# Modeling the influence of Twitter in reducing and increasing the spread of influenza epidemics

- Hai-Feng Huo
^{1}Email author and - Xiang-Ming Zhang
^{1}

**Received: **27 October 2015

**Accepted: **7 January 2016

**Published: **27 January 2016

## Abstract

A more realistic mathematical influenza model including dynamics of Twitter, which may reduce and increase the spread of influenza, is introduced. The basic reproductive number is derived and the stability of the steady states is proved. The existence of Hopf bifurcation are also demonstrated by analyzing the associated characteristic equation. Furthermore, numerical simulations and sensitivity analysis of relevant parameters are also carried out. Our results show that the impact posed by the negative information of Twitter is not significant than the impact posed by the positive information of Twitter on influenza while the impact posed by the negative information of Twitter on the influenza virus is still extraordinary.

### Keywords

Positive effect and negative effect Twitter Basic reproductive number Equilibria Global stability Hopf bifurcation### Mathematics Subject Classification

92D25## Background

Twitter is an online social networking service that enables users to send and read short 140-character messages called “tweets”. The service has gained worldwide popularity rapidly since it launched in 2006 (Dijck 2011; Elsweiler and Harvey 2015; Dugue and Perez 2014). Roughly speaking, the site is largely used to daily chatter, conversation, and debate about information, or to allow ones to understand the world by Daily News (Java et al. 2007). Twitter’s users who are increasingly diverse in age, ethnicity and gender may see the messages of other followers and go along with these users (Sanderson 2014). Reciprocally, nicknames that follow a user have the ability to see the messages posted by other users (Dugue and Perez 2014). By doing this, it is fairly easy to make up a following of 20–30 people quickly (Launer 2013). At the same time, these companions who make full use of Twitter are strongly influenced by minority status, party leadership efforts, chamber, and member age (Lassen and Brown 2011). Twitter becomes so popular nowadays not only because of it can provide positive information (Tiernan 2014; Fu and Shen 2014; Roshanaei and Mishra 2015) but also the negative information (Jin et al. 2014; Dugue and Perez 2014; Alowibdi et al. 2015; Roshanaei and Mishra 2015) it still can be provided. As of June 2013, Twitter had 218 million monthly active users who collectively expressed around 500 million tweets a day (Elsweiler and Harvey 2015). Undoubtedly, Twitter has become a more and more powerful tool for spreading and mining messages in our daily life (Fu and Shen 2014).

Social networking sites play a vital role in medicine and other walks of modern life. Public Health Organizations (WHO), Centers for Disease Control and Prevention (CDC), the Food and Drug Administration (FDA), and the American Red Cross always increasingly advocate public to take advantage of social media programs included Twitter, Facebook, and similar internet sites to disseminate important health information. For example, the CDC made full use of Twitter to post messages for preventing flu to help slow the spread of H1N1 influenza in 2009, growing from 2500 followers to 370,000 followers during the 2009 outbreak (Currie 2009). It is observed that information that users of Twitter shared took advantages or disadvantages for spreading of infectious diseases by reminding them to stay at home when they are sick, teaching users the effectiveness of regular hand-washing, and raising awareness about vaccines or misleading their do some irrational things.

Influenza has always a far-reaching influence on our lives, and many attempts have been made to investigate realistic mathematical models for researching the transmission dynamics of infectious diseases (Cui et al. 2008b; Xiao et al. 2013; Sahua and Dhara 2015; Wang et al. 2015; Kaur et al. 2014; Misra et al. 2011; Liu and Cui 2008; Cui et al. 2008a; Pawelek et al. 2014; Liu et al. 2007). Cui et al. (2008) proposed a SIS-type model to explore the influence of media coverage on the dissemination of emerging or reemerging infectious disease, and used a standard incidence \(\frac{{\beta SI}}{{S + I}}\) between susceptible individuals and infected individuals. Their results indicated that media coverage was critical for educating people in understanding the possibility of being infected by the disease. Xiao et al. (2013) developed a model with media coverage by including a piecewise smooth incidence rate to show that the reduction factor due to media coverage relies on both the number of cases and the rate of changes in case number. They demonstrated that the media impact resulted in a lower size of outbreak and delayed the epidemic peak. Liu and Cui (2008) considered a epidemic model with non-linear contact rate, \(\beta (I) = {\beta _1} - {\beta _2}\frac{I}{{m + I}}\), where \(\beta _1\) is the contact rate before media alert, and \(\beta (I)\) is the contact rate after media alert, and studied the basic reproductive number, the existence and stability of two equilibria. They showed that media and education played a crucial role in mounting infection awareness among the residents. An exponential incidence \(\beta (I) = \mu {e^{ - mI}}\) was applied to develop a three dimensional compartmental model Cui et al. (2008a). They analyzed dynamical behavior of the model; permanent oscillations are generated by a Hopf bifurcation. Pawelek et al. (2014) developed a simple mathematical model including the dynamics of “tweets”, and studied dynamics of the model. They showed that Twitter may serve as a good indicator of seasonal influenza epidemics. Liu et al. (2007) assumed that the total number of susceptible remains relatively unchanged as a result of the outbreak duration is extremely short, and incorporated a simple nonlinear incidence function \({\beta _0} = \beta {e^{ - {\alpha _1}E - {\alpha _2}I - {\alpha _3}H}}\), where *H* denotes hospitalized individuals. They illustrated the multiple outbreaks or the sustained periodic oscillations of emerging infectious diseases owing to the psychological impact.

It is well known that everything has two sides in reality. Massive media coverage is no exception. Alowibdi et al. (2015) focused specifically on the detection of inconsistent information involving user gender and user location; they shown that lying contained misleading, inconsistent, or false and deceptive information in online social networks is quite widespread. Roshanaei and Mishra (2015) compared the patterns of tweeting, replying and following by analysis of social engagement and psychological process in the positive and negative networks; their findings not only predicted positive and negative users but also provided the best recommendation for negative users through online social media. Unfortunately, most of the aforementioned model (Cui et al. 2008b; Sahua and Dhara 2015; Wang et al. 2015; Kaur et al. 2014; Misra et al. 2011; Liu and Cui 2008; Cui et al. 2008a; Pawelek et al. 2014; Liu et al. 2007) ignored the negative role of the media coverage. It has been observed that communications that people received or send through Twitter mislead the public to do some irrational things as well as benefited some people (Tiernan 2014; Fu and Shen 2014; Jin et al. 2014; Dugue and Perez 2014). Inspired by the documents (Cui et al. 2008a; Liu and Cui 2008; Liu et al. 2007; Pawelek et al. 2014), we introduce a more realistic mathematical influenza model, which incorporates the effects of Twitter in reducing and increasing the spread of influenza epidemics.

The rest of the paper is organized as follows: In “Basic properties” section, a more realistic \(SEI{T_1}{T_2}\) model is formulated, the basic reproductive number and stability of equilibria are also obtained. In “Analysis of the model” section, the Hopf bifurcation is studied. Numerical simulations are carried out in “Numerical simulation” section. Sensitivity analysis is conducted in “Sensitivity analysis” section. Some discussions and conclusions are given in the last section.

## Basic properties

### System description

*S*(

*t*), the number of susceptible individuals;

*E*(

*t*), the number of individuals exposed to the infected but not infectious;

*I*(

*t*), the infected who are infectious. All of them may tweet about influenza at the rates \(\mu _1\), \(\mu _2\), and \(\mu _3\), respectively, during an epidemic season. \(T_1(t)\) and \(T_2(t)\) represent the number of tweets that all of them provide positive and negative information about influenza at time

*t*, respectively. Our model is governed by the following system of five differential equations. A transfer diagram of our model is shown in Fig. 1 and the parameters description of our model are presented in Table 1.

The parameters description of the flu model

Parameter | Description |
---|---|

\(\beta\) | Transmission coefficient from the susceptible compartment to the exposed compartment |

\(\alpha\) | The coefficient that determines how effective the positive flu information can reduce the transmission rate |

\(\delta\) | The coefficient that determines how effective the negative flu information can increase the transmission rate |

\(\rho\) | Transmission coefficient from the exposed compartment to the infected compartment |

\(\gamma\) | The permanently recover rate |

\(\mu _{i},i=1,2,3\) | The rate that susceptible individuals, exposed individuals, and infectious individuals may tweet about influenza during an epidemic season respectively |

| The ratio that individuals may provide positive information about influenza during an epidemic season |

| The ratio that individuals may provide negative information about influenza during an epidemic season |

\(\tau\) | The rate that tweets become outdated |

*p*is the ratio that individuals may provide positive information about influenza during an epidemic season.

*q*is the ratio that individuals may provide negative information about influenza during an epidemic season. For simplicity, we assume that the ratio of positive/negative information for all three groups is same, that is,

*p*and

*q*. \(\mu _{i},i = 1,2,3\) is the rate that susceptible individuals, exposed individuals, and infectious individuals may tweet about influenza during an epidemic season, respectively. \(\tau\) is the rate that tweets become outdated in consequence of tweets that appeared earlier are less visible and have less effect on the public, and \(\beta\) is the disease transmission coefficient. The transmission coefficient \(\beta\) is reduced by a factor \({e^{ - \alpha {T_1}}}\) owing to the behavior change of the public after reading positive tweets about influenza, where \(\alpha\) determines how effective the disease positive twitter information can reduce the transmission coefficient, and is increased by a factor \({e^{ \delta {T_2}}}\) due to the behavior change of the public after reading negative tweets about influenza, where \(\delta\) determines how effective the disease negative twitter information can increase the transmission coefficient. Since we only consider the disease outbreak during extremely short time, we neglect the natural death and birth rates and further assume that the number of susceptible people is relatively constant (Liu et al. 2007). Therefore, the above system can be reduced as follows:

### The basic reproductive number

### The existence of equilibria

###
**Theorem 1**

*For the system (*1

*), there exist the following two equilibria:*

- 1.
*System*(1)*always exists the disease-free equilibrium*\({E_0} = (0,0,{T_1}^0,{T_2}^0)\),*where*\({T_1}^0\)*and*\({T_2}^0\)*are given by*(4). - 2.If \(R_{0}>1\)
*and*\(\alpha p - \delta q > 0\),*there exists the endemic equilibrium*\({E_*} = ({E^*},{I^*},{T_1}^*,{T_2}^*)\).*Where*\({E_*} = ({E^*},{I^*},{T_1}^*,{T_2}^*)\)*satisfies the following equalities*:$$\begin{aligned} {E^*} &= \frac{{\gamma \rho \ln ({R_0})}}{{(\alpha p - \delta q)(\gamma {\mu _2} + \rho {\mu _3})}},\nonumber \\ {I^*} &= \frac{{\tau \rho \ln ({R_0})}}{{(\alpha p - \delta q)(\gamma {\mu _2} + \rho {\mu _3})}},\nonumber \\ {T_1}^* &= \frac{{p\ln ({R_0})}}{{(\alpha p - \delta q)}} + {T_1}^0,\nonumber \\ {T_2}^* &= \frac{{q\ln ({R_0})}}{{(\alpha p - \delta q)}} + {T_2}^0. \end{aligned}$$(5)

###
*Proof*

- 1.
It is easy to know that system (1) always exists the disease-free equilibrium \(E_{0}=(0,0,\frac{{p{\mu _1}S}}{\tau },\frac{{q{\mu _1}S}}{\tau })\).

