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Research article

Modeling and optimization of the oyster mushroom growth using artificial neural network: Economic and environmental impacts


  • Received: 22 March 2022 Revised: 24 May 2022 Accepted: 27 May 2022 Published: 06 July 2022
  • The main aim of the study is to investigate the growth of oyster mushrooms in two substrates, namely straw and wheat straw. In the following, the study moves towards modeling and optimization of the production yield by considering the energy consumption, water consumption, total income and environmental impacts as the dependent variables. Accordingly, life cycle assessment (LCA) platform was developed for achieving the environmental impacts of the studied scenarios. The next step developed an ANN-based model for the prediction of dependent variables. Finally, optimization was performed using response surface methodology (RSM) by fitting quadratic equations for generating the required factors. According to the results, the optimum condition for the production of OM from waste paper can be found in the paper portion range of 20% and the wheat straw range of 80% with a production yield of about 4.5 kg and a higher net income of 16.54 $ in the presence of the lower energy and water consumption by about 361.5 kWh and 29.53 kg, respectively. The optimum condition delivers lower environmental impacts on Human Health, Ecosystem Quality, Climate change, and Resources by about 5.64 DALY, 8.18 PDF*m2*yr, 89.77 g CO2 eq and 1707.05 kJ, respectively. It can be concluded that, sustainable production of OM can be achieved in line with the policy used to produce alternative food source from waste management techniques.

    Citation: Tarahom Mesri Gundoshmian, Sina Ardabili, Mako Csaba, Amir Mosavi. Modeling and optimization of the oyster mushroom growth using artificial neural network: Economic and environmental impacts[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 9749-9768. doi: 10.3934/mbe.2022453

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  • The main aim of the study is to investigate the growth of oyster mushrooms in two substrates, namely straw and wheat straw. In the following, the study moves towards modeling and optimization of the production yield by considering the energy consumption, water consumption, total income and environmental impacts as the dependent variables. Accordingly, life cycle assessment (LCA) platform was developed for achieving the environmental impacts of the studied scenarios. The next step developed an ANN-based model for the prediction of dependent variables. Finally, optimization was performed using response surface methodology (RSM) by fitting quadratic equations for generating the required factors. According to the results, the optimum condition for the production of OM from waste paper can be found in the paper portion range of 20% and the wheat straw range of 80% with a production yield of about 4.5 kg and a higher net income of 16.54 $ in the presence of the lower energy and water consumption by about 361.5 kWh and 29.53 kg, respectively. The optimum condition delivers lower environmental impacts on Human Health, Ecosystem Quality, Climate change, and Resources by about 5.64 DALY, 8.18 PDF*m2*yr, 89.77 g CO2 eq and 1707.05 kJ, respectively. It can be concluded that, sustainable production of OM can be achieved in line with the policy used to produce alternative food source from waste management techniques.



    Since the pioneering work of Kermack and McKendrick [1], many mathematical models have been proposed attempting to gain a better understanding of disease transmission, especially for the control strategy and dynamical behavior of infectious diseases [2,3,4,5,6,7,8]. Simple models with assumption that individuals are well mixed, which implies each individual has the same probability to be infected, are beneficial in that one can obtain analytical results easily but may be lack of realism. Epidemic models with population structures, like age, sex and patch (such as communities, cities, or counties), may be a more realistic way to describe complex disease dynamics. As a matter of fact, the total population should be classified into different groups and the vital epidemic parameters should vary among different population groups. In addition, at different age stages, the effects of infectious transmission are various, which is another important and key factor that needs necessarily to be included in modeling this infectious transmission process. Thus, considering multi-group and age structure in epidemic models is very necessary and reasonable. Some recent developments on the transmission dynamics of multi-group and age structured epidemic models have been discussed in [8,9,10,11,12,13].

    Since that the population distribute heterogeneously in different spatial location in the real life and they will move or diffuse for many reasons, in epidemiology, there is increasing evidence that environmental heterogeneity and individual motility have significant impact on the spread of infectious diseases [14,15]. In recent years, global behavior of spatial diffusion systems, which are suitable for diseases such as the rabies and the Black Death, has been attracted extensive attention of researchers and has been one of the hot topics [16,17,18,19,20,21,22,23,24,25]. Among these works, few take age structure or multi-group into consideration. Yang et al. [24] proposed a novel model incorporated with both age-since-infection and spacial diffusion of brucellosis infection, and the basic reproduction number and global behaviors of this system were completely investigated. Fitzgibbon et al. [19] considered a diffusive epidemic model with age structure where the disease spreads between vector and host populations. Then, the existence of solutions of the model was studied based on semigroup theory and the asymptotic behavior of the solution was analyzed. Luo et al. [21] incorporated spatial heterogeneity in n-group reaction-diffusion SIR model with nonlinear incidence rate to investigate the global dynamics of the disease-free and endemic steady states for this model. Zhao et al. [25] modeled host heterogeneity by introducing multi-group structure in a time delay SIR epidemic model and showed that basic reproduction number determines the existence of traveling waves of this system. To determine how age structure, multi-group population and diffusion of individuals affect the consequences of epidemiological processes, Ducrot et al. [18] formulated a multi-group age-structured epidemic model with the classical Fickian diffusion and studied the existence of travelling wave solutions for this model.

    To the best of our knowledge, epidemic models established by researchers except for [18] only include one or two characteristics of multi-group, age structure and spatial diffusion. All the three characteristics are incorporated into epidemic model in [18], however, this model does not include the class of latent individuals. For some epidemic diseases like malaria, HIV/AIDS and West Nile virus, latent individuals may take days, months, or even years to become infectious. Moreover, the travel of latent individuals showing no symptoms can spread the disease geographically which makes disease harder to control. Motivated by the above discussion, in this paper, we investigate a diffusive version of multi-group epidemic system with age structure which is generalization of the model studied in [26] for the first time to allow for individuals moving around on the spatial habitat xΩRn with smooth boundary Ω. The organization of this paper is as follows. Firstly, we present our model in the next section. In section 3, some preliminaries including the positivity, boundedness, existence and uniqueness of solution, and the existence of compact global attractor of the associated solution semiflow, are established. In section 4, the sufficient conditions on the existence and global stability of disease-free and endemic steady states are stated and proved. In section 5, we conduct numerical simulations to illustrate the validity of our theoretical results. In section 6, a brief conclusion is given.

