# Dynamics Analysis of Avian Influenza A(H7N9) Epidemic Model

Authors: Yun Li; Peng Qin; Juping Zhang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

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.


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## Body

## 1. 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 live-poultry 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 Section2, 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.


## 2. 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 bySh, Ih, and Rh, respectively. Sfa and Ifa denote susceptible and infective poultry in farms and Sma and Ima 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.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:(1)dSfadt=Aa-daSfa-aSfa-βaSfaIfa,dIfadt=βaSfaIfa-daIfa-aIfa-αaIfa,dSmadt=aSfa-βmSmaIma-dmSma,dImadt=aIfa+βmSmaIma-dmIma-αmIma,dShdt=Ah-βhShIma-dhSh,dIhdt=βhShIma-dhIh-rIh-αhIh,dRhdt=rIh-dhRh,where Ah and Aa represent the birth rates of human and poultry, respectively. dh, da, and dm indicate the natural mortality rates of human, poultry of farms, and poultry of markets, respectively. αh, αa, and αm are the disease-related death rates of infected human, infected poultry of farms, and infected poultry of markets, respectively. βa is the transmission coefficient from infective poultry of farms to susceptible poultry of farms. βm is the contact rate from infective poultry of markets to susceptible poultry of markets. βh is the transmission rate from infected poultry of markets to susceptible human. r is the recovery rate of infected human. a is the proportion of poultry from farms to markets. All the parameters are nonnegative.LetSfa(t)+Ifa(t)=Nfa(t), Sma(t)+Ima(t)=Nma(t), Sh(t)+Ih(t)+Rh(t)=Nh(t). From system (1), these can find that(2)dNfadt=Aa-da+aNfa-αaIfa≤Aa-da+aNfa,(3)dNmadt=aNfa-dmNma-αmIma≤aNfa-dmNma,dNhdt=Ah-daNh-αhIh≤Ah-dhNh.Then, from (2), it follows that(4)Nfat≤Aaa+da+Nfa0-Aaa+dae-a+dat,and e-(a+da)t→0 as t→∞, so limt→∞Nfa(t)≤Aa/a+da.In the same way, from (3), it can be obtained that limt→∞Nma(t)≤aAa/dm(a+da), limt→∞Nh(t)≤Ah/dh. The feasible region of system (1) is(5)Ω=Sfa,Ifa,Sma,Ima,Sh,Ih,Rh∈R+7:Nfa≤Aaa+da,Nma≤aAadma+da,Nh≤Ahdh.


## 3. 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 E0=(Sfa0,0,Sma0,0,Sh0,0,0), where (6)Sfa0=Aada+a,Sma0=aAadmda+a,Sh0=Ahdh.According to the next generation matrix formulated in van den Driessche and Watmough [12], we can obtain (7)F=βaAaa+da000βmaAadma+da00βhAhdh0,V=αa+a+da00-adm+αm000dh+αh+r.Then (8)R01=βaAaa+da+αaa+da,R02=βmaAadma+dadm+αm.Hence, the basic reproduction number of system is as follows:(9)R0=max⁡R01,R02.Theorem 1.
The disease-free equilibriumE0 of system (1) always exists. If R0<1, E0 is locally asymptotically stable; if R0>1, it is unstable.Proof.
The Jacobian matrix at the disease-free equilibriumE0 is (10)JE0=-da-a-βaSfa0000000βaSfa0-da-a-αa00000a0-dm-βmSma00000a0βmSma0-dm-αm000000-βhSh0-dh00000βhSh00-dh-αh-r000000r-dh.The characteristic equation of the Jacobian matrix is (11)λ+da+aλ+da+a+αa-βaSfa0λ+dmλ+αm+dm-βmSma0λ+dhλ+dh+r+αhλ+dh=0.The eigenvalues of the characteristic equation are (12)λ1=-dh,λ2=-dh-r-αh,λ3=-dh,λ4=-da-a,λ5=-dm,λ6=-αa-a-da+βaAaa+da,λ7=-dm-αm+βmaAadma+da.Hence, if R01<1 and R02<1, all eigenvalues have negative real parts. Namely, if R0<1, the disease-free equilibrium E0 is locally asymptotically stable.
The discussion of the existence of positive equilibrium is as follows.
( 1 ) Let Ifa≠0; solving the second equation of system (1), one obtains that(13)Sfa∗∗=a+αa+daβa.Equation (13) is replaced with the first equation of system (1), which can be solved as follows:(14)Ifa∗∗=Aa-daSfa∗∗-aSfa∗∗βaSfa∗∗=βaAa-a+da+αada+aβaa+da+αa=a+daR01-1βa.Again, let Ima=0; the fourth equation of system (1) appears as follows: its left is equal to aIfa∗∗, but the right is 0. Both sides are not equal, so there is no Ima=0. Let Ima≠0, the third equation of system (1) can be solved as follows:(15)Sma∗∗=aSfa∗∗βmIma∗∗+dm.Substituting (15) into the fourth equation of system (1) gives(16)βmSma∗∗Ima∗∗+aIfa∗∗-dmIma∗∗-αmIma∗∗=aIfa∗∗βmIma∗∗+dm+aβmSfa∗∗-dm+αmdm+βmIma∗∗Ima∗∗=0,and then(17)a1Ima∗∗2+b1Ima∗∗+c1=0,where (18)a1=-βmdm+αm,b1=aβmIfa∗∗+aβmSfa∗∗-dm2-dmαm,c1=admIfa∗∗.Due to a1<0, c1>0, Δ=b12-4a1c1>0 is always constant and there is a unique positive root. From (17), we obtain(19)Ima∗∗=-aIfa∗∗βmSma∗∗-dm-αm=-b1+b12-4a1c12a1.Substituting (19) into the fifth, sixth, and seventh equations of system (1) gives(20)Sh∗∗=AhβhIma∗∗+dh,Ih∗∗=βhSh∗∗Ima∗∗dh+r+αh,Rh∗∗=rIh∗∗dh.(2) Let Ifa=0, from the first equation of (1), we can obtain(21)Sfa∗=Aada+a.Again, let Ima≠0. Substituting Ifa=0 into the fourth equation in (1) gives(22)Sma∗=dm+αmβm.Combining (21) and (22), from the third equation of system (1), it is known that(23)Ima∗=aSfa∗-dmSma∗βmSma∗=aAadm+αma+da-dmβm=dmR02-1βm.Substituting (23) into the fifth, sixth, and seventh equations of system (1) gives(24)Sh∗=AhβhIma∗+dh,Ih∗=βhSh∗Ima∗dh+r+αh,Rh∗=rIh∗dh.Hence,(i)
whenR01<1, there are two cases:whenR02<1, the disease-free equilibrium E0=(Sfa0,0,Sma0,0,Sh0,0,0) is obtained,whenR02>1, the boundary equilibrium E∗=(Sfa∗,0,Sma∗,Ima∗,Sh∗,Ih∗,Rh∗) is obtained;(ii)
whenR01>1, there are two cases:whenR02<1, the positive equilibrium does not exist,whenR02>1, the endemic equilibrium E∗∗=(Sfa∗∗,Ifa∗∗,Sma∗∗,Ima∗∗,Sh∗∗,Ih∗∗,Rh∗∗) is obtained.In conclusion, we can obtain the following theorem.Theorem 2.
For system (1), if R01<1, R02>1, there is the unique boundary equilibrium E∗; if R01>1, R02>1, there is the unique endemic equilibrium E∗∗.


## 4. Stability of Equilibria

We note that the variableRh 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)dSfadt=Aa-daSfa-aSfa-βaSfaIfa,dIfadt=βaSfaIfa-daIfa-aIfa-αaIfa,dSmadt=aSfa-βmSmaIma-dmSma,dImadt=aIfa+βmSmaIma-dmIma-αmIma,dShdt=Ah-βhShIma-dhSh,dIhdt=βhShIma-dhIh-rIh-αhIh.


### 4.1. Stability of the Disease-Free Equilibrium

The discussion of global stability of the disease-free equilibriumE0 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(26)V11=Sfa-Sfa0-Sfa0lnSfaSfa0+Ifa,and then the derivative of V11 along solutions of system (25) is (27)dV11dt=Sfa-Sfa0SfadSfadt+dIfadt=Sfa-Sfa0SfaAa-daSfa-aSfa-βaSfaIfa+βaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa0SfadaSfa0-Sfa+aSfa0-Sfa+βaSfa0Ifa-Ifada+a+αa=-a+daSfaSfa-Sfa02+a+da+αaR01-1Ifa,and if R0<1, we get dV11/dt≤0. Thus,Ω 1 = { ( S f a , I f a ) ∈ R + 2 : d V 11 / d t = 0 } = { ( S f a , I f a ) ∈ R + 2 : S f a = S f a 0 ,   I f a = 0 } = { E f a 0 }. According to Lasalle’s invariance principle [13, 14], Efa0 is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the disease-free steady state(28)dSmadt=aSfa0-βmSmaIma-dmSma,dImadt=βmSmaIma-dmIma-αmIma.We define a Lyapunov function(29)V12=Sma-Sma0-Sma0ln⁡SmaSma0+Ima.Calculating the derivative of V12 along solutions of system (28), it follows that (30)dV12dt=Sma-Sma0SmadSmadt+dImadt=Sma-Sma0SmaaSfa0-dmSma-βmSmaIma+βmSmaIma-dmIma-αmIma=Sma-Sma0SmadmSma0-Sma+βmSma0Ima-dmIma-αmIma=-dmSmaSma-Sma02+dm+αmR02-1Ima,and if R0<1, dV12/dt(28)≤0. Thus, Ω2={(Sma,Ima)∈R+2:dV12/dt=0}={(Sma,Ima)∈R+2:Sma=Sma0,Ima=0}={Ema0}. According to Lasalle’s invariance principle [13, 14], Ema0 is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the disease-free steady state(31)dShdt=Ah-dhSh,dIhdt=-dhIh-rIh-αhIh.We define a Lyapunov function(32)V13=Sh-Sh0-Sh0ln⁡ShSh0+Ih,and then the derivative of V13 along solutions of system (31) is (33)dV13dt=Sh-Sh0ShdShdt+dIhdt=Sh-Sh0ShAh-dhSh-dhIh-rIh-αhIh=-dhShSh-Sh02-r+dh+αhIh≤0,and, thus, Ω3={(Sh,Ih)∈R+2:dV13/dt=0}={(Sh,Ih)∈R+2:Sh=Sh0,Ih=0}={Eh0}. According to Lasalle’s invariance principle [13, 14], Eh0 is globally asymptotically stable. In summary, the following theorem can be obtained.Theorem 3.
For system (1), if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.


