Processing math: 64%
Research article Special Issues

The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy


  • Received: 07 December 2022 Revised: 05 January 2023 Accepted: 05 January 2023 Published: 18 January 2023
  • The global spread of COVID-19 has not been effectively controlled. It poses a significant threat to public health and global economic development. This paper uses a mathematical model with vaccination and isolation treatment to study the transmission dynamics of COVID-19. In this paper, some basic properties of the model are analyzed. The control reproduction number of the model is calculated and the stability of the disease-free and endemic equilibria is analyzed. The parameters of the model are obtained by fitting the number of cases that were detected as positive for the virus, dead, and recovered between January 20 and June 20, 2021, in Italy. We found that vaccination better controlled the number of symptomatic infections. A sensitivity analysis of the control reproduction number has been performed. Numerical simulations demonstrate that reducing the contact rate of the population and increasing the isolation rate of the population are effective non-pharmaceutical control measures. We found that if the isolation rate of the population is reduced, a short-term decrease in the number of isolated individuals can lead to the disease not being controlled at a later stage. The analysis and simulations in this paper may provide some helpful suggestions for preventing and controlling COVID-19.

    Citation: Yujie Sheng, Jing-An Cui, Songbai Guo. The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5966-5992. doi: 10.3934/mbe.2023258

    Related Papers:

    [1] Zhibo Cheng, Lisha Lv, Jie Liu . Positive periodic solution of first-order neutral differential equation with infinite distributed delay and applications. AIMS Mathematics, 2020, 5(6): 7372-7386. doi: 10.3934/math.2020472
    [2] Xin Long . Novel stability criteria on a patch structure Nicholson's blowflies model with multiple pairs of time-varying delays. AIMS Mathematics, 2020, 5(6): 7387-7401. doi: 10.3934/math.2020473
    [3] Lini Fang, N'gbo N'gbo, Yonghui Xia . Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory. AIMS Mathematics, 2022, 7(3): 3788-3801. doi: 10.3934/math.2022210
    [4] Jing Ge, Xiaoliang Li, Bo Du, Famei Zheng . Almost periodic solutions of neutral-type differential system on time scales and applications to population models. AIMS Mathematics, 2025, 10(2): 3866-3883. doi: 10.3934/math.2025180
    [5] Ping Zhu . Dynamics of the positive almost periodic solution to a class of recruitment delayed model on time scales. AIMS Mathematics, 2023, 8(3): 7292-7309. doi: 10.3934/math.2023367
    [6] Yanshou Dong, Junfang Zhao, Xu Miao, Ming Kang . Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. AIMS Mathematics, 2023, 8(9): 21828-21855. doi: 10.3934/math.20231113
    [7] Shihe Xu, Zuxing Xuan, Fangwei Zhang . Analysis of a free boundary problem for vascularized tumor growth with time delays and almost periodic nutrient supply. AIMS Mathematics, 2024, 9(5): 13291-13312. doi: 10.3934/math.2024648
    [8] Ramazan Yazgan . An analysis for a special class of solution of a Duffing system with variable delays. AIMS Mathematics, 2021, 6(10): 11187-11199. doi: 10.3934/math.2021649
    [9] Yongkun Li, Xiaoli Huang, Xiaohui Wang . Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. AIMS Mathematics, 2022, 7(4): 4861-4886. doi: 10.3934/math.2022271
    [10] Ardak Kashkynbayev, Moldir Koptileuova, Alfarabi Issakhanov, Jinde Cao . Almost periodic solutions of fuzzy shunting inhibitory CNNs with delays. AIMS Mathematics, 2022, 7(7): 11813-11828. doi: 10.3934/math.2022659
  • The global spread of COVID-19 has not been effectively controlled. It poses a significant threat to public health and global economic development. This paper uses a mathematical model with vaccination and isolation treatment to study the transmission dynamics of COVID-19. In this paper, some basic properties of the model are analyzed. The control reproduction number of the model is calculated and the stability of the disease-free and endemic equilibria is analyzed. The parameters of the model are obtained by fitting the number of cases that were detected as positive for the virus, dead, and recovered between January 20 and June 20, 2021, in Italy. We found that vaccination better controlled the number of symptomatic infections. A sensitivity analysis of the control reproduction number has been performed. Numerical simulations demonstrate that reducing the contact rate of the population and increasing the isolation rate of the population are effective non-pharmaceutical control measures. We found that if the isolation rate of the population is reduced, a short-term decrease in the number of isolated individuals can lead to the disease not being controlled at a later stage. The analysis and simulations in this paper may provide some helpful suggestions for preventing and controlling COVID-19.



    Based on the experimental data which were observed and summarized by Nicholson [1], Gurney et al. [2] presented a classic biological dynamical system model

    N(t)=δN(t)+pN(tτ)eaN(tτ). (1.1)

    Here, N(t) is the size of the population at time t, p is the maximum per capita daily egg production, 1a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time. The research on the Nicholson's blowfly model and its modifications has realized a remarkable progress in the past fifty years and an abundance of results on the existence of positive solutions, persistence, oscillation, stability, periodic solutions, almost periodic solutions, pseudo almost periodic solutions, etc. (see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]) have been obtained. Furthermore, Berezansky et al. [22] systematically collected and compared the results in the above-mentioned studies and put forward several open problems that have been partially answered in recent works [23,24,25,26,27,28,29,30], such as Nicholson's blowfly model with impulsive perturbation, harvesting term and nonlinear density-dependent mortality term. On the other hand, Liu [31] considered the following cooperative Nicholson's blowfly equation with a patch structure:

    xi(t)=nj=1aijxj(t)+βxi(tri)exi(tri)dixi(t),i=1,2,,n. (1.2)

    Also, Berezansky et al. [32] studied the following cooperative Nicholson-type delay differential system

    {x1(t)=a1x1(t)+b1x2(t)+c1x1(tτ)ex1(tτ),x2(t)=a2x2(t)+b2x1(t)+c2x2(tτ)ex2(tτ). (1.3)

    In this paper, we will discuss the following competitive and cooperative Nicholson's blowfly system

    {x1(t)=δ1(t)x1(t)+a1(t)x2(t)+nj=1c1j(t)x1(tτ1j(t))eb1j(t)x1(tτ1j(t))k1(t)x1(t)x2(t),x2(t)=δ2(t)x2(t)+a2(t)x1(t)+nj=1c2j(t)x2(tτ2j(t))eb2j(t)x2(tτ2j(t))k2(t)x1(t)x2(t) (1.4)

    where δi,ai,bij,cij,τij,ki:R1[0,+) are almost periodic functions i=1,2,j=1,2,,n. x1(t),x2(t) denote the sizes of the different populations at time t, cij denotes the maximum per capita daily egg production of xi, 1bij represents the size at which the population xi reproduces at its maximum rate, δi is the per capita daily adult death rate of xi, and τij denotes the generation time of xi. a1(t) represents the rate at which x2 contributes to x1 and a2(t) represents the rate at which x1 contributes to x2 at time t. ki(t) denotes the death rate of xi due to the competition between x1 and x2 at time t. The cooperative terms a1(t)x2(t) and a2(t)x1(t) and the competitive terms k1(t)x1(t)x2(t) and k1(t)x1(t)x2(t) reflect the degree at which they cooperate and compete with each other, respectively.

    Recently, there have been wide-ranging results obtained on competitive and cooperative systems added the literatures [33,34,35,36,37,38] due to its extensive applicability. However, to the best of our knowledge, few results are presented in literatures about the existence of positive almost periodic solutions for competitive and cooperative Nicholson's blowfly system. In the real world, since the competition is inevitable and there exists an almost periodically changing environment, it is worth studying the positive almost periodic solution for competitive and cooperative Nicholson's blowfly system. Based on the above idea, we shall consider the existence and exponential convergence of positive almost periodic solutions of system (1.4) which possesses obvious dynamics significance.

    For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on R1, let g+ and g be defined as follows:

    g=inftR1g(t),g+=suptR1g(t).