    In 2015, Liu et al. [26] introduced age-of-latent and age-of-relapse into epidemic model which is appropriate for diseases such as tuberculosis and herpes virus infection. For these diseases, latent individuals may take days, months, or even years to become infectious and the treatment efficacy may decline with time for recovered individuals then cause recurrence of disease. In order to study the global dynamics o for these diseases, they formulated the following SEIR epidemic system with continuous age-dependent latency and relapse

    {dS(t)dt=ΛμS(t)βS(t)I(t),e(t,a)t+e(t,a)a=σ(a)e(t,a)(μ+δ1)e(t,a),dI(t)dt=0σ(a)e(t,a)da(μ+δ2+c)I(t)+0γ(b)r(t,b)db,r(t,b)t+r(t,b)b=γ(b)r(t,b)μr(t,b),e(t,0)=βS(t)I(t),r(t,0)=cI(t), (2.1)

    for t0 and with initial conditions

    {S(0)=S0>0,e(0,a)=e0(a)L1+(0,+),I(0)=I0>0,r(0,b)=r0(b)L1+(0,+),

    for a,b0, where L1+(0,+) is the space of functions on (0,+) that are nonnegative and Lebesgue integrable. At time t, the densities of susceptible individuals, latent individuals with latent age a, infectious individuals and removed individuals with relapse age b are denoted by S(t), e(t,a), I(t), r(t,b), respectively. σ(a) and γ(b) denote the conversional rate from the latent class and the relapse rate in the removed class, which depend on age a and age b, respectively. Furthermore, β is the transmission rate of the disease between susceptible and infectious individuals, Λ is the density of the recruitment into the susceptible class (including the births and immigration), μ is the natural death rate of all individuals, δ1 and δ2 are the additional death rate induced by the infectious diseases, and c is the recovery rate from the infectious class. All parameters are assumed to be positive.

    It is clear that the variations of different epidemic parameters between or within different groups can be well realized according to the description of multi-group epidemic models. Hence, Liu and Feng [27] extended model (2.1) to the situation in which the population is divided into n groups according to different contact patterns and derived the following multi-group SEIR epidemic model

    {dSk(t)dt=ΛkμkSk(t)nj=1βkjSk(t)Ij(t),ek(t,a)t+ek(t,a)a=σk(a)ek(t,a)(μk+δ1k)ek(t,a),dIk(t)dt=0σk(a)ek(t,a)da(μk+δ2k+ck)Ik(t)+0γk(b)rk(t,b)db,rk(t,b)t+rk(t,b)b=γk(b)rk(t,b)μkrk(t,b),ek(t,0)=nj=1βkjSk(t)Ij(t),rk(t,0)=ckIk(t), (2.2)

    for t0 and with initial conditions

    {Sk(0)=S0k>0,ek(0,a)=e0k(a)L1+(0,+),Ik(0)=I0k>0,rk(0,b)=r0k(b)L1+(0,+),

    for a,b>0. Λk, μk and ck denote the recruitment rate of the susceptible class, the per-capita natural death rate and the recovery rate from the infectious class in group k, respectively. βkj denotes the transmission rate of the disease between susceptible individuals in group k and infectious individuals in group j. δ1k and δ2k denote the additional death rates of exposed and infectious individuals induced by the infectious diseases in group k, respectively. σk(a) denotes the conversional rate from the latent class in group k, which depends on age a and γk(b) denotes the relapse rate from the removed class into the infectious class in group k, which depends on age b.

    Spatial diffusion is an intrinsic characteristic for investigating the roles of spatial heterogeneity on diseases mechanisms and transmission routes and can lead to rich dynamics. Based on this fact, we generalize (2.2) by taking account of the case that individuals move or diffuse around on the spatial habitat xΩRn with smooth boundary Ω. Let Sk(t,x) and Ik(t,x) be the densities of susceptible individuals and infectious individuals at time t and location xΩ in group k, respectively, where the habitat Ω is bounded and connected. And let ek(t,a,x) and rk(t,b,x) denote the densities of individuals in the latent class with age a and the removed class with age b at time t and location x in group k, respectively. Hence, the n-group diffusive SEIR epidemic model with age-dependent latent and relapse has the following form

    {Sk(t,x)t=d1kSk(t,x)+ΛkμkSk(t,x)nj=1βkjSk(t,x)Ij(t,x),ek(t,a,x)t+ek(t,a,x)a=d2kek(t,a,x)σk(a)ek(t,a,x)(μk+δ1k)×ek(t,a,x),Ik(t,x)t=d3kIk(t,x)+0σk(a)ek(t,a,x)da(μk+δ2k+ck)Ik(t,x)+0γk(b)rk(t,b,x)db,rk(t,b,x)t+rk(t,b,x)b=d4krk(t,b,x)γk(b)rk(t,b,x)μkrk(t,b,x),ek(t,0,x)=nj=1βkjSk(t,x)Ij(t,x),rk(t,0,x)=ckIk(t,x), (2.3)

    for xΩ, a,bR+=(0,+), with the homogeneous Neumann boundary conditions

    Sk(t,x)ν=ek(t,a,x)ν=Ik(t,x)ν=rk(t,b,x)ν=0,xΩ,

    and initial functions

    Sk(0,x)=S0k(x),ek(0,a,x)=e0k(a,x),Ik(0,x)=I0k(x),rk(0,b,x)=r0k(b,x).

    d1k, d2k, d3k, d4k denote the diffusion coefficients of susceptible individuals, exposed individuals, infectious individuals and removed individuals in group k, respectively. And the other parameters have the same biological meanings as in (2.2). The homogeneous Neumann boundary conditions imply that there is no population flux across the boundary Ω.

    We define the functional spaces X=C(ˉΩ,R) and Y=L1(R+,X) for model (2.3) equipped, respectively, with the norms

    |ϕ|X=supxˉΩ|ϕ(x)|,φY=0|φ(a,)|Xda,

    for ϕX, φY. The positive cones are denoted by X+ and Y+. In addition, we define a vector space Z=(C([0,T],X))2n with the norm

    ψZ=maxisup0tT|ψi(t,)|X,ψ=(ψ1,ψ2,...,ψ2n)Z.

    Throughout this paper, for convenience, we always denote S=(S1,S2,...,Sn), e=(e1,e2,...,en), I=(I1,I2,...,In), r=(r1,r2,...,rn), and S0=(S01,S02,...,S0n), e0=(e01,e02,...,e0n), I0=(I01,I02,...,I0n), r0=(r01,r02,...,r0n). We also denote (y1,y2,...,yn)T>(z1,z2,...,zn)T as yi>zi for all i=1,2,...,n. For each i=1,2,3,4, we suppose that Tik:C(ˉΩ,R)C(ˉΩ,R) is the C0 semigroup generated by dikΔ subjects to the Neumann boundary condition in group k. From subsection 2.1 in [28], we have

    (Tik(t)[ϕ])(x)=ΩΓik(t,x,y)ϕ(y)dy,

    for all t>0 and ϕC(ˉΩ,R), where Γik(t,x,y) is the Green function. We have that Tik, i=1,2,3,4, k=1,2,...,n are compact and strongly positive for each t>0 by the Corollary 7.2.3 in [29]. Integrating the second equation in model (2.3) along the characteristic line ta=c, where c is a constant, we obtain

    ek(t,a,x)={ΩΓ2k(a,x,y)ek(ta,0,y)dyπ1k(a),ta,ΩΓ2k(a,x,y)e0k(at,y)dyπ1k(a)π1k(at),t<a, (2.4)

    where π1k(a)=ea0[μk+δ1k+σk(s)]ds. Similarly,

    rk(t,b,x)={ΩΓ4k(b,x,y)rk(tb,0,y)dyπ2k(b),tb,ΩΓ4k(b,x,y)r0k(bt,y)dyπ2k(b)π2k(bt),t<b, (2.5)

    where π2k(b)=eb0[μk+γk(s)]ds. To study the asymptotic behaviors of the dynamics of model (2.3), we require the following assumptions on the model parameters.