- 2.By letting the right-hand sides of (1) equal zero, namely,$$\begin{aligned}&\beta SI{e^{ - \alpha {T_1} + \delta {T_2}}} - \rho E = 0,\nonumber \\&\rho E - \gamma I = 0,\nonumber \\&p{\mu _1}S + p{\mu _2}E + p{\mu _3}I- \tau {T_1} = 0,\nonumber \\&q{\mu _1}S + q{\mu _2}E + q{\mu _3}I - \tau {T_2} = 0. \end{aligned}$$(6)

So we can obtain endemic equilibrium \({E_*} = ({E^*},{I^*},{T_1}^*,{T_2}^*)\). It is clear that the endemic equilibrium exists if and only if \(R_{0}>1\) and \(\alpha p - \delta q > 0\). This completes the proof of Theorem 1. \(\square\)

## Analysis of the model

In this section we will discuss the stability of equilibria of the system (1).

### Stability of the disease-free equilibrium

###
**Theorem 2**

*If*
\(R_{0} < 1\)
*and*
\(\alpha p - \delta q \ge 0\), *then the disease-free equilibrium*
\(E_{0}\)
*is globally asymptotically stable*.

###
*Proof*

*V*(

*t*) and using \(\beta S{e^{ - \alpha {T_1}^0 + \delta {T_2}^0}} = \gamma {R_0}\), we have

*V*(

*t*) is bounded and non-increasing. Therefore, \(\mathop {\lim }\nolimits_{t \rightarrow \infty } V(t)\) exists. Note that \(\frac{{dV(t)}}{{dt}} = 0\) if and only if \(E(t) = I(t) = 0\), \(T_1={T_1}^0\) and \(T_2={T_2}^0\). By LaSalle Invariance Principle (LaSalle 1987), the disease-free equilibrium \(E_{0}\) is globally attracting when \(\alpha p - \delta q \ge 0\) and \({R_0} < 1\). Together with the local asymptotic stability, we show that \(E_{0}\) is globally asymptotically stable when \(\alpha p - \delta q \ge 0\) and \({R_0} < 1\). This completes the proof of Theorem 2. \(\square\)

###
*Remark 1*

When \(R_{0} < 1\) and \(\alpha p - \delta q < 0\), globally asymptotically stability of the disease-free equilibrium \(E_{0}\) is not been established. Figure 2b seems to support the idea that the disease-free equilibrium of system (1) is still global asymptotically stable even in this case.

### Stability of the endemic equilibrium

###
**Theorem 3**

*The endemic equilibrium*\({E_*}\)

*is locally asymptotically stable if and only if one of the following statements is satisfied*:

- 1.
\(R_{0}>1\), \(\alpha p - \delta q > 0\), and \({\mu _3} \le {\mu _3}^*\), where \({\mu _3}^* = \frac{{\rho + \tau }}{\rho }{\mu _2}\);

- 2.
\(R_{0}>1\), \(\alpha p - \delta q > 0\), \({\mu _3} > {\mu _3}^*\), and \(\beta < min\{ {\beta ^*},{\beta ^{**}}\}\), where \(\beta ^{*}\)

*and*\(\beta ^{**}\)*are given by*(25)*and*(27),*respectively*.

###
*Proof*

**Matlab**, we can obtain two solutions of this unary quadratic equation, in other words,

Let \({\mu _3}^* = \frac{{\rho + \tau }}{\rho }{\mu _2}\).

If \({\mu _3} \le {\mu _3}^* ({\rm i.e.,} (\rho + \tau ){\mu _2} - \rho {\mu _3} \ge 0 )\), then \(\varphi >0\).

For \({\mu _3} > {\mu _3}^*\) and \(\beta < {\beta ^*}\), if \({a_3}({a_1}{a_2} - {a_3}) - {a_1}^2{a_4}>0\) holds, we must make \({\left( {\gamma {H^*}(\alpha p - \delta q)} \right) _2} < \frac{{\tau [{{(\rho + \gamma )}^2} + \tau (\rho + \gamma )]}}{{\rho {\mu _3} - (\rho + \tau ){\mu _3}}}\) hold constantly.