    It will be assumed that

    δi>0,bij>0,ri=max1jn{τ+ij}>0i=1,2,j=1,2,,n. (1.5)

    Let Rn(Rn+) be the set of all (nonnegative) real vectors; we will use x=(x1,x2,,xn)TRn to denote a column vector, in which the symbol (T) denotes the transpose of a vector. we let |x| denote the absolute-value vector given by |x|=(|x1|,|x2|,,|xn|)T and define ||x||=max1in|xi|. For a matrix A=(aij)n×n, AT, A1, |A| and ρ(A) denote the transpose, inverse, the absolute-value matrix and the spectral radius of A respectively. A matrix or vector A0 means that all entries of A are greater than or equal to zero. A>0 can be defined similarly. For matrices or vectors A and B, AB (resp. A>B) means that AB0 (resp. AB>0). Denote C=2i=1C([ri,0],R1)  and  C+=2i=1C([ri,0],R1+) as Banach spaces equipped with the supremum norm defined by

    ||φ||=suprit0max1i2|φi(t)|   for all  φ(t)=(φ1(t),φ2(t))TC(orC+).

    If xi(t) is defined on [t0ri,ν) with t0,νR1 and i=1,2, then we define xtC as xt=(x1t,x2t)T where xit(θ)=xi(t+θ) for all θ[ri,0] and i=1,2.

    The initial conditions associated with system (1.4) are of the following form:

    xt0=φ, φ=(φ1,φ2)TC+. (1.6)

    We write xt(t0,φ)(x(t;t0,φ)) for a solution of the initial value problems (1.4) and (1.6). Also, let [t0,η(φ)) be the maximal right-interval of the existence of xt(t0,φ).

    The remaining part of this paper is organized as follows. In Section 2, we shall give some definitions and preliminary results. In Section 3, we shall derive sufficient conditions for checking the existence, uniqueness and exponential convergence of the positive almost periodic solution of (1.4). In Section 4, we shall give an example and numerical simulation to illustrate the results obtained in the previous section.

    In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

    Definition 2.1. [39,40] Let u(t):R1Rn be continuous in t. u(t) is said to be almost periodic on R1, if for any ε>0, the set T(u,ε)={δ:|u(t+δ)u(t)|<ε  for all  tR1} is relatively dense, i.e., for any ε>0, it is possible to find a real number l=l(ε)>0, such that for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)u(t)|<ε,  for all  tR1.

    Definition 2.2. [39,40] Let xRn and Q(t) be an n×n continuous matrix defined on R1. The linear system

    x(t)=Q(t)x(t) (2.1)

    is said to admit an exponential dichotomy on R1 if there exist positive constants k,α, projection P and the fundamental solution matrix X(t) of (2.1) satisfying

    X(t)PX1(s)keα(ts)for allts,X(t)(IP)X1(s)keα(st)for all ts.

    Definition 2.3. A real n×n matrix K=(kij) is said to be an M-matrix if kij0,i,j=1,,n,ij and K10.

    Set

    B={φ|φ=(φ1(t),φ2(t))T  is an almost periodic vector function on R1}.

    For any φB, we define an induced module φB=suptR1max1i2|φi(t)|; then, B is a Banach space.

    Lemma 2.1. [39,40] If the linear system (2.1) admits an exponential dichotomy, then the almost periodic system

    x(t)=Q(t)x+g(t) (2.2)

    has a unique almost periodic solution x(t), and

    x(t)=tX(t)PX1(s)g(s)ds+tX(t)(IP)X1(s)g(s)ds. (2.3)

    Lemma 2.2. [39,40] Let ci(t) be an almost periodic function on R1 and

    M[ci]=limT+1Tt+Ttci(s)ds>0,i=1,2,,n.

    Then the linear system

    x(t)=diag(c1(t),c2(t),,cn(t))x(t)

    admits an exponential dichotomy on R1.

    Lemma 2.3. [41,42] Let A0 be an n×n matrix and ρ(A)<1; then, (InA)10, where In denotes the identity matrix of size n.

    Lemma 2.4. Suppose that there exist positive constants Ei1 and Ei2 such that

    Ei1>Ei2,nj=1c+1jδ1b1je+a+1E21δ1<E11,nj=1c+2jδ2b2je+a+2E11δ2<E21, (2.4)
    a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21>E121min1jmb1j, (2.5)
    a2δ+2E12+nj=1c2jδ+2E21eb+2jE21k+2δ+2E11E21>E221min1jmb2j, (2.6)

    where i=1,2. Let

    C0:={φ|φC,Ei2<φi(t)<Ei1,  for all  t[ri,0], i=1,2}.

    Moreover, assume that x(t;t0,φ) is the solution of (1.4) with φC0. Then,

    Ei2<xi(t;t0,φ)<Ei1,  for all   t[t0,η(φ)),i=1,2, (2.7)

    and η(φ)=+.

    Proof. We rewrite the system (1.4) as

    x(t)=f(t,xt),

    where x(t)=(x1(t),x2(t))T, f(t,φ)=(f1(t,φ),f2(t,φ))T, f1(t,φ)=δ1(t)φ1(0)+a1(t)φ2(0)+nj=1c1j(t)φ1(τ1j(t))eb1j(t)φ1(τ1j(t))k1(t)φ1(0)φ2(0), f2(t,φ)=δ2(t)φ2(0)+a2(t)φ1(0)+nj=1c2j(t)φ2(τ2j(t))eb2j(t)φ2(τ2j(t))k2(t)φ1(0)φ2(0), φ(t)=(φ1(t),φ2(t))TC0. It is obvious that f:R1×C0R2 is continuous and C0C is open. Let ϕ,ψC0; then, considering that supu0|1ueu|=1 and the inequality

    |xexyey|=|1(x+θ(yx))ex+θ(yx)||xy|       |xy|  where  x,y[0,+),0<θ<1, (2.8)

    we obtain

    |f1(t,ϕ)f1(t,ψ)|δ1(t)|ϕ1(0)ψ1(0)|+a1(t)|ϕ2(0)ψ2(0)|+nj=1c1j(t)×|ϕ1(τ1j(t))eb1j(t)ϕ1(τ1j(t))ψ1(τ1j(t))eb1j(t)ψ1(τ1j(t))|+k1(t)|ϕ1(0)ϕ2(0)ψ1(0)ψ2(0)|(δ+1+a+1)||ϕψ||+nj=1c1j(t)b1j(t)×|b1j(t)ϕ1(τ1j(t))eb1j(t)ϕ1(τ1j(t))b1j(t)ψ1(τ1j(t))eb1j(t)ψ1(τ1j(t))|+k+1(|ϕ1(0)||ϕ2(0)ψ2(0)|+|ψ2(0)||ϕ1(0)ψ1(0)|)(δ+1+a+1)||ϕψ||+nj=1c+1jb1j(t)×|b1j(t)ϕ1(τ1j(t))b1j(t)ψ1(τ1j(t))|+k+1(E11||ϕψ||+E21||ϕψ||)(δ+1+a+1+nj=1c+1j+k+1E11+k+1E21)||ϕψ||. (2.9)

    In the same way, we also get

    |f2(t,ϕ)f2(t,ψ)|(δ+2+a+2+nj=1c+2j+k+2E11+k+2E21))||ϕψ||. (2.10)

    Then (2.9) and (2.10) imply that f satisfies the Lipschitz condition in its second argument on each compact subset of R1×C0. Moreover, since φC+, it is easy to get that xt(t0,φ)C+forallt[t0,η(φ)) by using Theorem 5.2.1 from [43, p. 81]. Set x(t)=x(t;t0,φ) for all  t[t0,η(φ)).

    We claim that

    0xi(t)<Ei1  for all  t[t0,η(φ)),i=1,2. (2.11)

    By contradiction, assume that (2.11) does not hold. Then, there exists t1(t0,η(φ)) such that one of the following two cases must occur:

    (1)x1(t1)=E11,0xi(t)<Ei1for allt[t0ri,t1),i=1,2; (2.12)
    (2)x2(t1)=E21,0xi(t)<Ei1for allt[t0ri,t1),i=1,2. (2.13)

    In the sequel, we consider two cases.

    Case (ⅰ). Suppose that (2.12) holds. Considering the derivative of x1(t), together with (2.4) and the fact that supu0ueu=1e, we have

    0x1(t1)=δ1(t1)x1(t1)+a1(t1)x2(t1)+nj=1c1j(t1)x1(t1τ1j(t1))eb1j(t1)x1(t1τ1j(t1))k1(t1)x1(t1)x2(t1)δ1x1(t1)+a+1x2(t1)+nj=1c+1jb1j1eδ1E11+a+1E21+nj=1c+1jb1j1e=δ1(E11+nj=1c+1jδ1b1je+a+1E21δ1)<0,

    which is a contradiction.

    Case (ⅱ). Suppose that (2.13) holds. Considering the derivative of x2(t), together with (2.4) and the fact that supu0ueu=1e, we have

    0x2(t1)=δ2(t1)x2(t1)+a2(t1)x1(t1)+nj=1c2j(t1)x2(t1τ1j(t1))eb2j(t1)x2(t1τ1j(t1))k2(t1)x1(t1)x2(t1)δ2x2(t1)+a+2x1(t1)+nj=1c+2jb2j1eδ2E21+a+2E11+nj=1c+2jb2j1e=δ2(E21+nj=1c+2jδ2b2je+a+2E11δ2)<0,

    which is a contradiction. Together with Cases (ⅰ) and (ⅱ), (2.11) holds for t[t0,η(φ)).

    We next show that

    xi(t)>Ei2,  for all  t(t0,η(φ)),i=1,2. (2.14)

    Suppose, for the sake of contradiction, that (2.14) does not hold. Then, there exists t2(t0,η(φ)) such that one of the following two cases must occur:

    (1)x1(t2)=E12,Ei2<xi(t)<Ei1for allt[t0ri,t2),i=1,2; (2.15)
    (2)x2(t2)=E22,Ei2<xi(t)<Ei1for allt[t0ri,t2),i=1,2. (2.16)

    If (2.15) holds, from (2.5), (2.6), (2.11) and (2.15), we get

    Ei2<xi(t)<Ei1,  b+ijxi(t)b+ijEi2b+ij1min1jnbij1, (2.17)

    for all t[t0ri,t2),i=1,2, j=1,2,n. Calculating the derivative of x1(t), together with (2.5) and the fact that min1uκueu=κeκ, (1.4), (2.15) and (2.17) imply that

    0x1(t2)=δ1(t2)x1(t2)+a1(t2)x2(t2)+nj=1c1j(t2)x1(t2τ1j(t2))eb1j(t2)x1(t2τ1j(t2))k1(t2)x1(t2)x2(t2)δ+1x1(t2)+a1x2(t2)+nj=1c1j(t2)b+1jb+1jx1(t2τ1j(t2))eb+1jx1(t2τ1j(t2))k+1x1(t2)x2(t2)>δ+1E12+a1E22+nj=1c1jE11eb+1jE11k+1E11E21=δ+1(E12+a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21)>0,

    which is absurd and implies that (2.14) holds. If (2.16) holds, we can prove that (2.14) also holds in a similar way.

    It follows from (2.11) and (2.14) that (2.7) is true. From Theorem 2.3.1 in [44], we easily obtain η(φ)=+. This completes the proof.

    Theorem 3.1. Let (2.4) – (2.6) hold. Moreover, suppose that

    ρ(A1B)<1. (3.1)

    where

    A=(δ100δ2),B=(nj=1c+1je2+k+1E21a+1+k+1E11a+2+k+2E21nj=1c+2je2+k+2E11).

    Then, there exists a unique positive almost periodic solution of system (1.4) in the region B={φ|φB,Ei2φi(t)Ei1,  for all  tR1,i=1,2,,n}.

    Proof. For any ϕB, we consider the following auxiliary system

    {x1(t)=δ1(t)x1(t)+a1(t)ϕ2(t)+nj=1c1j(t)ϕ1(tτ1j(t))eb1j(t)ϕ1(tτ1j(t))k1(t)ϕ1(t)ϕ2(t)x2(t)=δ2(t)x2(t)+a2(t)ϕ1(t)+nj=1c2j(t)ϕ2(tτ2j(t))eb2j(t)ϕ2(tτ2j(t))k2(t)ϕ1(t)ϕ2(t). (3.2)

    Since M[δi]>0 (i=1,2), it follows from Lemma 2.2 that the linear system

    xi(t)=δi(t)xi(t),i=1,2 (3.3)

    admits an exponential dichotomy on R1. Thus, by Lemma 2.1, we obtain that the system (3.2) has exactly one almost periodic solution xϕ(t)=(xϕ1(t),xϕ2(t))T:

    {xϕ1(t)=tetsδ1(u)du[a1(s)ϕ2(s)+nj=1c1j(s)ϕ1(sτ1j(s))eb1j(s)ϕ1(sτ1j(s))k1(s)ϕ1(s)ϕ2(s)]dsxϕ2(t)=tetsδ2(u)du[a2(s)ϕ1(s)+nj=1c2j(s)ϕ2(sτ2j(s))eb2j(s)ϕ2(sτ2j(s))k2(s)ϕ1(s)ϕ2(s)]ds. (3.4)

    Define a mapping T:BB by setting

    T(ϕ(t))=xϕ(t),  ϕB.

    Since B={φ|φB,Ei2φi(t)Ei1,  for all  tR1,i=1,2}, it is obvious that B is a closed subset of B. For i=1,2 and any ϕB, from (2.4), (3.4) and the fact that supu0ueu=1e, we have

          xϕ1(t)tetsδ1(u)du[a+1E21+nj=1c1j(s)1b1j(s)e]ds1δ1[a+1E21+nj=1c+1jb1je]=nj=1c+1jδ1b1je+a+1E21δ1<E11      for all  tR1,                               (3.5)

    and

          xϕ2(t)tetsδ2(u)du[a+2E11+nj=1c2j(s)1b2j(s)e]ds1δ2[a+2E11+nj=1c+2jb2je]=nj=1c+2jδ2b2je+a+2E11δ2<E21      for all  tR1.                               (3.6)

    In view of the fact that min1uκueu=κeκ, from (2.5)–(2.7) and (3.4), we obtain

    xϕ1(t)tetsδ1(u)du[a1E22+nj=1c1j(s)1b+1jb+1jϕ1(sτ1j(s))eb+1jϕ1(sτ1j(s))k+1ϕ1(s)ϕ2(s)]ds1δ+1[a1E22+nj=1c1jE11eb+1jE11k+1E11E21]=a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21>E12  for all  tR1, (3.7)

    and

    xϕ2(t)tetsδ2(u)du[a2E12+nj=1c2j(s)1b+2jb+2jϕ2(sτ2j(s))eb+2jϕ1(sτ2j(s))k+2ϕ1(s)ϕ2(s)]ds1δ+2[a2E12+nj=1c2jE21eb+2jE21k+2E11E21]=a2δ+2E12+nj=1c2jδ+2E21eb+2jE21k+2δ+2E11E21>E22  for all  tR1. (3.8)

    Therefore, (3.5)–(3.8) show that the mapping T is a self-mapping from B to B.

    Let φ,ψB; for i=1,2, we get

    suptR1|(T(φ(t))T(ψ(t)))1|=suptR1|tetsδ1(u)du[a1(s)(φ2(s)ψ2(s))+nj=1c1j(s)(φ1(sτ1j(s))eb1j(s)φ1(sτ1j(s))ψ1(sτ1j(s))eb1j(s)ψ1(sτ1j(s)))k1(s)(φ1(s)φ2(s)ψ1(s)ψ2(s))]ds|=suptR1|tetsδ1(u)du[a1(s)(φ2(s)ψ2(s))+nj=1c1j(s)b1j(s)×(b1j(s)φ1(sτ1j(s))eb1j(s)φ1(sτ1j(s))b1j(s)ψ1(sτ1j(s))eb1j(s)ψ1(sτ1j(s)))k1(s)(φ1(s)φ2(s)φ1(s)ψ2(s)+φ1(s)ψ2(s)ψ1(s)ψ2(s))]ds|, (3.9)

    and

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2} | \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_{ 2}(s)(\varphi_{1} (s)-\psi_{1} (s)) \\ & &+\sum^n_{j = 1}c_{ 2j}(s)(\varphi_{2} (s-\tau_{2j}(s))e^{-b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))}-\psi_{2} (s-\tau_{2j}(s))e^{-b_{2j}(s)\psi_{2} (s-\tau_{2j}(s))}) \\ & &- k_{2}(s)(\varphi_{1}(s)\varphi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_{ 2}(s)(\varphi_{1} (s)-\psi_{1} (s))+\sum^n_{j = 1}\frac{c_{ 2j}(s)}{b_{ 2j}(s)}\times \\ & &(b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))e^{-b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))}-b_{ 2j}(s)\psi_{2} (s-\tau_{2j}(s))e^{-b_{2j}(s)\psi_{2} (s-\tau_{2j}(s))}) \\ & &- k_{2}(s)(\varphi_{1}(s)\varphi_{2}(s)-\varphi_{1}(s)\psi_{2}(s)+\varphi_{1}(s)\psi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| . \end{eqnarray} (3.10)

    Since

    b_{ij}(s)\varphi_{i} (s-\tau_{ ij}(s))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq n}b_{ij}^{-}}\geq 1,\ \mbox{ for all } \ s\in R^{1}, i = 1, 2, j = 1, 2, \cdots, n,

    and

    \ b_{ij}(s)\psi_{i} (s-\tau_{i j}(s))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq n}b_{ij}^{-}}\geq 1, \ \mbox{ for all }\ s\in R^{1}, i = 1, 2, j = 1, 2, \cdots, n.

    According to (1.4), (2.5), (3.5), (3.7) and (3.9), together with \sup\limits_{u\geq 1}|\frac{1-u}{e^{u}}| = \frac{1}{e^{2}} and the inequality

    \begin{align} &|xe^{-x}-ye^{-y}| = |\frac{1-(x+\theta (y-x))}{e^{x+\theta (y-x)}}|| x -y | \\&\ \ \ \ \ \ \ \ \\ \ \leq\frac{1}{e^{2}}| x -y | \ \mbox{ where } \ x, y\in [1, +\infty) , 0 < \theta < 1, \end{align} (3.11)

    we have

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{1} | \\ &\leq & \frac{a^{+}_{1}}{ \delta^{-}_{1}}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| \\ & &+\sup\limits_{t \in R^{1}}\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}\sum^n_{j = 1}c^{+}_{ 1j}\frac{1}{e^{2}} |\varphi_{1} (s-\tau_{1j}(s)) - \psi _{1}(s-\tau_{1j}(s))| ds \\ & &+\sup\limits_{t \in R^{1}}\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}k_1(s)( |\varphi_{1} (s)||\varphi_2(s)-\psi_2(s)|+|\psi_{2} (s)||\varphi_1(s)-\psi_1(s)|) ds \\ &\leq&\frac{a^{+}_{1}}{\delta^{-}_{1}}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+\sum^n_{j = 1}\frac{c^{+}_{ 1j}}{\delta^{-}_1e^{2}}\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|\\ &&+\frac{k_1^+}{\delta^{-}_{1}}E_{11}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| +\frac{k_1^+}{\delta^{-}_{1}}E_{21}\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|, \\ & = &(\sum^n_{j = 1}\frac{c^{+}_{ 1j}}{\delta^{-}_1e^{2}}+\frac{k_1^+}{\delta^{-}_{1}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|+ (\frac{a^{+}_{1}}{\delta^{-}_{1}}+\frac{k_1^+}{\delta^{-}_{1}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|. \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \end{eqnarray} (3.12)

    Similarly, we also get

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2} | \\ &\leq & (\sum^n_{j = 1}\frac{c^{+}_{ 2j}}{\delta^{-}_2e^{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+ (\frac{a^{+}_{2}}{\delta^{-}_{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|. \end{eqnarray} (3.13)

    Hence

    \begin{eqnarray} &&( \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2}|)^T \\ &\leq & ( (\sum\limits_{j = 1}^n\frac{c^{+}_{1j}}{ \delta^{-}_{1}e^2}+\frac{k_1^+}{\delta^{-}_{1}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|+ (\frac{a^{+}_{1}}{ \delta^{-}_{1}}+\frac{k_1^+}{\delta^{-}_{1}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|, \\ & &(\sum\limits_{j = 1}^n\frac{c^{+}_{2j}}{ \delta^{-}_{2}e^2}+\frac{k_2^+}{\delta^{-}_{2}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+ (\frac{a^{+}_{2}}{ \delta^{-}_{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|)^T \\ & = & F(\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|, \sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| )^T \\ & = & F(\sup\limits_{t\in R^{1}}|(\varphi(t)-\psi(t))_1|, \sup\limits_{t\in R^{1}}|(\varphi(t)-\psi(t))_2| )^T, \end{eqnarray} (3.14)

    where F = A^{-1}B. Let \mu be a positive integer. Then, from (3.14) we get

    \begin{eqnarray} &&( \sup\limits_{t \in R^{1}} |(T^\mu(\varphi(t))-T^\mu(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}} |(T^\mu(\varphi(t))-T^\mu(\psi(t)))_{2}|)^T \\ & = & ( \sup\limits_{t \in R^{1}} |(T(T^{\mu-1}(\varphi(t)))-T(T^{\mu-1}(\psi(t))))_{1}|, \\ &\ &\sup\limits_{t \in R^{1}}|(T(T^{\mu-1}(\varphi(t)))-T(T^{\mu-1}(\psi(t))))_{2}|)^T \\ &\leq& F( \sup\limits_{t \in R^{1}} |(T^{\mu-1}(\varphi(t))-T^{\mu-1}(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}}|(T^{\mu-1}(\varphi(t))-T^{\mu-1}(\psi(t)))_{2}|)^T\\ &\ & \vdots\\ &\leq& F^\mu( \sup\limits_{t \in R^{1}} |(\varphi(t)-\psi(t))_{1}|, \sup\limits_{t \in R^{1}}|(\varphi(t)-\psi(t))_{2}|)^T\\ & = & F^\mu( \sup\limits_{t \in R^{1}} |\varphi_{1}(t)-\psi_{1}(t)|, \sup\limits_{t \in R^{1}}|\varphi_{2}(t)-\psi_{2}(t)|)^T. \end{eqnarray} (3.15)

    Since \rho(F)<1, we obtain

    \lim\limits_{\mu\rightarrow +\infty}F^\mu = 0,

    which implies that there exist a positive integer N and a positive constant r<1 such that

    \begin{align} F^N = (A^{-1}B)^N = (g_{ij})_{2\times 2} \; \; \mbox{and}\; \; \sum\limits_{j = 1}^2 g_{ij}\leq r, i = 1, 2. \end{align} (3.16)

    In view of (3.15) and (3.16), we have

    \begin{eqnarray} |(T^N(\varphi(t))-T^N(\psi(t)))_{i}|&\leq& \sup\limits_{t\in R^{1}}|(T^N(\varphi(t))-T^N(\psi(t)))_{i}| \\ &\leq& \sum\limits_{j = 1}^2 g_{ij} \sup\limits_{t \in R^{1}} |\varphi_{j}(t)-\psi_{j}(t)|\\ &\leq& \sup\limits_{t \in R^{1}} \max\limits_{1\leq j\leq 2}|\varphi_{j}(t)-\psi_{j}(t)|\sum\limits_{j = 1}^2 g_{ij}\\ &\leq& r \| \varphi(t)-\psi(t)\|_B, \end{eqnarray}

    for all t\in R, i = 1, 2. It follows that

    \begin{align} \|T^N(\varphi(t))-T^N(\psi(t))\|_B = \sup\limits_{t\in R^{1}} \max\limits_{1\leq i\leq 2}|(T^N(\varphi(t))-T^N(\psi(t)))_i|\leq r\|\varphi(t)-\psi(t)\|_B. \end{align} (3.17)

    This implies that the mapping T^N:B^*\rightarrow B^* is a contraction mapping.

    By the fixed point theorem for Banach space, T possesses a unique fixed point \varphi^{*}\in B^{*} such that T\varphi^{*} = \varphi^{*}. By (3.2), \varphi^{*} satisfies (1.4). So \varphi^{*} is an almost periodic solution of (1.4) in B^{*}. The proof of Theorem 3.1 is now completed.

    Theorem 3.2. Let x^{*}(t) be the positive almost periodic solution of system (1.4) in the region B^{*} . Suppose that (2.4)–(2.6) and (3.1) hold. Then, the solution x(t; t_{0}, \varphi) of (1.4) with \varphi \in C^{0} converges exponentially to x ^{* }(t) as t\rightarrow +\infty.

    Proof. Since \rho(A^{-1}B) < 1, it follows from Theorem 3.1 that system (1.4) has a unique almost periodic solution x^*(t) = (x^*_1(t), x^*_2(t))^T in the region B^{*} . Set x(t) = x(t; t_{0}, \varphi), x^*(t) = x^*(t; t_{0}, \varphi^*) and y_{i}(t) = x_{i}(t)-x_{i}^{*}(t), where \varphi, \varphi^*\in C^{0}, t\in [t_{0}-r_{i}, +\infty), \ i = 1, 2. Then, we have the following:

    \begin{align} \left\{ \begin{array}{rcl} y_{1} '(t)& = & -\delta_1(t) y_{1}(t)+ a_{1}(t)y_{2}(t)+\sum\limits_{j = 1}^nc_{1j}(t)(x_{1}(t-\tau_{1j}(t))e^{-b_{1j}(t)x_{1}(t-\tau_{1j}(t))}\\ &&-x_{1}^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_{1}^*(t-\tau_{1j}(t))})-k_1(t)(x_1(t)x_2(t)-x_1^*(t)x_2^*(t)), \\ y_{2} '(t) & = & -\delta_2(t) y_{2}(t)+ a_{2}(t)y_{1}(t)+\sum\limits_{j = 1}^n c_{2j}(t)(x_{2}(t-\tau_{2j}(t))e^{-b_{2j}(t)x_{2}(t-\tau_{2j}(t))}\\ &&-x_{2}^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_{2}^*(t-\tau_{2j}(t))})-k_2(t)(x_1(t)x_2(t)-x_1^*(t)x_2^*(t)). \end{array} \right. \end{align} (3.18)

    Again from \rho(A^{-1}B) < 1, it follows from Lemma 2.3 that I_2-A^{-1}B is an M-matrix; we obtain that there exists a constant \bar{\mu}>0 and a vector \xi = (\xi_1, \xi_2)^T>(0, 0)^T such that

    (I_2-A^{-1}B)\xi > (\bar{\mu}, \bar{\mu})^T.

    Therefore,

    \left\{ \begin{array}{rcl} && \xi_1-(\sum\limits_{j = 1}^n\frac{c_{1j}^+}{\delta_1^-e^2}+\frac{k_1^+}{\delta_1^-}E_{21}) \xi_1-(\frac{a_1^+}{\delta_1^-}+\frac{k_1^+}{\delta_1^-}E_{11}) \xi_2 > \overline{\mu}, \\ && \xi_2-(\sum\limits_{j = 1}^n\frac{c_{2j}^+}{\delta_2^-e^2}+\frac{k_2^+}{\delta_2^-}E_{11}) \xi_2-(\frac{a_2^+}{\delta_2^-}+\frac{k_2^+}{\delta_2^-}E_{21}) \xi_1 > \overline{\mu}, \\ \end{array} \right.

    which implies that

    \begin{align} \left\{ \begin{array}{rcl} && (-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2 < -\delta_1^-\overline{\mu}, \\ && (-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1 < -\delta_2^-\overline{\mu}. \\ \end{array} \right. \end{align} (3.19)

    We can choose a positive constant \eta<1 such that

    \begin{align} \left\{ \begin{array}{rcl} && \eta\xi_1+(-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}e^{\eta r_1}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2 < p > 0, \\ && \eta\xi_2+(-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}e^{\eta r_2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1 < p > 0. \\ \end{array} \right. \end{align} (3.20)

    In the sequel, we consider the following Lyapunov function:

    \begin{align} V_i(t) = |y_i(t)|e^{\eta(t-t_0)}, \; \; i = 1, 2. \end{align} (3.21)

    In view of (2.5)–(2.7), for i\in \{1, 2\} and j\in \{1, 2, \cdots, n\}, we obtain

    b_{ij}(t)x_i (t-\tau_{ij}(t))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq m}b_{ij}^{-}}\geq 1, \ \mbox{ for all } \ t\in [t_{0}-r_{i}, +\infty),

    and

    b_{ij}(t)x_i^*(t-\tau_{ij}(t)) \geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq m}b_{ij}^{-}}\geq 1,\ \mbox{ for all } \ t\in R^{1},

    which, together with (3.11) and (3.18), imply that

    \begin{array}{rcl} D^-V_1(t) &\leq & -\delta_1(t)|y_1(t)|e^{\eta(t-t_0)}+a_{1}(t)|y_2(t)|e^{\eta(t-t_0)}+\sum\limits_{j = 1}^nc _{1j}(t)e^{\eta(t-t_0)}\times\\ &&|x_1(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1(t-\tau_{1j}(t))}-x_1^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1^*(t-\tau_{1j}(t))}|\\ &&+ k_1(t)e^{\eta(t-t_0)}|x_1(t)x_2(t)-x_1^*(t)x_2^*(t)|+\eta|y_1(t)|e^{\eta(t-t_0)} \\ &\leq& (\eta-\delta_1(t))V_1(t)+a_1^+V_2(t)+\sum\limits_{j = 1}^n\frac{c _{1j}(t)}{b_{1j}(t)}e^{\eta(t-t_0)}\times\\ &&|b_{1j}(t)x_1(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1(t-\tau_{1j}(t))}\\ & &-b_{1j}(t)x_1^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1^*(t-\tau_{1j}(t))}|\\ &&+ k_1(t)|x_1(t)||x_2(t)-x_2^*(t)|e^{\eta(t-t_0)} +k_1(t)|x_2^*(t)||x_1(t)-x_1^*(t)|e^{\eta(t-t_0)}\\ &\leq& (\eta-\delta_1^-)V_1(t)+a_1^+V_2(t)+\sum\limits_{j = 1}^n\frac{c _{1j}(t)}{e^2}|y_1(t-\tau_{1j}(t))|e^{\eta(t-t_0)}\\ &&+ k_1^+E_{11}V_2(t)+k_1^+E_{21}V_1(t)\\ &\leq&(\eta-\delta_1^-+k_1^+E_{21})V_1(t)+\sum\limits_{j = 1}^n\frac{c _{1j}^+}{e^2}e^{\eta r_1}V_1(t-\tau_{1j}(t))\\ &&+ (a_1^++k_1^+E_{11})V_2(t), \end{array} (3.22)

    and

    \begin{array}{rcl} D^-V_2(t) &\leq & -\delta_2(t)|y_2(t)|e^{\eta(t-t_0)}+a_{2}(t)|y_1(t)|e^{\eta(t-t_0)}+\sum\limits_{j = 1}^nc _{2j}(t)e^{\eta(t-t_0)}\times\\ &&|x_2(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2(t-\tau_{2j}(t))}-x_2^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2^*(t-\tau_{2j}(t))}|\\ &&+ k_2(t)e^{\eta(t-t_0)}|x_1(t)x_2(t)-x_1^*(t)x_2^*(t)|+\eta|y_2(t)|e^{\eta(t-t_0)} \\ &\leq& (\eta-\delta_2(t))V_2(t)+a_2^+V_1(t)+\sum\limits_{j = 1}^n\frac{c _{2j}(t)}{b_{2j}(t)}e^{\eta(t-t_0)}\times\\ &&|b_{2j}(t)x_2(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2(t-\tau_{2j}(t))}\\ & &-b_{2j}(t)x_2^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2^*(t-\tau_{2j}(t))}|\\ &&+ k_2(t)|x_1(t)||x_2(t)-x_2^*(t)|e^{\eta(t-t_0)} +k_2(t)|x_2^*(t)||x_1(t)-x_1^*(t)|e^{\eta(t-t_0)}\\ &\leq& (\eta-\delta_2^-)V_2(t)+a_2^+V_1(t)+\sum\limits_{j = 1}^n\frac{c _{2j}(t)}{e^2}|y_2(t-\tau_{2j}(t))|e^{\eta(t-t_0)}\\ &&+ k_2^+E_{11}V_2(t)+k_2^+E_{21}V_1(t)\\ &\leq&(\eta-\delta_2^-+k_2^+E_{11})V_2(t)+\sum\limits_{j = 1}^n\frac{c _{2j}^+}{e^2}e^{\eta r_2}V_2(t-\tau_{2j}(t))\\ &&+ (a_2^++k_2^+E_{21})V_1(t). \end{array} (3.23)

    Let \varsigma>1 denote an arbitrary real number such that

    \varsigma\xi_i > ||\varphi-\varphi^*|| = \sup\limits_{-r_i\leq s\leq0}\max\limits_{1\leq i\leq 2}|\varphi_i(s)-\varphi^*_i(s)| > 0 , \; i = 1, 2.

    It follows from (3.21) that

    V_i(t) = |y_i(t)|e^{\eta(t-t_0)} < \varsigma\xi_i , \; \; \mbox{for all}\; t\in[t_0-r_i, t_0], \; i = 1, 2.

    We claim that

    \begin{align} V_i(t) = |y_i(t)|e^{\eta(t-t_0)} < \varsigma\xi_i , \; \; \mbox{for all}\; t > t_0, \; i = 1, 2. \end{align} (3.24)

    In contrast, there must exist i\in \{1, 2\} and r^*>t_0 such that

    \begin{align} V_i(r^*) = \varsigma\xi_i, \; \mbox{and}\; V_j(t) < \varsigma\xi_j, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2. \end{align} (3.25)

    Thus,

    \begin{align} V_1(r^*)-\varsigma\xi_1 = 0, \; \mbox{and}\; V_j(t)-\varsigma\xi_j < 0, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2, \end{align} (3.26)

    or

    \begin{align} V_2(r^*)-\varsigma\xi_2 = 0, \; \mbox{and}\; V_j(t)-\varsigma\xi_j < 0, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2. \end{align} (3.27)

    Together with (3.20), (3.22), (3.23), (3.26) and (3.27), we obtain

    \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \begin{array}{rcl} 0&\leq&D^-(V_1(r^*)-\varsigma\xi_1) \\ & = &D^-(V_1(r^*))\\ &\leq&(\eta-\delta_1^-+k_1^+E_{21})V_1(r^*)+\sum\limits_{j = 1}^n\frac{c _{1j}^+}{e^2}e^{\eta r_1}V_1(r^*-\tau_{1j}(r^*))\\ &&+ (a_1^++k_1^+E_{11})V_2(r^*)\\ &\leq&\varsigma[\eta\xi_1+(-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}e^{\eta r_1}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2]\\ & < &0, \end{array} (3.28)

    or

    \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \begin{array}{rcl} 0&\leq&D^-(V_2(r^*)-\varsigma\xi_2) \\ & = &D^-(V_2(r^*))\\ &\leq&(\eta-\delta_2^-+k_2^+E_{11})V_2(r^*)+\sum\limits_{j = 1}^n\frac{c _{2j}^+}{e^2}e^{\eta r_2}V_2(r^*-\tau_{2j}(r^*))\\ &&+ (a_2^++k_2^+E_{21})V_1(r^*)\\ &\leq&\varsigma[\eta\xi_2+(-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}e^{\eta r_2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1]\\ & < &0, \end{array} (3.29)

    which are both contradictory. Hence, (3.24) holds. Let M>1 such that

    \begin{align} \varsigma\xi_i\leq M||\varphi-\varphi^*||, \; i = 1, 2. \end{align} (3.30)

    In view of (3.24) and (3.30), we get

    |x_i(t)-x_i^*(t)| = |y_i(t)|\leq \varsigma\xi_ie^{-\eta (t-t_0)}\leq M||\varphi-\varphi^* || e^{-\eta (t-t_0)}, \; \; \mbox{for all }\; \; t > t_0\; \; i = 1, 2.

    This completes the proof.

    Corollary 3.1. Let (2.4)–(2.6) hold. Suppose that I_2-A^{-1}B is an M-matrix. Then system (1.4) has exactly one almost periodic solution x^*(t). Moreover, the solution x(t; t_{0}, \varphi) of (1.4) with \varphi \in C^{0} converges exponentially to x ^{* }(t) as t\rightarrow +\infty.

    Proof. Since I_2-A^{-1}B is an M-matrix, it follows that there exists a vector d = (d_1, d_2)^T>(0, 0)^T such that

    \begin{align} (I_2-A^{-1}B)d > 0, \end{align} (3.31)

    hence

    \begin{align} \left\{ \begin{array}{rcl} && -\delta_1^-d_1+(\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}+k_1^+E_{21}) d_1+(a_1^++k_1^+E_{11}) d_2 < 0 \\ && -\delta_2^-d_2+(\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}+k_2^+E_{11}) d_2+(a_2^++k_2^+E_{21}) d_1 < 0. \\ \end{array} \right. \end{align} (3.32)

    For any matrix norm ||\cdot|| and nonsingular matrix D, ||A||_D = ||D^{-1}AD|| also defines a matrix norm. Let D = diag(d_1, d_2). Then (3.32) implies that the row norm of matrix D^{-1}A^{-1}BD is less than 1. Hence \rho(A^{-1}B) < 1. Corollary 3.1 follows immediately from Theorems 3.1 and 3.2.

    In this section, we give an example and present a numerical simulation to demonstrate the results obtained in previous sections.

    Example 4.1. Consider the following competitive and cooperative Nicholson's blowfly system:

    \begin{align} \left\{ \begin{array}{rcl} x_{1} '(t) & = & -(18+\cos^{2}\sqrt{5}t) x_{1} (t)+(1+0.7\sin ^{2} t)e^{e-2} x_{2} (t)\\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{2}t|) x_{1} (t-e^{|\sin t|+|\sin \sqrt{2}t|})e^{- x _{1} (t-e^{|\sin t|+|\sin\sqrt{2} t|})} , \ \ \\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{5}t|) x _{1} (t-e^{|\cos \sqrt{3} t|+|\cos t|})e^{- x _{1} (t-e^{|\cos \sqrt{3}t|+|\cos t|})}\\&\; &- 0.1e^{-2}\cos ^{2} t x _{1} (t)x _{2} (t) \\ x_{2} '(t) & = & -(18+\sin^{2}\sqrt{5}t) x_{2} (t)+(1+0.7\cos ^{2} t)e^{e-2} x_{1} (t)\\&\; &+ e^{e-1}(9.5+0.005|\cos \sqrt{2}t|) x_{2} (t-e^{|\cos t|+|\cos \sqrt{7} t|})e^{- x _{2} (t-e^{|\cos t|+|\cos \sqrt{7} t|})} , \ \ \\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{6}t|) x _{2} (t-e^{|\cos \sqrt{7}t|+|\cos \sqrt{3} t|})e^{- x _{2} (t-e^{|\cos \sqrt{7}t|+|\cos \sqrt{3} t|})}\\&\; &- 0.1e^{-2}\sin ^{2} t x _{1} (t)x _{2} (t) . \end{array} \right. \end{align} (4.1)

    Obviously, \delta_{i}^{-} = 18, \delta_{i}^{+} = 19, a_i^- = e^{e-2}, a_i^+ = 1.7e^{e-2}, b_{ij}^{-} = b_{ij} ^{+} = 1, \ c_{ij}^{-} = 9.5e^{e-1}, \ c_{ij}^{+} = 9.505e^{e-1}, k_i^+ = 0.1e^{-2} and r_i = e^2 (i, j = 1, 2). Let E_{i1} = e and E_{i2} = 1 for i = 1, 2; we obtain

    A = \left( \begin{array}{ccc} 18 & 0 \\ 0 & 18 \end{array} \right)\; \; , B = \left( \begin{array}{ccc} 19.01e^{e-3}+0.1e^{-1} & 1.7e^{e-2}+0.1e^{-1} \\ 1.7e^{e-2}+0.1e^{-1} & 19.01e^{e-3}+0.1e^{-1} \end{array} \right)
    \begin{align} \sum\limits_{j = 1}^2 \frac{c^{+}_{1j} }{\delta_{1} ^{-}b_{1j}^{-}e}+\frac{a_1^+E_{21}}{\delta_{1} ^{-}} = \frac{19.01e^{e-2}+1.7e^{e-1}}{18} < e \end{align} (4.2)
    \begin{align} \sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{\delta_{2} ^{-}b_{2j}^{-}e}+\frac{a_2^+E_{11}}{\delta_{2} ^{-}} = \frac{19.01e^{e-2}+1.7e^{e-1}}{18} < e \end{align} (4.3)
    \begin{align} \frac{a_{1}^{-}}{\delta_{1} ^{+}}E_{22}+\sum\limits_{j = 1}^n \frac{c^{-}_{1j}}{\delta_{1} ^{+}} E_{11} e^{- b_{1j}^{+}E_{11}}-\frac{k_{1}^+}{\delta_{1} ^{+}}E_{11}E_{21} = \frac{19+e^{e-2}-0.1}{19} > 1 \end{align} (4.4)
    \begin{align} \frac{a_{2}^{-}}{\delta_{2} ^{+}}E_{12}+\sum\limits_{j = 1}^n \frac{c^{-}_{2j}}{\delta_{2} ^{+}} E_{21} e^{- b_{2j}^{+}E_{21}}-\frac{k_{2}^+}{\delta_{2} ^{+}}E_{11}E_{21} = \frac{19+e^{e-2}-0.1}{19} > 1 \end{align} (4.5)
    \begin{align} \rho(A^{-1}B)\approx 0.9946 < 1. \end{align} (4.6)

    Then (4.2) - (4.6) imply that the competitive and cooperative Nicholson's blowfly system (4.1) satisfies (2.4)–(2.6) and (3.1). Hence, from Theorems 3.1 and 3.2, system (4.1) has a positive almost periodic solution

    x^{*}(t )\in B^{*} = \{\varphi|\varphi \in B, 1\leq \varphi_{i}(t) \leq e, \ \mbox{ for all } \ t\in R, i = 1, 2 \} .

    Moreover, if \varphi\in C^{0} = \{\varphi|\varphi \in C, 1 < \varphi_{i}(t) < e, \ for all \ t\in [-e^{2}, 0], i = 1, 2 \}, then x(t; t_{0}, \varphi) converges exponentially to x^{*}(t) as t\rightarrow +\infty. The fact is verified by the numerical simulation illustrated in Figure 1.

    Figure 1.  Numerical solution x(t) = (x_1(t), x_2(t))^T of system (4.1) for the initial value \varphi(t)\equiv (1.5, 2.5)^T.

    Remark 4.1. To the best of our knowledge, few authors have considered the problems related to positive almost periodic solutions of competitive and cooperative Nicholson's blowfly systems. Therefore, the main results in [31,32] and the references therein can not be appled to prove that all solutions of (4.1) with initial the value \varphi\in C^{0} converge exponentially to the positive almost periodic solution. This implies that the results in this paper are new and this complements previously obtained results.

    This article investigated a class of competitive and cooperative Nicholson's blowfly system. Unlike what has been done for some known cooperative Nicholson's blowfly systems [31,32], we have introduced the competitive terms to describe two distinct blowfly populations that compete with each other. By constructing invariant sets and applying the fixed point theorem, we derived some sufficient conditions to ensure that the addressed system has a unique exponential stable positive almost periodic solution. Inspired by the latest Nicholson's blowfly models [3,4,5,6,7,8,9,10], our future works will be devoted to competitive and cooperative Nicholson's blowfly systems involving distinct delays, distributed delays and mixed delays.

    The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Scientific Research Fund of Zhejiang Provincial of China (grant nos. LY18A010019, LY16A010018).

    The author declares no conflicts of interest.