    Assumption 2.1. For each k,j=1,2,...,n,

    (H1)d1k,d2k,d3k,d4k,Λk,μk,δ1k,δ2k,ck>0.

    (H2)βkjR+, and the n-dimensional square matrix (βkj)n×n is irreducible.

    (H3)σk(),γk()L(R+,R+), ˉσk>0 and ˉγk>0, where ˉσk:=esssupaR+σk(a), ˉγk:=esssupbR+γk(b).

    We define Ak(t,x)=ek(t,0,x), Bk(t,x)=rk(t,0,x) for (t,x)R+×Ω, and let A=(A1,A2,...,An), B=(B1,B2,...,Bn) and C=(A,B). Positiveness of the solutions of model (2.3) is given below.

    Theorem 3.1. Suppose that there exists a solution (S(t,),e(t,,),I(t,),r(t,,))Xn×Yn×Xn×Yn of (2.3) corresponding to (S0,e0,I0,r0)Xn+×Yn+×Xn+×Yn+ with an interval of existence [0,T], T>0. Then

    (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+,

    for all t[0,T].

    Proof. From the first equation of (2.3), we have

    Sk(t,x)=ˆFSk(t,x)+t0eta[μk+nj=1βkjIj(τ,x)]dτΛkΩΓ1k(ta,x,y)dyda,

    where ˆFSk(t,x)=et0[μk+nj=1βkjIj(τ,x)]dτΩΓ1k(t,x,y)S0k(y)dy. The positivity of Λk and S0k ensures Sk(t,x)>0 for each (t,x)[0,T]×Ω. The positivity of C which means the positivity for Ak and Bk, k=1,2,...,n is established by constructing Picard sequences as follows.

    Solving equation Ik for system (2.3), we have

    Ik(t,x)=FIk(t,x)+t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)×[0σk(b)ek(a,b,y)db+0γk(b)rk(a,b,y)db]dyda, (3.1)

    where FIk(t,x)=e(μk+δ2k+ck)tΩΓ3k(t,x,y)I0k(y)dy. For (t,x)[0,T]×Ω, by (2.4) and (2.5), we obtain

    0σk(b)ek(t,b,y)db=FAk(t,y)+t0σk(b)π1k(b)ΩΓ2k(b,y,z)Ak(tb,z)dzdb, (3.2)

    where FAk(t,y)=0σk(b+t)π1k(b+t)π1k(b)ΩΓ2k(b+t,y,z)e0k(b,z)dzdb, and

    0γk(b)rk(t,b,y)db=FBk(t,y)+t0γk(b)π2k(b)ΩΓ4k(b,y,z)Bk(tb,z)dzdb, (3.3)

    where FBk(t,y)=0γk(b+t)π2k(b+t)π2k(b)ΩΓ4k(b+t,y,z)r0k(b,z)dzdb. From (3.1)–(3.3) and the definitions of Ak and Bk, we have

    Ak(t,x)=nj=1βkjSk(t,x){FIj(t,x)+t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)[FAj(a,y)+a0σj(b)π1j(b)ΩΓ2j(b,y,z)Aj(ab,z)dzdb+FBj(a,y)+a0γj(b)π2j(b)×ΩΓ4j(b,y,z)Bj(ab,z)dzdb]dyda},

    and

    Bk(t,x)=ck{FIk(t,x)+t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)[FAk(a,y)+a0σk(b)×π1k(b)ΩΓ2k(b,y,z)Ak(ab,z)dzdb+FBk(a,y)+a0γk(b)π2k(b)×ΩΓ4k(b,y,z)Bk(ab,z)dzdb]dyda}.

    Let

    A(0)k(t,x)=nj=1βkjSk(t,x){FIj(t,x)+t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)×[FAj(a,y)+FBj(a,y)]dyda},

    and

    B(0)k(t,x)=ck{FIk(t,x)+t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)×[FAk(a,y)+FBk(a,y)]dyda}.

    Then it is obvious that A(0)k(t,x)>0, B(0)k(t,x)>0. Now we assume that A(m)k(t,x)>0, B(m)k(t,x)>0 (mN) for e0k>0, r0k>0 and (t,x)[0,T]×Ω. Then

    A(m+1)k(t,x)=A(0)k(t,x)+nj=1βkjSk(t,x){t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)[a0σj(b)×π1j(b)ΩΓ2j(b,y,z)A(m)j(ab,z)dzdb+a0γj(b)π2j(b)ΩΓ4j(b,y,z)×B(m)j(ab,z)dzdb]dyda},

    and

    B(m+1)k(t,x)=B(0)k(t,x)+ck{t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)[a0σk(b)π1k(b)×ΩΓ2k(b,y,z)A(m)k(ab,z)dzdb+a0γk(b)π2k(b)ΩΓ4k(b,y,z)×B(m)k(ab,z)dzdb]dyda}.

    From the positivity of βkj, σk and γk, together with the positivity of Γ2k, Γ3k and Γ4k, it follows that

    A(1)k(t,x)A(0)k(t,x)=nj=1βkjSk(t,x){t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)[a0σj(b)×π1j(b)ΩΓ2j(b,y,z)A(0)j(ab,z)dzdb+a0γj(b)π2j(b)ΩΓ4j(b,y,z)×B(0)j(ab,z)dzdb]dyda}>0,

    and

    B(1)k(t,x)B(0)k(t,x)=ck{t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)[a0σk(b)π1k(b)ΩΓ2k(b,y,z)×A(0)k(ab,z)dzdb+a0γk(b)π2k(b)ΩΓ4k(b,y,z)B(0)k(ab,z)dzdb]dyda}>0,

    which lead to C(1)(t,x)C(0)(t,x)>0 for (t,x)[0,T]×Ω. We assume that C(m)(t,x)C(m1)(t,x)>0 for all m2, that is, A(m)k(t,x)A(m1)k(t,x)>0 and B(m)k(t,x)B(m1)k(t,x)>0, k=1,2,...,n. Then,

    A(m+1)k(t,x)A(m)k(t,x)=nj=1βkjSk(t,x){t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)(a0σj(b)×π1j(b)ΩΓ2j(b,y,z)[A(m)j(ab,z)A(m1)j(ab,z)]dzdb+a0γj(b)×π2j(b)ΩΓ4j(b,y,z)[B(m)j(ab,z)B(m1)j(ab,z)]dzdb)dyda}>0,

    and

    B(m+1)k(t,x)B(m)k(t,x)=ck{t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)(a0σk(b)π1k(b)ΩΓ2k(b,y,z)×[A(m)k(ab,z)A(m1)k(ab,z)]dzdb+a0γk(b)π2k(b)ΩΓ4k(b,y,z)×[B(m)k(ab,z)B(m1)k(ab,z)]dzdb)dyda}>0.

