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Computational and Mathematical Methods in Medicine
Volume 2018, Article ID 7383170, 14 pages
https://doi.org/10.1155/2018/7383170
Research Article

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

1Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, China
2Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, Shanxi 030006, China
3Institute of Disease Control and Prevention of PLA, Beijing 100071, China

Correspondence should be addressed to Zhen Jin; ten.362@nhznij

Received 1 June 2018; Accepted 27 September 2018; Published 4 November 2018

Academic Editor: Konstantin Blyuss

Copyright © 2018 Juping Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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. The third outbreak occurred from November 2014 to June 2015. The fourth outbreak occurred from November 2015 to June 2016. And the fifth outbreak occurred from September 2016 to May 2017 (NHFPC [1]). The 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].

Figure 1: Provinces with avion influenza A (H7N9) from September 2016 to May 2017.

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 [815]. Liu et al. [16] constructed two avian influenza bird-to-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. The relationship between the global system and subsystem is analyzed. The 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. The 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.

2. Model Based on Poultry Transport Network

The 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. The relationship diagram of poultry transport and contacts between human and poultry are described in Figure 2. Let , , and be the total number of poultry in ith poultry farm, jth live-poultry wholesale market, and kth wet market at time t, respectively, where , , and are classified into two subclasses: susceptible and infective, denoted by and , and , and and , respectively. Suppose there are L poultry farms, M live-poultry wholesales, and K wet markets, namely, . And they are independent of each other. Let be the total number of human at time t. The human population is classified into three subclasses: susceptible, infective, and recovered, denoted by , , and , respectively. All new recruitments of human population and avian population are susceptible. The 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. The detailed description of dynamical transmission of H7N9 avian influenza is described in the following flowchart (Figure 3).

Figure 2: A possible network of H7N9 avian influenza.
Figure 3: Detailed transfer diagram on the dynamical transmission of H7N9 avian influenza.

The corresponding dynamical model can be seen in the following equation:

The interpretations of parameters of system (1) are described in Table 1. The parameters in system (1) are all nonnegative constants.

Table 1: Parameters of system (1).

The variation of the number of poultry in ith poultry farm isand thus,

Similarly, the variation of the number of poultry in backyard poultry farm isand thus,

The variation of the number of poultry in jth live-poultry wholesale market isand thus,

The variation of the number of poultry in kth wet market isand thus

The variation of the number of human isand thus,

For convenience, we denote the positive solution of system (1) by .

Let then G is a positively invariant for system (1).

In order to find the disease-free equilibrium of system (1), we consider

System (12) has the unique positive equilibrium , where , , , , and . Thus, is the disease-free equilibrium of system (1).

According to the concepts of the next generation matrix and reproduction number presented in [23, 24], we definewhere

Set , where represents the spectral radius of the matrix. Then, is called the reproduction number for system (1), where

If , then system (1) has the disease-free equilibrium , and is locally asymptotically stable.

Remark 1. If we do not consider backyard poultry farm, then system (1) becomesA similar analysis is available for the above system.

3. 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 , we obtain

If , system (17) has the positive equilibrium . If , system (17) has only the disease-free equilibrium .

Remark 2. (1)If , then each farm has the positive equilibrium.(2)If , then some of the poultry farms have the positive equilibrium, and the others have only the disease-free equilibrium.Consider the poultry of the backyard poultry farm subsystem, given by the third and fourth equations of system (1), as followsLet the right-hand side of system (19) equals to zero; when , we obtainIf , system (19) has the positive equilibrium . If , system (19) has only the disease-free equilibrium .
Consider the poultry of the live-poultry wholesale market subsystem, given by the fifth and sixth equations of system (1), as followsLet the right-hand side of system (21) equals to zero; when , we can divide it into two cases.
If and , then we haveIf , then system (21) has the positive equilibrium .
If or , then we obtainwhere Because , the solutions of the above equation areIf , system (21) has two positive equilibria (, ) and (, ). If and , system (21) has one positive equilibrium (, ). If , system (21) has no positive equilibrium.
Consider the poultry of the wet market (the retail live-poultry market) subsystem, given by the seventh and eighth equations of system (1), as follows:Let the right-hand side of system (26) equals to zero, when , we can divide it into two cases.
If and , then we haveIf , then system (26) has the positive equilibrium .
If or , then we havewhereBecause , the solutions of the above equation areIf , system (26) has two positive equilibria (, ) and (, ). If and , system (26) has one positive equilibrium (, ). If , system (26) has no positive equilibrium.
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 , we only need to analyze the first two equations of system (31). Let the right-hand side of system (31) equals to zero, when , if or , then we have

4. Analysis of the Full System (1)

We analyze the following equivalent system:

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.

Case 1. If , system (33) has only the disease-free equilibrium . Namely, when all poultry has no avian influenza, human will not be infected with avian influenza.

Case 2. If , , and , system (33) has the boundary equilibriumThis 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.

Case 3. If , , and , system (33) has the boundary equilibrium as described next.
If and , system (33) has one boundary equilibrium:If , system (33) has two boundary equilibria:This 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.

Case 4. If and , system (33) has the boundary equilibrium as described next.
If , , , and , system (33) has one boundary equilibrium:If , , and , system (33) has two boundary equilibria:If , , and , system (33) has two boundary equilibria:If and , 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 live-poultry market, and finally to humans, or direct transmission from backyard poultry to humans.

Case 5. If and , system (33) has the boundary equilibrium as described next.
If , , , and , system (33) has one boundary equilibrium:If , , and , system (33) has two boundary equilibria:If , , and , system (33) has two boundary equilibria:If and , system (33) has four boundary equilibriaWhen 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.

Case 6. If and , system (33) has the positive equilibrium as described next.
If , , , and , system (33) has one positive equilibrium:If , , and , system (33) has two positive equilibria:If , , and , system (33) has two positive equilibria: