Dynamics Analysis of Avian Influenza A(H7N9) Epidemic Model

The avian influenza A(H7N9) virus has certain fatal effects on human. In this paper, a mathematical model describing the transmission dynamics of avian influenza A(H7N9) between human and poultry is investigated. The basic reproduction number of the model is obtained by applying the method of the next generation matrix. Then the local and global stability of the equilibria are proven. At last, we use numerical simulations to verify the theoretical results.


Introduction
Infectious diseases are caused by a variety of bacteria, fungi, virus, and other pathogens, which can be transmitted between human and human, human and animals, animals and animals.Infectious diseases have always been the enemy of human; avian influenza is one of the most common diseases.According to the relevant studies, avian influenza was reported early in 1878 in Italy, which is caused by influenza A virus and is popular among animals.Meanwhile, it is transmitted to human by chickens, ducks, and other animals.The overwhelming majority of avian influenza virus cannot infect human; however, some avian influenza viruses are zoonotic [1].The clinical manifestations of human diseases caused by some avian influenza virus are not very obvious, which are not harmful enough to human health.However, certain virus such as H5N1 and H7N9 can cause serious human diseases.The virus subtype H5N1, a highly pathogenic avian influenza virus, was firstly detected in Hong Kong in 1997 and was transmitted from Asia to Europe and Africa in 2003 and 2004, resulting in the death of a large amount of poultry and humans [2,3].The virus subtype H7N9, a low pathogenic avian influenza virus, is transmitted mainly through contact.In March 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 Beijing, Shanghai, Jiangsu, Zhejiang, Anhui, Shandong, Henan, Taiwan, and Fujian [4,5].At present, infected humans of avian influenza A(H7N9) are still sporadic, and the ability of the virus to spread among humans has not yet been found.Since 2013, avian influenza A(H7N9) virus has been included in the main content of influenza virus research.The virus is not fatal to poultry, but it can cause severe respiratory diseases and certain lethality to human, which has attracted the attention of WHO [6].
At present, some scholars have studied avian influenza.In 2007, Iwami et al. [7] considered a dynamic model of avian influenza that might be transmitted by infected birds and infected humans with variant avian influenza.Che et al. [8] studied a model of highly pathogenic avian influenza with saturated contact rate.Initially, the health authorities controlled the outbreak of H7N9 by closing livepoultry markets, which can reduce the probability of human infected by reducing the contact between human and poultry.However, due to economic reasons, live-poultry markets cannot be permanently closed.Therefore, humans need to adopt other intervention strategies, such as screening poultry and killing infected poultry.Liu and Fang [9] established a dynamical model of avian influenza A(H7N9) that can spread between poultry and poultry, poultry and human, and human and human to evaluate the impact of these measures on avian influenza A(H7N9) epidemic.Chen and Wen [10] based on the bilinear disease incidence studied a model with mutant avian influenza A(H7N9) virus in 2015.In 2017, Liu et al. [11] proposed two avian influenza models with different growth rates of the avian population, one with logistic growth and the other with Allee effect.
The organization of this paper is as follows.In Section 2, consider that people mainly contact with poultry in wet markets.We construct an avian influenza A(H7N9) model of different groups in specific environment to study the transmission of avian influenza A(H7N9).In Sections 3 and 4, the basic reproduction number and the existence of feasible equilibria are studied.What is more, by using suitable Lyapunov functions, we demonstrate the global stability of equilibria.The numerical simulations used to verify the theoretical results and some conclusions are included in Sections 5 and 6.

