# Dynamics Analysis of an Avian Influenza A (H7N9) Epidemic Model with Vaccination and Seasonality

Authors: Juping Zhang; Yun Li; Zhen Jin; Huaiping Zhu
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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 Section2, 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 R0 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 R0<1 and at least a positive periodic solution when R0>1, 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 timet are denoted by Nb(t), Na(t), and Nh(t), respectively. The concentration of viruses in the environment is denoted by W(t), 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 Sh1(t),Sh2(t),Ih(t), and Rh(t), respectively. Sb(t) and Ib(t) denote the number of susceptible and infective individuals in the bird population. Sa(t),Va(t), and Ia(t) 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:(1)dSbtdt=Ab-βbSbIb-βwbtSbW-dbSb,dIbtdt=βbSbIb+βwbtSbW-dbIb-αbIb,dSatdt=Aa-βaSaIa-βwatSaW-daSa-paSa+ηaVa,dVatdt=paSa-σ1βaVaIa-σ2βwatVaW-daVa-ηaVa,dIatdt=βaSaIa+βwatSaW+σ1βaVaIa+σ2βwatVaW-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dSh1tdt=Ah-βh1Sh1Ia-βwh1tSh1W-dhSh1-ξhSh1,dSh2tdt=ξhSh1-βh2Sh2Ia-βwh2tSh2W-dhSh2,dIhtdt=βh1Sh1Ia+βwh1tSh1W+βh2Sh2Ia+βwh2tSh2W-dhIh-αhIh-γhIh,dRhtdt=γhIh-dhRh.The interpretations of the variables and parameters are shown in Table 1. All variables and parameters are nonnegative.Table 1
Description of parameters and variables.
Parameters Interpretation Value(week) Resource Ab The birth number of birds 1 ∗ 1 0 9 / 520 [15] db The natural mortality rate of birds 1/10/52 [13] αb The mortality rate of birds caused by H7N9 0.08 Assuming βb I b ( t )-to-Sb(t)transmission rate Parameter βwb(t) W ( t )-to-Sb(t)transmission rate Parameter βwb0  Parameter g1  0.0313 [15] g2  3.0624 [15] Aa The birth number of poultry 1.0438 ∗ 1 0 9 / 8 [13] da The natural mortality rate of poultry 1/8 [13] pa The vaccination rate of poultry [ 0,1 ] Assuming ηa The vaccine failure rate of poultry [ 0,1 ] Assuming σ1 Probability ofIa(t)-to-Va(t) 0.12 Assuming σ2 Probability ofW(t)-to-Va(t) 0.08 Assuming αa The mortality rate of poultry caused by H7N9 Parameter βa I a ( t )-to-Sa(t)transmission rate Parameter βwa(t) W ( t )-to-Sa(t)transmission rate Parameter βwa0  Parameter g3  0.0313 [15] g4  3.0624 [15] qb The discharging concentration of H7N9 virus byIb(t) 5 [15] qa The discharging concentration of H7N9 virus byIa(t) 5 [15] dw The natural mortality rate of virus 0.7 Assuming δw The effective disinfection rate Parameter A h The birth number of human 2.269741 ∗ 1 0 8 / 70 / 52 [13] dh The natural death rate of human 1/70/52 [13] ξh The proportion ofSh1(t)-to-Sh2(t) 0.0071 Assuming γh The recovery rate of human 2 [15] αh The mortality rate of human caused by H7N9 0.36 [13] βh1 I a ( t )-to-Sh1(t) transmission rate Parameter βwh1(t) W ( t )-to-Sh1(t) transmission rate Parameter βh10 The baseline transmission rate from poultry to ordinary humans Parameter βwh10  Parameter g5  0.0313 [15] g6  3.0624 [15] k1 Intervention intensity toSh1(t) Parameter t⋆ The initiated time for implementation of measures Parameter βh2 I a ( t )-to-Sh2(t) transmission rate Parameter βwh2(t) W ( t )-to-Sh2(t) transmission rate Parameter βh20 The baseline transmission rate from poultry to high-risk humans Parameter βwh20  Parameter g7  0.0313 [15] g8  3.0624 [15] k2 Intervention intensity toSh2(t) ParameterAccording to [15, 19, 20], the relationships between environmental transmission rates and temperature can be obtained(2)βwbt=fwbT=0,T>30∘C,βwb030-Te-g1T-g2,T≤30∘C.(3)βwat=fwaT=0,T>30∘C,βwa030-Te-g3T-g4,T≤30∘C.(4)βwh1t=fwh1T=0,T>30∘C,βwh1030-Te-g5T-g6,T≤30∘C, and(5)βwh2t=fwh2T=0,T>30∘C,βwh2030-Te-g7T-g8,T≤30∘C, where T=T0(1+ϕ1sin(ωt+ϕ2)) is the temperature, T0 is the average temperature, ω is a period, ϕ1 is the amplitude, and ϕ2 is the phase difference.We model the implementation of intervention strategies (initiated at timet⋆) by the following piecewise functions βh1 and βh2 [12]:(6)βh1t=βh10,t≤t⋆,βh10e-k1t-t⋆,t>t⋆,and(7)βh2t=βh20,t≤t⋆,βh20e-k2t-t⋆,t>t⋆.From system (1), these can find that(8)dNbdt=Ab-dbNb-αbIb≤Ab-dbNb.(9)dNadt=Aa-daNa-αaIa≤Aa-daNa.(10)dNhdt=Ah-daNh-αhIh≤Ah-dhNh.Then, from (8), it follows that(11)Nbt≤Abdb+Nb0-Abdbe-dbt,and e-dbt→0 as t→+∞, so limt→+∞Nb(t)≤Ab/db.In the same way, from (9) and (10), it can be obtained that limt→+∞Na(t)≤Aa/da, limt→+∞Nh(t)≤Ah/dh. The feasible region of system (1) is(12)Ω=Sb,Ib,Sa,Va,Ia,W,Sh1,Sh2,Ih,Rh∈R+10:Nb≤Abdb,Na≤Aada,Nh≤Ahdh,W≤qbAb/db+qaAa/dadw+δw.


## 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):(13)dSbtdt=Ab-βbSbIb-βwbtSbW-dbSb,dIbtdt=βbSbIb+βwbtSbW-dbIb-αbIb,dSatdt=Aa-βaSaIa-βwatSaW-daSa-paSa+ηaVa,dVatdt=paSa-σ1βaVaIa-σ2βwatVaW-daVa-ηaVa,dIatdt=βaSaIa+βwatSaW+σ1βaVaIa+σ2βwatVaW-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dSh1tdt=Ah-βh1Sh1Ia-βwh1tSh1W-dhSh1-ξhSh1,dSh2tdt=ξhSh1-βh2Sh2Ia-βwh2tSh2W-dhSh2,dIhtdt=βh1Sh1Ia+βwh1tSh1W+βh2Sh2Ia+βwh2tSh2W-dhIh-αhIh-γhIh. It is easy to see that system (13) always has the disease-free equilibrium E0=(Sb0,0,Sa0,Va0,0, 0,Sh10,Sh20,0), where (14)Sb0=Abdb,Sa0=Aada+ηadada+pa+ηa,Va0=paSa0da+ηa,Sh10=Ahdh+ξh,Sh20=ξhSh10dh.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 (15)Ft=βbSbIb+βwbtSbWβaSaIa+βwatSaW+σ1βaVaIa+σ2βwatVaW0βh1Sh1Ia+βwh1tSh1W+βh2Sh2Ia+βwh2tSh2W00000, and (16)Vt=αbIb+dbIbαaIa+daIaδwW+dwW-qbIb+qaIaαhIh+dhIh+γhIhβbSbIb+βwbtSbW+dbSb-AbβaSaIa+βwatSaW+daSa+paSa-ηaVa-Aaσ1βaVaIa+σ2βwatVaW+daVa+ηaVa-paSaβh1Sh1Ia+βwh1tSh1W+dhSh1+ξhSh1-Ahβh2Sh2Ia+βwh2tSh2W+dhSh2-ξhSh1. Then (17)FE0t=βbSb00βwbtSb000βaSa0+σ1βaVa0βwatSa0+σ2βwatVa0000000βh1Sh10+βh2Sh20βwh1tSh10+βwh2tSh200, and (18)VE0=αb+db0000da+αa00-qb-qaδw+dw0000dh+αh+γh.AssumeY(t,s),t≥s is the evolution operator of the linear ω-periodic system(19)dydt=-VE0y.That is, for each s∈R, the 4×4 matrix Y(t,s) satisfies(20)dYt,sdt=-VE0Yt,s,∀t≥s,Ys,s=I,where I is the 4×4 identity matrix. Thus, the monodromy matrix Φ-V(E0) of (19) is equal to Y(t,0),t≥0.In view of the periodic environment, we assume thatφ(s), ω-periodic in s, is the initial distribution of infectious individuals. Then F(E0)(s)φ(s) is the rate of new infections produced by the infected individuals who were introduced at time s. Given t≥s, then Y(t,s)F(E0)(s)φ(s) gives the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t.LetCω be the ordered Banach space of all ω-periodic functions from R to R4, which is equipped with the maximum norm • and the positive cone Cω+={φ∈Cω:φ(t)≥0,∀t∈R}. Then we can define a linear operator L: Cω→Cω by (Lφ)(t)=∫0+∞Y(t,t-ζ)F(E0)(t-ζ)φ(t-ζ)dζ. From Wang and Zhao [21], we call L the next infection operator and define the basic reproduction number as R0=ρ(L), where ρ is the spectral radius of L.Theorem 1.
For system (13), if R0<1, the disease-free equilibrium E0 is locally asymptotically stable; if R0>1, 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 R0<1, the disease-free equilibrium E0 is globally asymptotically stable.Proof.
Let(Sb,Ib,Sa,Va,Ia,W,Sh1,Sh2,Ih) 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 E0 as t→+∞.
The first, third, fourth, seventh, and eighth equations of system (13) with Va≤Na-Sa give the respective inequalities(21)dSbtdt≤Ab-dbSb,dSatdt≤Aa+ηaNa-daSa-paSa-ηaSa,dVatdt≤paNa-paVa-daVa-ηaVa,dSh1tdt≤Ah-dhSh1-ξhSh1,dSh2tdt≤ξhSh1-dhSh2.Hence, for any ε>0, it exists tε>0; when t≥tε, we have(22)Sb≤Sb0+ε,Sa≤Sa0+ε,Va≤Va0+ε,Sh1≤Sh10+ε,Sh2≤Sh20+ε.The second, fifth, sixth, and ninth equations of system (13), with Sb≤Sb0+ε, Sa≤Sa0+ε, Va≤Va0+ε, Sh1≤Sh10+ε, and Sh2≤Sh20+ε, give the respective inequalities(23)dIbtdt≤βbSb0+εIb+βwbtSb0+εW-dbIb-αbIb,dIatdt≤βaIa+βwatWSa0+ε+σ1βaIa+σ2βwatWVa0+ε-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dIhtdt≤βh1Ia+βwh1tWSh10+ε+βh2Ia+βwh2tWSh20+ε-dhIh-αhIh-γhIh.
We consider the auxiliary system of system (23), where the coefficient matrix is (24)Mε=βbε0βwbtε00βaε+σ1βaεβwatε+σ2βwatε000000βh1ε+βh2εβwh1tε+βwh2tε0.Clearly, if R0<1, it is known from Theorem 2.2 in [21] that ρ(ΦF-V(ω))<1. We can choose ε>0 small enough giving ρ(ΦF-V+Mε(ω))<1. It can be concluded from Lemma 2.1 in Zhang and Zhao [22] that there exists a positive, ω-periodic function f¯(t)=(Ib¯(t),Ia¯(t),W¯(t),Ih¯(t)) such that f^(t)=eΘtf¯(t) is a solution of the auxiliary system, where Θ=(1/ω)ln⁡(ρ(ΦF-V+Mε(ω))). Here, ρ(ΦF-V+Mε(ω)<1⇒Θ<0, which implies f^(t)→0 as t→+∞. Therefore, the zero solution of the auxiliary system is globally asymptotically stable. For any nonnegative initial value, there is a sufficiently large M. Applying the comparison theorem [23], it can be obtained that f(t)≤Mf^(t),∀t>0. Therefore, we obtain Ib(t)→0,Ia(t)→0,W(t)→0,Ih(t)→0 as t→+∞. By the theory of asymptotic autonomous systems [24], we get Sb(t)→Sb0,Ib(t)→0,Sa(t)→Sa0,Va(t)→Va0,Ia(t)→0,W(t)→0,Sh1(t)→Sh10,Sh2(t)→Sh20,Ih(t)→0 as t→+∞. Hence, if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.


### 4.2. The Existence of Positive Periodic Solutions

Define(25)X≔Ω,X0≔Sb,Ib,Sa,Va,Ia,W,Sh1,Sh2,Ih∈X:Ibt>0,Iat>0,Wt>0,Iht>0,∂X0≔X\X0.Let P:X→X be the Poincaré map associated with system (13); that is, P(z0)=u(ω,z0), ∀z0∈X, where ω is the period. u(t,z0) is the unique solution of system (13) with u(0,z0)=z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0)). It is easy to see that Pn(z0)=u(nω,z0),∀n≥0.Lemma 3.
For system (13), if R0>1, then there exists a μ>0 such that, for any z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 with z0-E0≤μ, we have limn→+∞supd[Pn(z0),E0]≥μ.Proof.
SinceR0>1, Theorem 1 implies that E0 is unstable; i.e., ρ(ΦF-V(ω))>1. We can choose ε1>0 small enough such that ρ(ΦF-V-Mε1(ω))>1, where (26)Mε1=βbε10βwbtε100βaε1+σ1βaε1βwatε1+σ2βwatε1000000βh1ε1+βh2ε1βwh1tε1+βwh2tε10.
To the contrary, if possible suppose that the limitlimn→+∞supd[Pn(z0),E0]<μ for some z0∈X0. Without loss of generality, we assume that d[Pn(z0),E0]<μ,∀n≥0. By the continuity of the solution with respect to the initial value, it follows that(27)ut,Pnz0-ut,E0<ε1,∀n≥0,∀t∈0,ω.
From the periodicity of the system, forε1>0, there exists tε1 such that, for all t>tε1, there holds(28)Sb≥Sb0-ε1,Sa≥Sa0-ε1,Va≥Va0-ε1,Sh1≥Sh10-ε1,Sh2≥Sh20-ε1.Then(29)dIbtdt≥βbSb0-ε1Ib+βwbtSb0-ε1W-dbIb-αbIb,dIatdt≥βaIa+βwatWSa0-ε1+σ1βaIa+σ2βwatW)Va0-ε1-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dIhtdt≥βh1Ia+βwh1tWSh10-ε1+βh2Ia+βwh2tWSh20-ε1-dhIh-αhIh-γhIh.
Now, consider the auxiliary system of (29); it can be concluded from Lemma 2.1 in Zhang and Zhao [22] that there exists a positive, ω-periodic function f¯(t)=(Ib¯(t),Ia¯(t),W¯(t),Ih¯(t)) such that f^(t)=eΘ1tf¯(t) is a solution of the auxiliary system, where Θ1=1/ωln⁡(ρ(ΦF-V-Mε1(ω))). Here, ρ(ΦF-V-Mε1(ω))>1⇒Θ1>0, which implies that, for nonnegative integer n, f^(nω)→+∞ as n→+∞. For any nonnegative initial value, there is a sufficiently small m>0. Applying the comparison theorem [23], it can be obtained that f(t)≥mf^(t),∀t>0. Thus, we obtain Ib(t)→+∞,Ia(t)→+∞,W(t)→+∞,Ih(t)→+∞ as t→+∞, which is a contradiction. This completes the proof.Lemma 4.
The following equation is established:(30)M∂≔z0∈∂X0:Pnz0∈∂X0,∀n≥0=Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0∈X:Sb0≥0,Sa0≥0,Va0≥0,Sh10≥0,Sh20≥0.Proof.
It is easy to know that(31)Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0⊆z0∈∂X0:Pnz0∈∂X0,∀n≥0.Proof by contradiction can be used to prove(32)Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0⊇z0∈∂X0:Pnz0∈∂X0,∀n≥0.If possible suppose that (33)z0=Sb0,Ib0,Sa0,Va0,Ia0,W0,Sh10,Sh20,Ih0∈z0∈∂X0:Pnz0∈∂X0,∀n≥0\Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0.Without loss of generality, we assume that Ib(nω)>0. By solving the general solution of system (13), we can derive Ib(t)>0,Ia(t)>0,W(t)>0,Ih(t)>0. Meaning, (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t))∉∂X0, which contradicts with (Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈∂X0. Therefore, the equation is established.Theorem 5.
IfR0>1, then there exists ε∗>0 such that any solution (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t)) of system (13) with initial value z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 satisfies(34)limt→+∞infIbt≥ε∗,limt→+∞infIat≥ε∗,limt→+∞infWt≥ε∗,limt→+∞infIht≥ε∗,and system (13) admits at least one positive periodic solution.Proof.
It is now proved that{Pn}n≥0 is uniformly persistent with respect to (X0,∂X0). For any z0∈X0, from the first equation of system (13), it follows that(35)Sbt=e-∫0tβbIbs~+βwbs~Ws~+dbds~Sb0+Ab∫0te∫0s1~βbIbs~+βwbs~Ws~+dbds~ds1~.Then, Sb(t)>0,∀t>0. Similarly, Sa(t)>0,Va(t)>0,Sh1(t)>0,Sh2(t)>0,∀t>0. As generalized to nonautonomous systems [25], the irreducibility of the cooperative matrix M~(t) implies that Ib(t)>0,Ia(t)>0,W(t)>0,Ih(t)>0,∀t>0, where (36)M~t=βbSb-db-αb0βwbtSb00βaSa+σ1βaVaβwatSa+σ2βwatVa0qbqa-dw-δw00βh1Sh1+βh2Sh2βwh1tSh1+βwh2tSh2-dh-αh-γh. Thus, both X and X0 are positively invariant. Clearly, ∂X0 is relatively closed in X.
The disease-free equilibriumE0 of system (13) is globally asymptotically stable. Lemmas 3 and 4 imply that E0 is a unique fixed point of P in M∂. Moreover, E0 is an isolated invariant set in X, and Ws(E0)⋂X0=ϕ. Note that every orbit in M∂ approaches E0 and E0 is acyclic in M∂. By [26], it follows that {Pn}n≥0 is uniformly persistent with respect to (X0,∂X0) and the solutions of system (13) are uniformly persistent with respect to (X0,∂X0); i.e., if R0>1, there exists ε∗>0 such that any solution (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t)) of system (13) with initial value z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 satisfies(37)limt→+∞infIbt≥ε∗,limt→+∞infIat≥ε∗,limt→+∞infWt≥ε∗,limt→+∞infIht≥ε∗.
Furthermore,P has a fixed point (Sb∗(0),Ib∗(0),Sa∗(0),Va∗(0), Ia∗(0),W∗(0),Sh1∗(0),Sh2∗(0),Ih∗(0))∈X0. Further, there exists some t¯∈[0,ω] such that Sb∗(t¯)>0. If it is not the case, Sb∗(t¯)≡0. Then, due to the periodicity of Sb∗(t), we have Sb∗(t)≡0, for all t≥0. From the first equation of system (13), we get 0=Ab>0, which is a contradiction. Thus ∀t∈[t¯,t¯+ω], we obtain(38)Sb∗t=e-∫t¯tβbIb∗s~+βwbs~W∗s~+dbds~Sb∗t¯+Ab∫t¯te∫t¯s1~βbIb∗s~+βwbs~W∗s~+dbds~ds1~>0.
The periodicity ofSb∗(t) implies that Sb∗(t)>0, ∀t≥0. Similarly, Ib∗(t)>0,Sa∗(t)>0,Va∗(t)>0,Ia∗(t)>0,W∗(t)>0,Sh1∗(t)>0,Sh2∗(t)>0,Ih∗(t)>0. Therefore, (Sb∗(t),Ib∗(t),Sa∗(t),Va∗(t),Ia∗(t),W∗(t),Sh1∗(t),Sh2∗(t),Ih∗(t)) is a positive ω-periodic solution of system (13).


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

Theorem 2.
For system (13), if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.Proof.
Let(Sb,Ib,Sa,Va,Ia,W,Sh1,Sh2,Ih) 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 E0 as t→+∞.
The first, third, fourth, seventh, and eighth equations of system (13) with Va≤Na-Sa give the respective inequalities(21)dSbtdt≤Ab-dbSb,dSatdt≤Aa+ηaNa-daSa-paSa-ηaSa,dVatdt≤paNa-paVa-daVa-ηaVa,dSh1tdt≤Ah-dhSh1-ξhSh1,dSh2tdt≤ξhSh1-dhSh2.Hence, for any ε>0, it exists tε>0; when t≥tε, we have(22)Sb≤Sb0+ε,Sa≤Sa0+ε,Va≤Va0+ε,Sh1≤Sh10+ε,Sh2≤Sh20+ε.The second, fifth, sixth, and ninth equations of system (13), with Sb≤Sb0+ε, Sa≤Sa0+ε, Va≤Va0+ε, Sh1≤Sh10+ε, and Sh2≤Sh20+ε, give the respective inequalities(23)dIbtdt≤βbSb0+εIb+βwbtSb0+εW-dbIb-αbIb,dIatdt≤βaIa+βwatWSa0+ε+σ1βaIa+σ2βwatWVa0+ε-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dIhtdt≤βh1Ia+βwh1tWSh10+ε+βh2Ia+βwh2tWSh20+ε-dhIh-αhIh-γhIh.
We consider the auxiliary system of system (23), where the coefficient matrix is (24)Mε=βbε0βwbtε00βaε+σ1βaεβwatε+σ2βwatε000000βh1ε+βh2εβwh1tε+βwh2tε0.Clearly, if R0<1, it is known from Theorem 2.2 in [21] that ρ(ΦF-V(ω))<1. We can choose ε>0 small enough giving ρ(ΦF-V+Mε(ω))<1. It can be concluded from Lemma 2.1 in Zhang and Zhao [22] that there exists a positive, ω-periodic function f¯(t)=(Ib¯(t),Ia¯(t),W¯(t),Ih¯(t)) such that f^(t)=eΘtf¯(t) is a solution of the auxiliary system, where Θ=(1/ω)ln⁡(ρ(ΦF-V+Mε(ω))). Here, ρ(ΦF-V+Mε(ω)<1⇒Θ<0, which implies f^(t)→0 as t→+∞. Therefore, the zero solution of the auxiliary system is globally asymptotically stable. For any nonnegative initial value, there is a sufficiently large M. Applying the comparison theorem [23], it can be obtained that f(t)≤Mf^(t),∀t>0. Therefore, we obtain Ib(t)→0,Ia(t)→0,W(t)→0,Ih(t)→0 as t→+∞. By the theory of asymptotic autonomous systems [24], we get Sb(t)→Sb0,Ib(t)→0,Sa(t)→Sa0,Va(t)→Va0,Ia(t)→0,W(t)→0,Sh1(t)→Sh10,Sh2(t)→Sh20,Ih(t)→0 as t→+∞. Hence, if R0<1, the disease-free equilibrium E0 is globally asymptotically stable.


## 4.2. The Existence of Positive Periodic Solutions

Define(25)X≔Ω,X0≔Sb,Ib,Sa,Va,Ia,W,Sh1,Sh2,Ih∈X:Ibt>0,Iat>0,Wt>0,Iht>0,∂X0≔X\X0.Let P:X→X be the Poincaré map associated with system (13); that is, P(z0)=u(ω,z0), ∀z0∈X, where ω is the period. u(t,z0) is the unique solution of system (13) with u(0,z0)=z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0)). It is easy to see that Pn(z0)=u(nω,z0),∀n≥0.Lemma 3.
For system (13), if R0>1, then there exists a μ>0 such that, for any z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 with z0-E0≤μ, we have limn→+∞supd[Pn(z0),E0]≥μ.Proof.
SinceR0>1, Theorem 1 implies that E0 is unstable; i.e., ρ(ΦF-V(ω))>1. We can choose ε1>0 small enough such that ρ(ΦF-V-Mε1(ω))>1, where (26)Mε1=βbε10βwbtε100βaε1+σ1βaε1βwatε1+σ2βwatε1000000βh1ε1+βh2ε1βwh1tε1+βwh2tε10.
To the contrary, if possible suppose that the limitlimn→+∞supd[Pn(z0),E0]<μ for some z0∈X0. Without loss of generality, we assume that d[Pn(z0),E0]<μ,∀n≥0. By the continuity of the solution with respect to the initial value, it follows that(27)ut,Pnz0-ut,E0<ε1,∀n≥0,∀t∈0,ω.
From the periodicity of the system, forε1>0, there exists tε1 such that, for all t>tε1, there holds(28)Sb≥Sb0-ε1,Sa≥Sa0-ε1,Va≥Va0-ε1,Sh1≥Sh10-ε1,Sh2≥Sh20-ε1.Then(29)dIbtdt≥βbSb0-ε1Ib+βwbtSb0-ε1W-dbIb-αbIb,dIatdt≥βaIa+βwatWSa0-ε1+σ1βaIa+σ2βwatW)Va0-ε1-daIa-αaIa,dWtdt=qbIb+qaIa-dwW-δwW,dIhtdt≥βh1Ia+βwh1tWSh10-ε1+βh2Ia+βwh2tWSh20-ε1-dhIh-αhIh-γhIh.
Now, consider the auxiliary system of (29); it can be concluded from Lemma 2.1 in Zhang and Zhao [22] that there exists a positive, ω-periodic function f¯(t)=(Ib¯(t),Ia¯(t),W¯(t),Ih¯(t)) such that f^(t)=eΘ1tf¯(t) is a solution of the auxiliary system, where Θ1=1/ωln⁡(ρ(ΦF-V-Mε1(ω))). Here, ρ(ΦF-V-Mε1(ω))>1⇒Θ1>0, which implies that, for nonnegative integer n, f^(nω)→+∞ as n→+∞. For any nonnegative initial value, there is a sufficiently small m>0. Applying the comparison theorem [23], it can be obtained that f(t)≥mf^(t),∀t>0. Thus, we obtain Ib(t)→+∞,Ia(t)→+∞,W(t)→+∞,Ih(t)→+∞ as t→+∞, which is a contradiction. This completes the proof.Lemma 4.
The following equation is established:(30)M∂≔z0∈∂X0:Pnz0∈∂X0,∀n≥0=Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0∈X:Sb0≥0,Sa0≥0,Va0≥0,Sh10≥0,Sh20≥0.Proof.
It is easy to know that(31)Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0⊆z0∈∂X0:Pnz0∈∂X0,∀n≥0.Proof by contradiction can be used to prove(32)Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0⊇z0∈∂X0:Pnz0∈∂X0,∀n≥0.If possible suppose that (33)z0=Sb0,Ib0,Sa0,Va0,Ia0,W0,Sh10,Sh20,Ih0∈z0∈∂X0:Pnz0∈∂X0,∀n≥0\Sb0,0,Sa0,Va0,0,0,Sh10,Sh20,0.Without loss of generality, we assume that Ib(nω)>0. By solving the general solution of system (13), we can derive Ib(t)>0,Ia(t)>0,W(t)>0,Ih(t)>0. Meaning, (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t))∉∂X0, which contradicts with (Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈∂X0. Therefore, the equation is established.Theorem 5.
IfR0>1, then there exists ε∗>0 such that any solution (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t)) of system (13) with initial value z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 satisfies(34)limt→+∞infIbt≥ε∗,limt→+∞infIat≥ε∗,limt→+∞infWt≥ε∗,limt→+∞infIht≥ε∗,and system (13) admits at least one positive periodic solution.Proof.
It is now proved that{Pn}n≥0 is uniformly persistent with respect to (X0,∂X0). For any z0∈X0, from the first equation of system (13), it follows that(35)Sbt=e-∫0tβbIbs~+βwbs~Ws~+dbds~Sb0+Ab∫0te∫0s1~βbIbs~+βwbs~Ws~+dbds~ds1~.Then, Sb(t)>0,∀t>0. Similarly, Sa(t)>0,Va(t)>0,Sh1(t)>0,Sh2(t)>0,∀t>0. As generalized to nonautonomous systems [25], the irreducibility of the cooperative matrix M~(t) implies that Ib(t)>0,Ia(t)>0,W(t)>0,Ih(t)>0,∀t>0, where (36)M~t=βbSb-db-αb0βwbtSb00βaSa+σ1βaVaβwatSa+σ2βwatVa0qbqa-dw-δw00βh1Sh1+βh2Sh2βwh1tSh1+βwh2tSh2-dh-αh-γh. Thus, both X and X0 are positively invariant. Clearly, ∂X0 is relatively closed in X.
