Boundedness and Large-Time Behavior Results for a Diffusive Epidemic Model

We consider a reaction-diffusion system modeling the spread of an epidemic disease within a population divided into the susceptible and infective classes. We first consider the question of the uniform boundedness of the solutions for which we give a positive answer. Then we deal with the asymptotic behavior of the solutions where in particular we are interested in reasonable conditions leading to the extinction of the infection disease as the time goes to infinity.


Introduction
In this paper, we consider the following reaction-diffusion system of equations: with homogeneous Neuman boundary conditions and the nonnegative and bounded initial data S(0,x) = S 0 (x), I(0,x) = I 0 (x) in Ω, (1.3) where Ω is an open bounded domain in R n with smooth boundary ∂Ω and outer normal ν(x).The constants d 1 , d 2 , Λ, μ are such that We assume that t → λ(t) is a nonnegative and bounded function in C(R + ) with 0 ≤ λ(t) ≤ λ and the nonlinearity f (ξ,η) is a nonnegative differentiable function in R + × R + such that there exist two increasing nonnegative functions ϕ and ψ in C 1 (R + ) with ) The reaction-diffusion system (1.1)-(1.3)may be viewed as a diffusive epidemic model where S and I represent the nondimensional population densities of susceptibles and infectives, respectively.In other words, system (1.1)- (1.3) is a model describing the spread of an infection disease (such as AIDS, e.g.) within a population assumed to be divided into the susceptible and infective classes as precised (for further motivation, see, e.g., [1][2][3], and the references therein).
A basic question arising in this context is the existence of global solutions in C(Ω) as well as their uniform boundedness to system (1.1)- (1.3).When Λ = 0 (which corresponds to the situation where there is no new supply in the susceptible class), a quite similar question was studied by many authors (see [4][5][6]) and a positive answer was first given by Haraux and Youkana [7] using the Lyapunov function techniques (see also [8]) and later on by Kanel (see, e.g., [9]) using useful properties inherent to the underlying Green function.
However when Λ > 0, these studies, while directly leading to conclude a global existence of the solutions, do not seem of a direct application concerning the uniform boundedness.To establish the uniform boundedness of the solutions in this case (i.e., when Λ > 0), it is worthwhile to mention the method developed by Morgan [10] which can be successfully applied to our case provided that |ϕ(η)| ≤ cη β , β > 0. Clearly the class considered in this work of ϕ satisfying the limit as it handles nonlinearities of a weakly exponential growth is larger than that required in [10] of nonlinearities of a polynomial growth.Indeed, it is easily observed for instance that ϕ(η) = e η α − 1, 0 < α < 1, satisfies this limit.Unfortunately for the nonlinearities ϕ not of a polynomial growth and satisfying this limit, the method in [10] cannot be applied.
In this paper, we first consider this problem of uniform boundedness of the solutions to system (1.1)-(1.3)by proving that the Lyapunov function argument proposed in [7] (or in [8]) can be adapted to our situation.Interestingly, we show that the same Lyapunov function is not necessarily nonincreasing as established in [7,8] but rather it satisfies a Lamine Melkemi et al. 3 differential inequality from which the uniform boundedness of the solutions is readily deduced.
Then we deal with the long-time behavior of the solutions as the time goes to +∞ where in particular we are concerned with reasonable conditions allowing to assure that (S,I) goes to the infection-free state (Λ/μ,0) of system (1.1)-(1.3)as t → +∞ in the sense where N > 0 is a positive constant independent of t of which the expression will be explicitly given in Lemma 2.3 in the next section.

Boundedness of the solutions
The basic existence theory for abstract semilinear differential equations directly leads to conclude a local existence result to system (1.1)-(1.3)(see, e.g., Henry [11] or Pazy [12]).Thus for nonnegative S 0 , I 0 in the class L ∞ (Ω), there exists a unique local nonnegative solution (S, On the other hand, using the comparison principle, one may also show that from which it follows that the solutions S and I of system (1.1)-(1.3)are global and uniformly bounded as soon as we can show that I is uniformly bounded in ]0,T * [.Following Haraux and Youkana [7], let us consider the function defined on ]0,T * [, where δ and ε are positive constants satisfying The main result of the paper can be stated as follows.
Theorem 2.1.For the solution (S,I) of system (1.1)- (1.3) in ]0,T * [, let L(t) be the function defined by (2.2) with δ and ε satisfying (2.3).Then there exists a nonnegative constant a such that Proof.Let (S,I) be the solution of system (1.1)-(1.3) in ]0,T * [.Differentiating L(t) defined by (2.2) with respect to t and using Green's formula, one obtains where We observe that G involves a quadratic form with respect to ∇S and ∇I, which is nonnegative since the constants δ and ε satisfying (2.3) are chosen in such a way that the discriminant is ≤ 0 so that one concludes that G ≤ 0 a.e. on ]0,T * [ (see [7]).On the other hand, H may be written as follows: Lamine Melkemi et al. 5 such that (2.10) Again from (2.3) where now one checks that from which it is obviously deduced that H 2 ≤ 0 and (2.17) be chosen on purpose in such a way that H 3 ≤ a.To sum up, one has exactly as the theorem claimed.
We are now ready to establish the global existence and uniform boundedness of the solutions of (1.1)-(1.3).
Theorem 2.2.If f satisfies conditions (1.5) and (1.6), the solutions S and I of system (1.1)- (1.3) with nonnegative and bounded initial data S 0 , I 0 are global and uniformly bounded on [0,+∞[.Proof.Let (S,I) be the solution of system (1.1)-(1.3) in ]0,T * [.Multiplying inequality (2.4) by e (σ/2)t and then integrating over [0,t], we deduce that there exists a positive constant C > 0 independent of t such that In this proof, we will make use of the result established in [13] from which the uniform boundedness of I is derived once, (where C 1 (p) is a positive constant independent of t) for some p > n/2.In this direction, we observe that and both ϕ(η) and η satisfy lim so that it is quite sufficient to establish that (where C 2 (p) is a positive constant independent of t) for some p > n/2.
To that purpose, let δ > 0 and ε > 0 be two positive numbers satisfying (2.3).It is readily seen from (1.6) that there exists η 0 ≥ 0 such that (2.24) from which one gets the following estimates: (2.25) Lamine Melkemi et al. 7 Hence, one merely lets in order to obtain (2.23) and thereby (2.20).As precised, the result established in [13] permits to deduce the uniform boundedness of the solutions of (1.1)-(1.3)and the theorem is completely proved.
In order to make assumption (H.2) stated in the introduction meaningful, we expose the following result where we establish the expression defining the positive constant N introduced in (H.2) as well as the property it enjoys.It will be soon observed that N depends on positive constants M(r,n) and C(r,n) issued from known embedding theorems.To be more precise, we refer the reader to the appendix where the existence of M(r,n) is shown in (P.2) of Lemma A.1 and that of C(r,n) is claimed in Lemma A.2.
We merely say here that these constants M(r,n) and C(r,n) are supposed to be available in the following lemma.
To keep the flow of the main objectives of this work, we postpone to the appendix the proof of this lemma which is rather technical and somewhat long.

Asymptotic behavior of the solutions
In this section, we deal with the large-time behavior of the solutions S and I of system (1.1)-(1.3)as t → +∞.Before stating the results, let us expose some notations and simple facts concluded from the results of the previous section.First, thanks to Theorem 2.2, let R > 0 be a positive constant independent of t such that and set for q ≥ 1 so that using the mean value theorem, one checks that for all (t,x) ∈ R + × Ω and q ≥ 1, On the other hand, let us observe that the application of the maximum principle directly implies that so that if we set one obtains and 0 ≤ J(t,x) ≤ (Λ/μ)(1 − e −μt ) + S 0 ∞ e −μt .We observe that both I and J satisfy a parabolic equation of the same kind, namely with The results of this section are based on the preliminary lemma below.
Our first result of this section regarding the asymptotic behavior can be stated as follows.
Theorem 3.2.Let assumption (H.1) hold and let (S,I) be the solution of (1.1)- (1.3)Let q > 1 be the dual number of p, that is We assume for simplicity that p > 1 and q < +∞ since the cases p = 1 and q = +∞ can be treated similarly.Integrating the parabolic equation satisfied by I over Ω and using Holder's inequality and (3.3), we get Therefore integrating over [0, t] and using again Holder's inequality, we obtain where ω is the unique positive root of σX q − A q ψ(K) α|Ω|  Remark 3.4.In the light of the proof of Theorem 3.3, it is clear that the constant N defined by (2.27) and required in assumption (H.2) might be replaced by any other positive constant, say N , such that for all (t,x) ∈ R + × Ω, where ε(t) is a nonnegative function with lim t→0 ε(t) = 0.
in [0,+∞[.Then as t → +∞, The second result of this section concerns also the large-time behavior of the solutions and can be stated as follows.Theorem 3.3.Let assumption (H.2) hold with N defined by expression (2.27) introduced in Lemma 2.3.Then as t → +∞,