Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 49, pp. 1–21.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STABILITY OF BISTABLE TRAVELING WAVEFRONTS FOR A

NONLOCAL DISPERSAL EPIDEMIC SYSTEM

YU-CAI HAO, GUO-BAO ZHANG

Abstract. This article concerns the stability of traveling wavefronts for a
nonlocal dispersal epidemic system. Under a bistable assumption, we first

construct a pair of upper-lower solutions and employ the comparison principle
to prove that the traveling wavefronts are Lyapunov stable. Then, applying

the squeezing technique combining with appropriate upper-lower solutions, we

show that the traveling wavefronts are globally exponentially stable. As a
corollary, the uniqueness of traveling wavefronts is obtained.

1. Introduction

In this article, we investigate the stability of traveling wavefronts of the nonlocal
dispersal epidemic system

∂u1(x, t)

∂t
= d1D1[u1](x, t)− αu1(x, t) + h(u2(x, t)),

∂u2(x, t)

∂t
= d2D2[u2](x, t)− βu2(x, t) + g(u1(x, t)),

(1.1)

where x ∈ R, t ∈ R, α and β are all positive constants, di ≥ 0, Di[·] model nonlocal
dispersal represented by the convolution operators

Di[ui](x, t) := Ji ∗ ui(x, t)− ui(x, t) =

∫
R
Ji(x− y)ui(y, t)dy − ui(x, t), i = 1, 2.

The variables u1(x, t) and u2(x, t) respectively stand for densities of the infectious
agents and the infectious human population at location x and at time t; −αu1 is
the natural death rate of the bacterial population and h(u2) is the contribution of
the infective humans to the growth rate of the bacteria; −βu2 is the natural dimin-
ishing rate of the infective population because of the finite mean duration of the
infectious population and g(u1) is the infection rate of the human population under
the assumption that the total susceptible human population is constant during the
evolution of the epidemic; d1 and d2 are diffusion coefficients.

The environmental pollution by an infective human population can lead to the
spread of the infectious diseases, which is regarded as one of the main factors of
relevant epidemics, such as cholera and malaria [2]. Capasso and Paveri-Fontana

2020 Mathematics Subject Classification. 35K57, 35B35, 92D30.
Key words and phrases. Epidemic system; nonlocal dispersal; bistable traveling waves;

stability.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted October 27, 2021. Published July 12, 2022.

1



2 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

[4] proposed a model to describe the spread of cholera epidemic which happened in
the European Mediterranean regions in 1973,

du1(t)

dt
= −αu1(t) + au2(t),

du2(t)

dt
= −βu2(t) + g(u1(t)),

(1.2)

where a > 0 is a constant.
By considering the mobility of the bacteria and neglecting the small mobility of

the infectious population, Capasso and Maddalena [3] gave the system

∂u1(x, t)

∂t
= d

∂2u1(x, t)

∂x2
− αu1(x, t) + au2(x, t),

∂u2(x, t)

∂t
= −βu2(x, t) + g(u1(x, t)).

(1.3)

Zhao and Wang [33] established the existence of monotone traveling waves and
the minimal wave speed of (1.3) with monostable nonlinearity. Xu and Zhao [26]
proved the existence, uniqueness and global exponential stability of traveling waves
of (1.3) with bistable nonlinearity.

In 2012, Hsu and Yang [10] studied the epidemic system

∂u1(x, t)

∂t
= d1

∂2u1(x, t)

∂x2
− αu1(x, t) + h(u2(x, t)),

∂u2(x, t)

∂t
= d2

∂2u2(x, t)

∂x2
− βu2(x, t) + g(u1(x, t)),

(1.4)

and proved the existence, uniqueness, monotonicity and asymptotic behaviour of
traveling wave solutions of (1.4) under the monostable assumptions. Wu and Hsu
[22] investigated the existence of entire solutions for delayed monostable epidemic
models (1.4) with and without the quasi-monotone condition. We also refer readers
to [28] for existence and stability of traveling waves of (1.4) with discrete diffusion.

Note that the Laplacian operator ∂2u
∂x2 , which is used to describe the diffusion

of the infectious agents in (1.3) and (1.4) only depicts a local and short range
diffusion process. However, in reality, the migration or diffusion of the individuals
is not just limited in a local or short range, more details can refer Lee et al. [11]
and Murray [16]. So it is not enough or very accurate to formulate the diffusion
process of individuals in a long range by Laplacian operator. To consider the spatial
migration and describe this model reasonably, the following nonlocal operator

(Du)(x, t) = (J ∗ u)(x, t)− u(x, t) =

∫
R
J(x− y)[u(y, t)− u(x, t)]dy

was introduced in [7, 32, 29, 31]. For example, Zhang and Wang [31] proposed the
nonlocal dispersal epidemic system with time delay

∂u1(x, t)

∂t
= d(J ∗ u1 − u1)(x, t)− αu1(x, t) + au2(x, t),

∂u2(x, t)

∂t
= −βu2(x, t) + g(u1(x, t− τ)).

(1.5)

In the quasi-monotone monostable case, Zhang and Wang [31] proved the existence
of traveling wavefronts of (1.5) in both isotropic dispersal case and anisotropic dis-
persal case by constructing appropriate upper and lower solutions. Later on, Zhang,



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 3

Li and Wu [32] considered system (1.5) without delay, and studied the multi-type
entire solutions, when g is monotone increasing monostable and bistable nonlin-
earity, respectively. More recently, Zhang, Li and Feng [29] proved the stability of
traveling waves of (1.5) in the quasi-monotone monostable case and the non-quasi-
monotone monostable case. To our best knowledge, the stability of traveling waves
of (1.1) and (1.5) is not addressed before in the bistable case.

In this article, we shall focus our attention on the stability of bistable traveling
wavefronts of system (1.1). Our main assumptions are as follows:

(A1) Ji ∈ C1(R,R), Ji(x) = Ji(−x) ≥ 0, x ∈ R and
∫
R Ji(x)dx = 1.

(A2) For every λ ∈ R,
∫
R Ji(x)e−λxdx < +∞.

(A3) g, h ∈ C1(R+,R+), g(0) = h(0) = 0, v∗i = g(u∗i )/β, h(g(u∗i )/β) = αu∗i ,
i = 1, 2, where u∗1 < u∗2 are two positive constants.

(A4) g′(0)h′(0) < αβ, g′(u∗2)h′(v∗2) < αβ and g′(u∗1)h′(v∗1) > αβ.
(A5) g′(0) = 0, h′(0) = 0, g′(u) > 0 for u ∈ (0, u∗2], h′(v) > 0 for v ∈ (0, v∗2 ].

A typical example of g and h satisfying the conditions (A3)-(A5) is g(u) = u2

1+u2

and h(v) = av3/2 with a > 0. The spatially homogeneous system associated with
system (1.1) is written as follows

du1(t)

dt
= −αu1(t) + h(u2(t)),

du2(t)

dt
= −βu2(t) + g(u1(t)).

(1.6)

By (A3), this system has three equilibria E− = (0, 0), E0 = (u∗1, v
∗
1), and E+ =

(u∗2, v
∗
2). By (A4), E0 is a saddle point, E− and E+ are stable nodes, and hence,

the system (1.1) is a bistable system.
A traveling wave solution (in short, traveling wave) of (1.1) has the special form

(u1(x, t), u2(x, t)) = (φ1(ξ), φ2(ξ)), ξ = x + ct, where c ∈ R is the wave speed and
(φ1(ξ), φ2(ξ)) is the wave profile. Moreover, we say (φ1(ξ), φ2(ξ)) is a traveling
wavefront if (φ1(ξ), φ2(ξ)) is monotone in ξ ∈ R. We want to find the traveling
wavefronts of (1.1) connecting E− and E+. It is well known that (1.1) has a
traveling wave solution φ(ξ) = (φ1(ξ), φ2(ξ)) which connects E− and E+ if and
only if φ(ξ) satisfies the wave profile system

cφ′1(ξ) = d1D1[φ1](ξ)− αφ1(ξ) + h(φ2(ξ)),

cφ′2(ξ) = d2D2[φ2](ξ)− βφ2(ξ) + g(φ1(ξ)),
(1.7)

with

lim
ξ→−∞

(φ1(ξ), φ2(ξ)) = E−, lim
ξ→+∞

(φ1(ξ), φ2(ξ)) = E+. (1.8)

By the abstract theory in Fang and Zhao [8], we know that under assumptions
(A1)–(A5), there exists a unique constant c ∈ R such that (1.1) has a unique
increasing traveling wavefront φ(·) = (φ1(·), φ2(·)) connecting E− and E+.

As mentioned before, the main goal of this article is to show the stability of
traveling wavefronts of (1.1). Hence, we need to provide further details on the
progress of stability of traveling waves in this direction. The stability of traveling
wave solutions of systems with Laplace diffusions and nonlocal dispersals has been
well studied in the past few years. We refer to [6, 9, 12, 13, 14, 15, 17, 18, 19,
20, 23, 24, 25, 26] for Laplace diffusions and [1, 27, 29, 30] for nonlocal dispersals.
As we know, there are three classical methods which have been used to prove



4 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

the stability of traveling wave solutions. The first one is spectral analysis, see
[17, 24] for the non-critical speed case and [9, 25] for the critical speed case. The
second one is the method of weighted energy together with the comparison principle,
see [12, 14, 15, 27, 30] and the references cited therein. The third one is the
squeezing technique developed by Chen [5]. We can refer to [13, 18, 20] for bistable
equations. Motivated by [5, 13, 18, 20], we generalize the squeezing technique to
nonlocal dispersal system (1.1) for proving the global stability of bistable traveling
wavefronts. We should remark that the method used here can be also applied
to (1.5) and the global stability of bistable traveling wavefronts can be similarly
obtained.

This article is organized as follows. In Section 2, we study the comparison
principle of the solutions of the initial value problem corresponding to (1.1), and
then show some properties of traveling wavefronts of (1.1). In Section 3, we prove
the Lyapunov stability traveling wavefronts of (1.1). In Section 4, we establish the
global stability and uniqueness of traveling wavefronts of (1.1).