### 4.2. Stability of the Boundary Equilibrium and the Endemic Equilibrium

After calculation, the Jacobian matrix of system (1) is given as (34)J=G0HK,where (35)G=-da-a-βaIfa-βaSfa00βaIfaβaSfa-da-a-αa00a0-βmIma-dm-βmSma0aβmImaβmSma-dm-αm,H=000-βhSh000βhSh0000,K=-βhIma-dh00βhIma-dh-r-αh00r-dh.The characteristic equation of the Jacobian matrix is (36)λ+da+a+βaIfaλ+da+a+αa-βaSfa+βa2SfaIfaλ+dm+βmImaλ+αm+dm-βmSma+βm2SmaImaλ+dh+βhImaλ+dh+r+αhλ+dh=0.(1) For the boundary equilibrium E∗=(Sfa∗,0,Sma∗,Ima∗,Sh∗,Ih∗,Rh∗), five eigenvalues are(37)λ1=-dh,λ2=-dh-r-αh,λ3=-dh-βhIma∗,λ4=-a-da,λ5=-αa-a-da+βaAaa+da.The remaining two eigenvalues λ6, λ7 depend on (38)λ2+λdmR02-1+dmαm+dmR02-1=0.Hence, if R01<1, R02>1, all eigenvalues have negative real parts.( 2 ) For the endemic equilibrium E∗∗=(Sfa∗∗,Ifa∗∗,Sma∗∗,Ima∗∗,Sh∗∗,Ih∗∗,Rh∗∗), three eigenvalues are(39)λ1=-dh,λ2=-dh-r-αh,λ3=-dh-βhIma∗∗.In addition, λ4, λ5 are satisfied:(40)λ2+λR01da+a+da+aα+a+daR01-1=0.The remaining two eigenvalues λ6, λ7 are satisfied:(41)λ2+λ2dm+αm-βmSma∗∗+βmIma∗∗+βmdmIma∗∗+βmαmIma∗∗+dmdm+αm-βmSma∗∗=0.It can be obtained that dm+αm-βmSma∗∗>0 by Ima∗∗>0. Hence, if R01>1,R02>1, all eigenvalues have negative real parts. According to the above discussion, we can derive the following theorem.Theorem 4.
For system (1), if R01<1, R02>1, the boundary equilibrium E∗ is locally asymptotically stable; if R01>1, R02>1, the endemic equilibrium E∗∗ is locally asymptotically stable.Consider the global stability of the boundary equilibrium and the endemic equilibrium.( 1 ) If R01<1, R02>1, there is the boundary equilibrium E∗.We firstly consider the poultry subsystem in farms and define a Lyapunov function(42)V21=Sfa-Sfa∗-Sfa∗ln⁡SfaSfa∗+Ifa,and then the derivative of V21 along solutions of system (25) is (43)dV21dt=Sfa-Sfa∗SfadSfadt+dIfadt=Sfa-Sfa∗SfaAa-daSfa-aSfa-βaSfaIfa+βaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa∗SfadaSfa∗-Sfa+aSfa∗-Sfa+βaSfa∗Ifa-Ifada+a+αa=-a+daSfaSfa-Sfa∗2+a+da+αaR01-1Ifa,and if R01<1, dV21/dt≤0, thus Ω4={(Sfa,Ifa)∈R+2:dV21/dt=0}={(Sfa,Ifa)∈R+2:Sfa=Sfa∗,Ifa=0}={Efa∗}. According to Lasalle’s invariance principle [13, 14], Efa∗ is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the disease-free steady state(44)dSmadt=aSfa∗-βmSmaIma-dmSma,dImadt=βmSmaIma-dmIma-αmIma.We define a Lyapunov function(45)V22=Sma-Sma∗-Sma∗ln⁡SmaSma∗+Ima-Ima∗-Ima∗ln⁡ImaIma∗,and then the derivative of V22 along solutions of system (44) is (46)dV22dt=Sma-Sma∗SmadSmadt+Ima-Ima∗ImadImadt=Sma-Sma∗SmaaSfa∗-dmSma-βmSmaIma+Ima-Ima∗ImaβmSmaIma-dmIma-αmIma=Sma-Sma∗SmadmSma∗-Sma+βmSma∗Ima∗-βmSmaIma+Ima-Ima∗ImaβmSmaIma-βmSma∗Ima=2-Sma∗Sma-SmaSma∗dmSma∗+βmSma∗Ima∗=2-Sma∗Sma-SmaSma∗dmSma∗+Sma∗dmR02-1.Since2-Sma∗/Sma-Sma/Sma∗≤0, if R02>1, dV22/dt(44)≤0, thus, Ω5={(Sma,Ima)∈R+2:dV22/dt=0}={(Sma,Ima)∈R+2:Sma=Sma∗,Ima=Ima∗}={Ema∗}. According to Lasalle’s invariance principle [13, 14], Ema∗ is globally asymptotically stable.Finally, considering the human subsystem with the avian components of markets already at the endemic steady state(47)dShdt=Ah-βhShIma∗-dhSh,dIhdt=βhShIma∗-dhIh-rIh-αhIh.We define a Lyapunov function(48)V23=Sh-Sh∗-Sh∗ln⁡ShSh∗+Ih-Ih∗-Ih∗ln⁡IhIh∗,and then the derivative of V23 along solutions of system (47) is (49)dV23dt=Sh-Sh∗ShdShdt+Ih-Ih∗IhdIhdt=Sh-Sh∗ShAh-βhShIma∗-dhSh+Ih-Ih∗IhβhShIma∗-dhIh-rIh-αhIh=Sh-Sh∗ShdhSh∗-dhSh+βhSh∗Ima∗-βhShIma∗+Ih-Ih∗IhβhShIma∗-βhSh∗Ima∗IhIh∗=dhSh∗2-Sh∗Sh-ShSh∗+βhSh∗Ima∗3-Sh∗Sh-IhIh∗-ShIh∗Sh∗Ih=dhSh∗2-Sh∗Sh-ShSh∗+βhSh∗dmR02-1βm3-Sh∗Sh-IhIh∗-ShIh∗Sh∗Ih.Since 2-Sh∗/Sh-Sh/Sh∗≤0,3-Sh∗/Sh-Ih/Ih∗-ShIh∗/Sh∗Ih≤0, if R02>1, dV23/dt(47)≤0, thus, Ω6={(Sh,Ih)∈R+2:dV23/dt=0}={(Sh,Ih)∈R+2:Sh=Sh∗,Ih=Ih∗}={Eh∗}. According to Lasalle’s invariance principle [13, 14], Eh∗ is globally asymptotically stable. In conclusion, if R01<1, R02>1, the boundary equilibrium E∗ is globally asymptotically stable.( 2 ) If R01>1, R02>1, there is the endemic equilibrium E∗∗.We firstly consider the poultry subsystem in farms and define a Lyapunov function(50)V31=Sfa-Sfa∗∗-Sfa∗∗ln⁡SfaSfa∗∗+Ifa-Ifa∗∗-Ifa∗∗ln⁡IfaIfa∗∗,and then the derivative of V31 along solutions of system (25) is(51)dV31dt=Sfa-Sfa∗∗SfadSfadt+Ifa-Ifa∗∗IfadIfadt=Sfa-Sfa∗∗SfaAa-daSfa-aSfa-βaSfaIfa+Ifa-Ifa∗∗IfaβaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa∗∗Sfada+aSfa∗∗-Sfa+βaSfa∗∗Ifa∗∗-βaSfaIfa-βaSfa∗∗Ifa∗∗Sfa∗∗Sfa+βaSfa∗∗Ifa+βaSfaIfa-βaSfa∗∗Ifa-βaSfaIfa∗∗+βaSfa∗∗Ifa∗∗=-a+daSfaSfa-Sfa∗∗2+βaSfa∗∗Ifa∗∗2-Sfa∗∗Sfa-SfaSfa∗∗=-a+daSfaSfa-Sfa∗∗2+Sfa∗∗a+daR01-12-Sfa∗∗Sfa-SfaSfa∗∗.Since 2-Sfa∗∗/Sfa-Sfa/Sfa∗∗≤0, if R01>1, dV31/dt≤0, thus, Ω7={(Sfa,Ifa)∈R+2:dV31/dt=0}={(Sfa,Ifa)∈R+2:Sfa=Sfa∗∗,Ifa=Ifa∗∗}={Efa∗∗}. According to Lasalle’s invariance principle [13, 14], Efa∗∗ is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the endemic steady state(52)dSmadt=aSfa∗∗-βmSmaIma-dmSma,dImadt=aIfa∗∗+βmSmaIma-dmIma-αmIma.We define a Lyapunov function(53)V32=Sma-Sma∗∗-Sma∗∗ln⁡SmaSma∗∗+Ima-Ima∗∗-Ima∗∗ln⁡ImaIma∗∗,and then the derivative of V32 along solutions of system (52) is(54)dV32dt=Sma-Sma∗∗SmadSmadt+Ima-Ima∗∗ImadImadt=Sma-Sma∗∗SmaaSfa∗∗-dmSma-βmSmaIma+Ima-Ima∗∗ImaaIfa∗∗+βmSmaIma-dmIma-αmIma=Sma-Sma∗∗SmadmSma∗∗-Sma+βmSma∗∗Ima∗∗-βmSmaIma+Ima-Ima∗∗ImaaIfa∗∗+βmSmaIma-aIfa∗∗+βmSma∗∗Ima∗∗ImaIma∗∗=aa+daR01-1βa2-Ima∗∗Ima-ImaIma∗∗+dmSma∗∗2-Sma∗∗Sma-SmaSma∗∗+βmSma∗∗Ima∗∗2-Sma∗∗Sma-SmaSma∗∗.Since 2-Sma∗∗/Sma-Sma/Sma∗∗≤0,2-Ima∗∗/Ima-Ima/Ima∗∗≤0, if R01>1,R02>1, Ima∗∗>0, dV32/dt(52)≤0, thus, Ω8={(Sma,Ima)∈R+2:dV32/dt=0}={(Sma,Ima)∈R+2:Sma=Sma∗∗,Ima=Ima∗∗}={Ema∗∗}. According to Lasalle’s invariance principle [13, 14], Ema∗∗ is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the endemic steady state(55)dShdt=Ah-βhShIma∗∗-dhSh,dIhdt=βhShIma∗∗-dhIh-rIh-αhIh.We define a Lyapunov function(56)V33=Sh-Sh∗∗-Sh∗∗ln⁡ShSh∗∗+Ih-Ih∗∗-Ih∗∗ln⁡IhIh∗∗,and then the derivative of V33 along solutions of system (55) is(57)dV33dt=Sh-Sh∗∗ShdShdt+Ih-Ih∗∗IhdIhdt=Sh-Sh∗∗ShAh-βhShIma∗∗-dhSh+Ih-Ih∗∗IhβhShIma∗∗-dhIh-rIh-αhIh=dhSh∗∗-Sh+βhSh∗∗Ima∗∗-βhShIma∗∗-βhSh∗∗Ima∗∗Sh∗∗Sh-dhSh∗∗Sh∗∗Sh+βhSh∗∗Ima∗∗+dhSh∗∗+βhShIma∗∗-βhSh∗∗Ima∗∗IhIh∗∗+βhSh∗∗Ima∗∗-βhShIma∗∗Ih∗∗Ih=dhSh∗∗2-Sh∗∗Sh-ShSh∗∗+βhSh∗∗Ima∗∗3-Sh∗∗Sh-IhIh∗∗-ShIh∗∗Sh∗∗Ih.Since 2-Sh∗∗/Sh-Sh/Sh∗∗≤0, 3-Sh∗∗/Sh-Ih/Ih∗∗-ShIh∗∗/Sh∗∗Ih≤0, if R02>1, Ima∗∗>0, dV33/dt(55)≤0, thus, Ω9={(Sh,Ih)∈R+2:dV33/dt=0}={(Sh,Ih)∈R+2:Sh=Sh∗∗,Ih=Ih∗∗}={Eh∗∗}. According to Lasalle’s invariance principle [13, 14], Eh∗∗ is globally asymptotically stable. Hence, the following theorem can be obtained.Theorem 5.
For system (1), if R01<1, R02>1, the boundary equilibrium E∗ is globally asymptotically stable; if R01>1, R02>1, the endemic equilibrium E∗∗ 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 (58)fx=1-x+ln⁡x,∀x>0,we have(59)f1=0,f′x=1x-1;thus, f(x)=1-x+ln⁡x≤0, and the equality holds only when x=1. Let x=Sfa/Sfa0>0; then Sfa-Sfa0-Sfa0ln⁡Sfa/Sfa0≥0. Hence, V11 is a Lyapunov function. In a similar way, V12,V13,V21,V22,V23,V31,V32,V33 are also Lyapunov functions.


