Analysis of a Stochastic Susceptible-Infective Epidemic Model in a Polluted Atmospheric Environment

Atmospheric pollution is deteriorating, which has affected the evolution of respiratory disease for the exposed human worldwide. Thus, exploring the influence of air pollution on the evolution of disease transmission dynamics is a significant issue. In this article, a stochastic susceptible-infective (SI) epidemic model in a polluted atmospheric environment is investigated.The existence and uniqueness of the global positive solution are established. In virtue of the aggregation methods and Lyapunov function, the sufficient conditions of disease extinction, persistence, and existence of the stationary distribution are established, respectively. Taking PM2.5 concentration as the air pollutant index, numerical simulations are carried out to support these results. Our results indicated that the disease transmission dynamics (extinction, persistence, and stationary distribution) are significantly associated with the environmental atmospheric pollution and fluctuation.


Introduction
Along with the rapid industrial development, pollution is increasingly serious; numerous environmental resources and populations have been polluted to different degrees.Atmosphere pollution (such as airborne fine particulate matters (PM) 2.5 , PM 10 , and SO 2 ) is deteriorating due to rapid urbanization and industrial development [1,2], which has affected the evolutionary dynamics of respiratory disease for the exposed human worldwide.In recent years, the epidemiological experiments and researches [3,4] indicated clearly that air pollution continues to have adverse effects on respiratory disease of humans.Therefore, exploring the effect of atmospheric pollution on the respiratory disease transmission dynamics is one of the meaningful issues, which may raise awareness of the effects of climate change on health and prompt action for public health measures.
To conduct experiments and public health decisionmaking for solving this worldwide issue, mathematical models become one of the powerful tools for investigating infectious diseases [5], which have been used to provide useful insights into control measures [6][7][8][9].For example, Wang and Ma [6] considered an SIS model in a polluted environment and obtained the persistence and periodic orbits of the model.Wang et al. [7] derived a toxin-dependent dynamic model that incorporates the birth rate with the toxindependent switching mode, the mortality rate, and infection rate with the toxin-dependent saturation effect.They analyzed the model by showing the positive invariance, existence and stability of equilibria, and bifurcations.Chauhan et al. [9] considered an SIR epidemic model with treatment affected by pollution.Zhao et al. [10] considered a stochastic susceptibleinfective epidemic model in a polluted environment, which incorporates both environmental fluctuations and pollution.Most recently, by assuming that () and () represent the number of susceptible and infective populations at time , respectively,  0 () is the toxin concentration in the organism at time , and   () is the toxin concentration in the atmospheric environment at time .Liu et al. [8] proposed a 2 Complexity Table 1: Biological meanings of the parameters in model (1).

Parameters
Biological meanings Unit The intrinsic growth rate in absence of the toxicant  −1  Carrying capacity of the population in absence of the toxicant person  Probability of infection - The population natural death rate The death rate caused by disease The dose-response rate due to uptake of pollution for the susceptible The dose-response rate due to uptake of pollution for the infected The organisms net uptake rate of pollution from environment The egestion rate of pollution in the organism (metabolism) The depuration rate of pollution in the organism The environmental pollution loss rate due to natural degradation The exogenous rate of atmospheric pollutant input into the environment susceptible-infective epidemic model in a polluted environment which takes the form where the biological meanings of the parameters are listed in Table 1.More assumptions of model formulation can be seen in [8] for details.
It is worth pointing out that aforementioned models do not take the environmental fluctuation factors into account.In fact, the infectivity of virus of the respiratory disease may suffer from environmental fluctuation (humidity [11], temperature [12], etc.).Thai et al. [13] identified a role for seasonality of absolute humidity in driving the epidemiology of influenza-like illness.Yang et al. [14] observed the nonlinear effects of temperature and relative humidity on the incidence of measles.Meanwhile, the atmosphere pollution timeseries (e.g., PM 2.5 ) exhibits remarkable stochastic fluctuation.Mathematically, stochastic processes driven by Brownian motion are used to describe this environmental fluctuation [15][16][17][18][19][20][21][22].Thus, it is reasonable to model the environmental fluctuation as a stochastic process.More precisely, we assume that the transmission coefficient  and dose-response rate   ,  = 1, 2, are subjected to where   (),  = 1, 2, 3, is a 1-dimensional standard Brownian motion defined in a complete probability space (Ω, F, {F  } ≥0 , ) with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all Pnull sets).Thus, we can further incorporate the environmental white noises into deterministic model (1): With initial data ,  = 1, 2, 3, denotes the intensities of the white noise.
The main purpose of this paper is to investigate the effect of the environmental fluctuation and atmospheric pollutant on the transmission dynamics (extinction, persistence, and stationary distribution) of model (3).Noting that model ( 3) is a nonautonomous model, it is difficult to establish the existence of stationary distribution.As far as we know, there is rare result with respect to this aspect.By using aggregation method, we transform model (3) into a fast time scale system (6) and a slower time scale system (7) and prove that almost every sample path of model ( 3) is uniformly Hölder continuous.Then by virtue of Hasminskii's methods, we establish the existence of stationary distribution of system (7), which is globally attractive to any solution of model (3).Therefore, the stationary distribution of (7) is the same as model (3).
The rest of this paper is organized as follows.Some related preliminaries are given in Section 2. In Section 3, we derive dynamic behaviors of stochastic model (7), such as extinction, existence of stationary distribution, and persistence.In Section 4, Taking PM 2.5 concentration as the atmospheric pollutant index, numerical simulations are carried out to support these results.Finally, a brief discussion is given in Section 5.