The Model
The exposure of infected poultry is one of the key factors in the human infection of avian influenza A(H7N9).In this section, we combine poultry with human to establish a mathematical model with the aim of understanding the spread of avian influenza A(H7N9) from poultry to human.Contagion occurs only between poultry and poultry as well as poultry and human; it cannot spread among humans.The human population is classified into three subclasses: susceptible, infected, and recovered, denoted by  ℎ ,  ℎ , and  ℎ , respectively.  and   denote susceptible and infective poultry in farms and   and   represent susceptible poultry and infective poultry of markets, respectively.The flowchart of avian influenza A(H7N9) transmission between poultry and human is described in Figure 1.
The dynamic model of avian influenza A(H7N9) is described as the following ordinary differential equations: where  ℎ and   represent the birth rates of human and poultry, respectively. ℎ ,   , and   indicate the natural mortality rates of human, poultry of farms, and poultry of markets, respectively. ℎ ,   , and   are the diseaserelated death rates of infected human, infected poultry of farms, and infected poultry of markets, respectively.  is the transmission coefficient from infective poultry of farms to susceptible poultry of farms.  is the contact rate from infective poultry of markets to susceptible poultry of markets. ℎ is the transmission rate from infected poultry of markets to susceptible human. is the recovery rate of infected human. is the proportion of poultry from farms to markets.All the parameters are nonnegative.

The Existence of Equilibria
By resolving these equations of system (1), it is easy to see that system (1) always has the disease-free equilibrium  0 = ( 0  , 0,  0  , 0,  0 ℎ , 0, 0), where According to the next generation matrix formulated in van den Driessche and Watmough [12], we can obtain Then Hence, the basic reproduction number of system is as follows: Theorem 1.The disease-free equilibrium  0 of system (1) always exists.If  0 < 1,  0 is locally asymptotically stable; if  0 > 1, it is unstable.
Proof.The Jacobian matrix at the disease-free equilibrium  0 is ) .
The characteristic equation of the Jacobian matrix is The eigenvalues of the characteristic equation are Hence, if  01 < 1 and  02 < 1, all eigenvalues have negative real parts.Namely, if  0 < 1, the disease-free equilibrium  0 is locally asymptotically stable.The discussion of the existence of positive equilibrium is as follows.
(1) Let   ̸ = 0; solving the second equation of system (1), one obtains that Equation ( 13) is replaced with the first equation of system (1), which can be solved as follows: Again, let   = 0; the fourth equation of system (1) appears as follows: its left is equal to  * *  , but the right is 0. Both sides are not equal, so there is no   = 0. Let   ̸ = 0, the third equation of system (1) can be solved as follows: Substituting ( 15) into the fourth equation of system (1) gives and then where Due to  1 < 0,  1 > 0, Δ =  2 1 − 4 1  1 > 0 is always constant and there is a unique positive root.From (17), we obtain Substituting ( 19) into the fifth, sixth, and seventh equations of system (1) gives (2) Let   = 0, from the first equation of (1), we can obtain Again, let   ̸ = 0. Substituting   = 0 into the fourth equation in (1) gives Combining ( 21) and ( 22), from the third equation of system (1), it is known that Substituting (23) into the fifth, sixth, and seventh equations of system (1) gives Hence, (i) when  01 < 1, there are two cases:

Stability of Equilibria
We note that the variable  ℎ does not appear in the first six equations of system (1).The last equation is independent of the first six equations; we can only consider the following subsystem of system (1): (25) 4.1.Stability of the Disease-Free Equilibrium.The discussion of global stability of the disease-free equilibrium  0 is as follows.
The poultry subsystem of farms, the poultry subsystem of markets, and the human subsystem are independent of each other.We firstly consider the poultry subsystem in farms and define a Lyapunov function and then the derivative of  11 along solutions of system ( 25) is and if  0 < 1, we get  11 / ≤ 0. Thus, According to Lasalle's invariance principle [13,14],  0  is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the disease-free steady state We define a Lyapunov function Calculating the derivative of  12 along solutions of system (28), it follows that and if  0 < 1,  12 / |(28) ≤ 0. Thus, According to Lasalle's invariance principle [13,14],  0  is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the disease-free steady state We define a Lyapunov function and then the derivative of  13 along solutions of system (31) is and, thus, According to Lasalle's invariance principle [13,14],  0 ℎ is globally asymptotically stable.In summary, the following theorem can be obtained.Theorem 3.For system (1), if  0 < 1, the disease-free equilibrium  0 is globally asymptotically stable.

Stability of the Boundary Equilibrium and the Endemic
Equilibrium.After calculation, the Jacobian matrix of system (1) is given as where The characteristic equation of the Jacobian matrix is In addition,  4 ,  5 are satisfied: The remaining two eigenvalues  6 ,  7 are satisfied: It can be obtained that   +   −    * *  > 0 by  * *  > 0. Hence, if  01 > 1,  02 > 1, all eigenvalues have negative real parts.According to the above discussion, we can derive the following theorem.Theorem 4. For system (1), if  01 < 1,  02 > 1, the boundary equilibrium  * is locally asymptotically stable; if  01 > 1,  02 > 1, the endemic equilibrium  * * is locally asymptotically stable.
Consider the global stability of the boundary equilibrium and the endemic equilibrium.
We firstly consider the poultry subsystem in farms and define a Lyapunov function and then the derivative of  21 along solutions of system ( 25) is ( We define a Lyapunov function and then the derivative of  22 along solutions of system ( 44) is  [13,14],  *  is globally asymptotically stable.
Finally, considering the human subsystem with the avian components of markets already at the endemic steady state We define a Lyapunov function and then the derivative of  23 along solutions of system ( 47) is Since According to Lasalle's invariance principle [13,14],  * ℎ is globally asymptotically stable.In conclusion, if  01 < 1,  02 > 1, the boundary equilibrium  * is globally asymptotically stable.
We firstly consider the poultry subsystem in farms and define a Lyapunov function and then the derivative of  31 along solutions of system ( 25) is ) . ( According to Lasalle's invariance principle [13,14],  * *  is globally asymptotically stable.
Next, considering the poultry subsystem of markets with the avian components of farms already at the endemic steady state We define a Lyapunov function and then the derivative of  32 along solutions of system ( 52) is  [13,14],  * *  is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the endemic steady state We define a Lyapunov function and then the derivative of  33 along solutions of system (55) is ) . ( According to Lasalle's invariance principle [13,14],  * * ℎ is globally asymptotically stable.Hence, the following theorem can be obtained.Theorem 5.For system (1), if  01 < 1,  02 > 1, the boundary equilibrium  * is globally asymptotically stable; if  01 > 1,  02 > 1, the endemic equilibrium  * * is globally asymptotically stable.Remark 6. Stability of the equilibrium depends on the Lyapunov functions.The quadratic form of Lyapunov functions is usually used in most references [15,16].Let us take the function in this paper we have

Discussion
Avian influenza infectious diseases caused by influenza virus can quickly spread in areas such as farms and markets.In this paper, an SI-SI-SIR dynamic model of avian influenza A(H7N9) is established by combining human and poultry.We get the basic reproduction number  0 ; it is the threshold which is endemic or not.If  0 < 1, there is only the disease-free equilibrium  0 , and it is globally asymptotically stable, which implies that the disease dies out.If  01 < 1,  02 > 1, at this time  0 > 1, there is the unique boundary equilibrium  * , which is globally asymptotically stable; namely, the disease will be sustained and lead to epidemic disease eventually.If  01 > 1,  02 < 1, at this time  0 > 1, there is no positive equilibrium; that is, the disease spreads in farms and it does not spread in markets, people will not be infected by the virus, and the disease will not be popular.If  01 > 1,  02 > 1, at this time  0 > 1, there is the endemic equilibrium  * * , which is globally asymptotically stable; that is, the disease will spread.From the above analysis, we can see that if artificial measures are taken to reduce the basic reproduction number to small enough value in the transmission system of avian influenza A(H7N9), the global stability point of the propagation dynamics process exists.Raise   and   by killing infected poultry or reduce   ,   , and  ℎ by closing farms and markets.Both of the two methods can reduce  0 or  ℎ , so as to control the occurrence and development of diseases.