Avian Influenza A (H7N9) Model Based on Poultry Transport Network in China.

In order to analyze the spread of avian influenza A (H7N9), we construct an avian influenza transmission model from poultry (including poultry farm, backyard poultry farm, live-poultry wholesale market, and wet market) to human according to poultry transport network. We obtain the threshold value for the prevalence of avian influenza A (H7N9) and also give the existence and number of the boundary equilibria and endemic equilibria in different conditions. We can see that poultry transport network plays an important role in controlling avian influenza A (H7N9). Finally, numerical simulations are presented to illustrate the effects of poultry in different places on avian influenza. In order to reduce human infections in China, our results suggest that closing the retail live-poultry market or preventing the poultry of backyard poultry farm into the live-poultry market is feasible in a suitable condition.


Introduction
Avian influenza A (H7N9) is a subtype of influenza viruses that have been detected in birds in the past. Until 2013 outbreak in China, no human infections with H7N9 viruses had ever been reported. But from March 31 to August 31, 2013, 134 cases had been reported in mainland China, resulting in 45 deaths [1]. However, the virus came back in November 2013 again. Afterwards the disease came back in November every year. In fact, the second outbreak occurred from November 2013 to May 2014. e third outbreak occurred from November 2014 to June 2015. e fourth outbreak occurred from November 2015 to June 2016. And the fifth outbreak occurred from September 2016 to May 2017 (NHFPC [1]). e disease causes a high death rate. In China, from March 2013 to May 2017, H7N9 has resulted in 1263 human cases including 459 deaths with a death rate of nearly 37%. In China, from September 2016 to May 2017, provinces with human cases are shown as Figure 1. H7N9 virus does not induce clinical signs in poultry and is classified as a low pathogenicity avian influenza virus (LPAIV) [2]. However, the virus can infect humans and most of the reported cases of human H7N9 infection have resulted in severe respiratory illness [3].
Jones et al. [4] demonstrated that interspecies transmission of H7N9 virus occurs readily between society finches and bobwhite quail but only sporadically between finches and chickens, and transmission occurs through shared water. Pantin-Jackwood et al. [3] showed that quail and chickens are susceptible to infection, shed large amounts of virus, and are likely important in the spread of the virus to humans, and it is therefore conceivable that passerine birds may serve as vectors for transmission of H7N9 virus to domestic poultry [4]. Zhang et al. [5] concluded that migrant birds are the original infection source. Many authors investigated the epidemic model which describes the transmission of avian influenza among birds and humans [8][9][10][11][12][13][14][15]. Liu et al. [16] constructed two avian influenza birdto-human transmission models with different growth laws of the avian population, one with logistic growth and the other with Allee effect, and analyzed their dynamical behavior. Lin et al. [17] developed three different SIRS models to fit the observed human cases between March 2013 and July 2015 in China and found that environmental transmission via viral shedding of infected chickens had contributed to the spread of H7N9 human cases in China. Chen and Wen [18] took into account gene mutation in poultry. Guo et al. [19] proposed and analyzed an SE-SEIS avian-human influenza model. Mu and Yang [20] analyzed an SI-SEIR avian-human influenza model with latent period and nonlinear recovery rate. Gourley et al. [21] analyzed the patchy model for the spatiotemporal distribution of a migratory bird species. Bourouiba et al. [22] investigated the role of migratory birds in the spread of H5N1 avian influenza among birds by considering a system of delay differential equations for the numbers of birds on patches, where the delays represent the flight times between patches. In China, in 2013, to control the outbreak, local authorities of the provinces and municipalities, such as Jiangsu, Shanghai, and Zhejiang, temporarily closed the retail live-poultry markets which proved to be an effective control measure. Data indicate that the novel avian influenza A (H7N9) virus was most likely transmitted from the secondary wholesale market to the retail live-poultry market and then to humans [6,7]. How is avian influenza A (H7N9) transmitted from live-poultry to human in China? In order to reveal the fact, the global network model of avian influenza A (H7N9) is constructed based on poultry transport network. e relationship between the global system and subsystem is analyzed. e corresponding risk indices are obtained. We study the impact of subsystems on the risk index of the global system. When the disease occurs, it can provide theoretical guidance for the global and local transport of poultry.
In this paper, we construct an avian influenza A (H7N9) transmission model from live poultry (including poultry farm, backyard poultry farm, live-poultry wholesale market, and wet market) to human for the heterogenous environments which affect the spread of H7N9. e remaining part of this paper is organized as follows: in Section 2, we first establish the model based on poultry transport network. We derive the threshold value of the model. In Sections 3 and 4, we discuss the different boundary and endemic equilibrium in the different thresholds. Section 5 gives the effect of different transmission rate on H7N9 by numerical simulation. Finally, concluding remarks are made in Section 6.

Model Based on Poultry Transport Network
e avian population is classified into poultry farm, backyard poultry farm, live-poultry wholesale market, and wet market (the retail live-poultry market). According to the present situation in China, the backyard poultry feeding is regarded as a large node, which is considered to be connected with all other nodes (except poultry farm) in network.
e relationship diagram of poultry transport and contacts between human and poultry are described in Figure 2. Let N i fa (t), N j pa (t), and N k ma (t) be the total number of poultry in ith poultry farm, jth live-poultry wholesale market, and kth wet market at time t, respectively, where N i fa (t), N i pa (t), and N i ma (t) are classified into two subclasses: susceptible and infective, denoted by S i fa (t) and I i fa (t), S j pa (t) and I j pa (t), and S k ma (t) and I k ma (t), respectively. Suppose there are L poultry farms, M live-poultry wholesales, and K wet markets, namely, i � 1, . . . , L; j � 1, . . . , M; k � 1, . . . , K. And they are independent of each other. Let N h (t) be the total number of human at time t. e human population is classified into three subclasses: susceptible, infective, and recovered, denoted by S h (t), I h (t), and R h (t), respectively. All new recruitments of human population and avian population are susceptible. e avian influenza virus is not contagious from an infective human to a susceptible human. It is only contagious from an infective avian to a susceptible avian and a susceptible human. An infected avian keeps in the state of disease and cannot recover, but an infected human can recover, and the recovered human has permanent immunity. We neglect death rates of the poultry individuals during the transport process. e detailed description of dynamical transmission of H7N9 avian influenza is described in the following flowchart ( Figure 3). e corresponding dynamical model can be seen in the following equation: Computational and Mathematical Methods in Medicine e interpretations of parameters of system (1) are described in Table 1. e parameters in system (1) are all nonnegative constants. e variation of the number of poultry in ith poultry and thus, Similarly, the variation of the number of poultry in backyard poultry farm N ba (t) is and thus, e variation of the number of poultry in jth live-poultry wholesale market N j pa (t) is and thus, e variation of the number of poultry in kth wet market and thus e variation of the number of human N h (t) is and thus, For convenience, we denote the positive solution , then G is a positively invariant for system (1).
In order to find the disease-free equilibrium of system (1), we consider All new recruitments of the avian in backyard poultry farm d f,b,p,m e natural death rate (including slaughter) of the avian in different places α f,b,p,m e disease-related death rate of the infected avian in different places β f,b,p,m e transmission rate from infective avian to susceptible avian in different places a ij e transport rate of individuals from ith poultry farm to jth live-poultry wholesale market b jk e transport rate of individuals from jth live-poultry wholesale market to kth wet market l j e transport rate of individuals from backyard poultry farm to jth live-poultry wholesale market c k e transport rate of individuals from backyard poultry farm to kth wet market A h All new recruitments of the human d h e natural death rate of the human β kh e transmission rate from the infective avian in kth wet market to the susceptible human β h e transmission rate from the infective avian in backyard farm to the susceptible human α h e disease-related death rate of the infected human c h e recovery rate of the infective human Computational and Mathematical Methods in Medicine 5 System (12) has the unique positive equilibrium According to the concepts of the next generation matrix and reproduction number presented in [23,24], we define where Set R 0 � ρ(FV −1 ), where ρ represents the spectral radius of the matrix. en, R 0 is called the reproduction number for system (1), where If R 0 < 1, then system (1) has the disease-free equilibrium E 0 , and E 0 is locally asymptotically stable.
Remark 1. If we do not consider backyard poultry farm, then system (1) becomes A similar analysis is available for the above system. 6 Computational and Mathematical Methods in Medicine

Analysis of Subsystems of System (1)
Consider the poultry of the poultry farm subsystem, given by the first two equations of system (1), as follows Let the right-hand side of system (17) equals to zero; when I i fa ≠ 0, we obtain If R i f0 > 1, system (17) has the positive equilibrium (S i * fa , I i * fa ). If R i f0 < 1, system (17) has only the disease-free equilibrium (S i0 fa , 0).

Remark 2.
(1) If min Consider the poultry of the backyard poultry farm subsystem, given by the third and fourth equations of system (1), as follows Let the right-hand side of system (19) equals to zero; when I ba ≠ 0, we obtain If R b0 > 1, system (19) has the positive equilibrium (S i * ba , I i * ba ). If R b0 < 1, system (19) has only the disease-free equilibrium (S 0 ba , 0). Consider the poultry of the live-poultry wholesale market subsystem, given by the fifth and sixth equations of system (1), as follows Let the right-hand side of system (21) equals to zero; when I j pa ≠ 0, we can divide it into two cases. If I i fa � 0 and I ba � 0, then we have If R j p0 > 1, then system (21) has the positive equilibrium (S j * pa , I j * pa ). If I i fa ≠ 0 or I ba ≠ 0, then we obtain where Because b 2 2 − 4b 1 b 3 > 0, the solutions of the above equation are  (21) has no positive equilibrium.
Consider the poultry of the wet market (the retail livepoultry market) subsystem, given by the seventh and eighth equations of system (1), as follows:
Consider the human subsystem, given by the last three equations of system (1), as follows: Since the first two equations of system (31) are independent of the variable R h , we only need to analyze the first two equations of system (31). Let the right-hand side of system (31) equals to zero, when I h ≠ 0, if I k ma ≠ 0 or I ba ≠ 0, then we have

Analysis of the Full System (1)
We analyze the following equivalent system:

Computational and Mathematical Methods in Medicine
For the sake of discussion, without loss of generality, we assume that a node has at least one link with the nodes in the next layer. So we have the following cases. 0). Namely, when all poultry has no avian influenza, human will not be infected with avian influenza.

Case 2.
If max

system (33) has the boundary equilibrium
is shows that avian influenza A (H7N9) virus is most likely transmitted from the retail live-poultry market to humans when poultry has no disease in other types of farms.
If min m0 < 1, system (33) has one boundary equilibrium: If min 1 ≤ k ≤ K R k2 m0 > 1, system (33) has two boundary equilibria: is shows that avian influenza A (H7N9) virus is most likely transmitted from the secondary wholesale market to the retail live-poultry market and then to humans [6,7]. And there may be two boundary equilibria.
If min m0 < 1, system (33) has one boundary equilibrium: If min p0 < 1, and min 1 ≤ k ≤ K R k2 m0 > 1, system (33) has two boundary equilibria: If min m0 > 1, system (33) has two boundary equilibria: If min p0 > 1 and min 1 ≤ k ≤ K R k2 m0 > 1, system (33) has four boundary equilibria: When the poultry of poultry farms has no avian influenza, and the poultry of backyard poultry farm has avian influenza, we can obtain four cases. In four cases, human is most likely transmitted from the backyard poultry farm to the secondary wholesale market then to the retail livepoultry market, and finally to humans, or direct transmission from backyard poultry to humans.
If min m0 < 1, system (33) has one boundary equilibrium: If min p0 < 1, and min 1 ≤ k ≤ K R k2 m0 > 1, system (33) has two boundary equilibria: If min m0 > 1, system (33) has two boundary equilibria: When the poultry of poultry farms has avian influenza, and the poultry of backyard poultry farm has no avian influenza, we can obtain four cases. In four cases, human is most likely transmitted from the poultry farm to the secondary wholesale market, then to the retail live-poultry market, and finally to humans.
If min When the poultry of poultry farms has avian influenza, and the poultry of backyard poultry farm has avian influenza, we can obtain four cases. In four cases, human is most likely transmitted from the poultry farm and backyard poultry farm to the secondary wholesale market, then to the retail live-poultry market, and finally to humans, or direct transmission from backyard poultry to humans.
Remark 3. If we assume that there is an edge between each node of the upper layer and each node of the next layer, that is, each node of the upper layer transport poultry to each node of the next layer in the network, when max most likely transmitted from the poultry wholesale market to the retail live-poultry market and finally to humans.
(3) e poultry of poultry farm has avian influenza, and the poultry of backyard poultry farm has no avian influenza, but there is a possible outbreak of avian influenza in poultry farm, and then avian influenza A (H7N9) virus is most likely transmitted from poultry market to poultry wholesale market, then to the retail live-poultry market, and finally to humans. (4) e poultry of poultry farm has no avian influenza, and the poultry of backyard poultry farm has avian influenza, but there is a possible outbreak of avian influenza in backyard poultry farm, and then avian influenza A (H7N9) virus is most likely transmitted from backyard poultry farm to poultry wholesale market, then to the retail live-poultry market, and finally to humans, or direct transmission from backyard poultry to humans. (5) e poultry of poultry farm and backyard poultry farm has avian influenza, but there is a possible outbreak of avian influenza in poultry farm and backyard poultry farm, and then avian influenza A (H7N9) virus is most likely transmitted from poultry farm and backyard poultry farm to poultry wholesale market, then to the retail live-poultry market, and finally to humans, or direct transmission from backyard poultry to humans.
Hence, the poultry of some nodes on network has avian influenza, and then all edges connected to the node should be cut off. It has a great inhibitory on preventing the spread of disease. So, the network of poultry transportation plays an important role in controlling avian influenza A (H7N9). Moreover, we find that there may have been avian influenza A (H7N9) among humans when there is avian influenza A (H7N9) in the retail live-poultry market, so closing the livepoultry market can reduce the spread of disease to humans at a certain time. In addition, we find that there may have been avian influenza A (H7N9) among humans when there is avian influenza A (H7N9) in the backyard poultry farm. But the spread of backyard poultry to human is quit complex. It can be either direct infection or indirect infection. In China, there are many backyard poultry, so there are still some difficulties in the prevention and control of avian influenza A (H7N9).

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.