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The risk index for an SIR epidemic model and spatial spreading of the infectious disease

  • Received: 05 May 2016 Accepted: 19 September 2016 Published: 01 October 2017
  • MSC : Primary: 35K51, 35R35; Secondary: 35B40, 92D25

  • In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number RDA0 for an associated model with Dirichlet boundary condition, we introduce the risk index RF0(t) for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if RF0(t0)q1 for some t0 and the disease is vanishing if RF0()<1, while if RF0(0)<1, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

    Citation: Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1565-1583. doi: 10.3934/mbe.2017081

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  • In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number RDA0 for an associated model with Dirichlet boundary condition, we introduce the risk index RF0(t) for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if RF0(t0)q1 for some t0 and the disease is vanishing if RF0()<1, while if RF0(0)<1, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.


    As we all know, Keller and Segel [1] first proposed the classical chemotaxis model (hereafter called K-S model), which has been widely applied in biology and medicine. The model can be given by the following:

    {v1t=Δv1χ(v1v2)+f(v1),   xΩ, t>0,τv2t=Δv2v2+v1,   xΩ, t>0, (1.1)

    where v1 is the cell density, v2 is the concentration of the chemical signal, and f(v1) is the logistic source function. For the case of τ=1 and f(v1)=0, it has been proven that the classical solutions to system (1.1) always remain globally bounded when n=1 [2]. A critical mass phenomenon of system (1.1) has been shown in a two-dimensional space. Namely, if the initial data v10 satisfies v10L1(Ω)<4πχ, then the solution (v1,v2) is globally bounded [3]. Alternatively, if the initial data v10 satisfies v10L1(Ω)>4πχ, then the solution (v1,v2) is unbounded in finite or infinite time, provided Ω is simply connected [4,5]. In particular, for a framework of radially symmetric solutions in a planar disk, the solutions blow up in finite time if v10L1(Ω)>8πχ [6]. When f(v1)=0, Liu and Tao [7] changed τv2t=Δv2v2+v1 to v2t=Δv2v2+g(v1) with 0g(v1)Kvα1 for K,α>0, and obtained the global well-posedness of model (1.1) provided that 0<α<2n. Later on, the equation τv2t=Δv2v2+v1 was changed to 0=Δv2ϖ(t)+g(v1) with ϖ(t)=1|Ω|Ωg(v1(,t)) for g(v1)=vα1. Winkler [8] deduced that for any v10, the model (1.1) is globally and classical solvable if α<2n; conversely, if α>2n, then the solutions are unbounded in a finite-time for any Ωv10=m>0. For τ=0, when f(v1)v1(cdv1) with c,d>0, Tello and Winkler [9] deduced the global well-posedness of model (1.1) provided that d>n2nχ. Afterwards, when f(v1)=cv1dvϵ1 with ϵ>1,c0,d>0, Winkler [10] defined a concept of very weak solutions and observed that these solutions are globally bounded under some conditions. For more results on (1.1), the readers can refer to [11,12,13,14].

    Considering the volume filling effect [15], the self-diffusion functions and chemotactic sensitivity functions may have nonlinear forms of the cell density. The general model can be written as follows:

    {v1t=(ψ(v1)v1ϕ(v1)v2)+f(v1),   xΩ, t>0,τv2t=Δv2v2+v1,   xΩ, t>0. (1.2)

    Here, ψ(v1) and ϕ(v1) are nonlinear functions. When τ=1 and f(v1)=0, for any Ωv10=M>0, Winkler [16] derived that the solution (v1,v2) is unbounded in either finite or infinite time if ϕ(v1)ψ(v1)cvα1 with α>2n,n2 and some constant c>0 for all v1>1. Later on, Tao and Winkler [17] deduced the global well-posedness of model (1.5) provided that ϕ(v1)ψ(v1)cvα1 with α<2n,n1 and some constant c>0 for all v1>1. Furthermore, in a high-dimensional space where n5, Lin et al. [18] changed the equation τv2t=Δv2v2+v1 to 0=Δv2ϖ(t)+v1 with ϖ(t)=1|Ω|Ωv1(x,t)dx, and showed that the solution (v1,v2) is unbounded in a finite time.

    Next, we introduce the chemotaxis model that involves an indirect signal mechanism. The model can be given by the following:

    {v1t=(ψ(v1)v1ϕ(v1)v2)+f(v1),   xΩ, t>0,τv2t=Δv2v2+w,   xΩ, t>0,τwt=Δww+v1,   xΩ, t>0. (1.3)

    For τ=1, when ψ(v1)=1,ϕ(v1)=v1 and f(v1)=λ(v1vα1), the conclusion in [19] implied that the system is globally classical solvable if α>n4+12 with n2. Furthermore, the authors in [20,21,22] extended the boundedness result to a quasilinear system. Ren [23] derived the global well-posedness of system (1.3) and provided the qualitative analysis of such solutions. For τ=0, when ψ(s)c(s+1)θ and |ϕ(s)|ds(s+1)κ1 with s0,c,d>0 and θ,κR, Li and Li [24] obtained that the model (1.3) is globally classical solvable. Meanwhile, they also provided the qualitative analysis of such solutions. More results of the system with an indirect signal mechanism can be found in [25,26,27,28].

    Considering that the cell or bacteria populations have a tendency to move towards a degraded nutrient, the authors obtain another well-known chemotaxis-consumption system:

    {v1t=Δv1χ(v1v2),   xΩ, t>0,v2t=Δv2v1v2,   xΩ, t>0, (1.4)

    where v1 denotes the cell density, and v2 denotes the concentration of oxygen. If 0<χ16(n+1)v20L(Ω) with n2, then the results of [29] showed that the system (1.4) is globally classical solvable. Thereafter, Zhang and Li [30] deduced the global well-posedness of model (1.4) provided that n2 or 0<χ16(n+1)v20L(Ω),n3. In addition, for a sufficiently large v10 and v20, Tao and Winkler [31] showed that the defined weak solutions globally exist when n=3. Meanwhile, they also analyzed the qualitative properties of these weak solutions.

    Based on the model (1.4), some researchers have considered the model that involves an indirect signal consumption:

    {v1t=Δv1χ(v1v2),   xΩ, t>0,v2t=Δv2v1v2,   xΩ, t>0,wt=δw+v1,   xΩ, t>0, (1.5)

    where w represents the indirect signaling substance produced by cells for degrading oxygen. Fuest [32] obtained the global well-posedness of model (1.5) provided that n2 or v20L(Ω)13n, and studied the convergence rate of the solution. Subsequently, the authors in [33] extended the boundedness conclusion of model (1.5) using conditions n3 and 0<v20L(Ω)πn. For more results on model (1.5), the readers can refer to [34,35,36,37,38,39].

    Inspired by the work mentioned above, we find that there are few papers on the quasilinear chemotaxis model that involve the nonlinear indirect consumption mechanism. In view of the complexity of the biological environment, this signal mechanism may be more realistic. In this manuscript, we are interested in the following system:

    {v1t=(ψ(v1)v1χϕ(v1)v2)+λ1v1λ2vβ1,   xΩ, t>0,v2t=Δv2wθv2,   xΩ, t>0,0=Δww+vα1,   xΩ, t>0,v1ν=v2ν=wν=0,   xΩ, t>0,v1(x,0)=v10(x),v2(x,0)=v20(x),   xΩ, (1.6)

    where ΩRn(n1) is a bounded and smooth domain, ν denotes the outward unit normal vector on Ω, and χ,λ1,λ2,θ>0,0<α1θ,β2. Here, v1 is the cell density, v2 is the concentration of oxygen, and w is the indirect chemical signal produced by v1 to degrade v2. The diffusion functions ψ,ϕC2[0,) are assumed to satisfy

    ψ(s)a0(s+1)r1 and 0ϕ(s)b0s(s+1)r2, (1.7)

    for all s0 with a0,b0>0 and r1,r2R. In addition, the initial data v10 and v20 fulfill the following:

    v10,v20W1,(Ω)  with v10,v200,0 in Ω. (1.8)

    Theorem 1.1. Assume that χ,λ1,λ2,θ>0,0<α1θ, and β2, and that ΩRn(n1) is a smooth bounded domain. Let ψ,ϕC2[0,) satisfy (1.7). Suppose that the initial data v10 and v20 fulfill (1.8). It has been proven that if r1>2r2+1, then the problem (1.6) has a nonnegative classical solution

    (v1,v2,w)(C0(ˉΩ×[0,))C2,1(ˉΩ×(0,)))2×C2,0(ˉΩ×(0,)),

    which is globally bounded in the sense that

    v1(,t)L(Ω)+v2(,t)W1,(Ω)+w(,t)W1,(Ω)C,

    for all t>0, with C>0.

    Remark 1.2. Our main ideas are as follows. First, we obtain the L bound for v2 by the maximum principle of the parabolic equation. Next, we establish an estimate for the functional y(t):=1pΩ(v1+1)p+12pΩ|v2|2p for any p>1 and t>0. Finally, we can derive the global solvability of model (1.6).

    Remark 1.3. Theorem 1.1 shows that self-diffusion and logical source are advantageous for the boundedness of the solutions. In this manuscript, due to the indirect signal substance w that consumes oxygen, the aggregation of cells or bacterial is almost impossible when self-diffusion is stronger than cross-diffusion, namely r1>2r2+1. We can control the logical source to ensure the global boundedness of the solution for model (1.6). Thus, we can study the effects of the logistic source, the diffusion functions, and the nonlinear consumption mechanism on the boundedness of the solutions.

    In this section, we first state a lemma on the local existence of classical solutions. The proof can be proven by the fixed point theory. The readers can refer to [40,41] for more details.

    Lemma 2.1. Let the assumptions in Theorem 1.1 hold. Then, there exists Tmax(0,] such that the problem (1.6) has a nonnegative classical solution (v1,v2,w) that satisfies the following:

    (v1,v2,w)(C0(ˉΩ×[0,Tmax))C2,1(ˉΩ×(0,Tmax)))2×C2,0(ˉΩ×(0,Tmax)).

    Furthermore, if Tmax<, then

    lim suptTmax(v1(,t)L(Ω)+v2(,t)W1,(Ω))=.

    Lemma 2.2. (cf. [42]) Let ΩRn(n1) be a smooth bounded domain. For any s1 and ϵ>0, one can obtain

    Ω|z|2s2|z|2νϵΩ|z|2s2|D2z|2+CϵΩ|z|2s,

    for all zC2(ˉΩ) fulfilling zν|Ω=0, with Cϵ=C(ϵ,s,Ω)>0.

    Lemma 2.3. (cf. [43]) Let ΩRn(n1) be a bounded and smooth domain. For s1, we have

    Ω|z|2s+22(4s2+n)z2L(Ω)Ω|z|2s2|D2z|2,

    for all zC2(ˉΩ) fulfilling zν|Ω=0.

    Lemma 2.4. Let ΩRn(n1) be a bounded and smooth domain. For any zC2(Ω), one has the following:

    (Δz)2n|D2z|2,

    where D2z represents the Hessian matrix of z and |D2z|2=ni,j=1z2xixj.

    Proof. The proof can be found in [41, Lemma 3.1].

    Lemma 2.5. (cf. [44,45]) Let a1,a2>0. The non-negative functions fC([0,T))C1((0,T)) and yL1loc([0,T)) fulfill

    f(t)+a1f(t)y(t),  t(0,T),

    and

    t+τty(s)dsa2,  t(0,Tτ),

    where τ=min{1,T2} and T(0,]. Then, one deduces the following:

    f(t)f(0)+2a2+a2a1,  t(0,T).

    In this section, we provide some useful Lemmas to prove Theorem 1.1.

    Lemma 3.1. Let β>1, then, there exist M,M1,M2>0 such that

    v2(,t)L(Ω)M for all t(0,Tmax), (3.1)

    and

    Ωv1M1 for all t(0,Tmax). (3.2)

    Proof. By the parabolic comparison principle for v2t=Δv2wθ1v2, we can derive (3.1). Invoking the integration for the first equation of (1.6), one has the following:

    ddtΩv1=λ1Ωv1λ2Ωvβ1  for all t(0,Tmax). (3.3)

    Invoking the Hölder inequality, we obtain the following:

    ddtΩv1λ1Ωv1λ2|Ω|β1(Ωv1)β. (3.4)

    We can apply the comparison principle to deduce the following:

    Ωv1max{Ωv10,(λ1λ2)1β1|Ω|}=M1. (3.5)

    Thereupon, we complete the proof.

    Lemma 3.2. For any γ>1, we have the following:

    ΩwγC0Ωvαγ1  for all  t(0,Tmax), (3.6)

    where C0=2γ1+γ>0.

    Proof. For γ>1, multiplying equation 0=Δww+vα1 by wγ1, one obtain the following:

    0=(γ1)Ωwγ2|w|2Ωwγ+Ωvα1wγ1Ωvα1wγ1Ωwγ  for all t(0,Tmax). (3.7)

    By Young's inequality, it is easy to deduce the following:

    Ωvα1wγ1γ12γΩwγ+2γ11γΩvαγ1. (3.8)

    Thus, we arrive at (3.6) by combining (3.7) with (3.8).

    Lemma 3.3. Let the assumptions in Lemma 2.1 hold. For any p>max{1,1θ1}, there exists C>0 such that

    12pddtΩ|v2|2p+12pΩ|v2|2p+14Ω|v2|2p2|D2v2|2CΩvθα(p+1)1+C, (3.9)

    for all t(0,Tmax).

    Proof. Using the equation v2t=Δv2wθ1v2, we obtain the following:

    v2v2t=v2Δv2v2(wθv2)=12Δ|v2|2|D2v2|2v2(wθv2), (3.10)

    where we used the equality v2Δv2=12Δ|v2|2|D2v2|2. Testing (3.10) by |v2|2p2 and integrating by parts, we derive the following:

    12pddtΩ|v2|2p+Ω|v2|2p2|D2v2|2+12pΩ|v2|2p=12Ω|v2|2p2Δ|v2|2+Ω|v2|2pΩ|v2|2p2v2(wθv2)=I1+12pΩ|v2|2p+I2. (3.11)

    Using Lemma 2.3 and (3.1), one has the following:

    Ω|v2|2p+2C1Ω|v2|2p2|D2v2|2  for all t(0,Tmax), (3.12)

    where C1=2(4p2+n)M2. In virtue of Lemma 2.2, Young's inequality, and (3.12), an integration by parts produces the following:

    I1+12pΩ|v2|2p=12Ω|v2|2p2Δ|v2|2+12pΩ|v2|2p=12Ω|v2|2p2|v2|2ν12Ω|v2|2p2|v2|2+12pΩ|v2|2p14Ω|v2|2p2|D2v2|2+C2Ω|v2|2pp12Ω|v2|2p4||v2|2|214Ω|v2|2p2|D2v2|2+14C1Ω|v2|2p+2+C312Ω|v2|2p2|D2v2|2+C3  for all t(0,Tmax), (3.13)

    with C2,C3>0. Due to |Δv2|n|D2v2|, we can conclude from (3.1) and the integration by parts that

    I2=Ω|v2|2p2v2(wθv2)=Ωwθv2(v2|v2|2p2)Ωwθv2(Δv2|v2|2p2+(2p2)|v2|2p2|D2v2|)Ω(n+2(p2))Mwθ|v2|2p2|D2v2|=C4Ωwθ|v2|2p2|D2v2|  for all t(0,Tmax), (3.14)

    with C4=(n+2(p2))M>0. Due to p>max{1,1θ1}, we have θ(p+1)>1. With applications of Young's inequality, (3.12), and Lemma 3.2, we obtain the following from (3.14):

    C4Ωwθ|v2|2p2|D2v2|18Ω|v2|2p2|D2v2|2+C5Ωw2θ|v2|2p218Ω|v2|2p2|D2v2|2+18C1Ω|v2|2p+2+C6Ωwθ(p+1)14Ω|v2|2p2|D2v2|2+C7Ωwθ(p+1)14Ω|v2|2p2|D2v2|2+C8Ωvθα(p+1)1, (3.15)

    with C5,C6,C7,C8>0. Substituting (3.13) and (3.15) into (3.11), we derive the following:

    12pddtΩ|v2|2p+12pΩ|v2|2p+14Ω|v2|2p2|D2v2|2C8Ωvθα(p+1)1+C3, (3.16)

    for all t(0,Tmax). Thereupon, we complete the proof.

    Lemma 3.4. Let the assumptions in Lemma 2.1 hold. If r1>2r2+1, then for any p>1, we obtain the following:

    1pddtΩ(v1+1)p+1pΩ(v1+1)p14Ω|v2|2p2|D2v2|2+(C+λ1+1p)Ω(v1+1)pλ2Ωvp+β11+C, (3.17)

    for all t(0,Tmax), with C>0.

    Proof. Testing the first equation of problem (1.6) by (v1+1)p1, one can obtain the following:

    1pddtΩ(v1+1)p+1pΩ(v1+1)p=(p1)Ω(v1+1)p2ψ(v1)|v1|2+1pΩ(v1+1)p+χ(p1)Ω(v1+1)p2ϕ(v1)v1v2+λ1Ωv1(v1+1)p1λ2Ωvβ1(v1+1)p1, (3.18)

    for all t(0,Tmax). In view of (1.7), the first term on the right-hand side of (3.18) can be estimated as follows:

    (p1)Ω(v1+1)p2ψ(v1)|v1|2(p1)a0Ω(v1+1)p+r12|v1|2. (3.19)

    For the second term on the right-hand side of (3.18), we can see that

    χ(p1)Ω(v1+1)p2ϕ(v1)v1v2χ(p1)b0Ωv1(v1+1)p+r22v1v2. (3.20)

    We can obtain the following from Young's inequality:

    χ(p1)b0Ωv1(v1+1)p+r22v1v2χ(p1)b0Ω(v1+1)p+r21v1v2(p1)a0Ω(v1+1)p+r12|v1|2+C1Ω(v1+1)p+2r2r1|v2|2, (3.21)

    with C1>0. Utilizing Young's inequality and (3.12), one has the following:

    C1Ω(v1+1)p+2r2r1|v2|218(4p2+n)M2Ω|v2|2(p+1)+C2Ω(v1+1)(p+1)(p+2r2r1)p14Ω|v2|2p2|D2v2|2+C2Ω(v1+1)(p+1)(p+2r2r1)p, (3.22)

    where C2>0. Due to r1>2r2+1, for any p>1>r12r22r2r1+1, we can obtain (p+1)(p+2r2r1)p<p. Applying Young's inequality, we obtain the following:

    C2Ω(v1+1)(p+1)(p+2r2r1)pC3Ω(v1+1)p+C3, (3.23)

    where C3>0. Hence, substituting (3.19)–(3.23) into (3.18), one obtains the following:

    1pddtΩ(v1+1)p+1pΩ(v1+1)p14Ω|v2|2p2|D2v2|2+(C3+λ1+1p)Ω(v1+1)pλ2Ωvp+β11+C4, (3.24)

    for all t(0,Tmax), where C4>0.

    Lemma 3.5. Let the assumptions in Lemma 2.1 hold. If r1>2r2+1, then for any p>max{1,1θ1}, we obtain the following:

    Ω(v1+1)p+Ω|v2|2pC, (3.25)

    where C>0.

    Proof. We can combine Lemma 3.3 with Lemma 3.4 to infer the following:

    ddt(1pΩ(v1+1)p+12pΩ|v2|2p)+1pΩ(v1+1)p+12pΩ|v2|2pC1Ωvθα(p+1)1+(C1+λ1+1p)Ω(v1+1)pλ2Ωvp+β11+C1, (3.26)

    where C1>0. Due to 0<α1θ and β2, we can obtain θα(p+1)p+1p+β1. Using Young's inequality, we can obtain the following:

    C1Ωvθα(p+1)1λ22Ωvp+β11+C2, (3.27)

    where C2>0. By the inequality (w+s)κ2κ(wκ+sκ) with w,s>0 and κ>1, we deduce the following:

    (C1+λ1+1p)Ω(v1+1)pλ22Ωvp+β11+C3, (3.28)

    where C3>0, where we have applied Young's inequality. Thus, we obtain the following:

    ddt(1pΩ(v1+1)p+12pΩ|v2|2p)+1pΩ(v1+1)p+12pΩ|v2|2pC4, (3.29)

    where C4>0. Therefore, we can obtain (3.25) by Lemma 2.5. Thereupon, we complete the proof.

    The proof of Theorem 1.1. Recalling Lemma 3.5, for any p>max{1,1θ1}, and applying the Lpestimates of elliptic equation, there exists C1>0 such that

    supt(0,Tmax)w(,t)W2,pα(Ω)C1  for all t(0,Tmax). (3.30)

    The Sobolev imbedding theorem enables us to obtain the following:

    supt(0,Tmax)w(,t)W1,(Ω)C2  for all t(0,Tmax), (3.31)

    with C2>0. Besides, using the well-known heat semigroup theory to the second equation in system (1.6), we can find C3>0 such that

    v2(,t)W1,(Ω)C3  for all t(0,Tmax). (3.32)

    Therefore, using the Moser-iteration[17], we can find C4>0 such that

    v1(,t)L(Ω)C4  for all t(0,Tmax). (3.33)

    Based on (3.31)–(3.33), we can find C5>0 that fulfills the following:

    v1(,t)L(Ω)+v2(,t)W1,(Ω)+w(,t)W1,(Ω)C5, (3.34)

    for all t(0,Tmax). According to Lemma 2.1, we obtain Tmax=. Thereupon, we complete the proof of Theorem 1.1.

    In this manuscript, based on the model established in [35], we further considered that self-diffusion and cross-diffusion are nonlinear functions, as well as the mechanism of nonlinear generation and consumption of the indirect signal substance w. We mainly studied the effects of diffusion functions, the logical source, and the nonlinear consumption mechanism on the boundedness of solutions, which enriches the existing results of chemotaxis consumption systems. Compared with previous results [29,32], the novelty of this manuscript is that our boundedness conditions are more generalized and do not depend on spatial dimension or the sizes of v20L(Ω) established in [32], which may be more in line with the real biological environment. In addition, we will further explore interesting problems related to system (1.6) in our future work, such as the qualitative analysis of system (1.6), the global classical solvability for full parabolic of system (1.6), and so on.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was partially supported by the National Natural Science Foundation of China (No. 12271466, 11871415).

    The authors declare there is no conflict of interest.

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