## 4.1. Stability of the Disease-Free Equilibrium

The discussion of global stability of the disease-free equilibriumE0 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(26)V11=Sfa-Sfa0-Sfa0lnSfaSfa0+Ifa,and then the derivative of V11 along solutions of system (25) is (27)dV11dt=Sfa-Sfa0SfadSfadt+dIfadt=Sfa-Sfa0SfaAa-daSfa-aSfa-βaSfaIfa+βaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa0SfadaSfa0-Sfa+aSfa0-Sfa+βaSfa0Ifa-Ifada+a+αa=-a+daSfaSfa-Sfa02+a+da+αaR01-1Ifa,and if R0<1, we get dV11/dt≤0. Thus,Ω 1 = { ( S f a , I f a ) ∈ R + 2 : d V 11 / d t = 0 } = { ( S f a , I f a ) ∈ R + 2 : S f a = S f a 0 ,   I f a = 0 } = { E f a 0 }. According to Lasalle’s invariance principle [13, 14], Efa0 is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the disease-free steady state(28)dSmadt=aSfa0-βmSmaIma-dmSma,dImadt=βmSmaIma-dmIma-αmIma.We define a Lyapunov function(29)V12=Sma-Sma0-Sma0ln⁡SmaSma0+Ima.Calculating the derivative of V12 along solutions of system (28), it follows that (30)dV12dt=Sma-Sma0SmadSmadt+dImadt=Sma-Sma0SmaaSfa0-dmSma-βmSmaIma+βmSmaIma-dmIma-αmIma=Sma-Sma0SmadmSma0-Sma+βmSma0Ima-dmIma-αmIma=-dmSmaSma-Sma02+dm+αmR02-1Ima,and if R0<1, dV12/dt(28)≤0. Thus, Ω2={(Sma,Ima)∈R+2:dV12/dt=0}={(Sma,Ima)∈R+2:Sma=Sma0,Ima=0}={Ema0}. According to Lasalle’s invariance principle [13, 14], Ema0 is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the disease-free steady state(31)dShdt=Ah-dhSh,dIhdt=-dhIh-rIh-αhIh.We define a Lyapunov function(32)V13=Sh-Sh0-Sh0ln⁡ShSh0+Ih,and then the derivative of V13 along solutions of system (31) is (33)dV13dt=Sh-Sh0ShdShdt+dIhdt=Sh-Sh0ShAh-dhSh-dhIh-rIh-αhIh=-dhShSh-Sh02-r+dh+αhIh≤0,and, thus, Ω3={(Sh,Ih)∈R+2:dV13/dt=0}={(Sh,Ih)∈R+2:Sh=Sh0,Ih=0}={Eh0}. According to Lasalle’s invariance principle [13, 14], Eh0 is globally asymptotically stable. In summary, the following theorem can be obtained.Theorem 3.
For system (1), if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.


## 4.2. Stability of the Boundary Equilibrium and the Endemic Equilibrium

After calculation, the Jacobian matrix of system (1) is given as (34)J=G0HK,where (35)G=-da-a-βaIfa-βaSfa00βaIfaβaSfa-da-a-αa00a0-βmIma-dm-βmSma0aβmImaβmSma-dm-αm,H=000-βhSh000βhSh0000,K=-βhIma-dh00βhIma-dh-r-αh00r-dh.The characteristic equation of the Jacobian matrix is (36)λ+da+a+βaIfaλ+da+a+αa-βaSfa+βa2SfaIfaλ+dm+βmImaλ+αm+dm-βmSma+βm2SmaImaλ+dh+βhImaλ+dh+r+αhλ+dh=0.(1) For the boundary equilibrium E∗=(Sfa∗,0,Sma∗,Ima∗,Sh∗,Ih∗,Rh∗), five eigenvalues are(37)λ1=-dh,λ2=-dh-r-αh,λ3=-dh-βhIma∗,λ4=-a-da,λ5=-αa-a-da+βaAaa+da.The remaining two eigenvalues λ6, λ7 depend on (38)λ2+λdmR02-1+dmαm+dmR02-1=0.Hence, if R01<1, R02>1, all eigenvalues have negative real parts.( 2 ) For the endemic equilibrium E∗∗=(Sfa∗∗,Ifa∗∗,Sma∗∗,Ima∗∗,Sh∗∗,Ih∗∗,Rh∗∗), three eigenvalues are(39)λ1=-dh,λ2=-dh-r-αh,λ3=-dh-βhIma∗∗.In addition, λ4, λ5 are satisfied:(40)λ2+λR01da+a+da+aα+a+daR01-1=0.The remaining two eigenvalues λ6, λ7 are satisfied:(41)λ2+λ2dm+αm-βmSma∗∗+βmIma∗∗+βmdmIma∗∗+βmαmIma∗∗+dmdm+αm-βmSma∗∗=0.It can be obtained that dm+αm-βmSma∗∗>0 by Ima∗∗>0. Hence, if R01>1,R02>1, all eigenvalues have negative real parts. According to the above discussion, we can derive the following theorem.Theorem 4.
For system (1), if R01<1, R02>1, the boundary equilibrium E∗ is locally asymptotically stable; if R01>1, R02>1, the endemic equilibrium E∗∗ is locally asymptotically stable.Consider the global stability of the boundary equilibrium and the endemic equilibrium.( 1 ) If R01<1, R02>1, there is the boundary equilibrium E∗.We firstly consider the poultry subsystem in farms and define a Lyapunov function(42)V21=Sfa-Sfa∗-Sfa∗ln⁡SfaSfa∗+Ifa,and then the derivative of V21 along solutions of system (25) is (43)dV21dt=Sfa-Sfa∗SfadSfadt+dIfadt=Sfa-Sfa∗SfaAa-daSfa-aSfa-βaSfaIfa+βaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa∗SfadaSfa∗-Sfa+aSfa∗-Sfa+βaSfa∗Ifa-Ifada+a+αa=-a+daSfaSfa-Sfa∗2+a+da+αaR01-1Ifa,and if R01<1, dV21/dt≤0, thus Ω4={(Sfa,Ifa)∈R+2:dV21/dt=0}={(Sfa,Ifa)∈R+2:Sfa=Sfa∗,Ifa=0}={Efa∗}. According to Lasalle’s invariance principle [13, 14], Efa∗ is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the disease-free steady state(44)dSmadt=aSfa∗-βmSmaIma-dmSma,dImadt=βmSmaIma-dmIma-αmIma.We define a Lyapunov function(45)V22=Sma-Sma∗-Sma∗ln⁡SmaSma∗+Ima-Ima∗-Ima∗ln⁡ImaIma∗,and then the derivative of V22 along solutions of system (44) is (46)dV22dt=Sma-Sma∗SmadSmadt+Ima-Ima∗ImadImadt=Sma-Sma∗SmaaSfa∗-dmSma-βmSmaIma+Ima-Ima∗ImaβmSmaIma-dmIma-αmIma=Sma-Sma∗SmadmSma∗-Sma+βmSma∗Ima∗-βmSmaIma+Ima-Ima∗ImaβmSmaIma-βmSma∗Ima=2-Sma∗Sma-SmaSma∗dmSma∗+βmSma∗Ima∗=2-Sma∗Sma-SmaSma∗dmSma∗+Sma∗dmR02-1.Since2-Sma∗/Sma-Sma/Sma∗≤0, if R02>1, dV22/dt(44)≤0, thus, Ω5={(Sma,Ima)∈R+2:dV22/dt=0}={(Sma,Ima)∈R+2:Sma=Sma∗,Ima=Ima∗}={Ema∗}. According to Lasalle’s invariance principle [13, 14], Ema∗ is globally asymptotically stable.Finally, considering the human subsystem with the avian components of markets already at the endemic steady state(47)dShdt=Ah-βhShIma∗-dhSh,dIhdt=βhShIma∗-dhIh-rIh-αhIh.We define a Lyapunov function(48)V23=Sh-Sh∗-Sh∗ln⁡ShSh∗+Ih-Ih∗-Ih∗ln⁡IhIh∗,and then the derivative of V23 along solutions of system (47) is (49)dV23dt=Sh-Sh∗ShdShdt+Ih-Ih∗IhdIhdt=Sh-Sh∗ShAh-βhShIma∗-dhSh+Ih-Ih∗IhβhShIma∗-dhIh-rIh-αhIh=Sh-Sh∗ShdhSh∗-dhSh+βhSh∗Ima∗-βhShIma∗+Ih-Ih∗IhβhShIma∗-βhSh∗Ima∗IhIh∗=dhSh∗2-Sh∗Sh-ShSh∗+βhSh∗Ima∗3-Sh∗Sh-IhIh∗-ShIh∗Sh∗Ih=dhSh∗2-Sh∗Sh-ShSh∗+βhSh∗dmR02-1βm3-Sh∗Sh-IhIh∗-ShIh∗Sh∗Ih.Since 2-Sh∗/Sh-Sh/Sh∗≤0,3-Sh∗/Sh-Ih/Ih∗-ShIh∗/Sh∗Ih≤0, if R02>1, dV23/dt(47)≤0, thus, Ω6={(Sh,Ih)∈R+2:dV23/dt=0}={(Sh,Ih)∈R+2:Sh=Sh∗,Ih=Ih∗}={Eh∗}. According to Lasalle’s invariance principle [13, 14], Eh∗ is globally asymptotically stable. In conclusion, if R01<1, R02>1, the boundary equilibrium E∗ is globally asymptotically stable.( 2 ) If R01>1, R02>1, there is the endemic equilibrium E∗∗.We firstly consider the poultry subsystem in farms and define a Lyapunov function(50)V31=Sfa-Sfa∗∗-Sfa∗∗ln⁡SfaSfa∗∗+Ifa-Ifa∗∗-Ifa∗∗ln⁡IfaIfa∗∗,and then the derivative of V31 along solutions of system (25) is(51)dV31dt=Sfa-Sfa∗∗SfadSfadt+Ifa-Ifa∗∗IfadIfadt=Sfa-Sfa∗∗SfaAa-daSfa-aSfa-βaSfaIfa+Ifa-Ifa∗∗IfaβaSfaIfa-daIfa-aIfa-αaIfa=Sfa-Sfa∗∗Sfada+aSfa∗∗-Sfa+βaSfa∗∗Ifa∗∗-βaSfaIfa-βaSfa∗∗Ifa∗∗Sfa∗∗Sfa+βaSfa∗∗Ifa+βaSfaIfa-βaSfa∗∗Ifa-βaSfaIfa∗∗+βaSfa∗∗Ifa∗∗=-a+daSfaSfa-Sfa∗∗2+βaSfa∗∗Ifa∗∗2-Sfa∗∗Sfa-SfaSfa∗∗=-a+daSfaSfa-Sfa∗∗2+Sfa∗∗a+daR01-12-Sfa∗∗Sfa-SfaSfa∗∗.Since 2-Sfa∗∗/Sfa-Sfa/Sfa∗∗≤0, if R01>1, dV31/dt≤0, thus, Ω7={(Sfa,Ifa)∈R+2:dV31/dt=0}={(Sfa,Ifa)∈R+2:Sfa=Sfa∗∗,Ifa=Ifa∗∗}={Efa∗∗}. According to Lasalle’s invariance principle [13, 14], Efa∗∗ is globally asymptotically stable.Next, considering the poultry subsystem of markets with the avian components of farms already at the endemic steady state(52)dSmadt=aSfa∗∗-βmSmaIma-dmSma,dImadt=aIfa∗∗+βmSmaIma-dmIma-αmIma.We define a Lyapunov function(53)V32=Sma-Sma∗∗-Sma∗∗ln⁡SmaSma∗∗+Ima-Ima∗∗-Ima∗∗ln⁡ImaIma∗∗,and then the derivative of V32 along solutions of system (52) is(54)dV32dt=Sma-Sma∗∗SmadSmadt+Ima-Ima∗∗ImadImadt=Sma-Sma∗∗SmaaSfa∗∗-dmSma-βmSmaIma+Ima-Ima∗∗ImaaIfa∗∗+βmSmaIma-dmIma-αmIma=Sma-Sma∗∗SmadmSma∗∗-Sma+βmSma∗∗Ima∗∗-βmSmaIma+Ima-Ima∗∗ImaaIfa∗∗+βmSmaIma-aIfa∗∗+βmSma∗∗Ima∗∗ImaIma∗∗=aa+daR01-1βa2-Ima∗∗Ima-ImaIma∗∗+dmSma∗∗2-Sma∗∗Sma-SmaSma∗∗+βmSma∗∗Ima∗∗2-Sma∗∗Sma-SmaSma∗∗.Since 2-Sma∗∗/Sma-Sma/Sma∗∗≤0,2-Ima∗∗/Ima-Ima/Ima∗∗≤0, if R01>1,R02>1, Ima∗∗>0, dV32/dt(52)≤0, thus, Ω8={(Sma,Ima)∈R+2:dV32/dt=0}={(Sma,Ima)∈R+2:Sma=Sma∗∗,Ima=Ima∗∗}={Ema∗∗}. According to Lasalle’s invariance principle [13, 14], Ema∗∗ is globally asymptotically stable.Finally, considering the human subsystem with the avian components already at the endemic steady state(55)dShdt=Ah-βhShIma∗∗-dhSh,dIhdt=βhShIma∗∗-dhIh-rIh-αhIh.We define a Lyapunov function(56)V33=Sh-Sh∗∗-Sh∗∗ln⁡ShSh∗∗+Ih-Ih∗∗-Ih∗∗ln⁡IhIh∗∗,and then the derivative of V33 along solutions of system (55) is(57)dV33dt=Sh-Sh∗∗ShdShdt+Ih-Ih∗∗IhdIhdt=Sh-Sh∗∗ShAh-βhShIma∗∗-dhSh+Ih-Ih∗∗IhβhShIma∗∗-dhIh-rIh-αhIh=dhSh∗∗-Sh+βhSh∗∗Ima∗∗-βhShIma∗∗-βhSh∗∗Ima∗∗Sh∗∗Sh-dhSh∗∗Sh∗∗Sh+βhSh∗∗Ima∗∗+dhSh∗∗+βhShIma∗∗-βhSh∗∗Ima∗∗IhIh∗∗+βhSh∗∗Ima∗∗-βhShIma∗∗Ih∗∗Ih=dhSh∗∗2-Sh∗∗Sh-ShSh∗∗+βhSh∗∗Ima∗∗3-Sh∗∗Sh-IhIh∗∗-ShIh∗∗Sh∗∗Ih.Since 2-Sh∗∗/Sh-Sh/Sh∗∗≤0, 3-Sh∗∗/Sh-Ih/Ih∗∗-ShIh∗∗/Sh∗∗Ih≤0, if R02>1, Ima∗∗>0, dV33/dt(55)≤0, thus, Ω9={(Sh,Ih)∈R+2:dV33/dt=0}={(Sh,Ih)∈R+2:Sh=Sh∗∗,Ih=Ih∗∗}={Eh∗∗}. According to Lasalle’s invariance principle [13, 14], Eh∗∗ is globally asymptotically stable. Hence, the following theorem can be obtained.Theorem 5.
For system (1), if R01<1, R02>1, the boundary equilibrium E∗ is globally asymptotically stable; if R01>1, R02>1, the endemic equilibrium E∗∗ 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 (58)fx=1-x+ln⁡x,∀x>0,we have(59)f1=0,f′x=1x-1;thus, f(x)=1-x+ln⁡x≤0, and the equality holds only when x=1. Let x=Sfa/Sfa0>0; then Sfa-Sfa0-Sfa0ln⁡Sfa/Sfa0≥0. Hence, V11 is a Lyapunov function. In a similar way, V12,V13,V21,V22,V23,V31,V32,V33 are also Lyapunov functions.


