Abstract

H7N9 virus in the environment plays a role in the dynamics of avian influenza A (H7N9). A nationwide poultry vaccination with H7N9 vaccine program was implemented in China in October of 2017. To analyze the effect of vaccination and environmental virus on the development of avian influenza A (H7N9), we establish an avian influenza A (H7N9) transmission model with vaccination and seasonality among human, birds, and poultry. The basic reproduction number for the prevalence of avian influenza is obtained. The global stability of the disease-free equilibrium and the existence of positive periodic solution are proved by the comparison theorem and the asymptotic autonomous system theorem. Finally, we use numerical simulations to demonstrate the theoretical results. Simulation results indicate that the risk of H7N9 infection is higher in colder environment. Vaccinating poultry can significantly reduce human infection.

1. Introduction

In general, the avian influenza virus does not infect human. However, in 1997, Hong Kong reported for the first time 18 cases of human infection with avian influenza A (H5N1), of which 6 cases died. It caused widespread concern worldwide [1–3]. Further, H5N1, H7N4, H7N7, H7N9, H9N2, and other avian influenza viruses with pathogenicity have great potential threat to human. Especially, the virus subtype H7N9 is mainly transmitted through the respiratory tract, infected poultry and their secretions, excreta, and water contaminated by the virus. In February 2013, 3 people were firstly infected, and by May 31, 132 cases were found, including 37 deaths, and the mortality rate even reached 30%. These cases are distributed in some provinces such as Shanghai, Jiangsu, Zhejiang, Anhui, Fujian, and so on [4–7]. At present, infected humans of avian influenza A (H7N9) are still sporadic, and it has not yet found the ability that the virus can spread among humans. Sporadic infections almost contact with poultry mainly in farms, live-poultry markets, wet markets, and other regions [8–12].

Zhang et al. [13] analyzed the source of infection of avian influenza A (H7N9). According to theirs analysis, the most probable transmission route of avian influenza A (H7N9) is that migratory birds carry the virus and transmit it to local birds through physical space transitions, which then transmit the virus to poultry and to human with direct or indirect contact with poultry. Vaidya and Wahl [14], considering the seasonal bird migration, studied the relationship between the time-varying environment-temperature and the decline of the avian influenza A (H7N9) virus in the environment. Xiao et al. [12] proposed and analyzed a mathematical model to mimic its transmission dynamics to assess the transmission potential of the novel avian influenza A (H7N9) virus. Xing et al. [15] fitted the dynamic model with the actual data and found that the main reason for the recurrence of avian influenza A (H7N9) in winter was temperature cycling. Che et al. [16] studied the model of highly pathogenic avian influenza with saturated contact rate. Li et al. [17] established an avian influenza A (H7N9) dynamics model with different groups in specific environment and studied the transmission of H7N9 avian influenza in the epidemic-prone areas, which reflected the impact of farms, live-poultry markets, and wet markets.

Note that birds and poultry are the natural storage hosts of avian influenza virus. The direct contact transmission between birds and poultry is very little and the cross infection between them happens through H7N9 virus carried in coexistence environment. Exposed and infected birds and poultry can shed virus into the environment due to secretions and excreta. The virus can survive for several weeks, or even months in the feces or contaminated environment under suitable conditions. So, environment transmission is indispensable during the process of the spread of influenza virus. The most easily infectious source of human infection is poultry with the virus, and the main routes of transmission are poultry-human and environment-human transmissions. In addition, the high-risk humans infected with avian influenza virus are mainly concentrated on humans who are often in contact with poultry, including slaughter, breeding, processing, trafficking in poultry, and low immunity groups. In order to reduce the production of sick poultry and the concentration of viruses in the environment, vaccinating poultry and disinfecting the environment are used.

The organization of this paper is as follows. In Section 2, we have constructed an avian influenza A (H7N9) epidemic model with vaccination and seasonality to study the spread of avian influenza A (H7N9). In Sections 3 and 4, the threshold value is obtained. By using the comparison theorem, and the asymptotic autonomous system theorem and so on, it is found that there exists a globally asymptotically stable disease-free equilibrium when and at least a positive periodic solution when , respectively. The numerical simulations used to illustrate the theoretical results and some conclusions are included in Sections 5 and 6.

2. Formulation of the Model

We combine birds, poultry, and viruses in the environment with human to establish a mathematical model of the spread of avian influenza A (H7N9). Their total numbers of birds, poultry, and human at any time are denoted by , , and , respectively. The concentration of viruses in the environment is denoted by , and the average number of viruses that causes a H7N9 individual case is called an infectious unit (IU) [13, 18]. The infectivity of virus in the environment to birds, poultry, or humans is more affected by the temperature, and we will consider it is periodic. The infectivities among birds, poultry, and human-poultry are related to themselves, which are less affected by the temperature, so periodicities are not considered. It is assumed that there is no transmission between humans and humans. Therefore, human individuals are infected by two ways: poultry-human and environment-human transmissions. The human population is classified into four subclasses: ordinary susceptible, high-risk susceptible, infected, and recovered, denoted by , and , respectively. and denote the number of susceptible and infective individuals in the bird population. , and denote the number of susceptible, vaccinated, and infective individuals in the poultry population, respectively. Transmission process of avian influenza A (H7N9) virus among these populations is described in Figure 1.

Figure 1 

Flowchart of avian influenza A (H7N9) transmission.

The dynamic model of avian influenza A (H7N9) is described as the following ordinary differential equations:The interpretations of the variables and parameters are shown in Table 1. All variables and parameters are nonnegative.

According to [15, 19, 20], the relationships between environmental transmission rates and temperature can be obtained and where is the temperature, is the average temperature, is a period, is the amplitude, and is the phase difference.

We model the implementation of intervention strategies (initiated at time ) by the following piecewise functions and [12]:and

From system (1), these can find thatThen, from (8), it follows thatand as , so .

In the same way, from (9) and (10), it can be obtained that , . The feasible region of system (1) is

3. The Basic Reproduction Number

The last equation is independent of the first nine equations of system (1); we can only consider the following subsystem of system (1): It is easy to see that system (13) always has the disease-free equilibrium , , where

From system (13), we know that human dynamic equations do not affect the basic reproduction number. But for the convenience of dynamic analysis, we add human dynamic equations when calculating the basic reproduction number. According to the method of Wang and Zhao [21], we can obtain and Then and

Assume is the evolution operator of the linear -periodic systemThat is, for each , the matrix satisfieswhere is the identity matrix. Thus, the monodromy matrix of (19) is equal to .

In view of the periodic environment, we assume that , -periodic in , is the initial distribution of infectious individuals. Then is the rate of new infections produced by the infected individuals who were introduced at time . Given , then gives the distribution of those infected individuals who were newly infected at time and remain in the infected compartments at time .

Let be the ordered Banach space of all -periodic functions from to , which is equipped with the maximum norm and the positive cone . Then we can define a linear operator : by . From Wang and Zhao [21], we call the next infection operator and define the basic reproduction number as , where is the spectral radius of .

Theorem 1. For system (13), if , the disease-free equilibrium is locally asymptotically stable; if , it is unstable.

4. The Stability of the Disease-Free Equilibrium and Existence of Periodic Solutions

4.1. The Stability of the Disease-Free Equilibrium

Theorem 2. For system (13), if , the disease-free equilibrium is globally asymptotically stable.

Proof. Let be a nonnegative solution of system (13). To complete the proof, it is sufficient to show that this nonnegative solution tends to the disease-free equilibrium as .
The first, third, fourth, seventh, and eighth equations of system (13) with give the respective inequalitiesHence, for any , it exists ; when , we haveThe second, fifth, sixth, and ninth equations of system (13), with , , , , and , give the respective inequalitiesWe consider the auxiliary system of system (23), where the coefficient matrix is Clearly, if , it is known from Theorem 2.2 in [21] that . We can choose small enough giving . It can be concluded from Lemma 2.1 in Zhang and Zhao [22] that there exists a positive, -periodic function such that is a solution of the auxiliary system, where . Here, , which implies as . Therefore, the zero solution of the auxiliary system is globally asymptotically stable. For any nonnegative initial value, there is a sufficiently large . Applying the comparison theorem [23], it can be obtained that . Therefore, we obtain as . By the theory of asymptotic autonomous systems [24], we get as . Hence, if , the disease-free equilibrium is globally asymptotically stable.

4.2. The Existence of Positive Periodic Solutions

DefineLet be the Poincaré map associated with system (13); that is, , , where is the period. is the unique solution of system (13) with . It is easy to see that .