The disease-free equilibriumE0 of system (13) is globally asymptotically stable. Lemmas 3 and 4 imply that E0 is a unique fixed point of P in M∂. Moreover, E0 is an isolated invariant set in X, and Ws(E0)⋂X0=ϕ. Note that every orbit in M∂ approaches E0 and E0 is acyclic in M∂. By [26], it follows that {Pn}n≥0 is uniformly persistent with respect to (X0,∂X0) and the solutions of system (13) are uniformly persistent with respect to (X0,∂X0); i.e., if R0>1, there exists ε∗>0 such that any solution (Sb(t),Ib(t),Sa(t),Va(t),Ia(t),W(t),Sh1(t),Sh2(t),Ih(t)) of system (13) with initial value z0=(Sb(0),Ib(0),Sa(0),Va(0),Ia(0),W(0),Sh1(0),Sh2(0),Ih(0))∈X0 satisfies(37)limt→+∞infIbt≥ε∗,limt→+∞infIat≥ε∗,limt→+∞infWt≥ε∗,limt→+∞infIht≥ε∗.
Furthermore,P has a fixed point (Sb∗(0),Ib∗(0),Sa∗(0),Va∗(0), Ia∗(0),W∗(0),Sh1∗(0),Sh2∗(0),Ih∗(0))∈X0. Further, there exists some t¯∈[0,ω] such that Sb∗(t¯)>0. If it is not the case, Sb∗(t¯)≡0. Then, due to the periodicity of Sb∗(t), we have Sb∗(t)≡0, for all t≥0. From the first equation of system (13), we get 0=Ab>0, which is a contradiction. Thus ∀t∈[t¯,t¯+ω], we obtain(38)Sb∗t=e-∫t¯tβbIb∗s~+βwbs~W∗s~+dbds~Sb∗t¯+Ab∫t¯te∫t¯s1~βbIb∗s~+βwbs~W∗s~+dbds~ds1~>0.
The periodicity ofSb∗(t) implies that Sb∗(t)>0, ∀t≥0. Similarly, Ib∗(t)>0,Sa∗(t)>0,Va∗(t)>0,Ia∗(t)>0,W∗(t)>0,Sh1∗(t)>0,Sh2∗(t)>0,Ih∗(t)>0. Therefore, (Sb∗(t),Ib∗(t),Sa∗(t),Va∗(t),Ia∗(t),W∗(t),Sh1∗(t),Sh2∗(t),Ih∗(t)) is a positive ω-periodic solution of system (13).


## 5. Numerical Simulations

In this section, based on the initial value of(39)z01=999999900,100,1043798853,1043,104,100,226960414,13618,11,57,z02=999999500,500,1043788518,10438,1044,300,226942176,31776,57,91,z03=999999000,1000,1043765555,31314,3131,500,226451720,522040,113,227,the temperature of the Yangtze River Delta, i.e., T=20.8286[1-0.6273sin((π/26)t+19.9451)], and other parameter values from Table 1 [12, 13, 15, 19, 20] are taken as some examples to simulate the stability of the disease-free equilibrium and the existence of positive periodic solutions of system (1), and the time-series diagram is given. At last, when parameters take different values, the time-variation diagrams of Ih are given.Example 1.
Take parametersβb=85/100∗1.09546∗10-11, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-10, βwa0=85/100∗1.3579∗10-10, αa=0.18,δw=0.8∗7, βh10=85/100∗9.0876∗10-16, βh20=60/100∗1.5954∗10-15, βwh10=85/100∗9.0876∗10-16, βwh20=60/100∗1.5954∗10-15, k1=0.35, k2=0.48, t⋆=7. Figure 2 shows the time-variation diagram of system (1) state variables. It is found that if R0<1, the curve tends to be stable with time, and the disease-free equilibrium E0 is globally asymptotically stable.Figure 2
IfR0=0.8575<1, the time-variation diagram of system (1) state variables.Example 2.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, αa=0.18, δw=0.018∗7, βh10=85/100∗9.0876∗10-16, βh20=60/100∗1.5954∗10-15, βwh10=85/100∗9.0876∗10-16, βwh20=60/100∗1.5954∗10-15, k1=0.35, k2=0.48, t⋆=7. As shown in Figure 3, it is found that if R0>1, the state variables of system (1) change periodically with time.Figure 3
IfR0=2.92>1, the time-variation diagram of system (1) state variables.Example 3.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, αa=0.18, δw=0.018∗7, βh10=85/100∗9.0876∗10-16, βh20=60/100∗1.5954∗10-15, βwh10=85/100∗9.0876∗10-16, βwh20=60/100∗1.5954∗10-15, k1=0.35, k2=0.48, t⋆=7 and let T=25∘C,15∘C,5∘C. Figure 4 shows the curve-trend diagram of Ih with time. It is found that the lower the temperature, the larger Ih.Figure 4
The curve-trend diagram ofIh with time, when T takes different values.Example 4.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, αa=0.18, δw=0.8∗7, βh20=60/100∗1.5954∗10-15, βwh20=60/100∗1.5954∗10-15, k2=0.48, t⋆=7 and let k1=0.35, βh10=85/100∗3.0904∗10-14, βwh10=85/100∗9.0876∗10-15; βh10=85/100∗3.0904∗10-15, βwh10=85/100∗9.0876∗10-15; βh10=85/100∗3.0904∗10-14, βwh10=85/100∗9.0876∗10-16 or βh10=85/100∗3.0904∗10-16, βwh10=85/100∗9.0876∗10-16, k1=0.007,0.085,0.35. Figure 5 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of βh1 or βwh1 over a period of time.Figure 5
