Loading [MathJax]/jax/output/SVG/jax.js
Research article

Consumers' stance on food waste in the Czech Republic, Poland, and Slovakia

  • Received: 23 May 2022 Revised: 21 July 2022 Accepted: 26 July 2022 Published: 16 August 2022
  • This work focuses on the area of food waste from the subjective perspective of the consumers themselves. The key source of data is a questionnaire survey with a sample selection of 3,429 respondents from the Czech Republic, Poland, and Slovakia, which are countries with historical and cultural ties. This survey was orchestrated in 2019 and it aimed to uncover the consumers' stances on this area. For every country involved, the results proved that the most common reason for food waste is that the food spoils when stored. One's stance on food waste is influenced by identifiers such as age, education, economic activity, and perceived income. A significant difference between the surveyed countries can be seen in the fact that Poles have over a 10× greater chance of wasting the smallest amount of food. Descriptive statistics, contingency analysis, and logistic regression were used to analyze the data. It is evident that subsequent research is necessary in this area, ideally with a united methodology for every country in the European Union.

    Citation: Naďa Hazuchová, Jana Stávková, Agnieszka Siedlecka, Ľudmila Nagyová. Consumers' stance on food waste in the Czech Republic, Poland, and Slovakia[J]. AIMS Agriculture and Food, 2022, 7(3): 637-658. doi: 10.3934/agrfood.2022040

    Related Papers:

    [1] Hongfan Lu, Yuting Ding, Silin Gong, Shishi Wang . Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19. Mathematical Biosciences and Engineering, 2021, 18(4): 3197-3214. doi: 10.3934/mbe.2021159
    [2] Sarita Bugalia, Jai Prakash Tripathi, Hao Wang . Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy. Mathematical Biosciences and Engineering, 2021, 18(5): 5865-5920. doi: 10.3934/mbe.2021295
    [3] A. Q. Khan, M. Tasneem, M. B. Almatrafi . Discrete-time COVID-19 epidemic model with bifurcation and control. Mathematical Biosciences and Engineering, 2022, 19(2): 1944-1969. doi: 10.3934/mbe.2022092
    [4] Xinyu Liu, Zimeng Lv, Yuting Ding . Mathematical modeling and stability analysis of the time-delayed SAIM model for COVID-19 vaccination and media coverage. Mathematical Biosciences and Engineering, 2022, 19(6): 6296-6316. doi: 10.3934/mbe.2022294
    [5] Yuting Ding, Gaoyang Liu, Yong An . Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion. Mathematical Biosciences and Engineering, 2022, 19(2): 1154-1173. doi: 10.3934/mbe.2022053
    [6] Qingwen Hu . A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences and Engineering, 2018, 15(4): 863-882. doi: 10.3934/mbe.2018039
    [7] Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006
    [8] Tao Chen, Zhiming Li, Ge Zhang . Analysis of a COVID-19 model with media coverage and limited resources. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307. doi: 10.3934/mbe.2024233
    [9] Honghua Bin, Daifeng Duan, Junjie Wei . Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay. Mathematical Biosciences and Engineering, 2023, 20(7): 12194-12210. doi: 10.3934/mbe.2023543
    [10] Huan Dai, Yuying Liu, Junjie Wei . Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays. Mathematical Biosciences and Engineering, 2020, 17(4): 4127-4146. doi: 10.3934/mbe.2020229
  • This work focuses on the area of food waste from the subjective perspective of the consumers themselves. The key source of data is a questionnaire survey with a sample selection of 3,429 respondents from the Czech Republic, Poland, and Slovakia, which are countries with historical and cultural ties. This survey was orchestrated in 2019 and it aimed to uncover the consumers' stances on this area. For every country involved, the results proved that the most common reason for food waste is that the food spoils when stored. One's stance on food waste is influenced by identifiers such as age, education, economic activity, and perceived income. A significant difference between the surveyed countries can be seen in the fact that Poles have over a 10× greater chance of wasting the smallest amount of food. Descriptive statistics, contingency analysis, and logistic regression were used to analyze the data. It is evident that subsequent research is necessary in this area, ideally with a united methodology for every country in the European Union.



    COVID-19 is a respiratory infectious disease and was first reported in Wuhan, China. It is caused by a novel coronavirus named SARS-CoV-2. SARS-COV-2 appeared after severe acute respiratory syndrome coronavirus (SARS) and Middle East respiratory syndrome coronavirus (MERS) [1]. Coronaviruses are enveloped nonsegmented positive-sense RNA viruses and are broadly distributed in mammals, including humans. They can destroy the human respiratory systems and cause severe acute respiratory syndrome [2]. Infected individuals usually cough, have a fever, and have difficulty breathing. Some of them even die of COVID-19. Moreover, there are many ways to spread COVID-19, such as direct transmission, aerosol transmission and contact transmission. This caused COVID-19 to spread rapidly worldwide. As a global infectious disease, COVID-19 has caused more than one million deaths. We attempt to establish a model to analyze the spread of COVID-19 and provide some advice to make COVID-19 controllable.

    In recent investigations, many researchers have studied different epidemic models applied to the spread of COVID-19. In ref. [3] and ref. [4], Varotsos et al. established an SPRD model that included people that were susceptible, infected, recovered and dead. The model was based on COVID-19 data and was established to predict COVID-19 spread. We think there is no need to include people who died as a main part of the model because the proportion of people who died is very small. In ref. [5], León et al. developed an SEIARD mathematical model to investigate the COVID-19 outbreak in Mexico. The SEIARD model included infection with symptoms, infection without symptoms (asymptomatic), recovery from symptomatic infection and recovery from asymptomatic infection. If we consider only the infectivity of COVID-19, both infection with symptoms and infection without symptoms (asymptomatic) can infect the suspected cases. There is no need to divide them into infected with symptoms and infected without symptoms (asymptomatic). Both the recovery from symptomatic infection and the recovery from asymptomatic infection have very small possibilities for reinfection. There is no need to divide them into recovered from symptomatic infection and recovered from asymptomatic infection. In ref. [6], an SEIR epidemic model was developed for COVID-19. In ref. [7] and ref. [8], two SIR epidemic models were developed for COVID-19. For COVID-19, many countries have taken quarantine measures, which play an important part in the spread of COVID-19. These authors did not consider the influence of these quarantine measures. The quarantine measures taken by many countries make the proportion of quarantined people large. These quarantine measures can prevent infected individuals from infecting suspected individuals. In addition, exposure to air pollution can damage the heart and lungs and increase vulnerability to more serious coronavirus effects [9]. These quarantine measures can reduce the influence of air pollution. We think it is necessary to consider quarantined people because of the importance of these quarantine measures and the number of quarantined people. We attempt to construct an SIQR epidemic model including quarantined people to describe the spread of COVID-19.

    In the spreading process of COVID-19, there is a time delay from infection to isolation. There are some studies about other epidemics with time delays. Liu et al. [10] developed an SEIRU epidemic model with a time delay before an infected person can transmit the infection to another person. They evaluated the effect of the latency period on the dynamics of COVID-19. Xu [11] and Yang et al. [12] also developed epidemic models with time delays before an infected person can transmit the infection to another person. Although this kind of time delay exists, human beings have difficulty changing this kind of time delay. In addition, Wang et al. [13] and Liu et al. [14] made time delays in their models the sojourn times in an infective state. Lu et al. [15] presented an SIQR model with a time delay from infection to recovery. The influence of the time delay from infection to recovery was discussed in detail. According to their conclusions, the smaller the time delay from infection to isolation was, the better COVID-19 was controlled. We decide to focus on the time delay from infection to isolation and discuss the effect of the time delay from infection to isolation on the spread of COVID-19, which is the main difference between our work and the works of others.

    Stability and bifurcation have great significance to epidemic models, and some studies have analyzed stability and bifurcation for some epidemic models. In ref. [16], Greenhalgh investigated the stability of some SEIRS epidemiological models with vaccination and temporary immunity. Greenhalgh proposed that there was a threshold parameter R0, and the disease could persist if and only if R0 exceeded one. In addition, disease-free equilibrium always existed and was locally stable if R0<1 and unstable if R0>1. However, for R0>1, the endemic equilibrium was unique and locally asymptotically stable. Based on this conclusion, we think it is necessary for us to calculate the threshold parameter R0 in our model. In ref. [17], Xie et al. investigated the global stability of endemic equilibrium of an SIS epidemic model in complex networks. They proved that the endemic equilibrium was globally asymptotically stable by using a combination of the Lyapunov function method and a monotone iterative technique. However, the Hopf bifurcation analysis was not mentioned in this paper. In refs. [18,19], two SIS models were proposed. In ref. [20], an SIQRS model was proposed. The stability of the models was analyzed, but bifurcation analysis was not mentioned. In refs. [21,22,23,24], the stability and bifurcation of some SIR models were analyzed by different authors. In refs. [25,26,27,28], different SIR models were developed. The stability and bifurcation of them were analyzed. In ref. [29], an SIR epidemic model with time delay was proposed. In ref. [30], an SIR model with information-dependent vaccination was proposed. We compare the methods they used to analyze the stability and bifurcation of these SIR models. Then, we use the multiple time scales method, which was also used in ref. [21]. The multiple time scales method is systematic. It is a standard method for calculating the normal form of Hopf bifurcation. It can be directly applied to the original nonlinear dynamical system, which is described by ordinary differential equations and delayed differential equations, without application of the center manifold theory [31].

    It is obvious that the isolation measures taken by many countries play an important role in controlling the spread of COVID-19. The time delay also has a considerable influence on the spread. Thus, we attempt to develop one mathematical model with a time delay to investigate the spread of COVID-19 and analyze the stability of this model. In some studies such as refs. [3,4,5], the authors forecasted how the epidemic situation changed with detailed data. However, we decide to focus on the stability of equilibria in our model, calculate the critical time delay from infection to isolation and analyze the dynamical property of our model to discuss the tendency of the spread of COVID-19.

    The main objective of this paper includes constructing the DDE model for COVID-19 and analyzing the stability of the model, the existence of Hopf bifurcation and the stability of the bifurcating periodic solution. The rest of the paper is organized as follows. In Section 2, we construct the DDE model according to the characteristics of the spread of COVID-19. In Section 3, we analyze the existence and stability of the equilibria and the existence of Hopf bifurcation associated with the COVID-19 model with a time delay. We calculate the critical time delay from infection to isolation and analyze the dynamic properties of our model. In Section 4, we derive the normal form of the Hopf bifurcation for the above model and analyze the stability of the bifurcating periodic solution. In Section 5, we present the results of simulations to verify the correctness of our analysis. Finally, the conclusion is drawn in Section 6.

    Suspected individuals can be infected with COVID-19 by infected individuals. After individuals are infected with COVID-19, some of them will be quarantined, and some of them will die of COVID-19. The quarantined individuals will not infect others. For COVID-19, we ignore the effect of reinfection because the probability of reinfection is very small. In addition, we ignore the probability of transforming the suspected individuals into recovered individuals directly since the probability is also small. We divide people into four kinds (suspected, infected, quarantined and recovered) based on their infectivity. The specific conversion between the four kinds of people is given in Figure 1.

    Figure 1.  Flow chart for the model SIQR.

    The variables and parameters in Figure 1 are given in Table 1.

    Table 1.  The definitions of parameters and variables.
    Symbol Definition
    S The number of suspected people
    I The number of infected people
    Q The number of quarantined people
    R The number of recovered people
    Λ The population growth
    μ The natural mortality
    α1 The rate of COVID-19 death of infected people
    α2 The rate of COVID-19 death of quarantined people
    β The transmission rate from S to I
    ε The transmission rate from I to Q
    γ1 The transmission rate from I to R
    γ2 The transmission rate from Q to R

     | Show Table
    DownLoad: CSV

    For COVID-19, there is a latent period after the suspected individuals are infected. In the latent period, the infected individuals have no symptoms. However, these asymptomatic infected individuals can also infect suspected individuals. If we consider only their infectivity, there is little difference between asymptomatic infected individuals and symptomatic infected individuals. Thus, both of them can be divided into infected. Additionally, some suspected individuals may be quarantined, and they will not be infected. This means that the number of suspected persons who can be infected will decrease. Considering the influence of the suspected individuals who are quarantined, we can reduce the transmission rate from S to I, and there is no need to include the suspected individuals who are quarantined in the model.

    Based on Figure 1, we construct the DDE model (2.1):

    {S(t)=ΛμS(t)βS(t)I(t),I(t)=βS(t)I(t)εI(tτ)μI(t)α1I(t)γ1I(t),Q(t)=εI(tτ)γ2Q(t)α2Q(t)μQ(t),R(t)=γ1I(t)+γ2Q(t)μR(t), (2.1)

    where the definitions of parameters and variables are presented in Figure 1 and Table 1, and τ>0 denotes the time delay from infection to isolation.

    The initial condition of system (2.1) is ϕS(θ)0,ϕI(θ)0,ϕQ(θ)0,ϕR(θ)0, where Φ=(ϕS(θ),ϕI(θ),ϕQ(θ),ϕR(θ))C([τ,0],R4+0) for θ[τ,0], and C([τ,0],R4+0) is the Banach space of continuous functions mapping interval [τ,0] into R4+0. For the system (2.1), R4+0={(S,I,Q,R)|S0,I0,Q0,R0}.

    According to the initial condition of system (2.1), we present a theorem addressing the nonnegativity and the boundedness of the solution to system(2.1).

    Theorem 2.1. If S(0)0,I(0)0,Q(0)0,R(0)0, the solution S(t), I(t), Q(t), R(t) to system (2.1) with τ=0 is nonnegative and bounded when t>0.

    Proof. First, we prove S(t)0 when t0 under the initial condition of system (2.1).

    We assume that S(t) is not always nonnegative for t0 and make t1 the first time that S(t1)=0, S(t1)<0. According to the first equation of system (2.1), we can obtain S(t1)=Λ>0. The two conclusions we obtain are contradictory.

    Therefore, S(t)0 when t>0. In the same way, I(t)0, Q(t)0, R(t)0 when t>0. The solution to system (2.1) is positive when t>0.

    Then, let N(t)=S(t)+I(t)+Q(t)+R(t), and N(t) represent the total number of people at time t.

    Add four equations of Eq. (2.1), and we obtain N(t)=Λμ(S(t)+I(t)+Q(t)+R(t)). Then, we can obtain limtsupN(t)=Λμ. Thus, the solution S(t), I(t), Q(t), R(t) to system (2.1) is bounded when t>0.

    Remark 1: We prove if S(0)0,I(0)0,Q(0)0,R(0)0, the solution S(t), I(t), Q(t), R(t) to system (2.1) with τ=0 is positive. It is not easy for us to prove that the solution to system (2.1) is positive when τ>0. However, according to our numerical simulation, we can find that when system (2.1) is stable, the solution to system (2.1) is always positive, which is not contradictory to the positivity of the solution to system (2.1).

    Next, we consider the dynamic phenomena of system (2.1).

    In this section, the system (2.1) is considered, and we first determine the equilibria of the above system. Obviously, the system (2.1) has two equilibria:

    E1=(S1,I1,Q1,R1),E2=(S2,I2,Q2,R2), (3.1)

    with

    S1=Λμ,I1=0,Q1=0,R1=0,S2=ε+μ+α1+γ1β,I2=Λε+μ+α1+γ1μβ,Q2=εα2+γ2+μ(Λε+μ+α1+γ1μβ),R2=1μ(γ1+εγ2α2+γ2+μ)(Λε+μ+α1+γ1μβ). (3.2)

    Transferring the equilibria Ek(k=1,2) to the origin, we make w=S+Sk, x=I+Ik, y=Q+Qk, z=R+Rk. We can obtain S=wSk, I=xIk, Q=yQk and R=zRk. Substituting these equations into the model (2.1), we use S, I, Q and R to represent w, x, y and z. We obtain the following model:

    {S(t)=μS(t)βS(t)I(t)βSkI(t)βS(t)Ik,I(t)=βS(t)I(t)+βSkI(t)+βS(t)IkεI(tτ)μI(t)α1I(t)γ1I(t),Q(t)=εI(tτ)γ2Q(t)α2Q(t)μQ(t),R(t)=γ1I(t)+γ2Q(t)μR(t),k=1,2. (3.3)

    In this subsection, we first demonstrate the stability of the equilibrium E1=(S1,I1,Q1,R1)=(Λμ,0,0,0). We obtain the characteristics equation of the linearized system of Eq. (3.3) at the equilibrium E1 as follows:

    (λ+μ)2(λ+μ+α2+γ2)(λ+μ+α1+γ1βS1+εeλτ)=0. (3.4)

    When τ=0, Eq. (3.4) becomes:

    (λ+μ)2(λ+μ+α2+γ2)(λ+μ+α1+γ1βS1+ε)=0. (3.5)

    We show the following assumption:

    (H1) βS1(ε+μ+α1+γ1)<0.

    Under (H1), the four roots of Eq. (3.5) have negative real parts due to μ>0,α1>0,γ1>0, and the equilibrium E1 is locally asymptotically stable when τ=0.

    Similarly, for the stability of the other equilibrium E2=(S2,I2,Q2,R2), we obtain the characteristic equation of the linearized system of Eq. (3.3) at the equilibrium E2 as follows:

    (λ+μ)(λ+μ+α2+γ2)[(λε+εeλτ)(λ+μ+βI2)+β2S2I2]=0. (3.6)

    When τ=0, Eq. (3.6) becomes:

    (λ+μ)(λ+μ+α2+γ2)[λ2+λ(μ+βI2)+β2S2I2]=0. (3.7)

    If (H1) is not satisfied, the four roots of Eq. (3.7) have negative real parts due to μ>0,α2>0,γ2>0, and the equilibrium E2 of system (2.1) is locally asymptotically stable when τ=0.

    We can find that E1 is the disease-free equilibrium, and E2 is the endemic equilibrium. We calculate the basic reproduction number R0, which is the number of suspected individuals who are infected by the same infectious individual and can estimate the infectiousness of an infectious disease. According to the system (2.1), we can obtain the new infections matrix F and the transition matrix V.

    F=[0βS(t)I(t)00],V=[Λ+μS(t)+βS(t)I(t)εI(t)+μI(t)+α1I(t)+γ1I(t)εI(t)+γ2Q(t)+α2Q(t)+μQ(t)γ1I(t)γ2Q(t)+μR(t)].

    Then, we make F0 represent the derivative of F at E1 and V0 represent the derivative of V at E1:

    F0=[00000βS10000000000],V0=[μβS1000ε+μ+α1+γ1000εγ2+α2+μ00γ1γ2 μ].

    The inverse of V0 is:

    V10=[1μβS1μ(ε+μ+α1+γ1)0001ε+μ+α1+γ1000ε(ε+μ+α1+γ1)(γ2+α2+μ)1γ2+α2+μ001μ[γ1ε+μ+α1+γ1+εγ2(ε+μ+α1+γ1)(γ2+α2+μ)]γ2μ(γ2+α2+μ)1μ.].

    And we can obtain:

    F0V10=[00000βS1ε+μ+α1+γ10000000000].

    The maximum eigenvalue of F0V10 is R0:

    R0=ρ(F0V10)=βS1ε+μ+α1+γ1.

    We compare R0 with (H1), and we find that if R0<1, the inequality of (H1) exists constantly, and if R0>1, (H1) is not satisfied. This represents if R0<1, the disease-free equilibrium E1 is locally asymptotically stable when τ=0 and if R0>1, the endemic equilibrium E2 is locally asymptotically stable when τ=0.

    For equilibrium E1, when τ>0, let λ=iω0(ω0>0) be a root of the following equation:

    λ+μ+α1+γ1βS1+εeλτ=0. (3.8)

    Substituting λ=iω0 into Eq. (3.8) and separating the real and imaginary parts, we have

    {ω0=εsin(ω0τ),βS1(μ+α1+γ1)=εcos(ω0τ). (3.9)

    Thus, we obtain:

    {Q1sin(ω0τ)=ω0ε,P1cos(ω0τ)=βS1(μ+α1+γ1)ε. (3.10)

    Adding the square of two equations (3.9), we can obtain

    ω20=ε2[βS1(μ+α1+γ1)]2. (3.11)

    Thus, ω0=ε2[βS1(μ+α1+γ1)]2 when ε2[βS1(μ+α1+γ1)]20, and substituting it into Eq. (3.10). Since ω0>0 and ε0, we can obtain Q1>0 and obtain:

    τ(j)1=1ω0[arccos(P1)+2jπ],Q10, (3.12)

    where Q1,P1 are given in Eq. (3.10), j=0,1,2.

    Let λ(τ)=α(τ)+iω(τ) be the root of Eq. (3.8) satisfying α(τ(j)1)=0,ω(τ(j)1)=ω0,j=0,1,2. Then, we have the transversality conditions

    Re(dλdτ)1|λ=iω0,τ=τ(j)1=1ε2>0,j=0,1,2.

    Thus, the system (2.1) undergoes a Hopf bifurcation at equilibrium E1 of system (2.1) when τ=τ(j)1,j=0,1,2,.

    For the other equilibrium E2, when τ>0, let λ=iω(ω>0) be the root of the following equation:

    (λε+εeλτ)(λ+μ+βI2)+β2S2I2=0. (3.13)

    Substituting it into the equation and separating the real and imaginary parts, we have

    {ω2+με+εβI2β2S2I2=(ωε)sin(ωτ)+(με+βεI2)cos(ωτ),μωεω+βωI2=(ωε)cos(ωτ)+(με+βεI2)sin(ωτ). (3.14)

    Equation (3.14) can be presented as:

    {Q2sin(ωτ)=(ω2+ε(βI2+μ)β2S2I2)ω(ε(βI2+μ))(βI2+μ)ωε[ω2+(βI2+μ)2],P2cos(ωτ)=(ω2+ε(βI2+μ)β2S2I2)(βI2+μ)+(ε(βI2+μ))ω2ε[ω2+(βI2+μ)2]. (3.15)

    Adding the square of two equations Eq. (3.14), let z=ω2, and

    h(z)=z2+c1z+c0=0, (3.16)

    where c1=(βI2+μ)22β2S2I2,c0=(β2S2I2)22β2S2I2(βI2+μ)ε.

    Therefore, we have the following assumption:

    (H2) c214c0>0,c1<0,c0>0.

    (H3) c0<0.

    If (H2) holds, then Eq. (3.16) has two positive roots, z1 and z2 (z1<z2). If (H3) holds, then Eq. (3.16) has only one positive root z3. Without loss of generality, substituting ωk=zk(k=1,2,3) into Eq. (3.15), we obtain:

    τ(j)2,k={1ωk[arccos(P2,k)+2jπ],Q2,k0,1ωk[2πarccos(P2,k)+2jπ],Q2,k<0,j=0,1,2, (3.17)

    where Q2,P2 are given in Eq. (3.15) and Q2,k=Q2|ω=ωk,τ=τ(j)2,k,P2,k=P2|ω=ωk,τ=τ(j)2,k,k=1,2,3.

    Note that since Q1=ω0ε is always nonnegative, we can obtain τ(j)1 directly. However, whether Q2 is positive depends on the parameters of Eq. (3.15), and we need to obtain τ(j)2,k, (j=1,2,3) according to the value of Q2.

    Furthermore, let λ(τ)=α(τ)+iωk(τ) be the root of Eq. (3.17) satisfying α(τ(j)2,k)=0, ω(τ(j)2,k)=ωk(k=1,2,3;j=0,1,2).

    If (H2) or (H3) holds and zk=ω2k(k=1,2,3), then we can deduce Re(dτdλ)|τ=τ(j)2,k:

    Re(dτdλ)|τ=τ(j)2,k=ωk2[(βI2+μ)22β2S2I2]ε2[ωk2+(βI2+μ)2]=ωk2h(zk)ε2[ωk2+(βI2+μ)2]0,k=1,2,3.

    There is a Hopf bifurcation at equilibrium E2 of system (2.1) when τ=τ(j)2,k,j=0,1,2.

    Then, we present a stability theorem of system (2.1)'s equilibria E1, E2 and the existence of Hopf bifurcation.

    Theorem 3.1. We consider the model (2.1):

    (1) If (H1) holds, the equilibrium E1 of the model (2.2) undergoes a Hopf bifurcation when τ=τ(j)1,j=0,1,2, and we obtain the following. When τ[0,τ(0)1), the equilibrium E1 is locally asymptotically stable. When τ[τ(0)1,+), the equilibrium E1 is locally asymptotically unstable.

    (2) Under the condition that (H1) is not satisfied, and if (H2) or (H3) holds, the equilibrium E2 of the model (2.1) undergoes a Hopf bifurcation when τ=τ(j)2,k(k=1,2,3), and we obtain the following.

    (a) If (H2) holds, h(z) has two positive roots, z1 and z2; then we assume z1<z2, and we obtain h(z1)<0,h(z2)>0. Then, mN, which can make 0<τ(0)2,2<τ(0)2,1<τ(1)2,2<τ(1)2,1<<τ(m1)2,1<τ(m)2,2<τ(m+1)2,2. When τ[0,τ(0)2,2)ml=1(τ(l1)2,1,τ(l)2,2), the equilibrium of the model is locally asymptotically stable. When τm1l=0(τ(l)2,2,τ(l)2,1)(τ(m)2,2,+), the equilibrium is locally asymptotically unstable.

    (b) If (H3) holds, h(z) has only one positive root z3, when τ[0,τ(0)2,3), the equilibrium E2 is locally asymptotically stable. When τ(τ(0)2,3,+), the equilibrium E2 is unstable.

    Without loss of generality, we denote the critical value τ=τ when the characteristic Eq. (3.4) and Eq. (3.6) have eigenvalue λ=iωk,k=1,2, at which system (2.1) undergoes a Hopf bifurcation at equilibrium Ek=(Sk,Ik,Qk,Rk),k=1,2.

    First, we transform the equilibrium to the origin; then, we use the multiple time scales method, and system (2.1) can be written as

    ˙X(t)=AX(t)+BX(tτ)+F(X(t),X(tτ)), (4.1)

    where X(t)=(^Sk(t),^Ik(t),^Qk(t),^Rk(t))T, X(tτ)=(^Sk(tτ),^Ik(tτ),^Qk(tτ),^Rk(tτ))T,

    A=[μβIkβSk00βIkβSk(μ+α1+γ1)0000(γ2+α2+μ)00γ1γ2μ],B=[00000ε000ε000000],F(X(t),X(tτ))=[FSFIFQFR]=[ΛμSkβSkIkβSkIkβSkIk+βSkIk(ε+μ+α1+γ1)IkεIk(μ+α2+γ2)Qkγ1Ik+γ2QkμRk].

    We make ˜˙X(t)=AX(t)+BX(tτ) the linear system of system (4.1). We assume that hk(k=1,2) is the eigenvector of the eigenvalue λ=iωk(k=1,2) of the linear system and that hk is the eigenvector of the eigenvalue λ=iωk of the linear system. We make ˜I the unit matrix, and hk satisfies (λ˜IABeλτ)hk=0, where λ=iωk, and hk satisfies (λ˜IABeλτ)Thk=0, where λ=iωk. Additionally, hk,hk=¯hkThk=1, and we can obtain

    hk=(hk1,hk2,hk3,hk4)T=(βSkλ+μ+βIk,1,εeλτλ+μ+α2+γ2,γ1λ+μ+γ2εeλτ(λ+μ)(λ+μ+α2+γ2))T,hk=(hk1,hk2,hk3,hk4)T=dk(βIkλ+μ+βIk,1,0,0)T, (4.2)

    where dk=(β2SkIk(μ+βIk)2+ωk2+1)1,k=1,2.

    We use the multiple time scale method to deduce the normal form of the Hopf bifurcation and assume the solution to Eq. (4.1) is

    X(t)=X(T0,T1,T2,)=k=1ϵkXk(T0,T1,T2,), (4.3)

    where X(T0,T1,T2,)=[S(T0,T1,T2,),I(T0,T1,T2,),Q(T0,T1,T2,),R(T0,T1,T2,)]T, Xk(T0,T1,T2,)=[Sk(T0,T1,T2,),Ik(T0,T1,T2,),Qk(T0,T1,T2,),Rk(T0,T1,T2,)]T, k=1,2, Ti=ϵit,i=0,1,2, and Ti is the scaling transform in the time direction.

    The derivative with respect to t is

    ddt=T0+ϵT1+ϵ2T2+=D0+ϵD1+ϵ2D2+,

    where Di=Ti,i=0,1,2,.

    Note that Xj=(Sj,Ij,Qj,Rj)T=Xj(T0,T1,T2,),Xj,τc=(Sj,τc,Ij,τc,Qj,τc,Rj,τc)T=Xj(T0τc,T1,T2,),j=1,2,.

    Then, we can obtain

    X(t)=ϵD0X1+ϵ2D1X1+ϵ3D2X1+ϵ2D0X2+ϵ3D1X2+ϵ3D0X3+. (4.4)

    We consider that τ is the bifurcation parameter, and we set τ=τc+ϵτε, where τc=τ(j)k(k=1,2) is the critical value of the Hopf bifurcation, τε is the perturbation parameter, and ϵ is the dimensionless parameter.

    Using Taylor series expansion of X(tτ), we obtain that

    X(tτ)=ϵX1,τc+ϵ2X2,τc+ϵ3X3,τcϵ2τεD0X1,τcϵ3τεD0X2,τcϵ2τcD1X1,τcϵ3τεD1X1,τcϵ3τcD2X1,τcϵ3τcD1X2,τc+, (4.5)

    where Xj,τc=Xj(T0τc,T1,T2,),j=1,2,.

    Then, substituting the solution with the multiple scales Eqs. (4.3)-(4.5) into Eq. (4.1) and balancing the coefficients of ϵn(n=1,2,3), we obtain a set-ordered linear differential equation:

    {D0Sk1+(μ+βIk)Sk1+βSkIk1=0,D0Ik1βIkSk1+(μ+α1+γ1)Ik1βSkIk1+εIk1,τc=0,D0Qk1+(μ+α2+γ2)Qk1εQk1,τc=0,D0Rk1γ1Ik1γ2Qk1+μRk1=0,k=1,2. (4.6)

    Since ±iωk(k=1,2) are the eigenvalues of the characteristic Eq. (4.1), the solution to Eq. (4.6) can be expressed in the form of Eq. (4.7):

    X1(T1,T2,T3,)=G1(T1,T2,T3,)eiωkT0hk+ˉG1(T1,T2,T3,)eiωkT0ˉhk,k=1,2. (4.7)

    where G1 and ˉG1 represent the coordinates on the center manifold, and the linear part of the center manifold can be presented as equation (4.7).

    Next, for the ϵ2-order terms, we can obtain the following equations:

    {D0Sk2+(μ+βIk)Sk2+βSkIk2=D1Sk1βSk1Ik1,D0Ik2βIkSk2+(μ+α1+γ1)Ik2βSkIk2+εIk2,τc=ετεD0Ik1,τc+ετcD1Ik1,τcD1Ik1+βSk1Ik1,D0Qk2+(μ+α2+γ2)Qk2εQk2,τc=ετεD0Qk1,τcετcD1Qk1,τcD1Qk1,D0Rk2γ1Ik2γ2Qk2+μRk2=D1Rk1,k=1,2. (4.8)

    We substitute Eq. (4.7) into the right expression of Eq. (4.8), and the coefficients before eiωkT0 are denoted by the vector m1. According to the solvability condition hk,m1=0, we can solve GT1 as follows:

    G1T1=MkτεG1, (4.9)

    where Mk=εeiωkτc[βIk(iωk+μ+βIk)(μ+βIk)2+ωk2βSkiωk+μ+βIk+1ετceiωkτc]1,k=1,2.

    We assume:

    Sk2=fk1eiωkT0G1+ˉfk1eiωkT0ˉG1+gk1e2iωkT0G12+ˉgk1e2iωkT0ˉG12+lk1G1ˉG1,Ik2=fk2eiωkT0G1+ˉfk2eiωkT0ˉG1+gk2e2iωkT0G12+ˉgk2e2iωkT0ˉG12+lk2G1ˉG1,Qk2=fk3eiωkT0G1+ˉfk3eiωkT0ˉG1+gk3e2iωkT0G12+ˉgk3e2iωkT0ˉG12+lk3G1ˉG1,Rk2=fk2eiωkT0G1+ˉfk4eiωkT0ˉG1+gk4e2iωkT0G12+ˉgk4e2iωkT0ˉG12+lk4G1ˉG1. (4.10)

    Substituting Eq. (4.10) into Eq. (4.8), we obtain:

    fk1=U1τε(iωk+μ+α2+γ2εeiωkτc)[(Mkhk1)(iωk+μ+α1+γ1βSk+εeiωkτc)βSk(iωkεeiωkτchk2+ετcMkhk2Mkhk2)](iωk+μ),fk2=U1τε(iωk+μ+α2+γ2εeiωkτc)[(iωk+μ+βIk)(iωkεeiωkτchk2+ετcMkhk2Mkhk2)+βIk(Mkhk1)](iωk+μ),fk3=τε(iωk+μ)[(iωk+μ+βIk)(iωk+μ+α1+γ1βSk+εeiωkτc)(iωk+μ+α2εeiωkτc+γ2)+β2SkIk(iωk+μ+α2+γ2εeiωkτc)]U1,fk4=τε{(iωk+μ+βIk)[(Mkhk4)(iωk+μ+α1+γ1βSk+εeiωkτc)(iωk+μεeiωkτc+α2+γ2)+γ1(iωkεeiωkτchk2+ετcMkhk2Mkhk2)(iωk+μ+α2+γ2εeiωkτc)+γ2(iωk+μ+α1+γ1βSk+εeiωkτc)(Mkhk3ετεMkeiωkτchk3iωkεeiωkτchk3)]+βIk[βSk(iωk+μ+α2+γ2εeiωkτc)(Mkhk4)+γ1(iωkεeiωkτchk2+ετcMkhk2Mkhk2)(iωk+μ+α2+γ2εeiωkτc)+γ2βSk(Mkhk3iωkεMkeiωkτchk3εeiωkτchk3)]}U1,gk1=(2iωk+μ)(2iωk+μ+α2+γ2εe2iωkτc)[(βhk1hk2)(2iωk+μ+α1+γ1+εe2iωkτcβSk)+β2Skhk1hk2]V1,gk2=(2iωk+μ)(2iωk+μ+α2+γ2εe2iωkτc)[βhk1hk2(2iωk+μ+βIk)β2Ikhk1hk2]V1,gk3=0,gk4=(2iωk+μ+α2+γ2εe2iωkτc)[(2iωk+μ+βIk)(βhk1ˉhk2+βhk2ˉhk1)+βIk(βhk1ˉhk2βhk2ˉhk1)]γ1V1,lk1=μ(μ+α2+γ2ε)[(βhk1ˉhk2βhk2ˉhk1)(μ+α1+γ1βSk+ε)+βSk(βhk1ˉhk2+βhk2ˉhk1)]W1,lk2=W1μ(μ+α2+γ2ε)[(βhk1ˉhk2+βhk2ˉhk1)(μ+βIk)+βIk(βhk1ˉhk2βhk2ˉhk1)],lk3=0,lk4=(μ+α2+γ2ε)γ1[(βhk1ˉhk2+βhk2ˉhk1)(μ+βIk)+βIk(βhk1ˉhk2βhk2ˉhk1)]W1,U=(iωk+μ)(iωk+μ+α2+γ2εeiωkτc)[(iωk+μ+βIk)(iωk+μ+α1+γ1βSk+εeiωkτc)+β2SkIk],V=(2iωk+μ)(2iωk+μ+α2+γ2εe2iωkτc)[(2iωk+μ+βIk)(2iωk+μ+α1+γ1βSk+εe2iωkτc)+β2SkIk],W=μ(μ+α2+γ2ε)[(μ+βIk)(μ+α1+γ1βSk+ε)+β2SkIk],k=1,2. (4.11)

    Next, balancing the ϵ3-order terms, we can obtain equations as follows:

    {D0S3+(μ+βIk)S3+βSkI3=D2S1D1S2βS2I1βS1I2,D0I3βIkS3+(μ+α1+γ1)I3βSkI3+εI3,τc=D2I1D1I2+βS2I1+βS1I2+ε[τε(D0I2,τcD1I1,τc)τc(D2I1,τc+D1I2,τc)],D0Q3+(μ+α2+γ2)Q3εQ3,τc=ε(τεD0Q2,τcτεD1Q1,τcτcD2Q1,τcτcD1Q2,τc)D2Q1D1Q2,D0R3γ1I3γ2Q3+μR3=D2R1D1R2,k=1,2. (4.12)

    Substituting Eq. (4.7) and Eq. (4.10) into the right expression of Eq. (4.12), and the coefficients before eiωkT0 are denoted by the vector m2. According to the solvability condition h1,m2=0 and note that τ2ε is small enough for small unfolding parameter τε, we ignore the term τ2εG1. Then, we have:

    G1T2=HkχkG21ˉG1, (4.13)

    where

    Hk=1βIk(μ+βIkiωk)(μ+βIk)2+ωk2hk1+hk2+ετceiωkτchk2,χk=βIk(μ+βIkiωk)(μ+βIk)2+ωk2βgk1ˉhk2βIk(μ+βIkiωk)(μ+βIk)2+ωk2βgk2ˉhk2+βgk1ˉhk2+βgk2ˉhk2,k=1,2,

    where gk are given in Eq. (4.11) and hk are given in Eq. (4.2).

    Let G1(G1/ε), and we can obtain the normal form of Hopf bifurcation of system (2.1) as:

    G1=MkτεG1+HkχkG2G1, (4.14)

    where Mk is given in Eq. (4.9), and Hk,χk are given in Eq. (4.13).

    Let G=reiθk(k=1,2) and substitute it into Eq. (4.14), and we can obtain the normal form of Hopf bifurcation in polar coordinates:

    {r=Re(Mk)τεr+Re(Hkχk)r3,θk=Im(Mk)τε+Im(Hkχk)r2, (4.15)

    where Mk is given in Eq. (4.9), and Hk,χk are given in Eq. (4.13) (k=1,2).

    According to the normal form of bifurcation in polar coordinates, there is a theorem as follows:

    Theorem 4.1. For system (4.15), if Re(Mk)τεRe(Hkχk)<0(k=1,2) holds, then system (2.1) exists periodic solutions near equilibrium Ek:

    (1) If Re(Mk)τε<0, the bifurcating periodic solutions are unstable.

    (2) If Re(Mk)τε>0, the bifurcating periodic solutions are locally asymptotically stable.

    In this section, according to the natural mortality in China and the data presented in ref. [32], we choose Λ=100,μ=0.00713,ε=0.8,γ1=0.1,γ2=0.1,α1=0.025,α2=0.025,β=0.00001. According to Eq. (3.2), we obtain S1=Λμ=14025,I1=0,Q1=0,R1=0,S2=93213,I2=605.719,Q2=3667.4,R2=59932. Obviously, assumption (H1) holds and equilibrium E1 is locally asymptotically stable when τ=0. However, the equilibrium E2 is unstable for I2<0.

    Remark 2: If I2<0, the number of infected individuals is less than zero, which is inconsistent with the facts. Additionally, according to our theoretical analysis, when I2<0, the equilibrium E2 is unstable, which is consistent with the facts.

    Substituting these parameter values into Eqs. (3.10)-(3.12), by using MATLAB, we can obtain ω0=0.8000,Q1=0.0102,P1=0.9999,τ(0)1=1.9509.

    Thus, the equilibrium E1 is locally asymptotically stable when τ[0,τ(0)1), and Hopf bifurcation occurs near the equilibrium E1 when τ=τ(0)1. According to Eq. (4.9) and Eq. (4.13), we obtain Re(M1)<0, Re(H1χ1)>0. The bifurcating periodic solution is unstable due to Theorem 4.1. We find that if the time delay from infection to isolation is over the critical value, the epidemic will not be controlled under a stable situation, and the epidemic situation will be more severe.

    When τ=0, we choose the initial value (1500,1000,100,200), and the solution corresponding to a locally asymptotically stable equilibrium is shown in Figure 2. Thus, when τ=0, once the individuals are infected, they will be quarantined and they will not infect others, and the epidemic situation can be controlled into a stable situation.

    Figure 2.  When τ=0, the equilibrium E1 of system (2.1) is locally asymptotically stable.

    When τ=1.5(0,τ(0)1), we choose initial values (1,500,1,000,100,200), and the equilibrium E1 of system (2.1) is locally asymptotically stable. See Figure 3.

    Figure 3.  When τ=1.5, the equilibrium E1 of system (2.1) is locally asymptotically stable.

    When τ=2(τ(0)1,+), we choose the initial values (1,500,1,000,100,200), and the equilibrium E1 is unstable, as shown in Figure 4. This means that if τ is larger than the critical value τ(0)1, the epidemic situation cannot be controlled into a stable state.

    Figure 4.  When τ=2, the equilibrium E1 of system (2.1) is unstable.

    According to Figures 2-4, we find that the time delay from infection to isolation has a great influence on the epidemic situation in E1, and with the increasing time delay, the epidemic situation will become increasingly severe.

    According to the natural mortality in China and the data presented in ref. [32], we choose another group of parameters, namely, Λ=200,μ=0.00713,ε=0.8,γ1=0.05,γ2=0.05,α1=0.025,α2=0.025,β=0.0001. According to Eq. (3.2), we obtain S1=28,050,I1=0,Q1=0,R1=0,S2=8,821.300,I2=155.424,Q2=1,513.9,R2=11,703. We can find that assumption (H1) does not hold and the equilibrium E1 is unstable. However, the equilibrium E2 is locally asymptotically stable when τ=0.

    Substituting these parameter values into Eq. (3.16), we can obtain c1=0.0269,c2=0.00030938, and Eq. (3.16) has only one positive root, z1=0.0356. According to Eq. (3.15)and Eq. (3.17), we can obtain ω2=0.1887,Q2=0.9892,P1=0.1463,τ(0)2,3=0.7782 by using MATLAB.

    Thus, the equilibrium E2 is locally asymptotically stable for τ[0,τ(0)2,3), and Hopf bifurcation occurs near the equilibrium E2 when τ=τ(0)2,3. According to Eq. (4.9) and Eq. (4.13), we obtain Re(M1)>0,Re(H1χ1)>0. Thus, Re(M1)τε<0, the bifurcation solution is unstable. In E2, the epidemic situation cannot be controlled under a stable situation.

    When τ=0, we choose the initial values (1,000,50,2,000,1,000), and the solution corresponding to a locally asymptotically stable equilibrium is shown in Figure 5.

    Figure 5.  When τ=0, the equilibrium E2 of system (2.1) is locally asymptotically stable.

    When τ=0.5(0,τ(0)2,3), we choose the initial value (1,000,50,2,000,1,000), and the equilibrium E2 of system (2.1) is locally asymptotically stable. See Figure 6.

    Figure 6.  When τ=0.5, the equilibrium E2 of system (2.1) is locally asymptotically stable.

    When τ(τ(0)2,3,+),τ(0)2,3=0.7782, the equilibrium E2 is unstable. We choose τ=0.8>τ(0)2,3=0.7782 and the initial value (1,000,50,2,000,1,000). See Figure 7.

    Figure 7.  When τ=0.8, the equilibrium E2 of system (2.1) is unstable.

    According to numerical simulations, we can find that the smaller τ is, the better the control effect is. The results of the numerical simulation are consistent with this fact. Then, we provide some explanations for the above results.

    Remark 3: For the two groups of parameter values that are Λ=100,μ=0.00713,ε=0.8,γ1=0.1,γ2=0.1,α1=0.025,α2=0.025,β=0.00001 and Λ=200,μ=0.00713,ε=0.8,γ1=0.05,γ2=0.05,α1=0.025,α2=0.025,β=0.0001, we have the following conclusions.

    For the first group of parameter values, we can control τ<1.9509, and the epidemic situation can gradually become stable. However, if τ>1.9509, the epidemic cannot be controlled into a stable situation. This means that if we can make the time delay from infection to isolation within 1.9509, we can make the number of infected individuals gradually tend to zero, and the epidemic situation will be controllable. However, if we cannot make the time delay from infection to isolation within 1.9509, the epidemic situation will be uncontrollable.

    For the second group of parameter values, we can control τ<0.7782, and the epidemic situation can be gradually stable. If τ>0.7782, the epidemic cannot be controlled into a stable situation. This means that if we can make the time delay from infection to isolation within 0.7782, we can make the number of infected individuals gradually tend to stabilize, and the epidemic situation will be controllable. However, if we cannot make the time delay from infection to isolation within 0.7782, the epidemic situation will be uncontrollable.

    Remark 4: According to our theoretical analysis, with the increase in τ, the epidemic situation will be increasingly severe, and COVID-19 will further threaten the health of humans. To control the epidemic situation, we need to take some measures to shorten the time from infection to isolation. In addition, for epidemic situations in different areas, we can deduce the corresponding parameter values and use the model we constructed to obtain the stability determination and the strategies to control epidemic situations.

    Our work is reproducible. For different regions, we can calculate the different critical time delays by changing variables such as the transmission rate from S to I, the transmission rate from I to Q and the transmission rate from I to R. We can obtain the critical time delay based on Eqs. (3.10)-(3.12) and Eqs. (3.15)-(3.17). Then, according to Theorem 3.1, we know that the equilibria are locally asymptotically stable if the time delay is within the critical time delay. Finally, we can obtain the normal form of Hopf bifurcation by using the multiple time scales method and determine whether the epidemic situation can be controlled into a stable situation based on Theorem 4.1. The time delay from infection to isolation is the most important variable in our model. We calculate the critical time delay based on the parameter values. We choose parameter values according to the natural mortality in China and the data presented in ref. [32]. Then, we change the value of the time delay from infection to isolation, and we know that the equilibria are locally asymptotically stable if the time delay is within the critical time delay through numerical simulations.

    In this paper, we constructed a DDE according to the characteristics of the COVID-19 epidemic phenomenon based on the SIQR model. We considered the existence and stability of equilibria in the above delayed SIQR model. We also analyzed the existence and dynamic properties of Hopf bifurcation associated with both equilibria. We chose two groups of parameter values according to the natural mortality in China and the data presented in ref. [32]. Numerical simulations were carried out to verify the analytical results.

    According to the results of numerical simulations, the epidemic solution would be more severe with increasing the time delay from infection to isolation. In addition, we predicted the stability of epidemic solutions in different areas and provided effective strategies to control the epidemic.

    This study was funded by Fundamental Research Funds for the Central Universities of China. (Grant No. 2572019BC14), the Heilongjiang Provincial Natural Science Foundation of China (Grant No. LH2019A001) and College Students Innovations Special Project funded by Northeast Forestry University of China (No. 202010225035).

    All authors declare no conflicts of interest in this paper.



    [1] FAO (2014) Save food: Global initiative on food loss and waste reduction definitional framework of food loss (Working Paper).
    [2] Borma A (2017) Food waste–A global problem. SEA-Practical Application of Science, Romanian Foundation for Business Intelligence, Editorial Department 353–362.
    [3] Tukker A, Huppes G, Guinée J, et al. (2006) Environmental impact of products (EIPRO): Analysis of the life cycle environmental impacts related to the final consumption of the EU-25. EUROPEAN COMMISSION DIRECTORATE-GENERAL Joint Research Centre. Available from: http://ec.europa.eu/environment/ipp/pdf/eipro_report.pdf.
    [4] Quested TE, Marsh E, Stunell D, et al. (2013) Spaghetti soup: The complex world of food waste behaviours. Resour Conserv Recy 79: 43–51. https://doi.org/10.1016/j.resconrec.2013.04.011 doi: 10.1016/j.resconrec.2013.04.011
    [5] Grizzetti B, Pretato U, Lassaletta L, et al. (2013) The contribution of food waste to global and European nitrogen pollution. Environ Sci Policy 33: 186–195. https://doi.org/10.1016/j.envsci.2013.05.013 doi: 10.1016/j.envsci.2013.05.013
    [6] Flanagana A, Priyadarshini A (2021) A study of consumer behaviour towards food-waste in Ireland: Attitudes, quantities and global warming potentials. J Environ Manage 284: 112046. https://doi.org/10.1016/j.jenvman.2021.112046 doi: 10.1016/j.jenvman.2021.112046
    [7] Martins IBA, Oliveira D, Rosenthal A, et al. (2019) Brazilian consumer's perception of food processing technologies: A case study with fruit juice. Food Res Int 125: 108555. https://doi.org/10.1016/j.foodres.2019.108555 doi: 10.1016/j.foodres.2019.108555
    [8] Jeswani HK, Figueroa-Torres G, Azapagic A (2021) The extent of food waste generation in the UK and its environmental impacts. Sustain Prod Consump 26: 532–547. https://doi.org/10.1016/j.spc.2020.12.021 doi: 10.1016/j.spc.2020.12.021
    [9] Gustavsson J, Cederberg C, Sonesson U, et al. (2011) Global food Losses and food waste: Extension, causes and prevention. Available from: http://www.fao.org/3/a-i2697e.pdf.
    [10] Porter SD, Reay DS (2016) Addressing food supply chain and consumption inefficiencies: potential for climate change mitigation. Reg Environ Change 16: 2279–2290. https://doi.org/10.1007/s10113-015-0783-4 doi: 10.1007/s10113-015-0783-4
    [11] Wohner B, Pauer E, Heinrich V, et al. (2019) Packaging-related food losses and waste: An overview of drivers and issues. Sustainability 11: 264. https://doi.org/10.3390/su11010264 doi: 10.3390/su11010264
    [12] Rezaei M, Liu B (2017) Food loss and waste in the food supply chain. International Nut and Dried Fruit Council 26–27.
    [13] Vilarino MV, Franco C, Quarrington C (2017) Food loss and waste reduction as an integral part of a circular economy. Front Environ Sci 5: 21. https://doi.org/10.3389/fenvs.2017.00021 doi: 10.3389/fenvs.2017.00021
    [14] FAO (2011) Global food losses and food waste: Extent, causes and preventions. Available from: https://www.fao.org/3/mb060e/mb060e00.htm.
    [15] Scherhaufer S, Moates G, Hartikainen H, et al. (2018) Environmental impacts of food waste in Europe. Waste Manage 77: 98–113. https://doi.org/10.1016/j.wasman.2018.04.038 doi: 10.1016/j.wasman.2018.04.038
    [16] Brancoli P, Rousta K, Bolton K (2017) Life cycle assessment of supermarket food waste. Resour Conserv Recy 118: 39–46. https://doi.org/10.1016/j.resconrec.2016.11.024 doi: 10.1016/j.resconrec.2016.11.024
    [17] USDA (2022) Food waste and its links to greenhouse gases and climate change. Available from: https://www.usda.gov/media/blog/2022/01/24/food-waste-and-its-links-greenhouse-gases-and-climate-change.
    [18] San-Epifanio LE, De Renobales Scheifler M (2015) Envisioning a future without food waste and food poverty: Societal challenges. Wageningen Academic Publishers. https://doi.org/10.3920/978-90-8686-820-9
    [19] Galli A, Wiedmann T, Ercin E, et al. (2012). Integrating ecological, carbon and water footprint into a "footprint family" of indicators: definition and role in tracking human pressure on the planet. Ecol Indic 16: 100–112. https://doi.org/10.1016/j.ecolind.2011.06.017 doi: 10.1016/j.ecolind.2011.06.017
    [20] Hoekstra AY, Mekonnen MM (2012) The water footprint of humanity. PNAS 109: 3232–3237. https://doi.org/10.1073/pnas.1109936109 doi: 10.1073/pnas.1109936109
    [21] Steen-Olsen K, Weinzettel J, Cranston G, et al. (2012) Carbon, land, and water footprint accounts for the European Union: Consumption, production, and displacements through international trade. Environ Sci Technol 46: 10883–10891. https://doi.org/10.1021/es301949t doi: 10.1021/es301949t
    [22] Jones CM, Kammen DM (2011) Quantifying carbon footprint reduction opportunities for U.S. households and communities. Environ Sci Technol 45: 4088–4095. https://doi.org/10.1021/es102221h doi: 10.1021/es102221h
    [23] Ridoutt BG, Pfister S (2010). Reducing humanity's water footprint. Environ Sci Technol 44: 6019–6021. https://doi.org/10.1021/es101907z doi: 10.1021/es101907z
    [24] Audsley E, Brander M, Chatterton JC, et al. (2010) How low can we go? An assessment of greenhouse gas emissions from the UK food system and the scope reduction by 2050. Report for the WWF and Food Climate Research Network. Available from: https://dspace.lib.cranfield.ac.uk/handle/1826/6503.
    [25] Leip A, Billen G, Garnier J, et al. (2015). Impacts of European livestock production: Nitrogen, sulphur, phosphorus and greenhouse gas emissions, land-use, water eutrophication and biodiversity. Environ Res Let 10: 115004. https://doi.org/10.1088/1748-9326/10/11/115004 doi: 10.1088/1748-9326/10/11/115004
    [26] Lesschen JP, van den Berg M, Westhoek HJ, et al. (2011) Greenhouse gas emission profiles of European livestock sectors. Anim Feed Sci Technol 166–167: 16–28. https://doi.org/10.1016/j.anifeedsci.2011.04.058 doi: 10.1016/j.anifeedsci.2011.04.058
    [27] Gaiani S, Caldeira S, Adorno V, et al. (2018). Food wasters: Profiling consumers' attitude to waste food in Italy. Waste Manage 72: 17–24. https://doi.org/10.1016/j.wasman.2017.11.012 doi: 10.1016/j.wasman.2017.11.012
    [28] Visschers VHM, Wickli N, Siegrist M (2016) Sorting out food waste behaviour: A survey on the motivators and barriers of self-reported amounts of food waste in households. J Environ Psychol 45: 66–78. https://doi.org/10.1016/j.jenvp.2015.11.007 doi: 10.1016/j.jenvp.2015.11.007
    [29] Ogunmoroti A, Liu M, Li MY, et al. (2022) Unraveling the environmental impact of current and future food waste and its management in Chinese provinces. Res Environ Sustainability 9: 100064. https://doi.org/10.1016/j.resenv.2022.100064 doi: 10.1016/j.resenv.2022.100064
    [30] Xue L, Liu G (2019) Introduction to global food losses and food waste, In: Saving Food: Production, supply chain, food waste, and food consumption, London: Elsevier Academic Press, 1–31. https://doi.org/10.1016/b978-0-12-815357-4.00001-8
    [31] United Nation, Department of economic and social affairs, sustainable development (2022) Transforming our world: The 2030 Agenda for sustainable development. Available from: https://sdgs.un.org/2030agenda.
    [32] Kummu M, de Moel H, Porkka M, et al. (2012) Lost food, wasted resources: Global food supply chain losses and their impacts on freshwater, cropland, and fertiliser use. Sci Total Environ 438: 477–489. https://doi.org/10.1016/j.scitotenv.2012.08.092 doi: 10.1016/j.scitotenv.2012.08.092
    [33] FAO (2022) Sustainable development goals. Available from: https://www.fao.org/sustainable-development-goals/indicators/1231/en/.
    [34] Kafa N, Jaegler A (2021) Food losses and waste quantification in supply chains: A systematic literature review. British Food J 123: 3502–3521. https://doi.org/10.1108/BFJ-09-2020-0879 doi: 10.1108/BFJ-09-2020-0879
    [35] de Oliveira MM, Lago A, Dal' Magro GP (2021) Food loss and waste in the context of the circular economy: a systematic review. J Clean Prod 294: 126284. https://doi.org/10.1016/j.jclepro.2021.126284 doi: 10.1016/j.jclepro.2021.126284
    [36] FAO (2019) Understanding the issue before taking action—Food loss and waste: what's the difference? Available from: https://www.fao.org/state-of-food-agriculture/2019/en/.
    [37] Buzby JC, Wells FH, Hyman J (2014) The estimated amount, value, and calories of postharvest food losses at the retail and consumer levels in the United States. USDA. Available from: https://www.ers.usda.gov/publications/pub-details/?pubid=43836https: //www.ers.usda.gov/publications/pub-details/?pubid = 43836.
    [38] Chang HH (2021) Is it unethical to waste food? Exploring consumer's ethical perspectives and waste intentions. Curr Psychol 1–15. https://doi.org/10.1007/s12144-020-01257-3 doi: 10.1007/s12144-020-01257-3
    [39] Lebersorger S, Schneider F (2011) Discussion on the methodology for determining food waste in household waste composition studies. Waste Manage 31: 1924–1933. https://doi.org/10.1016/j.wasman.2011.05.023 doi: 10.1016/j.wasman.2011.05.023
    [40] Parizeau K, Von Massow M, Martin R (2015) Household-level dynamics of food waste production and related beliefs, attitudes, and behaviours in Guelph, Ontario. Waste Manage 35: 207–217. https://doi.org/10.1016/j.wasman.2014.09.019 doi: 10.1016/j.wasman.2014.09.019
    [41] Williams H, Wikström F, Otterbring T, et al. (2012) Reasons for household food waste with special attention to packaging. J Clean Prod 24: 141–148. https://doi.org/10.1016/j.jclepro.2011.11.044 doi: 10.1016/j.jclepro.2011.11.044
    [42] Graham-Rowe E, Jessop DC, Sparks P (2019) Self-affirmation theory and pro-environmental behaviour: Promoting a reduction in household food waste. J Environ Psychol 62: 124–132. https://doi.org/10.1016/j.jenvp.2019.02.003 doi: 10.1016/j.jenvp.2019.02.003
    [43] Stefan V, van Herpen E, Tudoran AA, et al. (2013) Avoiding food waste by Romanian consumers: The importance of planning and shopping routines. Food Qual Prefer 28: 375–381. https://doi.org/10.1016/j.foodqual.2012.11.001 doi: 10.1016/j.foodqual.2012.11.001
    [44] Ehrlich PR, Holdren JP (1994) Population increase causes increased environmental damage, In: Environmental ethics: Reading in theory and application, Boston, London: Jones and Barlett Publishers.
    [45] Sanne C (2002) Willing consumers—or locked-in? Policies for sustainable consumption. Ecol Econ 42: 273–287. https://doi.org/10.1016/S0921-8009(02)00086-1 doi: 10.1016/S0921-8009(02)00086-1
    [46] Røpke I (1999) The dynamics of willingness to consume. Ecol Econ 28: 399–420. https://doi.org/10.1016/S0921-8009(98)00107-4 doi: 10.1016/S0921-8009(98)00107-4
    [47] Vieira VA (2008) Consumer behavior and managerial decision making. RAC 12: 253–255.
    [48] Fischer ARH (2015) Consumer behavior and food science, In: Reference module in food science, Elsevier. https://doi.org/10.1016/B978-0-08-100596-5.03335-7
    [49] Ananda J, Karunasena GG, Mitsis A, et al. (2021) Analysing behavioural and socio-demographic factors and practices influencing Australian household food waste. J Clean Prod 306: 127280. https://doi.org/10.1016/j.jclepro.2021.127280 doi: 10.1016/j.jclepro.2021.127280
    [50] Gatley A, Caraher M, Lang T (2014) A qualitative, cross cultural examination of attitudes and behaviour in relation to cooking habits in France and Britain. Appetite 75: 71–81. https://doi.org/10.1016/j.appet.2013.12.014 doi: 10.1016/j.appet.2013.12.014
    [51] Bloom J (2010) American wasteland: How America throws away nearly half of its food (and what we can do about it), Cambridge, MA: Da Capo Press.
    [52] Menzel P, D'aluisio F (2008) What the world eats, Berkeley, California: Tricycle Press.
    [53] Di Talia E, Simeone M, Scarpato D (2019) Consumer behaviour types in household food waste. J Clean Prod 214: 166–172. https://doi.org/10.1016/j.jclepro.2018.12.216 doi: 10.1016/j.jclepro.2018.12.216
    [54] Mou J, Benyoucef M (2021) Consumer behavior in social commerce: Results from a meta-analysis. Technol Forecast Soc Change 167: 120734. https://doi.org/10.1016/j.techfore.2021.120734 doi: 10.1016/j.techfore.2021.120734
    [55] Yadav PG, Rai J (2017) The generation Z and their social media usage: A review and a research outline. Glob J Enterprise Inform Syst 9: 110–116. https://doi.org/10.18311/gjeis/2017/15748 doi: 10.18311/gjeis/2017/15748
    [56] Hazuchová N, Antošová I, Stávková J (2020) Food wastage as a display of consumer behaviour. J Competitiveness 12: 51–66. https://doi.org/10.7441/joc.2020.02.04 doi: 10.7441/joc.2020.02.04
    [57] Jungowska J, Kulczyński B, Sidor A, et al. (2021) Assessment of factors affecting the amount of food waste in households run by Polish women aware of well-being. Sustainability 13: 976. https://doi.org/10.3390/su13020976 doi: 10.3390/su13020976
    [58] Farr-Wharton G, Foth M, Choi J (2014) Identifying factors that promote consumer behaviours causing expired domestic food waste. J Consum Behav 13: 393–402.
    [59] Eurostat (2021) EU statistics on income and living conditions microdata 2004–2019, release 1 in 2021, Available from: https://doi.org/10.2907/EUSILC2004-2019V.2.
    [60] Eurostat (2021) Mean consumption expenditure by type of household. Available from: https://ec.europa.eu/eurostat/databrowser/view/hbs_exp_t134/default/table?lang=en.
    [61] Setti M, Falasconi L, Segrè A, et al. (2016) Italian consumers' income and food waste behavior. British Food J 118: 1731–1746. https://doi.org/10.1108/BFJ-11-2015-0427 doi: 10.1108/BFJ-11-2015-0427
    [62] Ghinea C, Ghiuta OA (2019) Household food waste generation: young consumers behaviour, habits and attitudes. Int J Environ Sci Technol 16: 2185–2200. https://doi.org/10.1007/s13762-018-1853-1 doi: 10.1007/s13762-018-1853-1
    [63] Pellegrini G, Sillani S, Gregori M. et al. (2019) Household food waste reduction: Italian consumers' analysis for improving food management. British Food J 121: 1382–1397. https://doi.org/10.1108/bfj-07-2018-0425 doi: 10.1108/bfj-07-2018-0425
    [64] Rohm H, Oostindjer M, Aschemann-Witzel J, et al. (2017) Consumers in a sustainable food supply chain (COSUS): Understanding consumer behavior to encourage food waste reduction. Foods 6: 104. https://doi.org/10.3390/foods6120104 doi: 10.3390/foods6120104
    [65] Bogevska Z, Berjan S, El Bilali H, et al. (2021). Exploring food shopping, consumption and waste habits in North Macedonia during the COVID-19 pandemic. Socio-Econ Plan Sci 82: 101150. https://doi.org/10.1016/j.seps.2021.101150 doi: 10.1016/j.seps.2021.101150
    [66] Tomaszewska M, Bilska B, Kołożyn-Krajewska D (2022) The influence of selected food safety practices of consumers on food waste due to its spoilage. Int J Environ Res Public Health 19: 8144. https://doi.org/10.3390/ijerph19138144 doi: 10.3390/ijerph19138144
    [67] Silvennoinen K, Katajajuuri JM, Hartikainen H, et al. (2014) Food waste volume and composition in Finnish households. British Food J 116: 1058–1068. https://doi.org/10.1108/BFJ-12-2012-0311 doi: 10.1108/BFJ-12-2012-0311
    [68] Aschemann-Witzel J, De Hooge I, Amani P, et al. (2015). Consumer-related food waste: Causes and potential for action. Sustainability 7: 6457–6477. https://doi.org/10.3390/su7066457 doi: 10.3390/su7066457
    [69] Samotyja U (2022) Consumer understanding of the 'best before' and 'use by' dates. SJ GMU 122: 82–92. https://doi.org/10.26408/122.07 doi: 10.26408/122.07
    [70] Van Boxstael S, Devlieghere F, Berkvens D, et al. (2014) Understanding and attitude regarding the shelf life labels and dates on pre-packed food products by Belgian consumers. Food Control 37: 85–92. https://doi.org/10.1016/j.foodcont.2013.08.043 doi: 10.1016/j.foodcont.2013.08.043
    [71] Leib EB, Rice C, Neff R, et al. (2016) Consumer perceptions of date labels: National survey. Center for Health Law & Policy Innovation. Available from: https://chlpi.org/wp-content/uploads/2013/12/Consumer-Perceptions-on-Date-Labels_May-2016.pdf.
  • This article has been cited by:

    1. Zimeng Lv, Jiahong Zeng, Yuting Ding, Xinyu Liu, Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation, 2023, 31, 2688-1594, 1004, 10.3934/era.2023050
    2. Ruqi Li, Yurong Song, Haiyan Wang, Guo-Ping Jiang, Min Xiao, Reactive–diffusion epidemic model on human mobility networks: Analysis and applications to COVID-19 in China, 2023, 609, 03784371, 128337, 10.1016/j.physa.2022.128337
    3. S. Suganya, V. Parthiban, Optimal control analysis of fractional order delayed SIQR model for COVID-19, 2024, 1951-6355, 10.1140/epjs/s11734-024-01294-0
    4. Fatima Cherkaoui, Fatima Ezzahrae Fadili, Khalid Hilal, Stability analysis of a delayed fractional-order SIR epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate, 2024, 1982-6907, 10.1007/s40863-024-00470-3
    5. Xi-Xi 习习 Huang 黄, Min 敏 Xiao 肖, Leszek Rutkowski, Hai-Bo 海波 Bao 包, Xia 霞 Huang 黄, Jin-De 进德 Cao 曹, Mechanism analysis of regulating Turing instability and Hopf bifurcation of malware propagation in mobile wireless sensor networks, 2024, 33, 1674-1056, 060202, 10.1088/1674-1056/ad24d5
    6. Zafer ÖZTÜRK, Halis BİLGİL, Sezer SORGUN, Application of Fractional SPR Psychological Disease Model in Turkey and Stability Analysis, 2023, 6, 2636-8692, 49, 10.33187/jmsm.1196961
    7. Debao Gao, Xiangfeng Yang, Dynamic Analysis of the Project Investment Model with Time Delay and Density Constraints, 2023, 2023, 2314-4785, 1, 10.1155/2023/3791676
    8. Guo-Feng Feng, Jiaqi Chen, Bin Ge, A class of natural pinus koraiensis population system with time delay and diffusion term, 2024, 17, 1793-5245, 10.1142/S1793524523500195
    9. Zafer Öztürk, Halis Bilgil, Sezer Sorgun, Application of Fractional SIQRV Model for SARS-CoV-2 and Stability Analysis, 2023, 15, 2073-8994, 1048, 10.3390/sym15051048
    10. Youssra Hajri, Amina Allali, Saida Amine, A delayed deterministic and stochastic SIRICV model: Hopf bifurcation and stochastic analysis, 2024, 215, 03784754, 98, 10.1016/j.matcom.2023.07.027
    11. Chen Wang, Ruizhi Yang, Hopf bifurcation analysis of a pine wilt disease model with both time delay and an alternative food source, 2025, 33, 2688-1594, 2815, 10.3934/era.2025124
  • Reader Comments
  • © 2022 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(2564) PDF downloads(152) Cited by(2)

Figures and Tables

Figures(3)  /  Tables(9)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog