Uniform boundedness of the attractor in H2 of a non-autonomous epidemiological system

Uniform boundedness of the attractor in H2of a non-autonomous epidemiological system Mar´ıa Anguiano Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla. P.O. Box 1160, 41080-Sevilla (Spain) e-mail: [email protected] Abstract In this paper, we prove the uniform boundedness of the pullback attractor of a non-autonomous SIR (susceptible, infected, recovered) model from epidemiology considered in Anguiano and Kloeden [2]. We prove two uniform boundedness of this pullback attractor, firstly in the norm H1 0, and later, under appropriate additional assumptions, in the norm H2. Keywords: SIR epidemic model with diffusion; invariant sets; uniform boundedness in H2 Mathematics Subject Classifications (2010): 35B41 37B55 1 Introduction and setting of the problem Let us consider the following problem for a temporally-forced SIR (susceptible, infected, recovered) model with diffusion ∂S ∂t −∆S=aq(t)−aS +bI −γSI N, ∂I ∂t −∆I=−(a+b+c)I+γSI N, ∂R ∂t −∆R=cI −aR,                (1) with the Dirichlet boundary condition S(x, t) = I(x, t) = R(x, t) = 0 on ∂Ω×(t0,+∞),(2) and initial condition S(x, t0) = S0(x), I(x, t0) = I0(x), R(x, t0) = R0(x) for x∈Ω,(3) where Ω ⊂Rd,d≥1, is a bounded domain with a smooth boundary ∂Ω, N(t) = S(t) + I(t) + R(t) and t0∈R. We assume that the parameters a,b,cand γare positive constants such that γ+b 2+c 2< λ1, where λ1>0 is the first eigenvalue of the negative Laplacian with zero Dirichlet boundary condition in Ω. The temporal forcing term is given by a continuous function q:R→Rtaking positive bounded values, i.e. q(t)∈[q−, q+] for all t∈ Rwhere 0 < q−≤q+, such that q0∈L2 loc R;L2(Ω)and satisfies sup t0∈RZt0+1 t0 |q0(s)|2 L2(Ω) ds < ∞.(4) 1 Mar´ıa Anguiano Epidemiology is the study of the spread of diseases with the objective of tracing factors that are responsible for or contribute to their occurrence. Mathematical models are used extensively in the study of epidemiological phenomena. Most models for the transmission of infectious diseases (see for instance [1, 4]) descend from the classical SIR model of Kermack and McKendrick [7] established in 1927. Its classical form involves a system of autonomous ordinary differential equations for three classes, the susceptibles S, infectives Iand recovereds R, of a constant total population. There is a strong biological motivation to include time-dependent terms into epidemiological models, for instance temporally varying forcing is typical of seasonal variation of a disease [6, 9]. Our model (1)-(3) is a classical and well-known model from mathematical epidemiology in the form of the SIR equations, with diffusion, in which a temporal forcing term is considered. Several studies on this model have already been published. More precisely, in [2] we prove the existence and uniqueness of positive solutions of (1)-(3) for initial data in L2(Ω)3, and we establish that, if qtakes positive bounded values, the process associated to (1)-(3) has a unique pullback attractor A. In [3], we establish a regularity result for the unique positive solution to problem (1)-(3), and we prove some regularity results for the pullback attractor A. This study has motived the fact of investigating the problem considering in this paper. Moreover, as far as we know, there are no results in the literature concerning the uniform boundedness of the pullback attractor Aas we will consider in the present paper. The structure of the paper is as follows. In Section 2, we prove the uniform boundedness of the attractor A in H1 0(Ω)3. Then, under appropriate additional assumptions, the uniform boundedness in H2(Ω)3of Ais proved in Section 3. 2 Uniform boundedness of the pullback attractor in H1 0(Ω)3 We denote by (·,·) the inner product in L2(Ω), and by |·|L2(Ω) the associated norm. By ||·|| we denote the norm in H1 0(Ω), which is associated to the inner product ((·,·)) := (∇·,∇·).We will denote by h·,·i the duality product between H−1(Ω) and H1 0(Ω). In addition, X3denotes the space of functions (u1, u2, u3)∈L2(Ω)3with the scalar product ((u1, u2, u3),(v1, v2, v3)) = (u1, v1)+(u2, v2)+(u3, v3) , and norm |(u1, u2, u3)|L2(Ω) =|u1|L2(Ω) +|u2|L2(Ω) +|u3|L2(Ω) , for all (u1, u2, u3),(v1, v2, v3)∈X3, while Y3denotes the space of functions (u1, u2, u3)∈H1 0(Ω)3with the scalar product (((u1, u2, u3),(v1, v2, v3))) = ((u1, v1)) + ((u2, v2)) + ((u3, v3)) , and norm ||(u1, u2, u3)|| =||u1|| +||u2|| +||u3|| , for all (u1, u2, u3),(v1, v2, v3)∈Y3. Finally, let X+ 3be the subspace of non-negative functions in X3and Y+ 3be the subspace of non-negative functions in Y3. The globally defined nonnegative solutions of (1)–(3) generate a process in the Banach space X+ 3(see [2]), i.e., a family of mappings Ut,t0:X+ 3→X+ 3with t≥t0in Rsatisfying Ut0,t0x=x, Ut,t0x=Ut,r ◦Ur,t0x, for all t0≤r≤tand x∈X+ 3. In [2, Proposition 1] we established that the 2-parameter family of mappings Ut,t0:X+ 3→X+ 3,t0≤t, given by Ut,t0(S0, I0, R0)=(S(t), I(t), R(t)),(5) where (S(t), I(t), R(t)) is the unique positive solution of (1)–(3) with the initial value (S0, I0, R0), defines a continuous process on X+ 3. 2 Mar´ıa Anguiano Recall that a pullback attractor for the process Ut,t0(e.g., cf. [5]) in the space X+ 3is a family A={A(t), t ∈R} of nonempty compact subsets of X+ 3, which is invariant in the sense that Ut,t0A(t0) = A(t),for all t≥t0, and pullback attracts bounded subsets Dof X+ 3, i.e., distX+ 3(Ut,t0D, A(t)) →0 as t0→ −∞, where we denote by distX+ 3(·,·) the Hausdorff semi-distance in X+ 3. In [2, Theorem 6.2, Remark 6] we establish that the process associated to (1)–(3) has a unique pullback attractor A, which satisfies A(t)⊂Σ+ 3, t ∈R,(6) where Σ+ 3is a closed and bounded subset of X+ 3. We recall a lemma (see [8]) which is necessary for the proof of our results. Lemma 1 Let X, Y be Banach spaces such that Xis reflexive, and the inclusion X⊂Yis continuous. Assume that {un}is a bounded sequence in L∞(t0, T;X)such that un* u weakly in Lq(t0, T;X)for some q∈[1,+∞) and u∈C0([t0, T]; Y).Then, u(t)∈Xfor all t∈[t0, T ]and ku(t)kX≤sup n≥1 kunkL∞(t0,T ;X)∀t∈[t0, T]. Let A:H1 0(Ω) →H−1(Ω) be the linear operator associated with the negative Laplacian. The operator Ais symmetric, coercive and continuous. Since the space H1 0(Ω) is included in L2(Ω) with compact injection, as a consequence of the Hilbert-Schmidt Theorem there exists a nondecreasing sequence 0 < λ1≤λ2≤. . . of eigenvalues of Awith zero Dirichlet boundary condition in Ω, with limj→∞ λj= +∞and there exists an orthonormal basis of Hilbert {wj:j≥1}of L2(Ω) and orthogonal in H1 0(Ω) with Vn:= span {wj: 1 ≤j≤n}and {Vn:n∈N}densely embedded in H1 0(Ω), such that Awj=λjwjfor all j≥1. For each integer n≥1, we denote by (Sn(t), In(t), Rn(t)) = (Sn(t;t0, S0), In(t;t0, I0), Rn(t;t0, R0)) the Galerkin approximation of the solution (S(t;t0, S0), I(t;t0, I0), R(t;t0, R0)) of (1)-(3), which is given by Sn(t) = n X j=1 γ1 nj(t)wj, In(t) = n X j=1 γ2 nj(t)wj, Rn(t) = n X j=1 γ3 nj(t)wj, and is the solution of d dt (Sn(t), wj) = h∆Sn(t), wji+ (f1(Sn(t), In(t), Rn(t), t), wj), d dt (In(t), wj) = h∆In(t), wji+ (f2(Sn(t), In(t), Rn(t)), wj), d dt (Rn(t), wj) = h∆Rn(t), wji+ (f3(Sn(t), In(t), Rn(t)), wj), with initial data (Sn(t0), wj) = (S0, wj),(In(t0), wj)=(I0, wj),(Rn(t0), wj) = (R0, wj), for all wj∈Vn, where γ1 nj(t)=(Sn(t), wj), γ2 nj(t)=(In(t), wj), γ3 nj(t)=(Rn(t), wj). 3 Mar´ıa Anguiano We denote f1(Sn(t), In(t), Rn(t), t) := aq(t)−aSn(t) + bIn(t)−γSn(t)In(t) Nn(t), f2(Sn(t), In(t), Rn(t)) := −(a+b+c)In(t) + γSn(t)In(t) Nn(t), f3(Sn(t), In(t), Rn(t)) := cIn(t)−aRn(t), where Nn(t) = Sn(t) + In(t) + Rn(t). On the other hand, if we denote D(A) = v∈H1 0(Ω) : Av ∈L2(Ω), with the scalar product (v, w)D(A)= (Av, Aw)∀v, w ∈D(A), then D(A) is a Hilbert space, and D(A) is included in H1 0(Ω) with continuous and dense injection. Let D(A)+ be the subspace of non-negative functions in D(A). Remark 2 We note that if Ω⊂Rdis a bounded C2domain, then we have that D(A) = H2(Ω) ∩H1 0(Ω), and moreover the norm induced by (·,·)D(A)in D(A)and the norm of H2(Ω) are equivalent. Now, in our first main result, we prove the uniform boundedness of the attractor A(t) in H1 0(Ω)3. Theorem 3 Suppose that Ω⊂Rdis a bounded C2domain and assume that γ+b 2+c 2< λ1where λ1is the first eigenvalue of the operator Aon the domain Ωwith Dirichlet boundary condition. Then A(t)is uniformly bounded in tin H1 0(Ω)3. Proof. From the inequality (27) of [3], for any t≥t0we have |Sn(r)|2 L2(Ω) +|In(r)|2 L2(Ω) +|Rn(r)|2 L2(Ω) +Zr t0kSn(s)k2+kIn(s)k2+kRn(s)k2ds ≤C1|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +(t−t0),(7) for all r∈[t0, t], and all n≥1, where C1:= max 1,a 2(q+)2|Ω| min 1,2−λ−1 1(b+c+ 2γ). From (7) and (26) in [3] we now obtain that (r−t0)kSn(r)k2+kIn(r)k2+kRn(r)k2(8) ≤C1|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) + (t−t0) + (q+)2|Ω|(t−t0)2(2a2+a 2k1C) +k1C|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω)(t−t0), for any t≥t0, all r∈[t0, t],and all n≥1, where C:= (2λ1−b−c−2γ)−1and k1is a positive constant. In particular, from (8) we deduce kSn(r)k2+kIn(r)k2+kRn(r)k2≤C2|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) + 1, (9) for all r∈[t0+ 1, t0+ 2] ,and any n≥1, where C2:= max nC1+ 2k1C, 2C1+ 4(q+)2|Ω|2a2+a 2k1Co. 4 Mar´ıa Anguiano Using Lemma 3 in [3], we have that (Sn(·), In(·), Rn(·)) = (Sn(·;t0, S0), In(·;t0, I0), Rn(·;t0, R0)) converges weakly to the unique solution to (1)-(3) (S(·), I(·), R(·)) = (S(·;t0, S0), I(·;t0, I0), R(·;t0, R0)) in L2(t0, t; (Y+)3), for all t>t0. Thus, from (9) and Lemma 1, we in particular obtain kS(t0+1)k2+kI(t0+1)k2+kR(t0+1)k2≤C2|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +1, which together with (6) imply that A(t) is uniformly bounded in tin H1 0(Ω)3. 3 Uniform boundedness of the pullback attractor in H2(Ω)3 The aim of this section is to continue with the analysis of the model in the sense of proving that the attractor A(t) satisfies that is uniformly bounded in the space H2(Ω)3provided some additional assumptions are fulfilled. Our second main result is the following. Theorem 4 In addition to the assumptions in Theorem 3, assume moreover that q0∈L2 loc(R;L2(Ω)),and satisfies (4). Then A(t)is uniformly bounded in tin H2(Ω)3. Proof. From inequality (35) in [3], taking t=t0+ 3 and ε= 2,we have |S0 n(r)|2 L2(Ω) +|I0 n(r)|2 L2(Ω) +|R0 n(r)|2 L2(Ω) (10) ≤(4k3+ 1) Zt0+3 t0+1 |S0 n(θ)|2 L2(Ω) +|I0 n(θ)|2 L2(Ω) +|R0 n(θ)|2 L2(Ω)dθ +aZt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ, for all r∈[t0+ 2, t0+ 3], and any n≥1, where k3is a positive constant. Analogously, and if we take s=t0+ 1 and r=t=t0+ 3 in inequality (25) of [3], we, in particular, have Zt0+3 t0+1 |S0 n(θ)|2 L2(Ω) +|I0 n(θ)|2 L2(Ω) +|R0 n(θ)|2 L2(Ω)dθ (11) ≤ kSn(t0+ 1)k2+kIn(t0+ 1)k2+kRn(t0+ 1)k2 + 3(q+)2|Ω|(2a2+a 2k1C) +k1C|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω), for all n≥1, where k1is a positive constant and C:= (2λ1−b−c−2γ)−1. From (10) and (11), we obtain |S0 n(r)|2 L2(Ω) +|I0 n(r)|2 L2(Ω) +|R0 n(r)|2 L2(Ω) ≤(4k3+ 1) kSn(t0+ 1)k2+kIn(t0+ 1)k2+kRn(t0+ 1)k2 + (4k3+ 1) 3(q+)2|Ω|(2a2+a 2k1C) + aZt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ + (4k3+ 1) k1C|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω), for all r∈[t0+ 2, t0+ 3], and any n≥1. Owing to this inequality and (9), there exists a constant e C1>0 such that |S0 n(r)|2 L2(Ω) +|I0 n(r)|2 L2(Ω) +|R0 n(r)|2 L2(Ω) (12) ≤e C1|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +Zt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ + 1, 5 Mar´ıa Anguiano for all r∈[t0+ 2, t0+ 3], and any n≥1. From inequality (36) of [3], and thanks to (12), we have |∆Sn(r)|2 L2(Ω) +|∆In(r)|2 L2(Ω) +|∆Rn(r)|2 L2(Ω) ≤4e C1|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +Zt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ + 1 + 8a2(q+)2|Ω|+ 4k2|Sn(r)|2 L2(Ω) +|In(r)|2 L2(Ω) +|Rn(r)|2 L2(Ω), for all r∈[t0+ 2, t0+ 3], and any n≥1,where k2is a positive constant. Therefore, by (7) we obtain that there exists a constant e C2>0 such that |∆Sn(r)|2 L2(Ω) +|∆In(r)|2 L2(Ω) +|∆Rn(r)|2 L2(Ω) (13) ≤e C2|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +Zt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ + 1, for all r∈[t0+ 2, t0+ 3], and any n≥1. By Theorem 6 in [3], we have that (S(·;t0, S0), I(·;t0, I0), R(·;t0, R0)) ∈C[t0+ 2, t0+ 3]; Y+ 3. On the other hand, in the proof of Theorem 4 in [3], we proved that {(Sn(·;t0, S0), In(·;t0, I0), Rn(·;t0, R0))}is bounded in L2(t0, t; (D(A)+)3) for all t>t0. Then, in particular, we have that (Sn(·), In(·), Rn(·)) = (Sn(·;t0, S0), In(·;t0, I0), Rn(·;t0, R0)) converges weakly to the unique solution, (S(·), I(·), R(·)) = (S(·;t0, S0), I(·;t0, I0), R(·;t0, R0)), to (1)-(3) in L2(t0+ 2, t0+ 3; (D(A)+)3). Then, by Lemma 1, inequality (13) and the equivalence of the norms |∆v|L2(Ω) and kvkH2(Ω), we have that there exists a constant e C3>0 such that k(S(r;t0, S0), I(r;t0, I0), R(r;t0, R0))k2 H2(Ω)3(14) ≤e C3|S0|2 L2(Ω) +|I0|2 L2(Ω) +|R0|2 L2(Ω) +Zt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ + 1, for all r∈[t0+ 2, t0+ 3],any t0∈R,and (S0, I0, R0)∈X+ 3. Thus, from (14), and using (5), we deduce that there exists a constant e C4>0 such that kUt0+2,t0(S0, I0, R0)k2 H2(Ω)3≤e C4|(S0, I0, R0)|2 L2(Ω) +Zt0+3 t0+1 |q0(θ)|2 L2(Ω) dθ+1, for all t0∈R,(S0, I0, R0)∈X+ 3. From this inequality, and the fact that A(t0) = Ut0,t0−2A(t0−2), we obtain k(v1, v2, v3)k2 H2(Ω)3(15) ≤e C4 sup (w1,w2,w3)∈A(t0−2) |(w1, w2, w3)|2 L2(Ω) +Zt0+1 t0−1 |q0(θ)|2 L2(Ω) dθ+1!, for all (v1, v2, v3)∈ A(t0),and any t0∈R. Now, from (6) and (15), we have that there exists M > 0 such that sup (v1,v2,v3)∈A(t0) k(v1, v2, v3)kH2(Ω)3!2 ≤M+Zt0+1 t0−1 |q0(θ)|2 L2(Ω) dθ, for any t0∈R. Finally, the assumption (4) implies the uniform boundedness of A(t) in H2(Ω)3. 6 Mar´ıa Anguiano Acknowledgments The author has been supported by Junta de Andaluc´ıa (Spain), Proyecto de Excelencia P12-FQM-2466. References [1] R.M. Anderson, R.M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, 1992. [2] M. Anguiano, P.E. Kloeden, Asymptotic behavior of the nonautonomous SIR equations with diffusion, Communications on Pure and Applied Analysis 13 No. 1 (2014) 157-173. [3] M. Anguiano, H2-boundedness of the pullback attractor for the non-autonomous SIR equations with diffusion, Nonlinear Analysis 113 (2015) 180-189. [4] F. Brauer, P. van den Driessche, Jianhong Wu (editors), Mathematical Epidemiology, Springer Lecture Notes in Mathematics, vol. 1945, Springer-Verlag, Heidelberg, 2008. [5] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997) 307–341. [6] M. J. Keeling, P. Rohani, B. T. 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