The effect of media coverage on threshold dynamics for a stochastic SIS epidemic model.

Media coverage is one of the important measures for controlling infectious diseases, but the effect of media coverage on diseases spreading in a stochastic environment still needs to be further investigated. Here, we present a stochastic susceptible-infected-susceptible (SIS) epidemic model incorporating media coverage and environmental fluctuations. By using Feller's test and stochastic comparison principle, we establish the stochastic basic reproduction number R 0 s , which completely determines whether the disease is persistent or not in the population. If R 0 s ≤ 1 , the disease will go to extinction; if R 0 s = 1 , the disease will also go to extinction in probability, which has not been reported in the known literatures; and if R 0 s > 1 , the disease will be stochastically persistent. In addition, the existence of the stationary distribution of the model and its ergodicity are obtained. Numerical simulations based on real examples support the theoretical results. The interesting findings are that (i) the environmental fluctuation may significantly affect the threshold dynamical behavior of the disease and the fluctuations in different size scale population, and (ii) the media coverage plays an important role in affecting the stationary distribution of disease under a low intensity noise environment.

Introduction

Infectious diseases are caused by a variety of pathogens, which can transmit in human, animals or between human and animals, and are a major global cause of death resulting in a heavy economic burden all over the world. According to the world health organization (WHO)’s report [1], lower respiratory infections remained to be the most deadly infectious disease, causing 3.2 million deaths worldwide in 2015. Therefore, understanding the transmission process of infectious diseases can provide evidences for prevention and control, and reduce the economic burden of disease [2].

Once an infectious disease appears and outbreaks in one area, the department for disease control and prevention will take some actions to prevent the epidemic spreading. One strategy is to disseminate the correct preventive knowledge of diseases through the media with the least delay possible [3], [4], [5]. The mass media developing rapidly, especially, internet-based is becoming very popular, also take the profound influence to the information dissemination way as well as people’s life style. Kinds of mass media (such as TV, radio, billboards, Wechat, microblog, etc.) can improve the protection awareness and change the behaviors of the susceptible populations [6]. Generally, media education consequently changing the human behaviors plays a tremendous role in limiting the spread of infectious disease [3]. Human behavior change can result in the reduction in number of real susceptible individuals or effective contact rates. People became more aware of the protection against influenza and other types of infectious diseases during this period.

The media coverage is one of the most key factors to establish the prevention and control measure, some other factors like the medical treatment level [7] or climate change [8] can also affect the spread of infectious diseases. For preventing and controlling the infectious diseases, the influences of media coverage on the spread of the disease are mainly from the following two ways:

People can get disease related information (including the causes, the route of infection, etc.) through different media coverage (for example, television, Internet, newspapers and other media ways) from the beginning of the outbreak of disease. Thus, health education through media coverage may reduce the contact rate of human being as we observed during the spread of severe acute respiratory syndrome (SARS) during 2002 and 2004 [9].

Authoritative information, including the potential risk of the diseases and prevention measures before the outbreak of disease, can release to public directly. Such as H1N1 flu virus that broke out in Mexico in 2009, and before the disease outbreak spread widely across the globe, the centers for disease control and prevention (USA) provided an 9-1-1 public safety answering points (PSAPs) for management of patients with confirmed or suspected swine-origin H1N1 infection [10].

To evaluate the potential impact of media coverage on the infectious disease, mathematical modeling can provide a better understanding of the spreading mechanism of infectious diseases. For example, Cui et al. [11] investigated the impact of media coverage on the spread and control of infectious diseases (such as SARS) in a given region/area. Xiao et al. [12] analyzed a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. In particularly, when media coverage is present, social distancing mechanisms come into effect. The reporting by media can be assumed to be an increasing function of the number of infectious cases present, and consequently, the effective contact rate between susceptible and infectious individuals should be a decreasing function of the number of infectious cases present. Therefore the effective contact rate can be described by the following non-linear function [11]: where and are respectively the direct contact transmission coefficient and the maximum reduced contact rate due to the presence of protected individuals affected by the media coverage. The function satisfies:

(H) , , and .

In what follows, we assume that non-linear function takes the form , which represents that with more people understand the protection measures, the reduction effect reach its maximum saturation state and is the half-saturated constant.

Let and be the number of susceptible and infective individuals at time , respectively. Based on above assumptions, Cui et al. [9] proposed an SIS epidemic model incorporating media coverage which takes the following form: where is the recruitment rate of the population, the natural death rate of the population, is the recovery rate of infection individuals.

For deterministic model (1.1), its dynamics is completely determined by the basic reproduction number : if , the disease-free equilibrium is global asymptotic stability, if , the endemic equilibrium , which satisfies the following equations, is globally asymptotic stable [9],

On the other hand, in real life, the transmissions of infectious diseases are inevitable affected by the variations of environmental factors, such as temperature, humidity, etc. Recent experimental research result [13] showed that the virus transmission is closely related to the temperature and humidity. Semenza and Menne’s experimental results [14] presented that the effects of global climate change on infectious diseases are hypothetical until more is known about the degree of change in temperature and humidity that will occur. Thomas and Kevin [8] pointed out that global climate change may cause considerable uncertainty about the rates of change. Therefore, the environmental fluctuation (uncertainty change in temperature or humidity) may translate to affect the transmission rate of infectious diseases [15], [16], [17], [18], [19], [20], [21], [22], [23]. If we assume that the uncertainty change of transmission rate of infectious disease is subjected to a Gaussian white noise, with , where is a standard Brownian motion, and is the intensity of white noise, then we can rewrite model (1.1) as the following form:

Notice that as . Thus model (1.2) has the invariant set , on which model (1.2) is reduced to with initial value

In this paper, we pay attention on the question that how does the environmental fluctuation affect the threshold dynamics of the SIS epidemic model (1.3) incorporating media coverage. More precisely, the main aim is to obtain the completed threshold results and the existence of stationary distribution of stochastic epidemic model (1.3). As far as we know, there is rare result of model (1.3) with respect to a completely threshold and its statistical characters (i.e., the existence of stationary distribution).

As well known, there are different potential methods to introduce the environmental noise into a deterministic model. In [24], the authors used a white noise type that is directly proportional to and influences on , respectively. However, model (1.2) in this paper, we focus on the effect of environmental fluctuation on the key parameter in epidemic models. The diffusion term of model (1.2) is not only dependent on but also on in a nonlinear form . Moreover, the random noises for the two subpopulations ( and ) are assumed to be correlated, which corresponds to the real situation when the system is suffered from the same environmental factor (such as meteorological conditions etc.). Mathematically, the random perturbations of model (2.4) in [24] are described by two independent standard Brownian motions , while model (1.2) uses a degenerate type . Since Fokker–Planck equation corresponding to model (1.2) is of degenerate type, it is difficult to establish the existence of stationary distribution (unless the Markov semigroup theory is used for a two-dimensional system with degenerate type white noise, please see [16], [25]). Fortunately, there exists an invariant set for stochastic model (1.2), and thus it can be reduced to a one-dimensional system (1.3) with initial value (1.4). Then, using Feller’s test we can establish the complete threshold, and applying the Khasminskii’s method we can further prove the existence of stationary distribution.

The rest of this paper are organized as follows. In the next Section, by using of Feller’s test, we obtain the extinction results of infectious disease if . Then, in Sections 3, 4, by using of stochastic comparison principle and Khasminskii’s method, we prove the stochastic persistence and the existence of the stationary distribution of model (1.3) when . Some numerical simulations based on influenza parameters are carried out. Finally, a brief discussion and some biological implications are given in the last Section.

Disease extinction

To establish the extinction results of model (1.3), we first give the existence and unique result of global positive solution of model (1.3), and some useful lemmas.

For any given initial value (1.4) , there is a unique positive solution of model (1.3) on , and the solution will remain in almost surely.

Consider the following one-dimensional stochastic differential equations:

satisfying the following conditions:

(1) for any , where ,

(2) For any , there exists a such that .

[22], [26]

Assume that (1) and (2) hold, and let be a weak solution of system (2.1) in . For some fixed constant , the scale function is defined as . If and hold, then

We have the following result.

Let be the solution of model (1.3) with initial value (1.4) , if hold, then we have

For (1.3), we denote and Then, we can calculate that where is a constant, and then the scale function is Letting and yields Notice that implies . Letting , we can obtain That is to say, . It follows from Lemma 2.2 that . The proof is completed.

Now, we are in a position to prove the extinction fate of disease in the case when . To do this, we first present the following another useful lemma.

[27]

Assume that and system (2.1) admits a non-explosive solution which is unique in the sense of a probability law, denoted by . If the scale function and satisfy and , then in probability, namely, for any ,

Suppose that is the solution of model (1.3) with initial value (1.4) . If , then in probability.

Define . It is obvious that is a continuous and increasing function on . Utilizing the Itô formula to , we can get with , .

Taking , we can compute that Then, taking can result in . So we have Notice that , , and . Thus we have Similarly, we can obtain that and . According to Lemma 2.3, we complete the proof.

Theorem 2.1 implies that the disease will go to extinction if . Compared with the deterministic counterpart, the noise intensity plays an important role in helping the extinction of disease. In Theorem 2.2, a generalized result is given about the dynamical behavior of the disease when , which is not reported in references.

Stochastic persistence of disease

In this section, we shall investigate the stochastic persistence of model (1.3). It is proved that when , the disease will be stochastically persistent, which is one of the important issues from public health point of view.

Assume is the solution of model (1.3) with initial value (1.4) . If , then satisfies and where is the positive root in of

By the formula, we can obtain where : For , we have Note that , thus we can get Then there exists a such that and , thus

Next, we prove assertion (3.1). If it is not true, then there is a sufficiently small such that where . For every , there is a such that when , Clearly, we can choose small enough such that . According to (3.5)–(3.7), when we have By the large number theorem for martingales [28], there is an and such that for every : Now, for any fixed , it then follows from (3.4), (3.10) that for , we have This results in Thus, However, this contradicts with (3.10). We therefore must have the desired assertion (3.1).

Similarly, we can prove assertion (3.2). If it is not true, then there is a sufficiently small such that where . Hence, for every , there is a such that when , Now, for any fixed . It then follows from (3.4), (3.11) that for , That is, Whence This contradicts with (3.12), we thus have the desired assertion (3.2).  □

From Theorem 3.1, Theorem 2.1, Theorem 2.2, we know that evolution behaviors of the disease is completely determined by the stochastic basic reproduction number : if , the disease will go to extinction in probability 1; if , the disease will be persistent in some range, which provides for us a formula to estimate the fluctuation range of the infected number.

Stationary distribution and ergodic property

In this section, we shall further discuss the existence of stationary distribution and ergodic property in the case of , which can reflect the statistical characteristic of the sample trajectories. We first give the following useful definition and lemma.

See [29]

Let denote the probability measure induced by with initial value ; that is, If there is a probability measure on the measurable space such that we then say that model (1.3) has a stationary distribution.

See [30]

Let be a time-homogeneous solution of model (2.1) on . Assume that

(A1) In the domain and some neighborhood thereof, the diffusion is bounded away from zero;

(A2) If for all the mean time at which a path emerging from reaches the set is finite, and for every compact subset .

Then the Markov process has a stationary distribution , and for an integrable function ,

If , then model (1.3) admits a unique stationary distribution and the solution is ergodic.

To verify (A1) of Lemma 4.1, according to Zhu and Yin [31], let be a sufficiently large number and be a bounded open subset with a regular boundary such that where is the closure of . Define a -function : : where . By utilizing the Itô formula, we have where and It follows from (4.5)–(4.7) that Since , we can choose small enough such that . Thus for sufficiently , we have which implies that (A2) in Lemma 4.1 is satisfied. In addition, we can check that condition (A1) in Lemma 4.1 is also satisfied. As a consequence, model (1.3) admits a stationary distribution.  □

Theorem 4.1 shows that model (1.3) has a stationary distribution and has ergodic property: which provides a formula to estimate the average infected number in probability sense.

Numerical simulations analysis

In this section, we carry out some numerical simulations to give a further analysis of the effect of media coverage and noise on the epidemic characters. We will give the following two examples: we first give some simulations to support our threshold results obtained in previous sections; then, we discuss the impact of media coverage on the stationary distribution and explore the effect of noise on disease transmission in different population scale based on the real influenza parameters.

Based on the Milstein method [32], model (1.3) can be rewritten as the following discretized form: where are th realization of a Gaussian random normal variable .

Let us assume that , , , , , . The only difference in the following cases is the intensity of the white noise .

(i) In Fig. 1, we choose such that . It follows from Theorem 2.2 that the infected population will go to extinction, which is consistent with the multi-sample simulation results as shown in Fig. 1.

Fig. 1

The sample paths of stochastic model (1.3) and its corresponding deterministic model (1.1) when : (a) , (b) . The parameter values are the same as in Example 5.1 and the initial value is .

(ii) In Fig. 2, we choose , compute that . According to Theorem 2.4, we know that the infected population will also go to extinction in probability, which is also supported by Fig. 2.

Fig. 2

The sample paths of stochastic model (1.3) and its corresponding deterministic model (1.1) when : (a) , (b) . The parameter values are the same as in Example 5.1 and the initial value is .

(iii) In Fig. 3, we choose such that . It follows from Theorem 2.5 that the infected population is stochastically persistent, please see Fig. 3.

Fig. 3

(a) The sample paths of , and (b) the sample paths of of stochastic model (1.3) and its corresponding estimated range of the value of when . The parameter values is used in Example 5.1 and initial value is .

From Fig. 1, Fig. 2, Fig. 3, we can see that whether the infectious disease will be persistent or not is completely determined by the stochastic threshold of model (1.3). It follows from Fig. 1, Fig. 2 that will go to extinction in the case of , while its corresponding deterministic counterpart is still persistent as an epidemic. Thus, the large intensity environment noise may be benefit to the disease extinction. Moreover, we can observe that the disease will fluctuate around the given ranges between (3.1), (3.2) (see Fig. 3 for details), which can provide a fluctuation estimation of the infected level .

Next, we will further discuss the effect of media coverage on the stationary distribution of based on the real parameters of influenza (Flu).

Influenza, generally known as ‘the flu’, is an infectious disease caused by an influenza virus, which circulate in all parts of the world. Usually, the virus is spread through the air from coughs or sneezes [33]. Recently, SIS models have been widely used to describe the transmission of influenza, e.g., Mao et al. [28], [34], Li et al. [15], Li and Cui [35], Cai et al. [21] etc. Thus, in the following, we shall discuss the effect of media coverage from two aspects: the reported strength () and the audience scope () based on the following parameters of model (1.3) with respect to influenza.

We can calculate , according to Theorem 3.1, Theorem 4.1, the disease will be stochastically persistent and model (1.3) has a stationary distribution.

(i) Firstly, we explore the effect of media coverage on the stationary distribution. The only difference between the following cases is (the maximum reduced contact rate due to media coverage), which can be viewed as a related measurement of media coverage strength. From Fig. 4, we can know that with the increase of from 0.35 to 0.45, the infected number is decreased. The position of stationary distribution (denoted by the central tendency of infected number ) moves to left and the dispersion becomes smaller with the increase of . Thus, the media coverage plays an important role in determining the statistics of stationary distribution.

Fig. 4

The sample paths of stochastic model (1.3) and its corresponding probability density function (PDF) for different , the other parameter values is used in Example 5.2 and initial value is .

(ii) Secondly, due to the fact that the effect of media coverage may be affected by the audience scope, we discuss the effect of environmental fluctuation on different audience scope. In China, there are more than 100 large-scale cites and over 1000 middle-scale cites [39], and the total population of China in 2016 is about 1.3827 billion [37]. So we may assume that the average population number in large-scale cites is about , and in middle-scale cites, it is about . Therefore, for model (1.3), we choose two different size scale population and , which represent respectively two kinds of audience scopes of media coverage. The others parameters are the same as in Table 1. It can be obviously observed from Fig. 5 that the effect of same intensity environmental fluctuation () may result in a more stronger fluctuation of for the larger size scale population (), that is to say, the smaller size scale of the population is less sensitivity to the environmental fluctuation. In summary, the same intensity fluctuation caused by variation of environment (e.g., the climate change, season alternation, etc.) may produce diverse fluctuation effects in different size scale population.

Table 1

The parameters of model (1.3) based on the flu.

Parameters	Value	Unit	Source
β_1	0.1298	day−1	[36]
μ	0.007	day−1	[37]
γ	0.0714	day−1	[38]
b	0.1	day−1	[34]
N	13827000	person	assumed
β_2	0.03	day−1	assumed
σ	0.05	–	assumed

Fig. 5

The sample paths of stochastic model (1.3) for different size scale of population: (blue line) and (red line), the other parameter values are the same as in Example 5.2 and initial value is . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

From above simulation analysis, we can summarize the effect of on the average infected level as in Table 2. It can be seen from Table 2 that the decrease of susceptible number or the increase of media coverage is benefit to the control of infectious disease. Therefore, we can draw the conclusion that the media coverage may significantly affect the stationary distribution of and fluctuation effects in different size scale population.

Table 2

Effects of on the infected level .

	N↑	β_1↑	β_2↑	σ↑
Infected level I(t)	↑	↑	↓	↓

Discussion

The emerging and reemerging diseases have led to an extensive interest in investigation of infectious disease [40]. The medica coverage as an effective preventive measure plays an important role in controlling the disease spread [4]. Dynamical modeling of infectious disease has become a powerful tool to improve our understanding of the pattern of epidemic spread and develop better controlling strategies [41]. Meanwhile, in real situation, the environmental fluctuation should not be ignored. In this paper, we investigated a stochastic SIS epidemic model with both media coverage and environmental fluctuation. By using the Feller’s test method, we obtained that the dynamical behavior of model (1.3) is completely determined by the stochastic basic reproduction number . More precisely, if , the disease will go to extinction; if , the disease will be stochastically persistent in some ranges and model (1.3) admits a stationary distribution with ergodicity. Our theoretical and numerical simulation results showed that the environmental fluctuation , media coverage and population size scale may exert important influences on an epidemic:

The media coverage may play an important role in determining the persistence of the diseases and its statistical characteristics, such as the change of the position of central tendency of infected number with the strength of media coverage increasing. Although media coverage cannot completely eliminate the disease spread, it can decrease the contact rate of disease effectively during the early stage of infection [3]. Thus, taking media coverage as an effective preventive measure before the outbreak of the infectious diseases is reasonable in a fluctuation environment.

The environmental fluctuation may significantly affect the threshold dynamics of the disease and fluctuation effects in different size scale population. If we ignore the influence of the environmental fluctuation, the epidemic may be overestimated. In other words, the environmental stochasticity may be benefit to the disease extinction and result in diverse variation of the infected numbers for different size scale .

From the practical viewpoint, identifying the role of media coverage and stochasticity in the transmission of infectious disease may be conducive to the understanding of the potential control measures, and to configure the media coverage’s audience scope and coverage strength more reasonable.