Preliminaries
Due to the biological meanings of  0 and   , it is easy to have the following basic results for model (3) (see Liu et al. [22] for details).
From now on, we impose 0 <  <  +  and sup ≥0 () ≤ ℎ on models (3) and (7).For the last two equations of system (3), we assume that the following.( In order to explore the asymptotical dynamic behaviors of model (3), motivated by [17,20], we first apply aggregation method to transform model (3) into the following fast time scale system (under Assumption 2): and the following slower time scale system: Note that  0 () and   () of model ( 6) can be explicitly solved and have asymptotically properties in Lemma 3. From now on, we will pay our main attention to model (7) with initial value (4).We first recall the deterministic counterpart of model (7), which is given as Define a deterministic basic reproduction number of model (8) as follows: Model ( 8) always has a disease-free equilibrium point  1 = (( −  −  1  * 0 )/, 0), and the interior positive equilibrium point For these equilibrium points, we have the following stability results.
Proof.The proof is similar to that in Liu et al. [8] by applying Lyapunov method, and hence we omit it here.
Next, to investigate the dynamic behaviors and statistical characters of stochastic model (7), we shall give some preparations.Let () be a homogeneous Markov process defined in the   (which is a -dimensional Euclidean space) and be described by the following stochastic differential equation: The diffusion matrix is defined as follows: Assumption 5.There exists a bounded domain  ∈   with regular boundary Γ, having the following properties.
(i) In the domain  and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix () is bounded away form zero.
(ii) If  ∈   \, the mean time  at which a path emerging from  reaches the set  is finite, and sup ∈    < ∞ for every compact subset  ∈   .Theorem 6.If Assumption 5 holds, then the Markov process () has a stationary distribution (⋅).Let (⋅) be a function integrable with respect to the measure .Then for all  ∈   .
Remark 7. The proof of Theorem 6 is given in [23].Exactly, the existence of stationary distribution with density is referred to as Theorem 4.1 at p.119 and Lemma 9.4 at p.138 in [23].
To check (i) of Assumption 5, we need to prove that F is uniformly elliptical in , where F = () ⋅   + [(()  )]/2; namely, there is a positive number  such that (see Chapter 3, p.103 [24] and Rayleigh's principle in [25], Chapter 6, p.349]).To validate (ii) of Assumption 5, it is enough to show that there exist some neighborhood  and a nonnegative  2 -function  such that, for any   \ ,  is negative (see p.1163 of [26] for details).
Theorem 8.For any given initial value ( 4), there is a unique positive solution () = ((), ()) to model (7) on  ≥ 0, and the solution will remain in R 2 + with probability one.Moreover, there exist some positive constants   () such that Proof.The proof is deferred in Appendix.

Stochastic Endemic Dynamics of Model (7)
In this section, we shall establish the long-term dynamic behaviors of stochastic model (7), namely, the stochastic extinction of disease, persistence, and the existence of stationary distribution with ergodicity. Define For model (7), we have the following.
Proof.By virtue of the Itô's formula, one can get where Noticing that ∫ Therefore, we have lim →∞ () = 0, ..This completes the proof.
Remark 10.From Theorem 9, we know that if   0 < 1, then the disease will go to extinction; thus, we can view   0 as a disease control indicator in a stochastic environment.The pollution level ( 2 ,  * 0 ) and environmental fluctuation intensity ( 2 1 ,  2 3 ) are negative association with   0 ; that is to say,  2 ,  * 0 ,  2 1 ,  2 3 are benefit to the control of disease outbreak.Now, we are in a position to prove the existence of stationary distribution of model (3).Since system ( 7) is an autonomous system, we can use well-known Hasminskii's methods to prove the existence of its stationary distribution.Our proof can be outlined by the following two steps.
(i) First, we prove that, for any arbitrary solution of slower system (7), it is attractive to any solution of model (3).
(ii) Then, by means of Hasminskii's methods we prove the existence of stationary distribution of slower system (3).
Since all the solution of model (3) will tend to the stationary distribution of system (7), accordingly they have the same stationary distribution.
Lemma 11 (see [27]).Suppose that a stochastic process () on  ≥ 0 satisfies the condition for some positive constants , , and ; then there exist a continuous modification X() of (), which has the property that, for every  ∈ (0, /), and a positive random variable ℎ() such that In other words, almost every sample path of X() is locally but uniformly Hölder-continuous with exponent .Lemma 12. Let () be a solution of system (3) with initial value (4); then almost every sample path of () is uniformly continuous on  ≥ 0.
Proof.The first equation of model ( 3) is equivalent to the following stochastic integral equation: Then Making use of the moment inequality for stochastic integral (see Mao [21]), there exist 0 ≤  1 ≤  2 and  > 2 such that For 0 <  1 <  2 < ∞,  2 −  1 ≤ 1, 1/ + 1/ = 1, we obtain that where 2  1 ()}; then it follows from Lemma 11 that almost every sample path of () is locally but uniformly Hölder-continuous with exponent  for every  ∈ (0, (−2)/2), and therefore almost every sample path of () is uniformly continuous on  ≥ 0. Similarly, we can also prove that almost every sample path of () is uniformly continuous on  ≥ 0. This completes the proof.
Assumption 13.  > , which implies that the infection proportion in the total environmental population () is not larger than the intrinsic growth rate in absence of the toxicant (); otherwise, the population may go to extinction.Theorem 14.Let () = ((), ()) be the solution of system (7).If Assumption 13 holds, then () is globally attractive to any solution of model (3).
Theorem 15.Suppose that   0 > 1 and there exists a positive constant  such that where and  * = ( * ,  * ) is given in (8).Then, there is a stationary distribution (⋅) with respect to R 2 + for stochastic model (7) with initial value (4) and it has ergodic property.(34) and it can be easily verified that (, ) is a positive definite function for all (, ) ̸ = ( * ,  * ).By use of Itô formula to (24), we can get where Young's inequality is used in the second line from the bottom of (36).If  satisfies the following condition: then the ellipsoid − 1 ( −  * ) 2 −  2 ( −  * ) 2 +  = 0 lies in int(R 2 + ).Let  be a neighborhood of the ellipsoid with  ⊆   = R 2 + , so for  ∈  \   ,  < 0, which implies that condition (ii) of Assumption 5 is satisfied.
Moreover, model ( 7) can be rewritten in the following form: Here the diffusion matrix is

Complexity
In addition, there is for all ((), ()) ∈ ,  ∈ R 2 , which shows that condition (i) of Assumption 5 is also satisfied.By Theorem 9, we can draw the conclusion that stochastic system (17) has a stationary distribution (⋅) and it is ergodic.
Combining Theorems 14 and 15, we can have the following result on the existence of stationary distribution of model (3).
Theorem 16.Assuming that the conditions in Theorem 15 hold, then there is a stationary distribution (⋅) for model (3) with initial value (4) and it has ergodic property.
In addition, we have the following persistent result of model (7).
Theorem 17.Assuming that the conditions in Theorem 15 hold, model (7) is stochastic persistence in the mean; that is, Proof.Integrating (25) from 0 to t, we have where It then follows from (42) that lim sup In virtue of (46), we have lim sup Notice that Therefore, model ( 7) is stochastic and persistent in the mean.7) will be stochastic and persistent; red line: the trajectories of () for deterministic model (8).(b) Blue line: the sample path of () of model ( 7) will go to extinction; red line: the trajectories of () for deterministic model (8).

