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The effect of global travel on the spread of SARS

  • Received: 01 April 2005 Accepted: 29 June 2018 Published: 01 November 2005
  • MSC : 92D30.

  • The goal of this paper is to study the global spread of SARS. We propose a multiregional compartmental model using medical geography theory (central place theory) and regarding each outbreak zone (such as Hong Kong, Singapore, Toronto, and Beijing) as one region. We then study the effect of the travel of individuals (especially the infected and exposed ones) between regions on the global spread of the disease.

    Citation: Shigui Ruan, Wendi Wang, Simon A. Levin. The effect of global travel on the spread of SARS[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 205-218. doi: 10.3934/mbe.2006.3.205

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  • The goal of this paper is to study the global spread of SARS. We propose a multiregional compartmental model using medical geography theory (central place theory) and regarding each outbreak zone (such as Hong Kong, Singapore, Toronto, and Beijing) as one region. We then study the effect of the travel of individuals (especially the infected and exposed ones) between regions on the global spread of the disease.


    1. Introduction

    G. Caginalp introduced in [1] and [2] the following phase-field systems:

    ut+Δ2uΔf(u)=Δθ, (1.1)
    θtΔθ=ut, (1.2)

    where u is the order parameter and θ is the (relative) temperature. These equations model phase transition processes such as melting/solidification processes and have been studied, e.g., in [4] for a similar phase-field model with a memory term. Eqs. (1.1)-(1.2) consist of the coupling of the Cahn-Hilliard equation introduced in [18] with the heat equation and are known as the conserved phase-field model, in the sens that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity (see below). We refer the reader to, e.g., [6,7,8,9,10,11,13,14,16,17,19,20,22,23,24,25].

    Equations (1.1) and (1.2) are based on the total free energy

    ψ(u,θ)=Ω(12|u|2+F(u)uθ12θ2)dx, (1.3)

    where Ω is the domain occupied by the material (we assume that it is a bounded and smooth domain of Rn, n=2 or 3) and F=f (typically, F is the double-well potential F(s)=14(s21)2, hence f(s)=s3s). We then introduce the enthalpy H defined by

    H=θψ, (1.4)

    where denotes a variational derivative, so that

    H=u+θ. (1.5)

    The gouverning equations for u and θ are finally given by

    ut=Δuψ, (1.6)
    Ht=divq, (1.7)

    where q is the thermal flux vector. Assuming the classical Fourier law

    q=θ, (1.8)

    we obtain (1.1) and (1.2).

    Now, one drawback of the Fourier law is that it predicts that thermal signals propagate with an infinite speed, which violates causality (the so-called "paradox of heat conduction", see, e.g. [5]). Therefore, several modifications of (1.8) have been proposed in the literature to correct this unrealistic feature, leading to a second order in time equation for the temperature.

    In particular, we considered in [15] (see also [19] the Maxwell-Cattaneo law)

    (1+ηt)q=θ,η>0, (1.9)

    which leads to

    η2θt2+θtΔθ=η2ut2ut. (1.10)

    Green and Naghdi proposed in [21] an alternative treatment for a thermomechanical theory of deformable media. This theory is based on an entropy balance rather than the usual entropy inequality and is proposed in a very rational way. If we restrict our attention to the heat conduction, we recall that proposed three different theories, labelled as type Ⅰ, type Ⅱ and type Ⅲ, respectively. In particular, when type Ⅰ is linearized, we recover the classical theory based on the Fourier law. The linearized versions of the two other theories are decribed by the constitutive equation of type Ⅱ (see [12])

    q=kα,k>0, (1.11)

    where

    α(t)=tt0θ(τ)dτ+α0 (1.12)

    is called the thermal displacement variable. It is pertinent to note that these theories have received much attention in the recent years.

    If we add the constitutive equation (1.9) to equation (1.7), we then obtain the following equations for α (note that αt=θ):

    2αt2kΔα=ut. (1.13)

    Our aim in this paper is to study the model consisting the equation (1.1) (θ=αt) and the temperature equation (1.13). In particular, we obtain the existence and the uniqueness of tne solutions.


    2. Setting of the problem

    We consider the following initial and boundary value problem (for simplificity, we take k=1):

    ut+Δ2uΔf(u)=Δαt, (2.1)
    2αt2Δα=ut, (2.2)
    u=Δu=α=0onΓ, (2.3)
    u|t=0=u0,α|t=0=α0,αt|t=0=α1, (2.4)

    where Γ is the boundary of the spatial domain Ω.

    We make the following assumptions:

    f is of class C2(R),f(0)=0, (2.5)
    f(s)c0,c00,sR, (2.6)
    f(s)sc1F(s)c2c3,c1>0,c2,c30,sR, (2.7)

    where F(s)=s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3s satisfies these assumptions.

    We futher assume that

    u0H10(Ω)H2(Ω). (2.8)

    Remark 2.1. We take here, for simplicity, Dirichlet boundary conditions. However, we can obtain the same results for Neumann boundary conditions, namely,

    uν=Δuν=αν=0onΓ, (2.9)

    where ν denotes the unit outer normal to Γ. To do so, we rewrite, owing to (2.1) and (2.2), the equations in the form

    ¯ut+Δ2¯uΔ(f(u)f(u))=Δ¯αt, (2.10)
    2¯αt2Δ¯α=¯ut, (2.11)

    where ¯v=vv, |v0|M1, |α0|M2, for fixed positive constants M1 and M2. Then, we note that

    v((Δ)12¯v2+v2)12,

    where, here, Δ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that

    .=1vol(Ω).,1H1(Ω),H1(Ω).

    Furthermore,

    v(¯v2+v2)12,
    v(¯v2+v2)12,

    and

    v(Δ¯v2+v2)12

    are norms in H1(Ω), L2(Ω), H1(Ω) and H2(Ω), respectively, which are equivalent to the usual ones. We further assume that

    |f(s)|ϵF(s)+cϵ,ϵ>0,sR, (2.12)

    which allows to deal with term f(u).

    We denote by . the usual L2-norm (with associated scalar product ((., .))) and set .1=(Δ)12., where Δ denotes the minus Laplace operator with Dirichlet boundary conditions. More generally, .X denotes the norm in the Banach space X.

    Throughout this paper, the same letters c, c and c denotes (generally positive) constants which may change from line to line, or even in a same line. Similary, the same letter Q denotes monotone increasing (with respect to each argument) functions which may change from line to line, or even in a same line.


    3. A priori estimates

    The estimates derived in this section are formal, but they can easily be justified within a Galerkin scheme.

    We rewrite (2.1) in the equivalent form

    (Δ)1utΔu+f(u)=αt. (3.1)

    We multiply (3.1) by ut and have, integrating over Ω and by parts,

    ddt(u2+2ΩF(u)dx)+2ut21=2((αt,ut)). (3.2)

    We then multiply (2.2) by αt and obtain

    ddt(α2+αt2)=2((αt,ut)). (3.3)

    Summing (3.2) and (3.3), we find a differential inequality of the form

    dE1dt+cut21c,c>0, (3.4)

    where

    E1=u2+2ΩF(u)dx+α2+αt2

    satisfies

    E1c(uH1(Ω)+ΩF(u)dx+α2H1(Ω)+αt2)c,c>0, (3.5)

    hence estimates on u,αL(0,T;H10(Ω)), on utL2(0,T;H1(Ω)) and on αtL(0,T;L2(Ω)).

    We multiply (3.1) by Δut to find

    12ddtΔu2+ut2=((Δf(u),ut))((Δαt,ut)),

    which yields, owing to (2.5) and the continuous embedding H2(Ω)C(¯Ω),

    ddtΔu2+ut2Q(uH2(Ω))2((Δαt,ut)). (3.6)

    Multiplying also (2.2) by Δαt, we have

    ddt(Δα2+αt2)=2((Δαt,ut)). (3.7)

    Summing then (3.6) and (3.7), we obtain

    ddt(Δu2+Δα2+αt2)+ut2Q(uH2(Ω)). (3.8)

    In particular, setting

    y=Δu2+Δα2+αt2,

    we deduce from (3.8) an inequation of the form

    yQ(y). (3.9)

    Let z be the solution to the ordinary differential equation

    z=Q(z),z(0)=y(0). (3.10)

    It follows from the comparison principle that there exists T0=T0(u0H2(Ω),α0H2(Ω),α1H1(Ω)) belonging to, say, (0,12) such that

    y(t)z(t),t[0,T0], (3.11)

    hence

    u(t)2H2(Ω)+α(t)2H2(Ω)+αt(t)2H1(Ω)Q(u0H2(Ω),α0H2(Ω),α1H1(Ω)),tT0. (3.12)

    We now differentiate (3.1) with respect to time and have, noting that 2αt2=Δαut,

    (Δ)1tutΔut+f(u)ut=Δαut. (3.13)

    We multiply (3.13) by tut and find, owing to (2.6)

    ddt(tut21)+32tut2ct(ut2+α2)+ut21,

    hence, noting that ut2cut1ut,

    ddt(tut21)+tut2ct(ut21+α2)+ut21. (3.14)

    In particular, we deduce from (3.4), (3.12), (3.14) and Gronwall's lemma that

    ut211tQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),t(0,T0]. (3.15)

    Multiplying then (3.13) by ut, we have, proceeding as above,

    ddtut21+ut2c(ut21+α2). (3.16)

    It thus follows from (3.4), (3.16) and Gronwall's lemma that

    ut21ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω))ut(T0)21,tT0, (3.17)

    hence, owing to (3.15),

    ut21ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),tT0. (3.18)