    Hence applying mathematical induction, we show that the sequence {C(m)}0 is monotonically increasing.

    Next, applying the contraction mapping principle, we show the sequence {C(m)}0 converges to C(t,x) for any (t,x)[0,T]×Ω as m approaches infinity. To this end, we define a variable

    ˆC(m)(t,x)=eλtC(m)(t,x),for someλR+.

    By the definitions of A(m)k and B(m)k, we have

    ˆA(m+1)k(t,x)=eλtA(0)k(t,x)+nj=1βkjSk(t,x){t0e(μj+δ2j+cj)aΩΓ3j(a,x,y)[ta0σj(b)×π1j(b)ΩΓ2j(b,y,z)eλ(a+b)ˆA(m)j(tab,z)dzdb+ta0γj(b)π2j(b)×ΩΓ4j(b,y,z)eλ(a+b)ˆB(m)j(tab,z)dzdb]dyda},

    and

    ˆB(m+1)k(t,x)=eλtB(0)k(t,x)+ck{t0e(μk+δ2k+ck)aΩΓ3k(a,x,y)[ta0σk(b)π1k(b)×ΩΓ2k(b,y,z)eλ(a+b)ˆA(m)k(tab,z)dzdb+ta0γk(b)π2k(b)×ΩΓ4k(b,y,z)eλ(a+b)ˆB(m)k(tab,z)dzdb]dyda}.

    For any mN,

    ˆA(m+1)kˆA(m)knj=1βkjˆSk{t0e(μj+δ2j+cj)(a)ΩΓ3j(a,x,y)[ta0σj(b)π1j(b)ΩΓ2j(b,y,z)×eλ(a+b)dzdb+ta0γj(b)π2j(b)ΩΓ4j(b,y,z)eλ(a+b)dzdb]dyda}׈C(m)ˆC(m1)Znj=1βkjˆSk(ˉσj+ˉγj)λ2ˆC(m)ˆC(m1)Z,

    where ˆSk=maxt[0,T]|Sk(t,)|X, and

    ˆB(m+1)kˆB(m)kck{t0e(μk+δ2k+ck)aΩΓ3k(a,x,y)[ta0σk(b)π1k(b)ΩΓ2k(b,y,z)×eλ(a+b)dzdb+ta0γk(b)π2k(b)ΩΓ4k(b,y,z)eλ(a+b)dzdb]dyda}׈C(m)ˆC(m1)Zck(ˉσk+ˉγk)λ2ˆC(m)ˆC(m1)Z.

    Hence,

    ˆC(m+1)ˆC(m)ZKλˆC(m)ˆC(m1)ZKmλˆC(1)ˆC(0)Z,

    where Kλ=max{Mλ,Nλ}, Mλ=maxk{nj=1βkjˆSk(ˉσj+ˉγj)λ2}, Nλ=maxk{ck(ˉσk+ˉγk)λ2}. Therefore, for any m1>m2, m1,m2N,

    ˆC(m1)ˆC(m2)ZKm1λ1KλˆC(1)ˆC(0)Z.

    We choose λ sufficiently large such that nj=1βkjˆSk(ˉσj+ˉγj)λ2<1 and ck(ˉσk+ˉγk)λ2<1 for all k=1,2,...,n, then Kλ<1. Hence, ˆC(m1)ˆC(m2)Z0 as m2 which implies that ˆC(m)ˆC and thus C(m)C as m. Furthermore, we have A(m)kAk and B(m)kBk for k=1,2,...,n as m. Since sequence {C(m)}0 is monotonically increasing, we obtain Ak and Bk are positive for k=1,2,...,n.

    By (2.4) and (2.5), together with the positivity of e0k, r0k, Ak and Bk, we conclude that ek(t,a,x) and rk(t,a,x) are positive. For the positivity of Ik, we prove this by contradiction. Suppose that there exist x0Ω and t0=inf{tR+|Ik(t,x0)=0} such that

    Ik(t0,x0)=0,Ik(t,x0)>0,Ik(t0,x0)t0,t[0,t0).

    By the third equation of system (2.3), we can easily obtain

    Ik(t0,x0)t=FAk(t0,x0)+FBk(t0,x0)+t00σk(a)π1k(a)ΩΓ2k(a,x0,y)Ak(t0a,y)dyda+t00γk(a)π2k(a)ΩΓ4k(a,x0,y)Bk(t0a,y)dyda>0.

    This leads to a contradiction. Hence, for any t[0,T], we have (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+.

    Let Nk(t)=Ω[Sk(t,x)+0ek(t,a,x)da+Ik(t,x)+0rk(t,b,x)db]dx denotes the total population at time t in group k and region Ω.

    Theorem 3.2. If

    lima+ek(t,a,x)=0,limb+rk(t,b,x)=0, (3.4)

    for all t>0, xΩ, the region Π defined by

    Π={(Sk,ek,Ik,rk)|NkΛkμk|Ω|},

    is positively invariant for system (2.3).

    Proof. Following condition (3.4) and the equations of system (2.3), we have

    Sk(t,x)t+0ek(t,a,x)tda+Ik(t,x)t+0rk(t,b,x)tdb=d1kSk(t,x)+d2k0ek(t,a,x)da+d3kIk(t,x)+d4k0rk(t,b,x)da+ΛkμkSk(t,x)nj=1βkjSk(t,x)Ij(t,x)0ek(t,a,x)ada0σk(a)×ek(t,a,x)da0(μk+δ1k)ek(t,a,x)da+0σk(a)ek(t,a,x)da(μk+δ2k+ck)Ik(t,x)+0γk(b)rk(t,b,x)db0rk(t,b,x)bdb0γk(b)rk(t,b,x)db0μkrk(t,b,x)db=d1kSk(t,x)+d2k0ek(t,a,x)da+d3kIk(t,x)+d4k0rk(t,b,x)da+ΛkμkSk(t,x)0(μk+δ1k)ek(t,a,x)da(μk+δ2k)Ik(t,x)0μkrk(t,b,x)db<d1kSk(t,x)+d2k0ek(t,a,x)da+d3kIk(t,x)+d4k0rk(t,b,x)da+ΛkμkSk(t,x)μk0ek(t,a,x)daμkIk(t,x)μk0rk(t,b,x)db.

    Noting the Neumann boundary conditions of system (2.3) and using the Gauss formula, we derive

    Ωd1kSk(t,x)dx=Ω0d2kek(t,a,x)dadx=Ωd3kIk(t,x)dx=Ω0d4krk(t,b,x)dadx=0.

    It follows that

    dNk(t)dt=Ω[Sk(t,x)t+0ek(t,a,x)tda+Ik(t,x)t+0rk(t,b,x)tdb]dx<Ω{Λkμk[Sk(t,x)+0ek(t,a,x)da+Ik(t,x)+0rk(t,b,x)db]}dx=Λk|Ω|μkNk(t).

    Thus if Nk(t)>Λkμk|Ω|, then dNk(t)dt<0. Moreover, we observe the ordinary differential equation

    dNk(t)dt=Λk|Ω|μkNk(t),

    with general solution

    Nk(t)=Λkμk|Ω|+[Nk(0)Λkμk|Ω|]eμkt,

    where Nk(0) means the initial value of total population in group k and region Ω. By applying the standard comparison theorem, we have for all t0,

    Nk(t)Λkμk|Ω|,ifNk(0)Λkμk|Ω|.

    Hence, Π is positive invariant for system (2.3).

    The existence and uniqueness of the solution of model (2.3) follow from Banach-Picard fixed point theorem.

    Theorem 3.3. Let initial functions satisfy (S0,e0,I0,r0)Xn+×Yn+×Xn+×Yn+. Then the system (2.3) has a unique solution (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+ for t[0,T].

    Proof. Solving equation Sk for system (2.3), we have

    Sk(t,x)=FSk(t,x)+t0eμk(ta)ΩΓ1k(ta,x,y)[ΛkAk(a,y)]dyda, (3.5)

    for (t,x)[0,T]×Ω, where FSk(t,x)=eμktΩΓ1k(t,x,y)S0k(y)dy. From (3.1)–(3.5) and the definitions of Ak and Bk, we have

    Ak(t,x)=nj=1βkj{FSk(t,x)+t0eμk(ta)ΩΓ1k(ta,x,y)[ΛkAk(a,y)]dyda}×{FIj(t,x)+t0e(μj+δ2j+cj)(ta)ΩΓ3j(ta,x,y)[FAj(a,y)+a0σj(b)×π1j(b)ΩΓ2j(b,y,z)Aj(ab,z)dzdb+FBj(a,y)+a0γj(b)π2j(b)×ΩΓ4j(b,y,z)Bj(ab,z)dzdb]dyda}:=F1k[C](t,x), (3.6)

    and

    Bk(t,x)=ck{FIk(t,x)+t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)[FAk(a,y)+a0σk(b)×π1k(b)ΩΓ2k(b,y,z)Ak(ab,z)dzdb+FBk(a,y)+a0γk(b)π2k(b)×ΩΓ4k(b,y,z)Bk(ab,z)dzdb]dyda}:=F2k[C](t,x), (3.7)

    where F1k, F2k:ZC([0,T],X) are nonlinear operators for each k=1,2,...,n. For the sake of convenience, we define for each (t,x)[0,T]×Ω,

    FCk(t,x)=FSk+t0eμk(ta)ΩΓ1k(ta,x,y)Λkdyda,FDk(t,x)=FIk(t,x)+t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)[FAk(a,y)+FBk(a,y)]dyda,Θ1(Ak)=t0eμk(ta)ΩΓ1k(ta,x,y)Ak(a,y)dyda,Θ2(Ak)=t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)a0σk(b)π1k(b)×ΩΓ2k(b,y,z)Ak(ab,z)dzdbdyda,Θ3(Bk)=t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)a0γk(b)π2k(b)×ΩΓ4k(b,y,z)Bk(ab,z)dzdbdyda.

    Then

    F1k[C]=nj=1βkj[FCkΘ1(Ak)][FDj+Θ2(Aj)+Θ3(Bj)],F2k[C]=ck[FDk+Θ2(Ak)+Θ3(Bk)].

    For any C, ˉCZ, we set ˜C=CˉC. Then, from the positivity of Ak and Bk proved in Theorem 3.1, we have

    F1k[C]F1k[ˉC]=nj=1βkj{FCk[Θ2(˜Aj)+Θ3(˜Bj)]FDjΘ1(˜Ak)Θ1(˜Ak)×[Θ2(Aj)+Θ3(Bj)]Θ1(ˉAk)[Θ2(˜Aj)+Θ3(˜Bj)]}nj=1βkjFCk[Θ2(˜Aj)+Θ3(˜Bj)]nj=1βkj|FCk(ˆΘ2+ˆΘ3)|˜CZ,

    and

    F2k[C]F2k[C]=ck[Θ2(˜Ak)+Θ3(˜Bk)]ck|ˆΘ2+ˆΘ3|˜CZ,

    where

    ˆΘ2=t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)a0σk(b)π1k(b)×ΩΓ2k(b,y,z)dzdbdyda,ˆΘ3=t0e(μk+δ2k+ck)(ta)ΩΓ3k(ta,x,y)a0γk(b)π2k(b)×ΩΓ4k(b,y,z)Bk(ab,z)dzdbdyda.

    Denote

    m1k(T)=nj=1βkj|FCk(T,)[ˆΘ2(T,)+ˆΘ3(T,)]|X,m2k(T)=ck|ˆΘ2(T,)+ˆΘ3(T,)|X,m(T)=max{m11(T),m12(T),...,m1n(T),m21(T),m22(T),...,m2n(T)},

    and

    F[C]=(F11,F12,...,F1n,F21,F22,...,F2n)[C]:ZZ.

    Clearly, we can choose T small enough such that m1k(T)<1 and m2k(T)<1 for all k=1,2,...,n. Consequently, we have m(T)<1. Then

    FCFˉCZm(T)CˉCZ.

    Hence, applying contraction operator theorem [30], we conclude that F has a unique fixed point C=(A1,A2,...,An,B1,B2,...,Bn). From Theorem 3.1, (2.4) and (2.5), together with Ak(t,x)=nj=1βkjSk(t,x)Ij(t,x) and Bk(t,x)=ckIk(t,x), we derive the existence and uniqueness of the solution (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+ for system (2.3).

    To further establish the global existence of the solution of system (2.3), we need the following lemma.

    Lemma 3.1. [31]. The following problem

    {ω(t,x)t=dωω(t,x)+Λμω(t,x),xΩ,ω(t,x)ν=0,xΩ,

    admits a unique positive steady state ω=Λμ, which is globally attractive in X.

    Theorem 3.4. Let initial functions satisfy (S0,e0,I0,r0)Xn+×Yn+×Xn+×Yn+. Then the system (2.3) has a unique solution (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+ for tR+.

    Proof. To extend the domain of existence from t[0,T] to tR+, it suffices to show that the solution does not blow up in finite time. In fact, by Theorem 3.1, we have that

    Sk(t,x)td1kSk(t,x)+ΛkμkSk(t,x),

    for all t>0 and xΩ. From Lemma 3.1 and the comparison principle, we get that Sk(t,x) is bounded above by the upper solution Λkμk.

    We now claim that ek(t,a,x)<+ for all t>0, a>0, xΩ and k=1,2,...,n. From (2.4), it is sufficient to show that ek(t,0,x)<+ for all t>0 and xΩ. Suppose on the contrary that there exist te>0 and xeΩ such that

    limtte0ek(t,0,xe)=+.

    We then have from (3.5) that

    limtte0Sk(t,xe)=,

    which implies that Sk(t,xe) is negative in the neighborhood of te. This contradicts to the positivity of Sk, which has been proved in Theorem 3.1. Furthermore, from ek(t,0,x)=nj=1βkjSk(t,x)Ij(t,x) and rk(t,0,x)=ckIk(t,x) in (2.4), we obtain Ik(t,x)<+ and rk(t,0,x)<+. And from (2.5), we get rk(t,a,x)<+ for all t>0, a>0, xΩ and k=1,2,...,n. Thus, blow up never occurs, and we obtain a solution (S(t,),e(t,,),I(t,),r(t,,))Xn+×Yn+×Xn+×Yn+ for tR+.

    Theorem 3.5. The solution semiflow Φ(t)=(S(t,),e(t,,),I(t,),r(t,,)):Xn+×Yn+×Xn+×Yn+(0,0,0,0)Xn+×Yn+×Xn+×Yn+(0,0,0,0) of model (2.3) has a compact and global attractor as condition (3.4) holds.

    Proof. According to Theorem 3.2, we know that system (2.3) is ultimately bounded, which implies that solution semiflow Φ(t) is point dissipative on Xn+×Yn+×Xn+×Yn+(0,0,0,0). By Theorem 2.6 in [15], we can get that Φ(t) is compact for any t>0. Thus, from Theorem 3.4.8 in [32], we further obtain that Φ(t) has a compact and global attractor in Xn+×Yn+×Xn+×Yn+(0,0,0,0).

    It is easy to see that model (2.3) has a unique disease-free steady state ˉE=(ˉS,0,0,0)T where ˉS=(ˉS1,ˉS2,...,ˉSn)T and ˉSk=Λkμk, k=1,2,...,n.

    We denote E=(S,e(),I,r())T as the space-independent endemic steady state of (2.3), where S=(S1,S2,...,Sn)T, e()=(e1(),e2(),...,en())T, I=(I1,I2,...,In)T and r()=(r1(),r2(),...,rn())T. Then, E satisfies

    {ΛkμkSknj=1βkjSkIj=0,dek(a)dt=[σk(a)+μk+δ1k]ek(a),0σk(a)ek(a)da(μk+δ2k+ck)Ik+0γk(b)rk(b)db=0,drk(b)dt=[γk(b)+μk]rk(b),ek(0)=nj=1βkjSkIj,rk(0)=ckIk. (4.1)

    We denote fk(I)=nj=1βkjIj:Rn+R+, hk(I)=Λkμk+fk(I):Rn+R+, k=1,2,...,n for brevity. By solving the Eq (4.1), we get

    Sk=hk(I),ek(a)=Skπ1k(a)fk(I),rk(b)=ckπ2k(b)Ik, (4.2)

    where π1k() and π2k() are given in (2.4) and (2.5), respectively. Substituting (4.2) into the third equation in (4.1) and rearranging it, we have

    Ik=Lkhk(I)fk(I)(μk+δ2k+ck)ckPk, (4.3)

    where Lk=0σk(a)π1k(a)da and Pk=0γk(b)π2k(b)db. Let us define a matrix-valued function M(x) on Rn to Rn×n, where M(x)ij=Liβijhi(x)(μi+δ2i+ci)ciPi, x=(x1,x2,...,xn)TRn. Then, (4.3) is equivalent to

    I=M(I)I. (4.4)

    On the existence of the endemic equilibrium E of system (2.3), we prove the following theorem.

    Theorem 4.1. Let M0=(ˉSiLiβij(μi+δ2i+ci)ciPi)n×n. If ρ(M0)>1, where ρ(M0) represents the spectral radius of M0, then system (2.3) has a space-independent steady state E.

    Proof. From (4.4), we only need to show that the nonlinear operator ˜M(x):=M(x)x, xRn+, has at least one positive fixed point xRn+. We define ˜M(x):=max1kn|˜M(x)k|, where ˜M(x)k denotes the k-th entry of vector ˜M(x). Then, ˜M(x) is monotone increasing with respect to xRn+ and uniformly bounded above by max1kn|ΛkLk(μk+δ2k+ck)ckPk|.

    It is obvious that ˜M(0)=0 and M(0) is the strong Fréchet derivative of ˜M() at the origin. Since M(0)=M0, we have ρ(M(0))>1. Thus, it follows from the Perron-Frobenius theorem (see [33]) that ρ(M(0)) is a simple eigenvalue of M(0) corresponding to a strictly positive eigenvector, and there exists no nonnegative eigenvector of M(0) corresponding to eigenvalue 1. Hence, we apply Theorem 4.11 of [34], to conclude that operator ˜M() has at least one positive fixed point xRn+.

    Theorem 4.2. For all k=1,2,...,n, if

    0αkσk(θ)π1k(θ)dθ<1,0γk(θ)π2k(θ)dθ<1, (4.5)

    where αk=nj=1ˉSjβjkμk+δ2k, π1k() and π2k() are given in (2.4) and (2.5), respectively, then the disease-free steady state ˉE is globally asymptotically stable.

    Proof. We construct a Lyapunov function

    V(t)=nk=1Vk(t),

    where

    Vk(t)=Ω[V1k(t,x)+V2k(t,x)+V3k(t,x)+V4k(t,x)]dx,
    V1k(t,x)=ˉSk[Sk(t,x)ˉSk1lnSk(t,x)ˉSk],V2k(t,x)=0χk(a)ek(t,a,x)da,V3k(t,x)=αkIk(t,x),V4k(t,x)=0ψk(b)rk(t,b,x)db,

    and

    χk(a)=aαkσk(θ)π1k(θ)π1k(a)dθ,ψk(b)=bαkγk(θ)π2k(θ)π2k(b)dθ. (4.6)

    Taking the derivation of V1k(t,x) along the trajectory of (2.3) with respect to t, we have

    V1kt=Sk(t,x)ˉSkSk(t,x)Sk(t,x)t=d1k[Sk(t,x)ˉSk]Sk(t,x)Sk(t,x)μk[Sk(t,x)ˉSk]2Sk(t,x)+ˉSknj=1βkjIj(t,x)Sk(t,x)nj=1βkjIj(t,x). (4.7)

    Recalling (2.4), we can rewrite V2k(t,x) as follows

    V2k(t,x)=t0χk(ta)ΩΓ2k(ta,x,y)Ak(a,y)dyπ1k(ta)da+0χk(a+t)ΩΓ2k(a+t,x,y)e0k(a,y)dyπ1k(a+t)π1k(a)da.

    Thus, we calculate V2kt along the solution of system (2.3) and get

    V2kt=χk(0)Ak(t,x)+0{χk(a)[μk+δ1k+σk(a)d2k]χk(a)}×ek(t,a,x)da. (4.8)

    Similarly,

    V3kt=αkd3kIk(t,x)+αk0σk(a)ek(t,a,x)da+αk0γk(b)rk(t,b,x)dbnj=1ˉSjβjkIk(t,x)αkckIk(t,x). (4.9)

    From (2.5), we get

    V4k(t,x)=t0ψk(tb)ΩΓ4k(tb,x,y)Bk(b,y)dyπ2k(tb)db+0ψk(b+t)ΩΓ4k(b+t,x,y)r0k(b,y)dyπ2k(b+t)π2k(b)db.

    Thus,

    V4kt=ψk(0)Bk(t,x)+0{ψk(b)[μk+γk(b)d4k]ψk(b)}rk(t,b,x)db. (4.10)

    Hence, combining (4.7)–(4.10), we calculate the derivative of Vk(t) along the solution trajectory of (2.3) as

    dVkdt=d1kˉSkΩ|Sk(t,x)|2S2k(t,x)dxΩμk[Sk(t,x)ˉSk]2Sk(t,x)dx+Ω[χk(0)1]×Ak(t,x)dx+Ω[ψk(0)αk]Bk(t,x)dx+Ω0{αkσk(a)[μk+δ1k+σk(a)d2k]χk(a)+χk(a)}ek(t,a,x)da+Ω0{αkγk(b)[μk+γk(b)d4k]ψk(b)+ψk(b)}×rk(t,b,x)db+ΩˉSknj=1βkjIj(t,x)dxΩnj=1ˉSjβjkIk(t,x)dx.

    Using (4.6), we yield that

    dVdt=nk=1d1kˉSkΩ|Sk(t,x)|2S2kdxnk=1Ωμk[Sk(t,x)ˉSk]2Sk(t,x)dx+nk=1Ω[χk(0)1]Ak(t,x)dx+nk=1Ω[ψk(0)αk]Bk(t,x)dx.

    Thus, from (4.5), we have

    χk(0)<1,ψk(0)<αk,

    which implies the global asymptotic stability of disease-free steady state ˉE.

    Theorem 4.3. If ρ(M0)>1, lima+ek(t,a,x)=0 and limb+rk(t,b,x)=0 for all t>0, xΩ, then the space-independent steady state E is globally asymptotically stable.

    Proof. We define g(x)=x1lnx, clearly, g(x) is always positive for x>0 and g(x)=11x. Consider a matrix D=(ˉβkj)n×n with entry ˉβkj=βkjLkSkIj and a digraph G=(U,H) which contains a set U={1,2,...,n} of vertices and a set H of arcs (k,j) leading from initial vertex k to terminal vertex j, then, we denote a weighted digraph as (G,D) for which each arc (j,k) is assigned a positive weight ˉβkj. Furthermore, we denote ˉD as the Laplacian matrix of matrix (G,D). Then, the irreducibility of matrix (βkj)n×n implies that ˉD is also irreducible. Let qk denote the cofactor of k-th diagonal element of ˉD. And we construct a Lyapunov function as the following form

    W(t)=nk=1qkWk(t),

    where

    Wk(t)=Ω[W1k(t,x)+W2k(t,x)+W3k(t,x)+W4k(t,x)]dx,
    W1k(t,x)=LkSkg[Sk(t,x)Sk],W2k(t,x)=0Ψ1k(a)ek(a)g[ek(t,a,x)ek(a)]da,W3k(t,x)=Ikg[Ik(t,x)Ik],W4k(t,x)=0Ψ2k(b)rk(b)g[rk(t,b,x)rk(b)]db,

    and

    Ψ1k(a)=aσk(s)π1k(s)π1k(a)ds,Ψ2k(b)=bγk(s)π2k(s)π2k(b)ds. (4.11)

    For convenience, we denote Jk(t,x)=nj=1βkjIj(t,x) and Jk=nj=1βkjIj. Taking the derivation of W1k along the trajectory of (2.3) with respect to t, we have

    W1kt=LkSk[1Sk1Sk(t,x)]Sk(t,x)t=Lk[1SkSk(t,x)]d1kSk(t,x)LkμkSk(t,x)[Sk(t,x)Sk]2+LkSkJk×[1+Jk(t,x)JkSkSk(t,x)Sk(t,x)Jk(t,x)SkJk]. (4.12)

    Calculating the derivative of W2k along the solution of system (2.3) yields

    W2kt=0Ψ1k(a)ek(a)tg[ek(t,a,x)ek(a)]da=0Ψ1k(a)[1ek(a)ek(t,a,x)]{d2kΔek(t,a,x)aek(t,a,x)[μk+δ1k+σk(a)]ek(t,a,x)}da=0Ψ1k(a)[1ek(a)ek(t,a,x)]d2kΔek(t,a,x)da0Ψ1k(a)ek(a)×[ek(t,a,x)ek(a)1][1ek(t,a,x)aek(t,a,x)+μk+δ1k+σk(a)]da=0Ψ1k(a)[1ek(a)ek(t,a,x)]d2kΔek(t,a,x)da0Ψ1k(a)ek(a)×ag[ek(t,a,x)ek(a)]da=0Ψ1k(a)[1ek(a)ek(t,a,x)]d2kΔek(t,a,x)da+Ψ1k(0)ek(0)×g[ek(t,0,x)ek(0)]+0g[ek(t,a,x)ek(a)]dda[Ψ1k(a)ek(a)]da=0Ψ1k(a)[1ek(a)ek(t,a,x)]d2kΔek(t,a,x)da+LkSkJkg[Sk(t,x)Jk(t,x)SkJk]+0σk(a)ek(a)[1ek(t,a,x)ek(a)+lnek(t,a,x)ek(a)]da. (4.13)

    Similarly,

    W3kt=[1IkIk(t,x)][d3kIk(t,x)+0σk(a)ek(t,a,x)da(μk+δ2k+ck)×Ik(t,x)+0γk(b)rk(t,b,x)db]=[1IkIk(t,x)]{d3kIk(t,x)+0σk(a)ek(a)[ek(t,a,x)ek(a)Ik(t,x)Ik]da+0γk(b)rk(b)[rk(t,b,x)rk(b)Ik(t,x)Ik]db}=[1IkIk(t,x)]d3kIk(t,x)+0σk(a)ek(a)[1Ik(t,x)Ik+ek(t,a,x)ek(a)Ikek(t,a,x)Ik(t,x)ek(a)]da+0γk(b)rk(b)[1Ik(t,x)Ik+rk(t,b,x)rk(b)Ikrk(t,b,x)Ik(t,x)rk(b)]db, (4.14)

    and

    W4kt=0Ψ2k(b)[1rk(b)rk(t,b,x)]d4kΔrk(t,b,x)db+PkckIkg[Ik(t,x)Ik]+0γk(b)rk(b)[1rk(t,b,x)rk(b)+lnrk(t,b,x)rk(b)]db. (4.15)