The curve-trend diagram ofIh with time, when βh1 or βwh1 takes different values.Example 5.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, αa=0.18, δw=0.8∗7, βh10=85/100∗3.0904∗10-16, βwh10=85/100∗9.0876∗10-16,k1=0.35, t⋆=7 and let k2=0.48, βh20=85/100∗3.0904∗10-12, βwh20=85/100∗9.0876∗10-16; βh20=85/100∗3.0904∗10-12, βwh20=85/100∗9.0876∗10-15; βh20=85/100∗3.0904∗10-15, βwh20=85/100∗9.0876∗10-15 or βh20=85/100∗3.0904∗10-15, βwh20=85/100∗9.0876∗10-15, k2=0.008,0.09,0.48. Figure 6 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of βh2 or βwh2 over a period of time.Figure 6
The curve-trend diagram ofIh with time, when βh2 or βwh2 takes different values.Example 6.
Take parameterspa=0.87, ηa=0.018∗7, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, αa=0.18, δw=0.8∗7, βh10=85/100∗3.0904∗10-16, βwh10=85/100∗9.0876∗10-16, k1=0.15, k2=0.28, t⋆=7, βh20=85/100∗3.0904∗10-15, βwh20=85/100∗9.0876∗10-15 and let βb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10; βb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-15; βb=85/100∗1.09546∗10-8, βwb0=85/100∗1.09546∗10-10. Figure 7 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of βb or βwb over a period of time.Figure 7
The curve-trend diagram ofIh with time, when βb or βwb takes different values.Example 7.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, pa=0.87, ηa=0.018∗7, αa=0.18, δw=0.8∗7, βh10=85/100∗3.0904∗10-16, βwh10=85/100∗9.0876∗10-16, k1=0.35, k2=0.48, t⋆=7, βh20=85/100∗3.0904∗10-15, βwh20=85/100∗9.0876∗10-15 and let βa=85/100∗3.0904∗10-5, βwa0=85/100∗1.3579∗10-10; βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10; βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-9. Figure 8 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of βa or βwa.Figure 8
The curve-trend diagram ofIh with time, when βa or βwa takes different values.Example 8.
Take parametersβb=85/100∗1.09546∗10-10, βwb0=85/100∗1.09546∗10-10, βa=85/100∗3.0904∗10-9, βwa0=85/100∗1.3579∗10-10, βh10=85/100∗3.0904∗10-16, βwh10=85/100∗9.0876∗10-16, k1=0.35, k2=0.48, t⋆=7,βh20=85/100∗3.0904∗10-15, βwh20=85/100∗9.0876∗10-15 and let pa, ηa, αa, and δw change each other. Figure 9 shows the curve-trend diagram of Ih with time. It is found that Ih will increase with the increase of ηa or the decrease of pa,αa,δw.Figure 9
The curve-trend diagram ofIh with time, when parameters take different values.


## 6. Discussion

Avian influenza is the potential threat to human health. The exposure of infected poultry is a key factor for human infection with avian influenza A (H7N9) virus. Avian influenza A (H7N9) virus mainly spreads from poultry to human through infected poultry and its secretions, excreta, and virus-contaminated water. Most humans contact with poultry mainly in farms, live-poultry markets, wet markets, and other areas. In this paper, aSI-SVI-W-SSIR dynamic model of avian influenza A (H7N9) is established by combining birds, poultry and human. We get the basic reproduction number R0; 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 R0>1, there is at least one positive periodic solution, that is, the disease will spread. Our numerical simulation suggests that raising αa by killing infected poultry, reducing βa, βh1, and βh2 by closing farms, live-poultry markets, and wet markets, reducing βwb, βwa, βwh1, and βwh2 by adjusting the temperature of the living environment, or increasing δw and pa by routine disinfection in areas prone to outbreak and vaccination of poultry can reduce Ih, so as to control the occurrence and development of diseases. These can provide some measures of the intensity of interventions for public-health management on avian influenza A (H7N9) during high-risk period. In addition, our simulation shows that the temperature of the environment has a great effect on the prevalence of epidemic. Therefore, at suitable temperature for disease outbreak, human should not be in contact with birds and live poultry as much as possible.



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*Source: 4161287-2019-03-17.xml*