    We now rewrite (3.1) in the forme

    Δu+f(u)=hu(t),u=0onΓ, (3.19)

    for tT0 fixed, where

    hu(t)=(Δ)1ut+αt (3.20)

    satisfies, owing to (3.4) and (3.18)

    hu(t)ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),tT0. (3.21)

    We multiply (3.19) by u and have, noting that f(s)sc, c0, sR,

    u2chu(t)2+c. (3.22)

    Then, multipying (3.19) by Δu, we find, owing to (2.6),

    Δu2c(hu(t)2+u2). (3.23)

    We thus deduce from (3.21)(3.23) that

    u(t)2H2(Ω)ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),tT0, (3.24)

    and, thus, owing to (3.12)

    u(t)2H2(Ω)ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),t0. (3.25)

    Returning to (3.7), we have

    ddt(Δα2+αt2)Δαt2+ut2. (3.26)

    Noting that it follows from (3.4), (3.16) and (3.18) that

    tT0(Δαt2+ut2)dτectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),tT0, (3.27)

    we finally deduce from (3.12) and (3.25)(3.27) that

    u(t)2H2(Ω)+α(t)2H2(Ω)+αt(t)2H1(Ω)ectQ(u0H2(Ω),α0H2(Ω),α1H1(Ω)),t0. (3.28)

    4. Existence and uniqueness of solutions

    We first have the following.

    Theorem 4.1. We assume that (2.5)(2.8) hold and (α0,α1)(H10(Ω)H2(Ω))×H10(Ω). Then, (2.1)(2.4) possesses at last one solution (u,α,αt) such that

    u,αL(0,T;H10(Ω)H2(Ω)),utL2(0,T;H1(Ω))andαtL(0,T;H10(Ω)).

    Proof. The proof is based on (3.28) and, e.g., a standard Galerkin scheme.

    We have, concerning the uniqueness, the following.

    Theorem 4.2. We assume that the assumptions of Theorem 4.1 hold. Then, the solution obtained in Theorem 4.1 is unique

    Proof. Let (u(1),α(1),α(1)t) and (u(2),α(2),α(2)t) be two solutions to (2.1)(2.3) with initial data (u(1)0,α(1)0,α(1)1) and (u(2)0,α(2)0,α(2)1), respectively. We set

    (u,α,αt)=(u(1),α(1),α(1)t)(u(2),α(2),α(2)t)

    and

    (u0,α0,α1)=(u(1)0,α(1)0,α(1)1)(u(2)0,α(2)0,α(2)1).

    Then, (u,α) satisfies

    ut+Δ2uΔ(f(u(1))f(u(2)))=Δαt, (4.1)
    2αt2Δα=ut, (4.2)
    u=α=0onΩ, (4.3)
    u|t=0=u0,α|t=0=α0,αt|t=0=α1. (4.4)

    We multiply (4.1) by (Δ)1ut and (4.2) by αt and have, summing the two resulting equations,

    ddt(u2+α2+αt2)+ut21(f(u(1))f(u(2)))2. (4.5)

    Furthermore,

    (f(u(1))f(u(2))=(10f(u(1)+s(u(2)u(1)))dsu)10f(u(1)+s(u(2)u(1)))dsu+10f(u(1)+s(u(2)u(1)))(u(1)+s(u(2)u(1)))dsuQ(u(1)0H2(Ω),u(2)0H2(Ω),α(1)0H1(Ω),α(2)0H1(Ω),α(1)1H1(Ω),α(2)1H1(Ω))×(u+|u||u(1)|+|u||u(2)|)Q(u(1)0H2(Ω),u(2)0H2(Ω),α(1)0H1(Ω),α(2)0H1(Ω),α(1)1H1(Ω),α(2)1H1(Ω))u. (4.6)

    We thus deduce from (4.5) and (4.6) that

    ddt(u2+α2+αt2)+ut21Q(u(1)0H2(Ω),u(2)0H2(Ω),α(1)0H1(Ω),α(2)0H1(Ω),α(1)1H1(Ω),α(2)1H1(Ω))u2. (4.7)

    In particular, we have a differential inequality of the form

    dE2dtQE2, (4.8)

    where

    E2=u2+α2+αt2

    satisfies

    E2c(u2H1(Ω)+α2H1(Ω)+αt2)c. (4.9)

    It follows from (4.8)(4.9) and Gronwall's lemma that

    u(t)2H1(Ω)+α(t)2H1(Ω)+αt(t)2ceQt(u02H1(Ω)+α02H1(Ω)+α12),t0, (4.10)

    hence the uniqueness, as well as the continuous dependence with respect to the initial data in the H1×H1×L2-norm.


    Acknowledgments

    The authors wish to thank the referees for their careful reading of the paper and useful comments.


    Conflict of interest

    All authors declare no conflicts of interest in this paper.


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