## 5. Numerical Simulations

In this section, the following parameter values are taken as some examples to simulate the stability of the disease-free equilibrium, the boundary equilibrium, and the endemic equilibrium of system (1), and the time-series diagram is given. At last, when βh has different parameter values, the time-variation diagram of Ih is given.Example 1.
Taking parametersAa=80, βa=0.0018, da=0.17, αa=0.25, a=0.14, βm=0.0027, dm=0.2, αm=0.35, Ah=50, βh=0.0018, dh=0.069, αh=0.63, and r=0.301, Figure 2 shows the time-variation diagram of system (25) state variables. It is found that if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.Figure 2
IfR0=0.88680353<1, the time-variation diagram of system (25) state variables is shown.Example 2.
Taking parametersAa=80, βa=0.0018, da=0.17, αa=0.25, a=0.14, βm=0.0056, dm=0.2, αm=0.35, Ah=50, βh=0.0018, dh=0.069, αh=0.63, and r=0.301, as shown in Figure 3, it is found that if R01<1, R02>1, the boundary equilibrium E∗ is globally asymptotically stable.Figure 3
IfR01=0.82949309<1 and R02=1.83929618>1, the time-variation diagram of system (25) state variables is shown.Example 3.
Taking parametersAa=80, βa=0.0045, da=0.17, αa=0.25, a=0.14, βm=0.0056, dm=0.2, αm=0.35, Ah=50, βh=0.0018, dh=0.069, αh=0.63, and r=0.301, Figure 4 shows the time-variation diagram of system (25) state variables. It is found that if R01>1, R02>1, the endemic equilibrium E∗∗ is globally asymptotically stable.Figure 4
IfR01=2.07373271>1 and R02=1.83929618>1, the time-variation diagram of system (25) state variables is shown.Example 4.
Taking parametersAa=80, βa=0.0045, da=0.17, αa=0.25, a=0.14, βm=0.0056, dm=0.2, αm=0.35, Ah=50, dh=0.069, αh=0.63, and r=0.301 and letting βh=0.00005,0.0018,0.02, Figure 5 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of βh.Figure 5
The curve-trend diagram ofIh with time, when βh has different parameter values.


## 6. 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 numberR0; it is the threshold which is endemic or not. If R0<1, there is only the disease-free equilibrium E0, and it is globally asymptotically stable, which implies that the disease dies out. If R01<1, R02>1, at this time R0>1, there is the unique boundary equilibrium E∗, which is globally asymptotically stable; namely, the disease will be sustained and lead to epidemic disease eventually. If R01>1, R02<1, at this time R0>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 R01>1, R02>1, at this time R0>1, there is the endemic equilibrium E∗∗, 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 αa and αm by killing infected poultry or reduce βa, βm, and βh by closing farms and markets. Both of the two methods can reduce R0 or Ih, so as to control the occurrence and development of diseases.



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*Source: 7485310-2018-05-23.xml*