Applications for PM 2.5 via Numerical Simulations
In this section, we shall apply the numerical simulations to illustrate the obtained theoretical results and to explore the effect of environmental noise and pollutant on the disease transmission dynamics with analytical result.With help of MATLAB (Mathworks, Inc., Natick, MA, USA) and Milstein's higher order method [29], which is a powerful tool for solving stochastic differential equations, we consider the following discretized equation of model (7) at  = ( + 1)△,  = 0, 1, . . ., △ = 0.001, where  , ,  = 1, 2, 3, are the (0, 1)-distribution independent Gaussian random variables.8) has a unique globally asymptotically stable endemic equilibrium  * (see Theorem 4).The following examples concentrate on the effect of environmental fluctuation on the long-term dynamical behaviors of stochastic model (7).The only difference between the following cases is the intensity of environmental noises.
(i) For  1 = 0.01,  2 =  3 = 0.015, we can obtain that   0 = 0.3701 < 1; according to Theorem 9, we know that the disease will go to extinction, which is supported by Figure 1.It can be seen that stochastic model (7) (blue lines in Figure 1) has completely different dynamic behaviors compared with its deterministic counterpart (8) (red lines in Figure 1).Thus,   7) will be stochastic and persistent; red line: the trajectories of () for deterministic model (8).(b) Blue line: the sample path of () of model (7) will be stochastic and persistent; red line: the trajectories of () for deterministic model (8).(c) The corresponding probability density function(PDF) of () of model (7).(d) The corresponding PDF of () of model (7). the environmental noises may suppress the outbreak of disease.
(ii) For  1 = 0.0001,  2 =  3 = 0.00015.We can computer that then we can further check that condition (32) holds for  = 5 and  > .It therefore follows from Theorem 15 that there is a stationary distribution for stochastic model (7) (as shown in Figure 2).If the intensity of environmental noise is not too large (  ,  = 1, 2, 3, satisfies (32)), then the sample path of stochastic model ( 7) fluctuated around deterministic model (8) and above the estimated lower bound ( = 26.6527, = 24.7130)(see (32) of Theorem 17).These results may provide insight into the disease dynamics that the slight environmental fluctuation cannot change the persistence of model (7); that is, the persistence of disease has robustness with respect to the environmental fluctuation.
(II) The Influence of Environmental Pollution  * on the Disease Transmission Dynamics.In the following, we shall carry out some numerical simulations in order to show how to derive some epidemiological insights from our analytic results.Taking the polluted atmosphere environment (indexed by PM 2.5 , the monthly average PM 2.5 concentrations in Ningxia are listed in Table 2) data,Yinchuan, China (the PM2.5 data source from [30], the data can be accessed from their website) as an example, we consider the effect of pollution level on the disease transmission dynamics.
The only difference in the following case is the annual mean concentrations of PM 2.5 ( * ).We choose  * = 44.07, * = 90, and  * = 117.75,respectively (the annual average PM 2.5 concentration) and assume that  = 0.5,  =  = 0.4.It can be observed from Figure 3 that, with the increase of environmental pollution level, the infected number decreases significantly.Moreover, we can find that the probability density function of () moves to left with the increase of  * .Therefore, the environmental pollution  * may play an important role in the disease transmission dynamics, which implies that the high pollution level may provide a reduction effect on infected number of diseases.These theoretical numerical results are consistent with the influenza infected peak number during 2013, 2014, and 2015, Yinchuan, China (the influenza data source from Ningxia Center for Diseases Prevention and Control [31], the data can be accessed from their website), as shown in Figure 3(d).

Discussions
There is no doubt that environmental pollution is an extremely serious problem, which has affected the evolutionary dynamics of respiratory disease (e.g., influenza-like illness [32] and rubella) for the exposed human worldwide; we should take strong measures to deal with them.In this article, a stochastic SI epidemic model is investigated.In virtue of the aggregation methods and Lyapunov function, the sufficient condition of disease extinction and existence of the stationary distribution are established, respectively.Taking PM 2.5 as the air pollutant index, the effect of pollution on the transmission dynamics is carried out by numerical simulations.These theoretical and simulation results indicated the following: (i) the fluctuations of transmission coefficient ( 2 1 ) may be associated with prevalence risk of the infectious disease.More precisely, the large intensity noise may suppress the outbreak of the disease (see Figure 1 and Theorem 9), while the small intensity noise may be responsible for the fluctuation of the positive equilibrium of corresponding deterministic counterpart (see Figure 2 and Theorem 16).Just as Britton et al. [33] pointed out that "a severe inadequacy of deterministic models in describing the persistence of infection process with demography: fluctuations in the prevalence of infection about the endemic level can often be large enough for transmission to be interrupted by stochastic fade-out."Therefore, the environmental fluctuation may significantly affect the transmission dynamics of infectious diseases.

Figure 1 :
Figure 1: (a) Blue line: the sample path of () of model (7) will be stochastic and persistent; red line: the trajectories of () for deterministic model(8).(b) Blue line: the sample path of () of model (7) will go to extinction; red line: the trajectories of () for deterministic model(8).

Figure 2 :
Figure 2: (a) Blue line: the sample path of () of model (7) will be stochastic and persistent; red line: the trajectories of () for deterministic model(8).(b) Blue line: the sample path of () of model(7) will be stochastic and persistent; red line: the trajectories of () for deterministic model(8).(c) The corresponding probability density function(PDF) of () of model(7).(d) The corresponding PDF of () of model(7).

Figure 3 :
Figure 3: (a) The sample paths of () to model (7) for different pollution level  * ; (b) the sample path of () to model (7) for different pollution level  * ; (c) the probability density function of () at time 500 for different pollution level  * ; (d) the infected cases of influenza in Yinchuan city during Nov. to Oct., respectively, in 2014,2015, and 2016.