently a Senior Engineer of the China Academy of Electronics and Information Technology and the Deputy Director of the National Engineering Laboratory for Risk Perception and Prevention (RPP), Beijing, China. He has over 20 publications primarily in cyberspace and data science. His current research interests include computer vision, machine learning, and knowledge engineering.
Dr. Liu is a selected candidate of the 5th Youth Talent Promotion Project of China Association for science and technology. He won the First Prize of the Shijingshan District Science and Technology Award in 2016. The artificial intelligence video analysis system that he led the team to develop has attracted more than 100 million RMB funds, which has been reported by People’s Daily.
Dong Zhang received the LL.M. degree in ideological and political education from Capital Normal University, Beijing, China, in 2002.
He is currently working with the Big Data Laboratory of Social Sciences, Shanghai Academy of Social Sciences, Shanghai, China. His research interests include mass communication and computing communication.
Hao Jiang received the B.Eng. degree in communication engineering and the M.Eng. and Ph.D. degrees in communication and information systems from Wuhan University, Wuhan, China, in 1999, 2001, and 2004, respectively.
He undertook his post-doctoral research work with LIMOS, Clermont-Ferrand, France, from 2004 to 2005. He was a Visiting Professor with the University of Calgary, Calgary, AB, Canada, and ISIMA, Blaise Pascal University, Clermont-Ferrand. He is currently a Professor with Wuhan University. He has authored over 60 articles in different journals and conferences. His research interests include mobile ad hoc networks and mobile big data.
Funding Statement
This work was supported in part by the National Natural Science Foundation of China under Grant U19B2004, in part by the Zhongshan City High-End Research Institution Innovation Project under Grant 181129112748101, and in part by the Guangdong Province “Major Project and Task List” Project under Grant 2019sdr002.
Contributor Information
Qi Nie, Email: nieqi@whu.edu.cn.
Yifeng Liu, Email: liuyifeng3@cetc.com.cn.
Dong Zhang, Email: 8100369@qq.com.
Hao Jiang, Email: jh@whu.edu.cn.
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