    [1] D. Cucinotta, M. Vanelli, WHO declares COVID-19 a pandemic, Acta Biomed., 91 (2020), 157–160. https://doi.org/10.23750/abm.v91i1.9397 doi: 10.23750/abm.v91i1.9397
    [2] S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. Intern. Med., 172 (2020), 577–582. https://doi.org/10.7326/M20-0504 doi: 10.7326/M20-0504
    [3] A. K. Singh, R. Gupta, A. Misra, Comorbidities in COVID-19: Outcomes in hypertensive cohort and controversies with renin angiotensin system blockers, Diabetes Metab. Syndr. Clin. Res. Rev., 14 (2020), 283–287. https://doi.org/10.1016/j.dsx.2020.03.016 doi: 10.1016/j.dsx.2020.03.016
    [4] Z. Xu, L. Shi, Y. Wang, J. Zhang, L. Huang, C. Zhang, et al., Pathological findings of COVID-19 associated with acute respiratory distress syndrome, Lancet Respir. Med., 8 (2020), 420–422. https://doi.org/10.1016/S2213-2600(20)30076-X doi: 10.1016/S2213-2600(20)30076-X
    [5] W. Guan, Z. Ni, Y. Hu, W. Liang, C. Ou, J. He, et al., Clinical characteristics of coronavirus disease 2019 in China, N. Engl. J. Med., 382 (2020), 1708–1720. https://doi.org/10.1056/NEJMoa2002032 doi: 10.1056/NEJMoa2002032
    [6] World Health Organization, Coronavirus disease 2019 (COVID-19) pandemic, (2020). https://www.who.int/emergencies/diseases/novel-coronavirus-2019 (accessed July 4, 2022).
    [7] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, et al., Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia, N. Engl. J. Med., 382 (2020), 1199–1207. https://doi.org/10.1056/NEJMoa2001316 doi: 10.1056/NEJMoa2001316
    [8] C. del Rio, P. N. Malani, COVID-19—new insights on a rapidly changing epidemic, Jama J. Am. Med. Assoc., 323 (2020), 1339–1340. https://doi.org/10.1001/jama.2020.3072 doi: 10.1001/jama.2020.3072
    [9] J. A. Cui, Y. Sun, H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equations, 20 (2008), 31–53. https://doi.org/10.1007/s10884-007-9075-0 doi: 10.1007/s10884-007-9075-0
    [10] Y. Li, J. A. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear. Sci. Numer. Simul., 14 (2009), 2353–2365. https://doi.org/10.1016/j.cnsns.2008.06.024 doi: 10.1016/j.cnsns.2008.06.024
    [11] J. Rui, Q. Wang, J. Lv, B. Zhao, Q. Hu, H. Du, et al., The transmission dynamics of middle east respiratory syndrome coronavirus, Travel Med. Infect. Dis., 45 (2022), 102243. https://doi.org/10.1016/j.tmaid.2021.102243 doi: 10.1016/j.tmaid.2021.102243
    [12] J. Li, P. Yuan, J. Heffernan, T. Zheng, N. Ogden, B. Sander, et al., Fangcang shelter hospitals during the COVID-19 epidemic, Wuhan, China, Bull. World Health Organ., 98 (2020), 830–841. https://doi.org/10.2471/BLT.20.258152 doi: 10.2471/BLT.20.258152
    [13] L. Wang, J. Wang, H. Zhao, Y. Shi, K. Wang, P. Wu, et al., Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2936–2949. https://doi.org/10.3934/mbe.2020165 doi: 10.3934/mbe.2020165
    [14] B. Yuan, R. Liu, S. Tang, A quantitative method to project the probability of the end of an epidemic: application to the COVID-19 outbreak in Wuhan, 2020, J. Theor. Biol., 545 (2022), 111149. https://doi.org/10.1016/j.jtbi.2022.111149 doi: 10.1016/j.jtbi.2022.111149
    [15] L. Xue, S. Jing, J. C. Miller, W. Sun, H. Li, J. G. Estrada-Franco, et al., A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy, Math. Biosci., 326 (2020), 108391. https://doi.org/10.1016/j.mbs.2020.108391 doi: 10.1016/j.mbs.2020.108391
    [16] C. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708–2724. https://doi.org/10.3934/mbe.2020148 doi: 10.3934/mbe.2020148
    [17] Z. Li, T. Zhang, Analysis of a COVID-19 epidemic model with seasonality, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01105-4 doi: 10.1007/s11538-022-01105-4
    [18] X. Wang, S. Wang, J. Wang, L. Rong, A multiscale model of COVID-19 dynamics, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01058-8 doi: 10.1007/s11538-022-01058-8
    [19] L. Xue, S. Jing, W. Sun, M. Liu, Z. Peng, H. Zhu, Evaluating the impact of the travel ban within mainland China on the epidemic of the COVID-19, Int. J. Infect. Dis., 107 (2021), 278–283. https://doi.org/10.1016/j.ijid.2021.03.088 doi: 10.1016/j.ijid.2021.03.088
    [20] S. Wang, Y. Pan, Q. Wang, H. Miao, A. N. Brown, L. Rong, Modeling the viral dynamics of SARS-CoV-2 infection, Math. Biosci., 328 (2020), 108438. https://doi.org/10.1016/j.mbs.2020.108438 doi: 10.1016/j.mbs.2020.108438
    [21] H. Wan, J. A. Cui, G. J. Yang, Risk estimation and prediction of the transmission of coronavirus disease-2019 (COVID-19) in the mainland of China excluding Hubei province, Infect. Dis. Poverty, 9 (2020). https://doi.org/10.1186/s40249-020-00683-6 doi: 10.1186/s40249-020-00683-6
    [22] K. S. Al-Basyouni, A. Q. Khan, Discrete-time COVID-19 epidemic model with chaos, stability and bifurcation, Results. Phys., 43 (2022), 106038. https://doi.org/10.1016/j.rinp.2022.106038 doi: 10.1016/j.rinp.2022.106038
    [23] A. Abbes, A. Ouannas, N. Shawagfeh, G. Grassi, The effect of the Caputo fractional difference operator on a new discrete COVID-19 model, Results Phys., 39 (2022), 105797. https://doi.org/10.1016/j.rinp.2022.105797 doi: 10.1016/j.rinp.2022.105797
    [24] S. He, J. Yang, M. He, D. Yan, S. Tang, L. Rong, The risk of future waves of COVID-19: modeling and data analysis, Math. Biosci. Eng., 18 (2021), 5409–5426. https://doi.org/10.3934/mbe.2021274 doi: 10.3934/mbe.2021274
    [25] P. Y. Liu, S. He, L. B. Rong, S. Y. Tang, The effect of control measures on COVID-19 transmission in Italy: comparison with Guangdong province in China, Infect. Dis. Poverty, 9 (2020). https://doi.org/10.1186/s40249-020-00730-2 doi: 10.1186/s40249-020-00730-2
    [26] H. Song, Z. Jia, Z. Jin, S. Liu, Estimation of COVID-19 outbreak size in Harbin, China, Nonlinear. Dyn., 106 (2021), 1229–1237. https://doi.org/10.1007/s11071-021-06406-2 doi: 10.1007/s11071-021-06406-2
    [27] X. Ma, X. F. Luo, L. Li, Y. Li, G. Q. Sun, The influence of mask use on the spread of COVID-19 during pandemic in New York City, Results Phys., 34 (2022), 105224. https://doi.org/10.1016/j.rinp.2022.105224 doi: 10.1016/j.rinp.2022.105224
    [28] J. K. K. Asamoah, E. Okyere, A. Abidemi, S. E. Moore, G. Q. Sun, Z. Jin, et al., Optimal control and comprehensive cost-effectiveness analysis for COVID-19, Results Phys., 33 (2022), 105177. https://doi.org/10.1016/j.rinp.2022.105177 doi: 10.1016/j.rinp.2022.105177
    [29] L. Masandawa, S. S. Mirau, I. S. Mbalawata, J. N. Paul, K. Kreppel, O. M. Msamba, Modeling nosocomial infection of COVID-19 transmission dynamics, Results Phys., 37 (2022), 105503. https://doi.org/10.1016/j.rinp.2022.105503 doi: 10.1016/j.rinp.2022.105503
    [30] C. Legarreta, S. Alonso-Quesada, M. De la Sen, Analysis and parametrical estimation with real COVID-19 data of a new extended SEIR epidemic model with quarantined individuals, Discrete Dyn. Nat. Soc., 2022 (2022), 1–29. https://doi.org/10.1155/2022/5151674 doi: 10.1155/2022/5151674
    [31] M. De la Sen, A. Ibeas, On an SE (Is) (Ih) AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic, Adv. Differ. Equations, 2021 (2021). https://doi.org/10.1186/s13662-021-03248-5 doi: 10.1186/s13662-021-03248-5
    [32] M. Rangasamy, N. Alessa, P. B. Dhandapani, K. Loganathan, Dynamics of a novel IVRD pandemic model of a large population over a long time with efficient numerical methods, Symmetry, 14 (2022), 1919. https://doi.org/10.3390/sym14091919 doi: 10.3390/sym14091919
    [33] A. K. Paul, M. A. Kuddus, Mathematical analysis of a COVID-19 model with double dose vaccination in Bangladesh, Results Phys., 35 (2022), 105392. https://doi.org/10.1016/j.rinp.2022.105392 doi: 10.1016/j.rinp.2022.105392
    [34] U. A. P. de León, E. Avila-Vales, K. Huang, Modeling COVID-19 dynamic using a two-strain model with vaccination, Chaos Solitons Fractals, 157 (2022), 111927. https://doi.org/10.1016/j.chaos.2022.111927 doi: 10.1016/j.chaos.2022.111927
    [35] X. Wang, H. Wu, S. Tang, Assessing age-specific vaccination strategies and post-vaccination reopening policies for COVID-19 control using SEIR modeling approach, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01064-w doi: 10.1007/s11538-022-01064-w
    [36] F. Zhang, Z. Jin, Effect of travel restrictions, contact tracing and vaccination on control of emerging infectious diseases: transmission of COVID-19 as a case study, Math. Biosci. Eng., 19 (2022), 3177–3201. https://doi.org/10.3934/mbe.2022147 doi: 10.3934/mbe.2022147
    [37] J. H. Buckner, G. Chowell, M. R. Springborn, Dynamic prioritization of COVID-19 vaccines when social distancing is limited for essential workers, Appl. Biol. Sci., 118 (2021). https://doi.org/10.1073/pnas.2025786118 doi: 10.1073/pnas.2025786118
    [38] S. Moore, E. M. Hill, M. J. Tildesley, L. Dyson, M. J. Keeling, Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study, Lancet Infect. Dis., 21 (2021), 793–802. https://doi.org/10.1016/S1473-3099(21)00143-2 doi: 10.1016/S1473-3099(21)00143-2
    [39] Z. Ma, Y. Zhou, C. Li, Qualitative and stability methods of ordinary differential equations (in Chinese), 2nd ed, Science Press, Beijing, 2015.
    [40] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
    [41] S. Guo, W. Ma, Remarks on a variant of Lyapunov-LaSalle theorem, Math. Biosci. Eng., 16 (2019), 1056–1066. https://doi.org/10.3934/mbe.2019050 doi: 10.3934/mbe.2019050
    [42] S. Guo, Y. Xue, X. Li, Z. Zheng, Dynamics of COVID-19 models with asymptomatic infections and quarantine measures, arXiv preprint, (2022). https://doi.org/10.21203/rs.3.rs-2291574/v1 doi: 10.21203/rs.3.rs-2291574/v1
    [43] Italian Ministry of Health, COVID-19 Vaccines Report, (2022). Available from: https://www.governo.it/it/cscovid19/report-vaccini/(accessed January 5, 2023).
    [44] United States Food and Drug Administration, FDA Briefing Document Pfizer-BioNTech COVID-19 Vaccine, (2020). Available from: https://www.fda.gov/media/144245/download.
    [45] Presidency of the Council of Ministers, DECREE-LAW No. 172 of December 18, 2020, (2020). Available from: https://www.normattiva.it/uri-res/N2Ls?urn:nir:stato:decreto.legge:2020-12-18;172!vig= (accessed October 9, 2022).
    [46] Governo Italiano, Council of Ministers Press Release No. 97, (2021). Available from: https://www.sitiarcheologici.palazzochigi.it/www.governo.it/febbraio%202021/node/16180.html (accessed October 8, 2022).
    [47] Italian Ministry of Health, OJ General Series No. 75, 27-03-2021, (2021). Available from: https://www.gazzettaufficiale.it/eli/id/2021/03/27/21A01967/sg (accessed October 9, 2022).
    [48] Italy Civil Protection Department, Italian COVID-19 data, (2022). Available from: https://github.com/pcm-dpc/COVID-19 (accessed January 5, 2023).
    [49] World Bank, Average life expectancy in Italy, (2020). Available from: https://data.worldbank.org/indicator/SP.DYN.LE00.IN?locations=IT (accessed January 5, 2023).
    [50] World Bank, Italy Birth rate, crude (per 1,000 people), (2020). Available from: https://data.worldbank.org/indicator/SP.DYN.CBRT.IN?locations=IT (accessed January 5, 2023).
    [51] Our World in Data, Italian COVID-19 vaccine dataset, (2022).
    [52] World Bank, Italian population data. Available from: https://data.worldbank.org/indicator/SP.POP.TOTL?locations=IT (accessed January 5, 2023).
    [53] N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of Malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
    [54] H. Tian, Y. Liu, Y. Li, C. H. Wu, B. Chen, M. U. G. Kraemer, et al., An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China, Science, 368 (2020), 638–642. https://doi.org/10.1126/science.abb6105 doi: 10.1126/science.abb6105
    [55] M. Duan, Z. Jin, The heterogeneous mixing model of COVID-19 with interventions, J. Theor. Biol., 553 (2022), 111258. https://doi.org/10.1016/j.jtbi.2022.111258 doi: 10.1016/j.jtbi.2022.111258
    [56] J. A. Cui, Y. Wu, S. Guo, Effect of non-homogeneous mixing and asymptomatic individuals on final epidemic size and basic reproduction number in a meta-population model, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-00996-7 doi: 10.1007/s11538-022-00996-7
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2912) PDF downloads(196) Cited by(8)

Figures and Tables

Figures(24)  /  Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog