Study on a Stochastic Avian Influenza Epidemic Model with Generalized Incidence Rate

Avian Influenza, with a high mortality rate in human population, is considered to be one of the most significant potential threats to human beings. Based on a recent avian influenza SI-SIR model with logistic growth for birds, we propose a stochastic model with generalized incidence rate. For the stochastic avian-only system, sufficient conditions for the extinction of infected birds are established, and the existence of a unique ergodic stationary distribution is also obtained. For the stochastic avian-human system, a threshold number 0 s R is established, and hence the extinction of disease is investigated. From the viewpoint of biology, the noise intensity in the infected birds plays a key role in the evolutionary dynamics. Moreover, we also analyze the asymptotic behavior around the endemic equilibrium of the corresponding deterministic model.


Introduction
Avian Influenza, an acute infectious disease caused by influenza A virus, is a complicated disease that can not only infect poultry but also infect humans who have direct exposure to infected birds or contaminated environments. Because humans generally lack immunity to avian influenza virus, the disease has a high mortality rate. For instance, an outbreak of the avian influenza AH7N9 in China caused 134 cases with 45 deaths, from March 31 to August 31, 2013 [1]. According to the data reported by the World Health Organization (WHO), 860 human infections have been reported worldwide since 2003, with about half of those people dying [2]. Hence, the WHO considers the disease to be one of the most significant potential threats to human beings. To better analyze the spread of avian influenza, an increasing amount of research has been studied from the viewpoint of epidemiology and biomathematics. Alexander et al. [3] introduced the pathogenicity of four avian influenza viruses for chickens, turkeys and ducks.
Grais et al. [4] and Ferguson et al. [5] took humans as the core and proposed some strategies to retard the spread of avian influenza in the population. Menach [6] regarded poultry farm as a unit to establish a model of infectious disease and focused on the transmission to humans. For more mathematical epidemic models on avian influenza, one can refer to [7] [8] [9] [10] and the references therein.
Recently, under the assumption that avian influenza virus does not spread from person to person and mutate, Liu et al. [1] proposed the following avian influenza bird-to-human transmission model  describes the net growth rate of the avian population, and two cases are studied in the reference [1]. In the present article, we pick the case that the avian population is subject to the logistic growth, then ( ) 1 a a a a a S g S r S k , where a r is the intrinsic growth rate, a k is the maximal carrying capacity of the avian population. The incidence rate of epidemic model plays a quite important role in describing the evolution of infectious disease, and system (1) chose bilinear ones, that is, rate of the form 1 SI I β α + [11], and incidence rate with media coverage effect of the form e mI SI β − [12]. In practice, the incidence function is frequently difficult to obtain because the details of disease transmission vary in different conditions. Therefore choosing generalized incidence rates may allow epidemic models to be more flexible in handling realistic data. For model (1) is plugged in, and the recovered population is removed since it has no effect on the dynamics of susceptible population and infective population. Furthermore, the transmission of influenza is disturbed by various noises in the environment, such as the unpredictable contact with infected ones, population mobility and meteorological factors. It is shown that environment fluctuations have a important effect on the development of infectious disease [13] [14] [15]. For instance, Meng et al. [14] showed that a large stochastic disturbance can cause infectious diseases to go to extinction, and Li et al. [15] also found that the average number of infected individuals always with the increase of noise intensity.  , , , 0, 1, 2, , Throughout the paper, we further assume that (H1) ( ) ( ) Our generalized incidence rates can be applied to some specific forms that have been frequently used, such as The article is dedicated to investigating the dynamics of the stochastic avian influenza epidemic model (3). The rest of the paper is organized as follows. In the next section, the existence and uniqueness of positive solution is proved for system (3). In Section 3, we discuss the dynamics of the avian-only subsystem, and obtain the sufficient conditions for the extinction of the disease as well as the existence of an ergodic stationary distribution. In Section 4, the dynamics of the avian-human system are discussed, and the asymptotic behavior of system (3) around the unique endemic equilibrium of system (2) is also investigated.

Dynamics of the Stochastic Avian-Only Subsystem
Since , , is an arbitrary twice continuously differential real-value function.
To begin with, we present the following fundamental theorem which guarantees the existence and uniqueness of positive solution for system (3).
Theorem 2.1. For any initial value and the solution will remain in 4 + R with probability one, namely, The proof is similar to those of [17] and hence is omitted. Consider the stochastic avian-only subsystem as follows It is obviously observed that the avian system is independent of the human system. Therefore in this section we will focus on the dynamics of the stochastic system (5), and our main goal is to discuss the extinction of disease and the existence of stationary distribution.
The threshold parameter 0 R of the deterministic system with respect to (5) can be computed by application of the next-generation matrix approach as ( ) a I t will tend to zero exponentially a.s. and the disease will tend to extinction with probability one.
The proof is similar to those of [17] and hence is omitted.
On the other hand, if we define where Q is a constant such that Proof. Consider the following auxiliary logistic equation with random perturbation x ∈ ∞ , then we can obtain ( ) ( ) Consequently, the condition of Theorem 1.16 in [18] follows from (7). Thus system (6) has the ergodic property, and the invariant density is given by where Q is a constant such that Let x(t) be the solution of (6) with the initial value ( ) ( ) 0 0 0 a x S = > , then the comparison theorem of stochastic differential equation [19] yields On the other hand, according to (9), we get Take the superior limit on both sides of (10), and note that 0 1 R < , then it follows that the distribution of the process ( ) a S t converges weakly to the measure with the density π . This completes the proof.
We now concentrate on verifying the existence of an ergodic stationary distribution for system (5). The following lemma is fundamental in the paper. And It is standard to verify that ( ) According to inequality (14), we can obtain ( ) a Choose a sufficiently small ε such that . According to the definition of 1 C and Inequality (15), we can obtain that According to Inequality (16) we can obtain that , similar to the case 3, it is easy to get According to inequality (16), we obtain that 1 LV ≤ − on 4 D ε . That is, the condition (B.2) holds.
The diffusion matrix of system (5) is given by

The Asymptotic Behavior of Stochastic Full System
For any initial value   S  I  S  I R + ∈ , the disease of system (3) will tend to extinction, almost surely.
The proof is similar to those of theorem 2.3 and hence is omitted. If the deterministic system (2) has an endemic equilibrium, then it means the disease will persist in the long term. Since stochastic system does not exist endemic equilibrium, it is interesting to investigate the asymptotic behavior of global positive solution of system (3) around endemic equilibrium.
where the basic inequality ( ) , , 3 x y z xyz

Conclusion
Most systems in the real world are disturbed by various stochastic factors, such as population mobility and meteorological factors including humidity, temperature and precipitation. Hence the effects of environmental fluctuation on the transmission of infectious diseases cannot be neglected. In this paper, we studied a stochastic avian-human influenza epidemic model with logistic growth for birds. To begin with, we proved the existence and uniqueness of global Hence it is interesting to note that a threshold number is difficult to obtain because of the logistic growth rate. For the full stochastic system, the disease will tend to extinction if 0 1 s R < . From the viewpoint of biology, 0 s R is a proper threshold parameter and the noise intensity in the infected avian population plays a key role. Moreover, we also discussed the asymptotic behavior and proved that the solution of the system (3) oscillates around corresponding endemic equilibrium under some conditions. Some interesting topics deserve further consideration. For instance, it has not been confirmed that avian influenza virus does not spread from person to person and mutate. Furthermore, the seasonal effect for the transmission of avian influenza is neglected in the present model. We hope to study the comprehensive impacts of seasonal variation and environmental noises in the future.