Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay

1 Introduction

In this article, we investigate the exponential stability of traveling waves in the following nonlocal dispersal delayed susceptible-infective-removed (SIR) epidemic model:

where d S , d I , d R , Λ , σ , μ , γ and ϱ are the positive constants. Here, S ( x , t ) , I ( x , t ) and R ( x , t ) stand for the densities of susceptible, infective, and removed individuals at position x and time t , respectively. The parameters d S , d I and d R describe the spatial motility of each class; the constant Λ > 0 represents the entering flux of the susceptible; γ > 0 is the recovery rate of the infective population; σ , μ , and ϱ are all positive parameters representing the death rates for all the susceptible, infective, and removed population, respectively; τ > 0 denotes the latent period of the disease. Moreover, J ( y ) denotes the probability distribution of rates of dispersal over distance y and ∫ R J ( x − y ) v ( y , t ) d y − v ( x , t ) can be interpreted as the net rate of increase due to dispersal of class v , where ∫ R J ( x − y ) v ( y , t ) d y is the standard convolution with space invariable x and v can be either S , I or R . In practical use, there are various types of the incidence term F ( S , I ) . The common types include bilinear incidence β S I , standard incidence β S I S + I + R and saturated incidence β S I 1 + α I .

In this article, we focus on the case of saturated incidence, and therefore, we assume that F ( S , I ) = β S I 1 + α I . Observing that the first two equations of (1.1) form a closed system and the function R can be derived as long as both S and I are solved, from now on, we only consider the first two equations of (1.1). Mathematically, for convenience, letting

scaling the spatial time variables, and absorbing the appropriate constant into u 1 and u 2 in ( 1.1 ) , we rewrite (1.1) in the following form (dropping the tildes on d , β and α for notational convenience).

Note that the infection-free equilibrium state (1, 0) always exists in system (1.2). Besides, when the basic reproduction number R 0 = β γ + μ > 1 , there also exists a positive endemic equilibrium state ( u 1 ∗ , u 2 ∗ ) , where

It is easy to see that ( 1 , 0 ) is unstable, and ( u 1 ∗ , u 2 ∗ ) is stable for the corresponding homogenous system for (1.2).

Throughout this article, we assume that ( 1.2 ) satisfies the initial conditions

We make the following conditions which are needed in the sequel.

1. J ∈ C 1 ( R ) , J ( x ) = J ( − x ) ≥ 0 , ∫ R J ( x ) d x = 1 , and J is compactly supported.

The theory of traveling wave solutions of reaction-diffusion systems has attracted much attention due to its significant nature in biology, chemistry, epidemiology and physics. For system (1.1), the spatial dynamics of some special cases have been extensively studied. System (1.1) is a nonlocal version of the following SIR epidemic model:

Yang et al. [1] derived the existence of a traveling wave connecting the disease-free steady state and the endemic steady state of (1.4) by the cross-iteration method and Schauder’s fixed point theorem. Later, by the upper-lower solution method and Schauder’s fixed point theorem, Li et al. [2] further obtained the existence and nonexistence of traveling waves of the subsystem

also, the minimal wave speed is established. When the natural death rate for all the susceptible, infective, and removed population are the same constant, system (1.4) is rewritten as

Applying the Schauder fixed point theorem to construct a family of solutions for the truncated problems and via the limiting argument, Fu [3] obtained the noncritical waves and critical waves of system (1.6). For the nonlocal system (1.1), Li et al. [4] proved the existence, nonexistence, and minimal wave speed of traveling waves; moreover, they discussed how the latency of infection and the spatial movement of the infective individuals affect the minimal wave speed.

Among the basic problems in the theory of traveling wave solutions, the stability of traveling wave solutions is an extremely important subject. Let us draw the background on the progress of the study in this subject. By the spectral analysis, Sattinger [5] considered a reaction-diffusion system without delay and proved that the traveling wavefronts were stable to perturbations in some exponentially weighted L ∞ spaces. Using the semigroup estimates, Kapitula [6] also studied a reaction-diffusion system without delay and obtained that the traveling wavefronts are stable in polynomially weighted L ∞ spaces. By the elementary super- and subsolution comparison and squeezing methods developed by Chen [7] (see also Wang et al. [8,9] for this technique), Smith and Zhao [10] studied the global asymptotic stability, Liapunov stability, and uniqueness of traveling wave solutions for a bistable quasimonotone delayed reaction-diffusion bistable equation on R . For the monostable case, since the unstable equilibrium, the study of the stability of traveling waves is more difficult than the bistable case. The first study of this case was obtained by Mei et al. [11] by using the weighted-energy method. They investigated a time-delayed diffusive Nicholson’s blowflies equation and proved that, under a weighted L 2 norm, if the solution is sufficiently close to a traveling wavefront initially, it converges exponentially to the wavefront as t → ∞ . By means of the weighted-energy method and the comparison principle, Lin and Mei [12] considered Nicholson’s blowflies equation with diffusion and found that the wavefront is time-asymptotically stable when the delay time is sufficiently small and the initial perturbation around the wavefront decays to zero exponentially in space as x → ∞ , but it can be large in other locations. In [13], Huang et al. used the anti-weighted-energy method developed by Chern et al. [14], considered a nonlocal dispersion equation with time delay, and proved that all noncritical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable when the initial perturbations around the waves are small. For the nonlocal version, by means of the weighted-energy method combined with the comparison principle, Lv and Wang [15] studied a nonlocal delayed reaction-diffusion equation and proved that traveling wavefronts are exponentially stable to perturbation in some exponentially weighted L ∞ spaces. Later, Yu et al. [16] extended this method to investigate the stability of invasion traveling waves for a competition system with nonlocal dispersals and proved that the invasion traveling waves are exponentially stable. For other related results on the stability of traveling wave solutions, one can refer to [17,18,19, 20,21,22, 23,24,25, 26,27].

Recently, Li et al. [28] investigated the delayed SIR epidemic model (1.4) and proved the exponential stability of traveling waves when the delay τ is less than some constant τ 0 by using the weighted-energy method and nonlinear Halanay’s inequality.

Encouraged by papers [11], [28], and [29], in this article, we will further consider the nonlocal dispersal delayed SIR epidemic model (1.1) by using the weighted energy method. But due to the effect of the nonlocal dispersal and the lack of quasi-monotonicity, we are not clear whether the weighted-energy method can also be used to solve the stability of traveling waves of this model. As a result, with the help of some technology, we successfully apply this method to prove that all noncritical traveling wave solutions with sufficiently large c ≫ 1 and arbitrarily large delays are exponentially stable. However, the shortcoming of this article is that we do not prove any nonlinear stability result for the slower waves with c > c min ( c can be arbitrarily close to c min ), where c min denotes critical wave speed, and particularly, the case for the critical traveling waves with c min . We leave this problem for further research.

The rest of this article is organized as follows. In Section 2, we give some preliminary lemmas and state our main stability result. In Section 3, we reformulate system (1.2) into the corresponding perturbed system around a given traveling wave solution and state the stability theorem of the new perturbed system. In Section 4, we devote to establish the a priori estimates, which are the core of this article.

2 Preliminaries and main result

First, we introduce some notations throughout this article. Let C > 0 denote a generic constant and C i > 0 ( i = 1 , 2 , … ) be a specific constant. I is an interval, typically I = R . Denote by L 2 ( I ) the space of square integrable functions defined on I and H k ( I ) ( k ≥ 0 ) the Sobolev space of the L 2 -function f ( x ) defined on the interval I whose derivatives d i d x i f ( i = 1 , 2 , … , k ) also belong to L 2 ( I ) . L w 2 ( I ) denotes the weighted L 2 -space with a weight function w ( x ) > 0 , and its norm is defined by

Let H w k ( I ) be the weighted Sobolev space with the norm

If T > 0 is a number and ℬ is a Banach space, we denote by C 0 ( [ 0 , T ] , ℬ ) the space of the ℬ -valued continuous function on [ 0 , T ] and by L 2 ( [ 0 , T ] , ℬ ) the space of the ℬ -valued L 2 -functions on [ 0 , T ] . The corresponding spaces of the ℬ -valued L 2 -functions on [ 0 , ∞ ) are defined similarly.

Throughout this article, we always assume that R 0 > 1 .

A traveling wave solution of (1.2) is a solution with the form

where c > 0 is called the wave speed, and ( ϕ 1 , ϕ 2 ) ∈ C 0 ( R , R ) is called the profile function. Furthermore, ( ϕ 1 , ϕ 2 ) with c > 0 satisfies

and the following asymptotic boundary conditions

By using the upper-lower solutions and Schauder’s fixed point theorem, in [4], Li et al. proved the existence of traveling wave solutions of system (1.2).

In order to state our stability result, we need a technical assumption.

1. (2.3) σ β 2 − β σ ( 2 α + 1 ) ( μ + γ ) + ( 3 α σ + β ) ( μ + γ ) 2 < 0 , 2 α σ 2 β + σ β 2 + β σ ( μ + γ ) − ( α σ + β ) ( μ + γ ) 2 > 0 .

Define two functions on η as follows:

and

According to (A2), it is easily checked

and

Therefore, by continuity, there exists η ∗ > 0 such that f 1 ( η ∗ ) > 0 and f 2 ( η ∗ ) > 0 . In addition, we also define other four functions on ξ as follows:

and

where ( ϕ 1 , ϕ 2 ) is a traveling wave solution given in Theorem 2.1. Then, we have the following lemma.

We define a weighted function as

Now, we present the corresponding stability theorem for the Cauchy problems (1.2) and (1.3) as follows.

3 Reformulation of the problem

Let ( u 1 ( x , t ) , u 2 ( x , t ) ) be the solution of Cauchy problems (1.2) and (1.3), and ( ϕ 1 ( x + c t ) , ϕ 2 ( x + c t ) ) be a given traveling wave solution of (1.2). Let ξ = x + c t and

Then, problems (1.2) and (1.3) can be reformulated as

and

with the initial condition

where

Let

endowed with the norm