    Hence, combining (4.12)–(4.15), we calculate the derivative of Wk(t) along the solution trajectory of (2.3) as

    dWkdt=Lkd1kSkΩ|Sk(t,x)|2S2k(t,x)dxΩ0d2kek(a)Ψ1k(a)|ek(t,a,x)|2e2k(t,a,x)dadxd3kIkΩ|Ik(t,x)|2I2k(t,x)dxΩ0d4krk(b)Ψ2k(b)|rk(t,b,x)|2r2k(t,b,x)dbdxLkμkΩ[Sk(t,x)Sk]2Sk(t,x)dx+PkckIkΩg[Ik(t,x)Ik]dx+LkSknj=1βkj×IjΩ[Jk(t,x)JkSkSk(t,x)lnSk(t,x)Jk(t,x)SkJk]dx+Ω0σk(a)ek(a)×[2Ik(t,x)IkIkek(t,a,x)Ik(t,x)ek(a)+lnek(t,a,x)ek(a)]dadx+Ω0γk(b)rk(b)×[2Ik(t,x)IkIkrk(t,b,x)Ik(t,x)rk(b)+lnrk(t,b,x)rk(b)]dbdx=Lkd1kSkΩ|Sk(t,x)|2S2k(t,x)dxΩ0d2kek(a)Ψ1k(a)|ek(t,a,x)|2e2k(t,a,x)dadxd3kIkΩ|Ik(t,x)|2I2k(t,x)dxΩ0d4krk(b)Ψ2k(b)|rk(t,b,x)|2r2k(t,b,x)dbdxLkμkΩ[Sk(t,x)Sk]2Sk(t,x)dxLkSkJkΩg[SkSk(t,x)]dxΩ0σk(a)×ek(a)g[Ikek(t,a,x)Ik(t,x)ek(a)]dadxΩ0γk(b)rk(b)g[Ikrk(t,b,x)Ik(t,x)rk(b)]dbdxLkSkJkΩ{g[Ik(t,x)Ik]g[Jk(t,x)Jk]}dx.

    By Theorem 2.3 in [35], the following identity holds

    nk=1qk{LkSkJkΩ[g(Ik(t,x)Ik)g(Jk(t,x)Jk)]dx}=0.

    Hence, together with the property of g, we have dWdt0. Furthermore, the strict equality holds only if Sk(t,x)=Sk, ek(t,a,x)=ek(a), Ik(t,x)=Ik, and rk(t,b,x)=rk(b). Thus, T={E}Ω is the largest invariant subset of {(S,e,I,r):dWdt=0}, and the Lyapunov-LaSalle invariance principle implies that the endemic equilibrium E is globally asymptotically stable when ρ(M0)>1.

    In this section, we present numerical examples to demonstrate the validity and applicability of our main results, Theorems 4.2 and 4.3. For simplicity, we consider the case of two groups, a normalized 1-dimensional space (Ω = [0, 3]) and a normalized maximum age (a = 2). Firstly, we fix some parameter values as follows,

    d11=0.2,d21=0.5,d31=0.4,d41=0.1,μ1=0.6,c1=0.006,d12=0.4,d22=0.1,d32=0.3,d42=0.2,μ2=0.7,c2=0.006.

    Then, we take the initial functions as

    S01(x)=0.8×(1+0.3×sinπx),e01(a,x)=1.2×ea,I01(x)=0.5×(1.5+0.2×sinπx),r01(a,x)=1.5×ea,S02(x)=0.7×(1+0.2×sinπx),e02(a,x)=1.3×ea,I02(x)=0.6×(1.5+0.3×sinπx),r02(a,x)=1.4×ea.

    Example 5.1. If we choose other parameters as

    Λ1=3,β11=0.04,β12=0.07,δ11=0.2,δ21=0.9,Λ2=2,β21=0.06,β22=0.05,δ12=0.5,δ22=0.8,σ1(a)=0.003×(1+sin(a5)π2),γ1(a)=0.1×(1+sin(a5)π2),σ2(a)=0.002×(1+sin(a5)π2),γ2(a)=0.2×(1+sin(a5)π2), (5.1)

    then we have

    0α1σ1(θ)π11(θ)dθ0.00051<1,0γ1(θ)π21(θ)dθ0.0817<1,0α2σ2(θ)π12(θ)dθ0.00027<1,0γ2(θ)π22(θ)dθ0.13319<1.

    In this case, from Theorem 4.2, we expect the disease-free steady state ˉE to be globally asymptotically stable. In fact, in Figure 1, the infective population I1(t,x) and I2(t,x) converge to zero over time.

    Figure 1.  Time evolution of infective population I1(t,x) and I2(t,x) for system (2.3) with parameters (5.1).

    Example 5.2. If we choose other parameters as

    Λ1=5,β11=0.99,β12=0.97,δ11=0.002,δ12=0.005,Λ2=7,β21=0.96,β22=0.98,δ21=0.009,δ22=0.008,σ1(a)=35×(1+sin(a5)π2),γ1(a)=0.1×(1+sin(a5)π2),σ2(a)=30×(1+sin(a5)π2),γ2(a)=0.2×(1+sin(a5)π2), (5.2)

    then we have ρ(M0)3.89217>1. In this case, from Theorem 4.3, we expect the space-independent steady state E to be globally asymptotically stable. In fact, in Figure 2, the infective population I1(t,x) and I2(t,x) converge to the positive distribution over time.

    Figure 2.  Time evolution of infective population I1(t,x) and I2(t,x) for system (2.3) with parameters (5.2).

    In this paper, as an additional structure of the system, we focus on the spatial diffusion of the population. Models with spatial diffusion allow individuals to move to adjacent positions through a random walk process, this is a key factor in considering the geographical spread of infectious diseases. Firstly, we propose the n-group diffusive SEIR epidemic model with age-dependent latent and relapse, it is a generalization of the model in [27] to a spatially diffusive system. Then, we investigate the positivity, boundedness, existence and uniqueness of solution and the existence of compact global attractor of the associated solution semiflow for this system. For these results, we use the method of constructing Picard sequences, Banach-Picard fixed point theorem and theories of partial functional differential equations. Thereafter, we establish the existence of disease-free and endemic steady states based on Perron-Frobenius theorem. we utilize appropriate Lyapunov functionals, graph-theoretical results and the LaSalle's invariance principle to prove the global stability of disease-free and endemic steady states. Thus, we presented the results of numerical simulations to verify the validity of our main theorems. This is important from the viewpoint of applications.

    In this epidemic model, we are concerned with two kinds of spatial heterogeneity: the patch structure and spatial diffusion. Furthermore, age-of-latent and age-of-relapse are included into the epidemic model which is appropriate for diseases such as tuberculosis and herpes virus infection. Dynamical results obtained in this paper provide theoretical foundation for seeking effective measures to prevent, control and study disease transmission.

    The expressions of basic reproduction number and endemic steady state depends on space are not analyzed in this paper owing to the complexity of model. In addition, how to improve the sufficient conditions that ensure the stabilities of steady states and make them be depended on basic reproduction number is also need to investigate. We leave these issues for future research.

    The authors are very grateful to the editor and the anonymous referees for their valuable comments and suggestions, which helped us to improve the presentation of this work significantly.

    This work was supported by the National Natural Science Foundation of China (Grant Number: 11971329).

    The authors declare there is no conflict of interest.



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