Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease
aa r X i v : . [ m a t h . NA ] M a r CONVERGENCE OF A FINITE VOLUME SCHEME FORNONLOCAL REACTION-DIFFUSION SYSTEMS MODELLINGAN EPIDEMIC DISEASE
MOSTAFA BENDAHMANE AND MAURICIO A. SEP ´ULVEDA
Abstract.
We analyze a finite volume scheme for nonlocal SIR model, whichis a nonlocal reaction-diffusion system modeling an epidemic disease. We es-tablish existence solutions to the finite volume scheme, and show that it con-verges to a weak solution. The convergence proof is based on deriving seriesof a priori estimates and using a general L p compactness criterion. Introduction
We consider a mathematical model describing an epidemic disease in a physicaldomain Ω ⊂ R d ( d = 1 , ,
3) over a time span (0 , T ), T >
0. In this modelwe consider the propagation of an epidemic disease in a simple population p = u + u + u , where u = u ( t, x ), u = u ( t, x ) and u = u ( t, x ) are the respectivedensities of susceptible (those who can catch the disease), infected (those who havethe disease and can transmit it) and of recover individuals (those who have beenexposed to the disease and will become infective after the lapse of an incubationperiod) at time t and location x . A prototype of a nonlinear system that governsthe spreading of a nonlocal SIR of epidemics with through a population in a spatialdomain is the following nonlocal reaction-diffusion system:(1.1) ∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = − σ ( u , u , u ) − µu ,∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = σ ( u , u , u ) − γu − µu ,∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = γu , in Q T , where Q T denotes the time-space cylinder (0 , T ) × Ω. We complete thesystem (1.1) with Neumann boundary conditions:(1.2) a i (cid:16)Z Ω u i dx (cid:17) ∇ u i · η = 0 on ∂ Ω × (0 , T ) , i = 1 , , , where η denotes the outer unit normal to the boundary ∂ Ω of Ω, and with initialdata:(1.3) u i (0 , x ) = u i, ( x ) , x ∈ Ω , i = 1 , , . Date : June 14, 2018.
Key words and phrases.
Finite volume scheme, reaction-diffusion system, weak solution, non-local SIR model.Partially supported by CMM, Universidad de Chile, and CI MA, Universidad de Concepci´on.
One of the simplest SIR models is the Kermack-McKendrick model [11] whichconsists in a systems of 3 × γ is thelength of latency period or duration of the exposed stage, and µ the natural mor-tality rate. The incidence term σ take the following form:(1.4) σ ( u, v, w ) = α uvu + v + w for some α > , which coincides with the classical model σ ( u, v, w ) = αuv when the total population P ( t ) remains constant. In fact, the well known SIR model which appears generallyin literature is renormalized which we do not suppose here. For technical reasons, weneed to extend the function σ ( u, v, w ) so that it becomes defined for all ( u, v, w ) ∈ R × R × R . We do this by setting σ ( u, v, w ) = (cid:26) σ ( u + , v + , w + ) if ( u, v, w ) = (0 , , , . In this work, the diffusion rates a > a > a > a i : R → R is a continuous function satisfying: there exist constants M i , C > M i ≤ a i and | a i ( I ) − a i ( I ) | ≤ C | I − I | for all I , I ∈ R , for i = 1 , , . Such equations with nolocal diffusion terms has already been studied from a theo-retical point of view by several authors. First, in 1997, M. Chipot and B. Lovat [4]studied the existence and uniqueness of the solutions for a scalar parabolic equationwith a nonlocal diffusion term. Existence-uniqueness and long time behaviour forother class of nonlocal nonlinear parabolic equations and systems are studied in[1, 14]. Liu and Jin made some experimental simulations in [13] in order to observespacial patterns in an epidemic model with constant removal rate of the infective.Before we define our finite volume scheme, let us state a relevant definition of aweak solution for the nonlocal SIR model.
Definition 1.1.
A weak solution of (1.1)-(1.3) is a triple u = ( u , u , u ) of func-tions such that u , u , u ∈ L (0 , T ; H (Ω)), − Z Ω u , ( x ) ϕ (0 , x ) dx − Z Z Q T u ∂ t ϕ dx dt + Z Z Q T a (cid:16)Z Ω u dx (cid:17) ∇ u · ∇ ϕ dx dt = − Z Z Q T ( σ ( u , u , u ) + µu ) ϕ dx dt, (1.6) − Z Ω u , ( x ) ϕ (0 , x ) dx − Z Z Q T u ∂ t ϕ dx dt + Z Z Q T a (cid:16)Z Ω u dx (cid:17) ∇ u · ∇ ϕ dx dt = Z Z Q T ( σ ( u , u , u ) − ( γ + µ ) u ) ϕ dx dt, (1.7) INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 3 − Z Ω u , ( x ) ϕ (0 , x ) dx − Z Z Q T u ∂ t ϕ dx dt + Z Z Q T a (cid:16)Z Ω u dx (cid:17) ∇ u · ∇ ϕ dx dt = Z Z Q T γu ϕ dx dt, (1.8)for all ϕ , ϕ , ϕ ∈ D ([0 , T ) × Ω).
Remark 1.
Note that we can easily check that Definition 1.1 makes sense. Further-more, observe that Definition 1.1 implies that ∂ t u i belongs to L (cid:0) , T ; ( H (Ω)) ′ (cid:1) ,so that u i ∈ C ([0 , T ]; L (Ω)) for i = 1 , , Remark 2.
A classical way to prove the existence of weak solutions in the sense of(1.6)-(1.8), is to use Faedo-Galerkin method like the system studied in [1] or in [14].On the other hand, the proof here, of convergence of the numerical scheme, impliesthe existence of weak solutions of (1.1)-(1.3). Additionally, a proof of uniquenessof the weak solution is given in the appendix.Following [9], we now give a precise definition of the finite volume scheme for thenonlocal SIR model. Let Ω be an open bounded polygonal connected subset of R with boundary ∂ Ω. Let Ω R be an admissible mesh of the domain Ω consisting ofopen and convex polygons called control volumes with maximum size (diameter) h .For all K ∈ Ω R , let by x K denote the center of K , N ( K ) the set of the neighborsof K i.e. the set of cells of Ω R which have a common interface with K , by N int ( K )the set of the neighbors of K located in the interior of Ω R , by N ext ( K ) the set ofedges of K on the boundary ∂ Ω. Furthermore, for all L ∈ N ( K ) denote by d ( K, L )the distance between x K and x L , by σ K,L the interface between K and L , by η K,L the unit normal vector to σ K,L outward to K . For all K ∈ Ω R , we denote by m ( K )the measure of K . The admissibility of Ω R implies that Ω = ∪ K ∈ Ω R K , K ∩ L = ∅ if K, L ∈ Ω R and K = L , and there exist a finite sequence of points ( x K ) K ∈ Ω R and the straight line x K x L is orthogonal to the edge σ K,L . We also need someregularity on the mesh: min K ∈ Ω R ,L ∈ N ( K ) d ( K, L )diam( K ) ≥ α for some α > H h (Ω) ⊂ L (Ω) the space of functions which are piecewise constanton each control volume K ∈ Ω R . For all u h ∈ H h (Ω) and for all K ∈ Ω R , wedenote by u K the constant value of u h in K . For ( u h , v h ) ∈ ( H h (Ω)) , we definethe following inner product: h u h , v h i H h = 12 X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u L − u K )( v L − v K ) , corresponding to Neumann boundary conditions. We define a norm in H h (Ω) by k u h k H h (Ω) = ( h u h , u h i H h ) / . Finally, we define L h (Ω) ⊂ L (Ω) the space of functions which are piecewise con-stant on each control volume K ∈ Ω R with the associated norm( u h , v h ) L h (Ω) = X K ∈ Ω R m ( K ) u K v K , k u h k L h (Ω) = X K ∈ Ω R m ( K ) | u K | , MOSTAFA BENDAHMANE AND M. A. SEP ´ULVEDA C. for ( u h , v h ) ∈ ( L h (Ω)) .Next, we let K ∈ Ω R and L ∈ N ( K ) with common vertexes ( a ℓ,K,L ) ≤ ℓ ≤ I with I ∈ N ⋆ . Next let T K,L (respectively T ext K,σ for σ ∈ N ext ( K )) be the open and convexpolygon with vertexes ( x K , x L ) ( x K respectively) and ( a ℓ,K,L ) ≤ ℓ ≤ I . Observe thatΩ = ∪ K ∈ Ω R (cid:16) ∪ L ∈ N ( K ) T K,L (cid:17) ∪ (cid:16) ∪ σ ∈ N ext ( K ) T ext K,σ (cid:17)!
The approximation ∇ h u h of ∇ u is defined by ∇ h u h ( x ) = ( m ( σ K,L ) d ( K,L ) ( u L − u K ) η K,L if x ∈ T K,L , x ∈ T ext K,σ , for all K ∈ Ω R .The next goal is to discretize the problem (1.1)-(1.3). We denote by D anadmissible discretization of Q T , which consists of an admissible mesh of Ω, a timestep ∆ t >
0, and a positive number N chosen as the smallest integer such that N ∆ t ≥ T . We set t n = n ∆ t for n ∈ { , . . . , N } .We approximate the nonlocal SIR model in the following way: Determine vectors( u ni,K ) K ∈ Ω R for i = 1 , ,
3, such that for all K ∈ Ω R and n ∈ { , . . . , N − } (1.9) u i, K = 1 m ( K ) Z K u i, ( x ) dx, i = 1 , , ,m ( K ) u n +11 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +11 ,L − u n +11 ,K )+ m ( K ) (cid:16) σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) + µu n +11 ,K (cid:17) = 0 , (1.10) m ( K ) u n +12 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +12 ,L − u n +12 ,K ) − m ( K ) (cid:16) σ ( u n, +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) − ( γ + µ ) u n +12 ,K (cid:17) = 0 , (1.11) m ( K ) u n +13 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +13 ,L − u n +13 ,K ) − m ( K ) γu n ,K = 0 . (1.12)To simplify the notation, it will always be understood that when h is sent to zerothen so is ∆ t , thereby assuming (without loss of generality) a functional relationshipbetween the spatial and temporal discretization parameters. This is not a realrestriction on the time step, but it allows us to write “ u i,h ” instead of “ u i,h, ∆ t ” for i = 1 , , h →
0” instead of “ h, ∆ t → u i,h ( t, x ) = u n +1 i,K , i = 1 , , , for all ( t, x ) ∈ ( n ∆ t, ( n + 1)∆ t ] × K , with K ∈ Ω R and n ∈ { , . . . , N − } . Tosimplify the notation, let us write u h for the vector ( u ,h , u ,h , u ,h ). Our mainresult is INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 5
Theorem 1.1.
Assume u i, ∈ L (Ω) for i = 1 , , . Then the finite volume solution u h , generated by (1.9) and (1.10) - (1.12) , converges along a subsequence to u =( u , u , u ) as h → , where u is a weak solution of (1.1) - (1.3) . The convergenceis understood in the following sense: u i,h → u i strongly in L ( Q T ) and a.e. in Q T , ∇ h u i,h → ∇ u i weakly in ( L ( Q T )) ,σ ( u ,h , u ,h , u ,h ) → σ ( u , u , u ) strongly in L ( Q T ) , for i = 1 , , . The remaining part of this paper is organized as follows. The proof of Theo-rem 1.1 is divided into Section 2 (existence of the scheme), Section 3 (basic a prioriestimates), Section 4 (space and time translation estimates), and Section 5 (conver-gence to a weak solution). In section 6 we give some numerical examples. Finallyin Appendix we prove the uniqueness of the solution using duality techniques.2.
Existence of the finite volume scheme
The existence of a solution to the finite volume scheme (1.9)-(1.12) will be ob-tained with the help of the following lemma proved in [12] and [16].
Lemma 2.1.
Let A be a finite dimensional Hilbert space with scalar product [ · , · ] and norm k·k , and let P be a continuous mapping from A into itself such that [ P ( ξ ) , ξ ] > for k ξ k = r > . Then there exists ξ ∈ A with k ξ k ≤ r such that P ( ξ ) = 0 . The existence for the finite volume scheme is given in
Proposition 2.2.
Let D be an admissible discretization of Q T . Then the problem (1.9) - (1.12) admits at least one solution ( u n ,K , u n ,K , u n ,K ) ( K,n ) ∈ Ω R ×{ ,...,N } .Proof. First we introduce the Hilbert spaces E h = ( H h (Ω) ∩ L h (Ω)) , under the norm k u h k E h := X i =1 k u i,h k H h (Ω) + X i =1 X K ∈ Ω R m ( K ) | u i,K | , where u h = ( u ,h , u ,h , u ,h ). Let Φ h = ( ϕ ,h , ϕ ,h , ϕ ,h ) ∈ E h and define thediscrete bilinear forms T h ( u nh , Φ h ) = X i =1 (cid:16) u ni,h , ϕ i,h (cid:17) ,b h ( u n +1 h , Φ h ) = X K ∈ Ω R m ( K ) (cid:16) σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) + µu n +11 ,K (cid:17) ϕ ,K − X K ∈ Ω R m ( K ) (cid:16) σ ( u n, +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) − ( γ + µ ) u n +12 ,K (cid:17) ϕ ,K , − γ X K ∈ Ω R m ( K ) u n ,K ϕ ,K , MOSTAFA BENDAHMANE AND M. A. SEP ´ULVEDA C. and a h ( u n +1 h , Φ h ) = X i =1 a i (cid:16) X K ∈ Ω h u ni,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +1 i,L − u n +1 i,K )( ϕ n +1 i,L − ϕ n +1 i,K ) . Multiplying (1.10), (1.11) and (1.12) by ϕ ,K , ϕ ,K , ϕ ,K , respectively, we get theequation1∆ t (cid:16) T h ( u n +1 h , Φ h ) − T h ( u nh , Φ h ) (cid:17) + a h ( u n +1 h , Φ h ) + b h ( u n +1 h , Φ h ) = 0 . Now we apply the Lemma 2.1 for proving the existence of u n +1 h for all K ∈ Ω R and n ∈ { , . . . , N } . We define the mapping P from E h into itself[ P ( u n +1 h ) , Φ h ] = 1∆ t ( T h ( u n +1 h , Φ h ) − T h ( u nh , Φ h ))+ a h ( u n +1 h , Φ h ) + b h ( u n +1 h , Φ h ) , for all Φ h ∈ E h . Note that it is easy to obtain from the discrete H¨older inequalitythe following bounds: a h ( u h , v h ) ≤ C k u h k E h k v h k E h ,T h ( u h , v h ) ≤ C k u h k E h k v h k E h ,b h ( u h , v h ) ≤ C k u h k E h k v h k E h , for all u h and v h in E h . This implies that a h , T h and b h are continuous. Thecontinuity of the mapping P follows from the continuity of the discrete forms a h ( · , · ), T h ( · , · ) and b h ( · , · ).Our goal now is to show that(2.1) [ P ( u n +1 h ) , u n +1 h ] > (cid:13)(cid:13) u n +1 h (cid:13)(cid:13) E h = r > , for a sufficiently large r . We observe that[ P ( u n +1 h ) , u n +1 h ] = 1∆ t X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) + a h ( u n +1 h , u n +1 h )+ b h ( u n +1 h , u n +1 h ) − t X i =1 X K ∈ Ω R m ( K ) u ni,K u n +1 i,K . (2.2) INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 7
It follows that from the definition of σ and (2.2) and Young’s inequality that[ P ( u n +1 h ) , u n +1 h ] ≥ t X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) + X i =1 M i (cid:13)(cid:13)(cid:13) u n +1 i,h (cid:13)(cid:13)(cid:13) H h (Ω) + X K ∈ Ω R m ( K ) (cid:16) σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) u n +1 , +1 ,K + µ (cid:12)(cid:12)(cid:12) u n +11 ,K (cid:12)(cid:12)(cid:12) (cid:17) − X K ∈ Ω R m ( K ) (cid:16) σ ( u n, +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) − ( γ + µ ) u n +12 ,K (cid:17) u n +12 ,K , − γ X K ∈ Ω R m ( K ) u n ,K u n +13 ,K − t X i =1 X K ∈ Ω R m ( K ) u ni,K u n +1 i,K ≥ t X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) + X i =1 M i (cid:13)(cid:13)(cid:13) u n +1 i,h (cid:13)(cid:13)(cid:13) H h (Ω) − t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) − C (∆ t, α ) X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n, +1 ,K (cid:12)(cid:12)(cid:12) − t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +13 ,K (cid:12)(cid:12)(cid:12) − C (∆ t, γ ) X K ∈ Ω R m ( K ) (cid:12)(cid:12) u n ,K (cid:12)(cid:12) − X i =1 t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) − C (∆ t ) X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ni,K (cid:12)(cid:12) . This implies that[ P ( u n +11 ,h ) , u n +11 ,h ] ≥ min n t , M , M , M o (cid:13)(cid:13) u n +1 h (cid:13)(cid:13) E h − n C (∆ t, α ) , C (∆ t, γ ) , C (∆ t ) o X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ni,K (cid:12)(cid:12) . (2.3)Finally, for given u n ,h , u n ,h and u n ,h , we deduce from (2.3) that (2.1) holds for r large enough (recall that (cid:13)(cid:13) u n +1 h (cid:13)(cid:13) E h = r ). Hence, we obtain the existence of at leastone solution to the scheme (1.9)-(1.12). (cid:3) Nonnegativity.
We have the following lemma.
Lemma 2.3.
Let ( u n ,K , u n ,K , u n ,K ) K ∈ Ω R ,n ∈{ ,...,N } be a solution of the finite vol-ume scheme (1.9) , (1.10) , (1.11) and (1.12) . Then, ( u n ,K , u n ,K , u n ,K ) K ∈ Ω R ,n ∈{ ,...,N } is nonnegative. MOSTAFA BENDAHMANE AND M. A. SEP ´ULVEDA C.
Proof.
Multiplying (1.10) by − ∆ tu n +11 ,K − , we find that − m ( K ) u n +11 ,K − ( u n +11 ,K − u n ,K ) + a (cid:16) X K ∈ Ω h u n ,K (cid:17) ∆ t X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +11 ,L − u n +11 ,K ) u n +11 ,K − − m ( K )∆ t (cid:16) σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) u n +11 ,K − + µu n +11 ,K (cid:17) u n +11 ,K − = 0 . (2.4)We know that u n +1 K = u n +1 K + − u n +1 K − and ( a + − b + )( a − − b − ) ≤ a, b ∈ R .With this, we deduce a (cid:16) X K ∈ Ω h u n ,K (cid:17) N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +11 ,L − u n +11 ,K ) u n +11 ,K − = − a (cid:16) X K ∈ Ω h u n ,K (cid:17) N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) ( u n +11 ,L − u n +11 ,K )( u n +11 ,L − − u n +11 ,K − ) ≥ a (cid:16) X K ∈ Ω h u n ,K (cid:17) N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) (cid:12)(cid:12)(cid:12) u n +11 ,L − − u n +11 ,K − (cid:12)(cid:12)(cid:12) ≥ , (2.5)and N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) + µu n +11 ,K (cid:17) u n +11 ,K − = − µ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +11 ,K − (cid:12)(cid:12)(cid:12) ≤ . (2.6)Let f ∈ C function. By using a Taylor expansion we find(2.7) f ( b ) = f ( a ) + f ′ ( a )( b − a ) + 12 f ′′ ( ξ )( b − a ) , for some ξ between a and b . Using the Taylor expansion (2.7) on the sequence f ( u n +11 ,K ) with f ( ρ ) = Z ρ − s ds , a = u n +11 ,K and b = u n ,K . We find(2.8) u n +11 ,K − ( u n +11 ,K − u n ,K ) = (cid:12)(cid:12) u n ,K − (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u n +11 ,K − (cid:12)(cid:12)(cid:12) − f ′′ ( ξ ) (cid:16) u n +11 ,K − u n ,K (cid:17) . We observe from the definition of f that f ′′ ( ρ ) ≥
0, which implies(2.9) u n +11 ,K − ( u n +11 ,K − u n ,K ) ≤ (cid:12)(cid:12) u n ,K − (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u n +11 ,K − (cid:12)(cid:12)(cid:12) . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 9
Now, using (2.5)-(2.9) to deduce from (2.4) N − X n =0 (cid:16) (cid:12)(cid:12)(cid:12) u n +11 ,K − (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) u n ,K − (cid:12)(cid:12) (cid:17) + a (cid:16) X K ∈ Ω h u n ,K (cid:17) N − X n =0 ∆ t (cid:12)(cid:12)(cid:12) u n +11 ,L − − u n +11 ,K − (cid:12)(cid:12)(cid:12) + µ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +11 ,K − (cid:12)(cid:12)(cid:12) ≤ . (2.10)This implies that(2.11) 12 (cid:18)(cid:12)(cid:12)(cid:12) u N ,K − (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u ,K − (cid:12)(cid:12)(cid:12) (cid:19) ≤ . Note that (2.12) is also true if we replace N by n ∈ { , . . . , N } , so we haveestablished(2.12) (cid:12)(cid:12)(cid:12) u n ,K − (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) u ,K − (cid:12)(cid:12)(cid:12) . Since u ,K is nonnegative, the result is u n +11 ,K − for all 0 ≤ n ≤ N − K ∈ Ω R .On the other hand, multiplying (1.11) by − ∆ tu n +12 ,K − , and along the same lines as u n +11 ,K , we obtain the nonnegativity of discrete solutions u n +12 ,K for all 0 ≤ n ≤ N − K ∈ Ω R . Finally, the nonnegativity of u n +13 ,K , is given by a maximum princi-ple proved in [8]. In fact the reaction terms m ( K ) γ u n ,K of the equation (1.12) donot depend on u n +13 ,K and it is a nonnegative term. We assume the nonnegativity of( u n ,K , u n ,K , u n ,K ) and we apply the discrete maximum principle for the third equa-tion (1.12) in order to prove the nonnegativity of u n +13 ,K . Then, using an inductionon n yields, we conclude. (cid:3) A priori estimates
The goal is to establish several a priori (discrete energy) estimates for the finitevolume scheme, which eventually will imply the desired convergence results.
Proposition 3.1.
Let ( u ni,K ) K ∈ Ω R ,n ∈{ ,...,N } , i = 1 , , , be a solution of the finitevolume scheme (1.9) - (1.12) . Then there exist constants C , C , C > , dependingon Ω , T , u i, and α such that (3.1) max n ∈{ ,...,N } X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ni,K (cid:12)(cid:12) ≤ C , (3.2) 12 N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) (cid:12)(cid:12)(cid:12) u n +1 i,K − u n +1 i,L (cid:12)(cid:12)(cid:12) ≤ C , and (3.3) N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16)(cid:12)(cid:12)(cid:12) σ ( u n ,K , u n +12 ,K , u n +13 ,K ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K ) (cid:12)(cid:12)(cid:12) (cid:17) ≤ C . for i = 1 , , . Proof.
We multiply (1.10), (1.11) and (1.12) by ∆ tu n +11 ,K , ∆ tu n +12 ,K and ∆ tu n +13 ,K ,respectively, and add together the outcomes. Summing the resulting equation over K and n yields E + E + E = E , where E = X i =1 N − X n =0 X K ∈ Ω R m ( K )( u n +1 i,K − u ni,K ) u n +1 i,K ,E = − X i =1 N − X n =0 ∆ t a i (cid:16) X K ∈ Ω h u ni,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +1 i,L − u n +1 i,K ) u n +1 i,K ,E = N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K ) + µu n +11 ,K (cid:17) u n +11 ,K + ( γ + µ ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) ! ,E = N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n ,K , u n +12 ,K , u n +13 ,K ) u n +12 ,K + γu n ,K u n +13 ,K (cid:17) . From the inequality “ a ( a − b ) ≥ ( a − b )”, we obtain E = X i =1 N − X n =0 X K ∈ Ω R m ( K )( u n +1 i,K − u ni,K ) u n +1 i,K ≥ X i =1 N − X n =0 X K ∈ Ω R m ( K ) (cid:18)(cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) u ni,K (cid:12)(cid:12) (cid:19) = 12 X i =1 X K ∈ Ω R m ( K ) (cid:16)(cid:12)(cid:12) u Ni,K (cid:12)(cid:12) − (cid:12)(cid:12) u i,K (cid:12)(cid:12) (cid:17) . Gathering by edges, we obtain E = − X i =1 N − X n =0 ∆ t a i (cid:16) X K ∈ Ω h u ni,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +1 i,L − u n +1 i,K ) u n +1 i,K ≥ X i =1 M i N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) (cid:12)(cid:12)(cid:12) u n +1 i,K − u n +1 i,L (cid:12)(cid:12)(cid:12) . Observe that from nonnegativity of ( u ni,K ) K ∈ Ω R ,n ∈{ ,...,N } for i = 1 , ,
3, we get E ≥ . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 11
We use Young’s inequality to deduce E = N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n ,K , u n +12 ,K , u n +13 ,K ) u n +12 ,K + γu n ,K u n +13 ,K (cid:17) ≤ α N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12) u n ,K (cid:12)(cid:12) + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +13 ,K (cid:12)(cid:12)(cid:12) ≤ ( α + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) + γ t X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ,K (cid:12)(cid:12) + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +13 ,K (cid:12)(cid:12)(cid:12) . Collecting the previous inequalities we obtain12 X i =1 X K ∈ Ω R m ( K )( (cid:12)(cid:12) u Ni,K (cid:12)(cid:12) − (cid:12)(cid:12) u i,K (cid:12)(cid:12) )+ X i =1 M i N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) (cid:12)(cid:12)(cid:12) u n +1 i,K − u n +1 i,L ) (cid:12)(cid:12)(cid:12) ≤ ( α + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) + γ t X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ,K (cid:12)(cid:12) + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +13 ,K (cid:12)(cid:12)(cid:12) , (3.4)which implies X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u Ni,K (cid:12)(cid:12) ≤ X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u i,K (cid:12)(cid:12) + ( α + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +12 ,K (cid:12)(cid:12)(cid:12) + γ t X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ,K (cid:12)(cid:12) + γ N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +13 ,K (cid:12)(cid:12)(cid:12) . (3.5)Clearly, X K ∈ Ω R m ( K ) (cid:12)(cid:12) u i,K (cid:12)(cid:12) ≤ k u i, k L (Ω) for i = 1 , , . In view of (3.5), this implies that there exist constants C , C > X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u Ni,K (cid:12)(cid:12) ≤ C + C X i =1 N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) . Note that (3.6) is also true if we replace N by n ∈ { , . . . , N } , so we haveestablished(3.7) X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n i,K (cid:12)(cid:12)(cid:12) ≤ C + C X i =1 n − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +1 i,K (cid:12)(cid:12)(cid:12) . By the discrete Gronwall inequality (see e.g. [10]), we obtain from (3.7)(3.8) X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n i,K (cid:12)(cid:12)(cid:12) ≤ C , for any n ∈ { , . . . , N } and some constant C >
0. Thenmax n ∈{ ,...,N } X i =1 X K ∈ Ω R m ( K ) (cid:12)(cid:12) u ni,K (cid:12)(cid:12) ≤ C . Moreover, we obtain from (3.4) and (3.8) the existence of a constant C > X i =1 N − X n =0 ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) (cid:12)(cid:12)(cid:12) u n +1 i,K − u n +1 i,L (cid:12)(cid:12)(cid:12) ≤ C . Finally a consequence of (1.4) and (3.1) is that k σ ( u ,h , u ,h , u ,h ) k L ( Q T ) ≤ C , for some constant C > (cid:3) Space and time translation estimates
In this section we derive estimates on differences of space and time translates ofthe function v h which imply that the sequence v h is relatively compact in L ( Q T ). Lemma 4.1.
There exists a constant
C > depending on Ω , T , u , , u , , u , , α , γ and µ such that (4.1) Z Z Ω ′ × (0 ,T ) | u i,h ( t, x + y ) − u i,h ( t, x ) | dx dt ≤ C | y | ( | y | + 2 h ) , i = 1 , , , for all y ∈ R with Ω ′ = { x ∈ Ω , [ x, x + y ] ⊂ Ω } , and (4.2) Z Z Ω × (0 ,T − τ ) | u i,h ( t + τ, x ) − u i,h ( t, x ) | dx dt ≤ C ( τ + ∆ t ) , i = 1 , , , for all τ ∈ (0 , T ) .Proof. The proof is similar to that found in, e.g, [8].
Proof of (4.1). Let y ∈ R , x ∈ Ω ′ , and L ∈ N ( K ). We set χ σ K,L = ( , if the line segment [ x, x + y ] intersects σ K,L , K and L, , otherwise . Next, the value c σ K,L is defined by c σ K,L = y | y | · η K,L with c σ K,L >
0. Observe that(4.3) Z Ω ′ χ σ K,L ( x ) dx ≤ m ( σ K,L ) | y | c σ K,L . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 13
Using this, we obtain | u ,h ( t, x + y ) − u ,h ( t, x ) | ≤ X σ K,L χ σ K,L ( x ) (cid:12)(cid:12)(cid:12) u n +11 ,L − u n +11 ,K (cid:12)(cid:12)(cid:12) . To simplify the notation, we write X σ K,L instead of X { ( K,L ) ∈ Ω R , K = L, m ( σ K,L ) =0 } . By the Cauchy-Schwarz inequality, we get | u ,h ( t, x + y ) − u ,h ( t, x ) | ≤ X σ K,L χ σ K,L ( x ) c σ K,L d ( K, L ) × X σ K,L (cid:12)(cid:12)(cid:12) u n +11 ,L − u n +11 ,K (cid:12)(cid:12)(cid:12) c σ K,L d ( K, L ) χ σ K,L ( x ) . (4.4)Note that(4.5) X σ K,L χ σ K,L ( x ) c σ K,L d ( K, L ) ≤ | y | + 2 h. Using (4.3), (4.4), and (4.5), we deduce
Z Z (0 ,T ) × Ω ′ | u ,h ( t, x + y ) − u ,h ( t, x ) | dx ≤ ( | y | + 2 h ) N − X n =0 ∆ t X σ K,L (cid:12)(cid:12)(cid:12) u n +11 ,L − u n +11 ,K (cid:12)(cid:12)(cid:12) c σ K,L d ( K, L ) Z Ω ′ χ σ K,L ( x ) dx ≤ | y | ( | y | + 2 h ) N − X n =0 ∆ t X σ K,L m ( σ K,L ) d ( K, L ) (cid:12)(cid:12)(cid:12) u n +11 ,L − u n +11 ,K (cid:12)(cid:12)(cid:12) . (4.6)Then, from (3.2) and (4.6), we deduce (4.1). Proof of (4.2). Let τ ∈ (0 , T ) and t ∈ (0 , T − τ ). We have B ( t ) = Z Ω | u ,h ( t + τ, x ) − u ,h ( t, x ) | dx. Set n ( t ) = E ( t/ ∆ t ) and n ( t ) = E (( t + τ ) / ∆ t ), where E ( x ) = n for x ∈ [ n, n + 1), n ∈ N .We get B ( t ) = X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n ( t )1 ,K − u n ( t )1 ,K (cid:12)(cid:12)(cid:12) . This implies B ( t ) = X L ∈ Ω R m ( K ) (cid:16) u n ( t )1 ,K − u n ( t )1 ,K (cid:17) X t< ( n +1)∆ t
0. We use (3.1) to deduce Z T − τ X t< ( n +1)∆ t
0. Hence Z T − τ B ( t ) ≤ τ C , for some constant C >
0. Reasoning along the same lines for u ,h yield (4.1) and(4.2) for u ,h and u ,h . This concludes the proof of the lemma. (cid:3) Convergence of the finite volume scheme
The next lemma is a consequence of Lemma 4.1 and Kolmogorov’s compactnesscriterion (see, e.g., [2], Theorem IV.25).
Lemma 5.1.
There exists a subsequence of u h = ( u ,h , u ,h , u ,h ) , not relabeled,such that, as h → ,(i) u i,h → u i strongly in L ( Q T ) and a.e. in Q T , (ii) ∇ h u i,h → ∇ u i weakly in ( L ( Q T )) , (iii) σ ( u ,h , u ,h , u ,h ) → σ ( u , u , u ) strongly in L ( Q T ) , (5.1) for i = 1 , , .Proof. The claim (i) in (5.1) follows from Lemma 4.1 and Kolmogorov’s compact-ness criterion (see, e.g., [2], Theorem IV.25). The proof of the claim (ii) will beomitted since it is similar to that of Lemma 4.4 in [3], we refer to the proof of thislemma for more details. The claim (iii) follows from Vitali theorem. (cid:3)
Our final goal is to prove that the limit functions u , u , u constructed in Lemma5.1 constitute a weak solution of the nonlocal system (1.1)-(1.3).We start by verifying (1.6). Let T be a fixed positive constant and ϕ ∈D ([0 , T ) × Ω). We multiply the discrete equation (1.10) by ∆ tϕ ( t n , x K ) for all K ∈ Ω R and n ∈ { , . . . , N } . Summing the result over K and n yields T + T + T = 0 , INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 17 where T = N − X n =0 X K ∈ Ω R m ( K )( u n +11 ,K − u n ,K ) ϕ ( t n , x K ) ,T = − N − X n =0 ∆ t a (cid:16) X K ∈ Ω h u n ,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +11 ,L − u n +11 ,K ) ϕ ( t n , x K ) ,T = N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K ) + µu n +11 ,K (cid:17) ϕ ( t n , x K ) . Doing integration-by-parts, keeping in mind that ϕ ( T, x K ) = 0 for all K ∈ Ω R ,we obtain T = − N − X n =0 X K ∈ Ω R m ( K ) u n +11 ,K ( ϕ ( t n +1 , x K ) − ϕ ( t n , x K )) − X K ∈ Ω R m ( K ) u ,K ϕ (0 , x K )= − N − X n =0 X K ∈ Ω R Z t n +1 t n Z K u n +11 ,K ∂ t ϕ ( t, x K ) dx dt − X K ∈ Ω R Z K u , ( x ) ϕ (0 , x K ) dx =: − T , − T , . Let us also introduce T ∗ = − N − X n =0 X K ∈ Ω R Z t n +1 t n Z K u n +11 ,K ∂ t ϕ ( t, x ) dx dt − Z Ω u , ( x ) ϕ (0 , x ) dx =: − T ∗ , − T ∗ , . Then T , − T ∗ , = X K ∈ Ω R Z K u , ( x )( ϕ (0 , x K ) − ϕ (0 , x )) dx. From the regularity of ϕ i , there exists a positive constant C such that | ϕ (0 , x K ) − ϕ (0 , x ) | ≤ C h, which implies (cid:12)(cid:12) T , − T ∗ , (cid:12)(cid:12) ≤ C h X K ∈ Ω R Z K | u , ( x ) | dx. (5.2)Sending h → h → (cid:12)(cid:12) T , − T ∗ , (cid:12)(cid:12) = 0 . Observe that T , − T ∗ , = N − X n =0 X K ∈ Ω R Z t n +1 t n Z K u n +11 ,K ∂ t ϕ ( t, x K ) dx dt − Z t n +1 t n Z K u n +11 ,K ∂ t ϕ ( t, x ) dx dt ! = N − X n =0 X K ∈ Ω R u n +11 ,K Z t n +1 t n Z K (cid:16) ∂ t ϕ ( t, x K ) − ∂ t ϕ ( t, x ) (cid:17) dx dt. Using the regularity of ∂ t ϕ and H¨older’s inequality, we obtain (cid:12)(cid:12) T , − T ∗ , (cid:12)(cid:12) ≤ C ( h ) N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:12)(cid:12)(cid:12) u n +11 ,K (cid:12)(cid:12)(cid:12) ! / , where C ( h ) > C ( h ) → h →
0. From (3.1) we deducelim h → (cid:12)(cid:12) T , − T ∗ , (cid:12)(cid:12) = 0 . Next, we define I D and T ∗ by I D = Z T a (cid:16)Z Ω u dx (cid:17)Z Ω ∇ u · ∇ ϕ dx dt,T ∗ = Z T a (cid:16)Z Ω u ,h dx (cid:17)Z Ω u ,h ∆ ϕ dx dt. Integration-by-parts yields I D = − Z T a (cid:16)Z Ω u dx (cid:17)Z Ω u ∆ ϕ dx dt. On the other hand, using the convergence results of Lemma 5.1, and taking into ac-count the assumption (1.5), it is easy to prove that a (cid:0)R Ω u ,h dx (cid:1) → a (cid:0)R Ω u dx (cid:1) strongly in L (0 , T ) and R Ω u ,h ∆ ϕ dx → − R Ω ∇ u · ∇ ϕ dx weakly in L (0 , T ), as h →
0. Thus, there holds T ∗ → − I D as h → . Note that T ∗ = N − X n =0 a (cid:16) X K ∈ Ω h u n ,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) u n +11 ,K Z t n +1 t n Z σ K,L ∇ ϕ · η K,L dγ = − N − X n =0 a (cid:16) X K ∈ Ω h u n ,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) ( u n +11 ,L − u n +11 ,K ) Z t n +1 t n Z σ K,L ∇ ϕ · η K,L dγ and T = − N − X n =0 ∆ ta (cid:16) X K ∈ Ω h u n ,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) u n +11 ,L − u n +11 ,K d ( K, L ) ϕ ( t n , x K )= 12 N − X n =0 a (cid:16) X K ∈ Ω h u n ,K (cid:17) ∆ t X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) × ( u n +11 ,L − u n +11 ,K )( ϕ ( t n , x L ) − ϕ ( t n , x K )) . Hence T + T ∗ = 12 N − X n =0 a (cid:16) X K ∈ Ω h u n ,K (cid:17) X K ∈ Ω R X L ∈ N ( K ) m ( σ K,L )( u n +11 ,L − u n +11 ,K ) × Z t n +1 t n ϕ ( t n , x L ) − ϕ ( t n , x K ) d ( K, L ) − m ( σ K,L ) Z t n +1 t n Z σ K,L ∇ ϕ · η K,L dγ ! . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 19
Since the straight line ( x K , x L ) is orthogonal to σ K,L , we have x K − x L = d ( K, L ) η K,L . This implies from the regularity of ϕ that ϕ ( t n , x L ) − ϕ ( t n , x K ) d ( K, L ) ≡ ∇ ϕ ( t n , x ) · η K,L with x between x K and x L , and so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t n +1 t n ϕ ( t n , x L ) − ϕ ( t n , x K ) d ( K, L ) − m ( σ K,L ) Z t n +1 t n Z σ K,L ∇ ϕ · η K,L dγ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∆ t h, (5.3)for some constant C >
0. Using (5.3) and (3.3), we deducelim h → T = − Z T a (cid:16)Z Ω u dx (cid:17)Z Ω u ∆ ϕ dx dt = Z T a (cid:16)Z Ω u dx (cid:17)Z Ω ∇ u ·∇ ϕ dx dt. Now, we show thatlim h → T = Z T Z Ω (cid:16) σ ( u , u , u ) + µu (cid:17) ϕ dx dt. For this purpose, we introduce T , := N − X n =0 X K ∈ Ω R (cid:16) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K )+ µu n +11 ,K (cid:17)Z t n +1 t n Z K (cid:16) ϕ ( t n , x K ) − ϕ ( t, x ) (cid:17) dx dt and T , := N − X n =0 X K ∈ Ω R Z t n +1 t n Z K (cid:16) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K ) − σ ( u , u , u ) (cid:17) ϕ ( t, x ) dx dt + N − X n =0 X K ∈ Ω R Z t n +1 t n Z K µ ( u n +11 ,K − u ) ϕ ( t, x ) dx dt. We have for all x ∈ K and t ∈ [ t n , t n +1 ] that(5.4) | ϕ ( t n , x K ) − ϕ ( t, x ) | ≤ C (∆ t + h ) , and thus, thanks to (3.1) and (3.3), | T , | ≤ C (∆ t + h ) N − X n =0 ∆ t X K ∈ Ω R m ( K ) (cid:16) σ ( u n +11 ,K , u n +12 ,K , u n +13 ,K ) + µu n +11 ,K (cid:17) ≤ C (∆ t + h ) . Hence, T , → h →
0. We also have | T , | ≤ C Z T Z Ω (cid:16) | σ ( u ,h , u ,h , u ,h ) − σ ( u , u , u ) | + | u ,h − u | (cid:17) dx dt. Therefore from this and (5.1), we deduce | T , | tends to zero as h → . This concludes the proof of (1.6). Reasoning along the same lines as above, weconclude that also (1.7) and (1.8) hold. Numerical Examples
Example 1. SIR model simulations.
In this section we consider a samplesquare domain Ω = (0 , × (0 ,
1) and we show the behavior of the solution fordifferent models of nonlocal functions a i , i = 1 , , Figure 1.
SIR model with a = a = a = 1 /
10: (a) Beginninginfected population ( t = 0 . sec. ); (b) Susceptible population( t = 0 . sec. ); (c) Infected population ( t = 0 . sec. ); (d) Recoverypopulation ( t = 0 . sec. ).We consider here a uniform mesh given by a Cartesian grid with N x × N y controlvolumes and choosing N x = N y = 300 for all simulations. Obviously, it is possibleto consider also unstructured meshes, but we will confine here to uniform mesh forsimplicity of the simulated models. The discretization in time is given by N t = 100time steps for T = 0 .
5. That is, δt = T /N t and m ( K ) = 1 / ( N x N y ). The parameterof the SIR model are given by α = 2 . µ = 0 .
01 and γ = 1 . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 21 (a) (b)(c) (d)
Figure 2.
SIR model with a = a = a = s/
10: (a) Begin-ning infected population; (b) Susceptible population; (c) Infectedpopulation; (d) Recovery population.The initial condition are given by u , ( x, y ) = ε ; u , ( x, y ) = B X j =1 sech( β ( x − x j ))sech( β ( y − y j )); u , ( x, y ) = 0 . . with ε = 0 . B = 5000, β = 2000, ( x , y ) = (0 . . . x , y ) = (0 . . . x , y ) = (0 . . . x , y ) = (0 . . . x , y ) = (0 . . . , / × [0 , /
2] whichwill diffuse the epidemic desease on the rest of the domain. Finally, we assume thatthere is no initially presence of recovery population.In a first simulation we consider a model which the diffusion rates do not dependon the population, that is a i , i = 1 , , . Figure 3.
Example 1: evolution in time of the diffusion terms a i (cid:18)Z Ω u i dx (cid:19) , with i = 1 , , a ( s ) = a ( s ) = a ( s ) = 0 . s ). Outside the nonlocal diffusion,we consider the same parameters for both simulations. We remark that secondsimulation with nonlocal diffusion violate the assumption (1.5). In fact, they cor-respond to degenerated parabolic cases. In order to ensure the convergence of ournumerical example, we replace in a practical way, the diffusion rate coefficients by(6.1) e a i ( s ) = M if s > Ms if ε s Mε if s < ε, with M = 10 and ε = 10 − . Figures 1 and 2 represent the simulation at time t = 0 .
025 for the localized infected population (see picture (a) of both figures) andthe simulation at time t = 0 . , × [0 ,
1] for each case. Finally, we observe in Figure 3, the evolution in timeof the diffusion a i (cid:18)Z Ω u i dx (cid:19) , with i = 1 , , u = 0 .
0. For this reason, it is not difficultto compute the Jacobian matrix for the SIR reaction term and to prove that thediffusion coefficients do not affect the sign of the real part of the eigenvalues. Inother, words, the Turing space is always empty in this case (see [17]), and thepopulations tend to their constant equilibrium state for long times.
INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 23
Example 2. An epidemic model of SARS with Patterns Formation.
Now, we consider the following modification of the SIR model(6.2) ∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = A − σ ( u , u , u ) − µu ,∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = σ ( u , u , u ) − γu − µu − H ( u ) ,∂ t u − a (cid:16)Z Ω u dx (cid:17) ∆ u = γu + H ( u ) − µu . In this modified SIR model, A is the recruitment rate of the population (such asgrowth rate of average population size, a recover becomes an susceptible, immi-grant and so on), and H ( u ) is the removal rate of infective individuals due to thetreatment. We suppose that the treated infectives becomes recovered when theyare treated in treatment, and H ( v ) = (cid:26) r if v > , , where r > σ defined in the introduction (1.4), and thenatural death rate µ is included for the recovery population. The detail aboutthis epidemic model can be found in [13, 18] where the authors adopt a bilinearincidence rate more simple than (1.4). One of the application to consider this modelis that it supposes that the capacity for the treatment of a disease in a communityis a constant r , in order to use the maximal treatment capacity to cure or isolateinfectives so that the disease is eradicated [18]. It can be used for example formathematical model to simulate the SARS outbreak in Beijing [19].The approximation of this modified SIR model using Finite Volume is very sim-ilar to (1.10)-(1.12). More precisely we have m ( K ) u n +11 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +11 ,L − u n +11 ,K )+ m ( K ) (cid:16) − A + σ ( u n +1 , +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) + µu n +11 ,K (cid:17) = 0 , (6.3) m ( K ) u n +12 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +12 ,L − u n +12 ,K ) − m ( K ) (cid:16) σ ( u n, +1 ,K , u n +1 , +2 ,K , u n +1 , +3 ,K ) − ( γ + µ ) u n +12 ,K − H ( u n +12 ,K ) (cid:17) = 0 , (6.4) m ( K ) u n +13 ,K − u n ,K ∆ t − a (cid:16) X K ∈ Ω h u n ,K (cid:17) X L ∈ N ( K ) m ( σ K,L ) d ( K, L ) ( u n +13 ,L − u n +13 ,K ) − m ( K ) (cid:16) γu n ,K + H ( u n +12 ,K ) − µu n +13 ,K (cid:17) = 0 . (6.5)The results of existence of solution of the Finite Volume scheme (Proposition 2.2),nonnegativity of the scheme (Lemma 2.3) and convergence main result (Theorem1.1) can be easy generalized using straightforward calculations to this modified SIRmodel (6.2) and his Finite Volume scheme (6.3)-(6.5). Analysis of spatial patterns.
The equilibrium state of the system (6.2) is notexactly the same described in [13] because we consider here a different incidenceterm than the given by Liu which takes the reduced form σ = αuv . In our case, thetwo positive equilibrium points are given by E = ( u , v , w ) and E = ( u , v , w )where v , = A − r R − A α ± p ( r α − A R − A α ) − A R α )2 α R (6.6) u , = A − r − R v , µ , w , = γ v , + rµ , (6.7)with R = µ + γ . The linear stability of the system (6.2) is obtained when the realpart of the eigenvalues of the Jacobian matrix J = − α vu + v + w + α uv ( u + v + w ) − µ − α uu + v + w + α uv ( u + v + w ) α uv ( u + v + w ) α vu + v + w − α uv ( u + v + w ) α uu + v + w − α uv ( u + v + w ) − γ − µ − α uv ( u + v + w ) γ − µ are negative. The eigenvalues of this Jacobian matrix are given by λ = − µ < λ and λ which are root of λ + (cid:18) (2 µ + γ ) + α v − uu + v + w (cid:19) λ + (cid:18) ( µ + γ ) µ + α ( µ + γ ) v − µ uu + v + w (cid:19) = 0 . Real part of λ and λ are negative if and only if all the coefficients of this polyno-mial function of degree 2 are positive, that is(6.8) u + v + wα > max (cid:26) u − v µ + γ , uµ + γ − vµ (cid:27) . We consider in this example the parameters A = 3, µ = 0 . α = 3 . r = 0 .
5, and γ = 0 . α of theincidence term). Replacing this parameter in (6.6)-(6.7), we have the equilibriumstate E = (7 . , . , . E = (4 . , . , . . It is easy to verify that the equilibrium point E verify the linear stability condition(6.8) and E corresponds to a instable point. Formation of spatial patterns resultsfrom the diffusion-induced instability when the real part of at least one of theeigenvalues of J = d d
00 0 d is positive [17]. If we suppose that d = d , choosing the parameters above and the the equilibrium point E , then Turingstabilities appear when (5 . d ) d + 3 . − . d < Simulations with local and nonlocal diffusion.
Similar the above Example 1,we consider here two simulations, one with a constant diffusion and another one withnonlocal diffusion. We take the same square domain of example 1, with a Cartesiangrid and choosing N x = N y = 300 for both simulations. The discretization intime is given by N t = 100 time steps for T = 2 .
5. That is, δt = T /N t and m ( K ) = 1 / ( N x N y ). The parameter of the SARS model are given by A = 3, µ = 0 . α = 3 . r = 0 .
5, and γ = 0 . INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 25 (ric) (a)(b) (c)
Figure 4.
Patter Formation for the SARS model (constant diffu-sion): (ric) Random initial condition; (a) Susceptible population;(b) Infected population; (c) Recovery population.In order to observe patterns formation, we consider the initial condition as follow: u , ( x, y ) = e u + ε ω ; u , ( x, y ) = e u + ε ω ; u , ( x, y ) = e u + ε ω ;where e u = 4 . , e u = 1 . , e u = 4 . E ), ε = ε = ε = 0 . ω i ∈ [0 ,
1] are random variables,with i = 1 , , a = 0 . a = 0 . a = 0 . t = 2 . (a)(b) (c) Figure 5.
Patter Formation for the SARS model (nonlocal dif-fusion): (a) Susceptible population; (b) Infected population; (c)Recovery population.On the other hand, in figure 5 we observe another simulations with a nonlocaldiffusion. In this simulation we consider the diffusion rate coefficients given by(6.9) e a i ( s ) = M if s > Ms if ε d i ( s − e u i ) Mε if s < ε, with M = 10 , ε = 10 − , d = d = 400000 and d = 400. The choice of d i isin order to take diffusion coefficients close to the values of the constant coefficientdiffusion of the first simulation (Figure 4) at time t = 0. In this case of nonlocaldiffusion, the coefficient diffusions are not constant in time, obtaining obviouslydifferent results than the first simulation of this example 2. Moreover, we remarkvery different behaviours between both simulations (compare Figure 4 and Figure5). Outside the nonlocal diffusion, we consider the same parameters for both sim-ulations in this example 2. Finally, we observe in Figure 6, the evolution in time INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 27 time [sec] diffusion a1diffusion a3 0 0.5 1 1.5 2 2.5020406080100120140160180200 time [sec] diffusion a2 Figure 6.
Example 1: evolution in time of the diffusion terms a i (cid:18)Z Ω u i dx (cid:19) , with i = 1 , ,
3; Left: Susceptible and recovery pop-ulations; Right: Infected population.of the diffusion a i (cid:18)Z Ω u i dx (cid:19) , with i = 1 , , Appendix A. Uniqueness of weak solutions.
In this appendix we prove uniqueness of weak solutions to our systems by usingduality technique (see e.g. [15]), thereby completing the well-posedness analysis.First, we consider ( u , u , u ) and ( v , v , v ) two solutions of the system (1.1)-(1.3). We set U i = u i − v i for i = 1 , ,
3, then U i satisfies(A.1) ∂ t U − (cid:16) a (cid:16)Z Ω u dx (cid:17) ∆ u − a (cid:16)Z Ω v dx (cid:17) ∆ v (cid:17) = − ( σ ( u , u , u ) − σ ( v , v , v )) − µU ,∂ t U − (cid:16) a (cid:16)Z Ω u dx (cid:17) ∆ u − a (cid:16)Z Ω v dx (cid:17) ∆ v (cid:17) = ( σ ( u , u , u ) − σ ( v , v , v )) − ( γ + µ ) U ,∂ t U − (cid:16) a (cid:16)Z Ω u dx (cid:17) ∆ u − a (cid:16)Z Ω v dx (cid:17) ∆ v (cid:17) = γU , ∇ u i · η = ∇ v i · η = 0 on Σ T , i = 1 , , ,U i ( x,
0) = 0 for x ∈ Ω , i = 1 , , . Now, we define the function ϕ i solution of the variational problem(A.2) Z Ω ∇ ϕ i ( t, · ) · ∇ φ dx = Z Ω U i ( t, · ) φ dx, for all φ ∈ H (Ω), such that R Ω φ dx = 0 i = 1 , , , for a.e. t ∈ (0 , T ). Since u i and v i are in L ( Q T ), then we get from the theoryof linear elliptic equations, the existence, uniqueness and regularity of solution ϕ i satisfying ϕ i ∈ C ([0 , T ]; H (Ω)) with Z Ω ϕ i ( t, · ) dx = 0 , for i = 1 , , . Note that from the boundary condition of ϕ i in (A.2) and U i (0 , · ) = 0 we deducethat(A.3) ∇ ϕ i (0 , · ) = 0 in L (Ω) for i = 1 , , . Multiplying the first, second and third equations in (A.1) by ψ , ψ , ψ ∈ L (0 , T ; H (Ω)),respectively, and integrating over Q t := (0 , t ) × Ω, we get X i =1 Z t h ∂ s U i , ψ i i ds + Z Z Q t (cid:16) a (cid:16)Z Ω u dx (cid:17) ∇ u − a (cid:16)Z Ω v dx (cid:17) ∇ v (cid:17) ·∇ ψ dx ds + Z Z Q t (cid:16) a (cid:16)Z Ω u dx (cid:17) ∇ u − a (cid:16)Z Ω v dx (cid:17) ∇ v (cid:17) ·∇ ψ dx ds + Z Z Q t (cid:16) a (cid:16)Z Ω u dx (cid:17) ∇ u − a (cid:16)Z Ω v dx (cid:17) ∇ v (cid:17) ·∇ ψ dx ds = − Z Z Q t (cid:16) ( σ ( u , u , u ) − σ ( v , v , v )) − µU (cid:17) ψ dx ds, + Z Z Q t (cid:16) ( σ ( u , u , u ) − σ ( v , v , v )) − ( γ + µ ) U (cid:17) ψ dx ds + Z Z Q t γU ψ dx ds. (A.4)Since ϕ i ∈ L (0 , T ; H (Ω)) we can take ψ i = ϕ i in (A.4) and we obtain from (A.2)and (A.3)(A.5) 2 Z t h ∂ s U i , ϕ i i ds = − Z t h ∂ s ∆ ϕ i , ϕ i i ds = Z Ω |∇ ϕ j ( t, x ) | dx − Z Ω |∇ ϕ i (0 , x ) | dx = Z Ω |∇ ϕ i ( t, x ) | dx. From definition of σ we obtain easily(A.6) | σ ( u , u , u ) − σ ( v , v , v ) | ≤ C X i =1 | u i − v i | , for some constant C > INITE VOLUME SCHEMES FOR NONLOCAL REACTION-DIFFUSION SYSTEMS 29
Using (A.2), (A.6), H¨older’s, Young’s, Sobolev poincar´e’s inequalities yields from(A.4) with ψ i = ϕ i X i =1 Z t h ∂ s U i , ϕ i i ds = − X i =1 Z Z Q t a i (cid:16)Z Ω u i dx (cid:17) U i ∆ ϕ i dx ds − X i =1 Z Z Q t (cid:16) a i (cid:16)Z Ω u i dx (cid:17) − a i (cid:16)Z Ω v i dx (cid:17) ∇ v i · ∇ ϕ i dx ds − Z Z Q t (cid:16) ( σ ( u , u , u ) − σ ( v , v , v )) − µU (cid:17) ϕ dx ds + Z Z Q t (cid:16) ( σ ( u , u , u ) − σ ( v , v , v )) − ( γ + µ ) U (cid:17) ϕ dx ds + Z Z Q t γU ϕ dx ds = − X i =1 M i Z Z Q t | U i | dx ds + X i =1 M i Z Z Q t | U i | dx ds + C X i =1 Z t k∇ v i k L (Ω) k∇ ϕ i k L (Ω) ds + (cid:16) X i =1 M i Z Z Q t | U i | dx ds + M Z Z Q t | U | dx ds (cid:17) + C Z t k∇ ϕ k L (Ω) ds + X i =1 M i Z Z Q t | U i | dx ds + C Z t k∇ ϕ k L (Ω) ds + M Z Z Q t | U | dx ds + C Z t k∇ ϕ k L (Ω) ds ≤ ( C + C + C + C ) X i =1 Z t ( k∇ v i k L (Ω) + 1) k∇ ϕ i k L (Ω) ds, (A.7)for some constants C , C , C , C > ∇ v i ∈ L ( Q T ) for i = 1 , , ϕ i = 0 , i = 1 , , , almost everywhere in Q T , ensuring the uniqueness of weak solutions. Acknowledgment.
This work has been supported by Fondecyt projects
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E-mail address : [email protected] (Mauricio A. Sepulveda) CI MA and Departamento de Ingenier´ıa Matem´aticaUniversidad de Concepci´onCasilla 160-C, Concepci´on, Chile
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