### Hopf bifurcation

###
**Theorem 4**

*A Hopf bifurcation occurrs when*
\(\beta\)
*increases and the curve*
\({\beta ^{**}}\)
*is crossed, where*
\({\beta ^{**}}\)
*is defined in equation* (26).

###
*Proof*

*x*,

*y*and a pair of complex roots \(a \pm bi\), where \(x<0,y<0\) and \(a,b \in R\). We yield

*Re*denotes the real part of a complex number. Calculating \(Re[P(a+bi)]=0\), we have

## Numerical simulation

*S*is set to 1 million and initially 10 people get exposed to the disease (Tracht et al. 2011; Pawelek et al. 2014). The other parameters are chosen to illustrate the theoretical results.

As is shown in the Figs. 3 and 4, we perform numerical simulations to illustrate the results showed in Theorem 3. Figure 3a describes a graph of the solution curve under the conditions of Theorem 2, and Fig. 3b reveals the phase diagram including *E*(*t*) and *I*(*t*) trajectories under the conditions of Theorem 3. For the purpose of simplicity, we assume \(\mu _1=\mu _2=0\), and \(\mu _3>0\); namely, only infectious individuals receive or send positive and negative information about influenza. It can be seen from the Fig. 3a that the results of numerical simulation fit in with the results of the theoretical analysis. Namely, the endemic equilibrium \({E_*}\) is locally asymptotically stable when \(R_{0}>1\), \(\alpha p - \delta q > 0\) and \({\mu _3} \le {\mu _3}^*\). Figure 4 shows that the endemic equilibrium \(E^{*}\) is locally asymptotically stable when \(R_{0}>1\) , \({\mu _3}>{\mu _3}^*\), and \(\beta <\beta ^{**}\),

## Sensitivity analysis

*p*,

*q*, and \(\tau\) on \(R_{0}\), we note that

Figure 6a illustrates the relationship between the basic reproductive number \(R_0\) and \(\mu _1\). According to Eq. (31), if \(\frac{{\partial {R_0}}}{{\partial {\mu _1}}}>0\) (i.e. \(\alpha p - \delta q<0\)), then \(R_0\) increases as \(\mu _1\) increases, and if \(\frac{{\partial {R_0}}}{{\partial {\mu _1}}}<0\) (i.e. \(\alpha p - \delta q>0\)), then \(R_0\) decreases as \(\mu _1\) increases. Figure 6b illustrates the relationship between the basic reproductive number \(R_0\) and \(\tau\). According to Eq. (32), when \(\frac{{\partial {R_0}}}{{\partial {\tau}}}>0\) (i.e. \(\alpha p - \delta q>0\)), \(R_0\) increases as \(\tau\) increases, and when \(\frac{{\partial {R_0}}}{{\partial {\tau}}}>0\) (i.e. \(\alpha p - \delta q<0\)), \(R_0\) decreases as \(\tau\) increases. By analyzing Eqs. (33) and (34), Fig. 6c distinctly demonstrates that the greater \(\beta\) increases, the more significant \(R_0\) grows, and the smaller \(\gamma\) decreases, the more remarkable \(R_0\) enlarges. Combining Fig. 6d and Eqs. (35) and (36), we can comprehensibly perceive that if \(\alpha\) increases, then \(R_0\) will decrease, and if \(\delta\) increases, then \(R_0\) will increase. Biologically, this means that to reduce influence of negative information and transmission rate or increase influence of positive and recover rate are vital essential for controlling influenza.

## Conclusions and discussions

*S*(

*t*) can be characterized by the following equation

## Declarations

### Authors’ contributions

HFH and XMZ designed the study, carried out the analysis and contributed to writing the paper. Both authors read and approved the final manuscript.

### Acknowledgements

This work is supported by the NNSF of China (11461041), the NSF of Gansu Province (148RJZA024), and the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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