2. Preliminaries

In this section, we present two lemmas that will be useful later. In view of [21,
Lemma 3.1], the solution semigroup of the following nonlocal dispersal equation

∂u(x, t)

∂t
= (J ∗ u− u)(x, t), x ∈ R, t > 0,

u(x, 0) = ϕ̂(x), x ∈ R.

is given by

P (t)[ϕ̂](x) = e−t
∞∑
m=0

tm

m!
am(ϕ̂)(x), (2.1)

where a0(ϕ̂)(x) = ϕ̂(x), am(ϕ̂)(x) =
∫
R J(x− y)am−1(ϕ̂)(y)dy, for all m ≥ 1.

Let χ = BUC(R,R2) be a Banach space of bounded and uniformly continuous
vector-valued function from R to R2 with the general norm ‖ · ‖ and

χ1 = {ϕ(x) ∈ χ : E− ≤ ϕ(x) ≤ E+, ∀x ∈ R},

where ϕ(x) = (ϕ1(x), ϕ2(x)). Define

Ti(t)[·](x) = P (dit)[·](x), x ∈ R, t > 0, i = 1, 2,

where P (dit) is defined as in (2.1) with J = Ji, i = 1, 2. Let T (t) = (T1(t), T2(t)).
It is easy to see that T (t) : χ1 → χ1 is a positive and analytic semigroup.

Now we consider the initial value problem

∂u1(x, t)

∂t
= d1D1[u1](x, t)− αu1(x, t) + h(u2(x, t)),

∂u2(x, t)

∂t
= d2D2[u2](x, t)− βu2(x, t) + g(u1(x, t)),

u1(x, 0) = ϕ1(x), u2(x, 0) = ϕ2(x), x ∈ R.

(2.2)

Integrating the first two equations of (2.2) with ϕ(x) := (ϕ1(x), ϕ2(x))T , it can be
derived that the initial value problem (2.2) is equivalent to the integral equation

u(x, t) = T (t)ϕ(x) +

∫ t

0

T (t− s)Q(u(x, s))ds, (2.3)



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 5

where

u(x, t) :=

(
u1(x, t)
u2(x, t)

)
, T (t) :=

(
T1 0
0 T2

)
,

Q(u(x, t)) :=

(
Q1(u(x, t))
Q2(u(x, t))

)
=

(
−αu1(x, t) + h(u2(x, t))
−βu2(x, t) + g(u1(x, t))

)
.

Definition 2.1. A continuous function u(x, t) = (u1(x, t), u2(x, t)) : R × [τ, T ] →
R2, τ < T , is called an upper (lower) solution of system (1.1) on [τ, T ) if

u(x, t) ≥ (≤)T (t− s)u(x, s) +

∫ t

s

T (t− r)Q(u(x, r))ds,

for any τ ≤ s < t < T .

Remark 2.2. If a continuous function u(x, t) = (u1(x, t), u2(x, t)) is C1 with t ≥ 0
and satisfies

∂u1(x, t)

∂t
≥ (≤)d1D1[u1](x, t)− αu1(x, t) + h(u2(x, t)),

∂u2(x, t)

∂t
≥ (≤)d2D2[u2](x, t)− βu2(x, t) + g(u1(x, t)),

(2.4)

then u(x, t) is an upper (lower) solution of (1.1).

Now we give the first lemma, i.e., the strong comparison principle.

Lemma 2.3. Let u(x, t) = (u1(x, t), u2(x, t)) and v(x, t) = (v1(x, t), v2(x, t)) be
two solutions of (2.2) with u(x, 0) = ϕ(x) and v(x, 0) = ϕ̃(x), respectively, where

ϕ(x) = (ϕ1(x), ϕ2(x)); ϕ̃(x) = (ϕ̃1(x), ϕ̃2(x)) ∈ C(R,R+),

with E− ≤ ϕ̃(x) ≤ ϕ(x) ≤ E+, x ∈ R. Then for any (x, t) ∈ R× (0,∞),

E− ≤ v(x, t) ≤ u(x, t) ≤ E+

and

ui(x, t)− vi(x, t) ≥ Ni(L, t− t0)

∫ y+1

y

(ui(z, t0)− vi(z, t0))dz ≥ 0, (2.5)

for L ≥ 0, x, y ∈ R satisfying |x− y| ≤ L and t > t0 ≥ 0, i = 1, 2, where

N1(L, t− t0) = C1(L)e−(α+d1)(t−t0)(t− t0),

N2(L, t− t0) = C2(L)e−(β+d2)(t−t0)(t− t0),

where Ci(L) = minx∈[−L−1,L+1] Ji(x), i = 1, 2.

Proof. The first assertion of the lemma can be proved by the properties of the
monotone semiflow, see [7, 27]. So we omit it here. We shall prove that the
inequality (2.5) holds.

Let wi(x, t) = ui(x, t) − vi(x, t), i = 1, 2. Then wi(x, t) ≥ 0 for (x, t) ∈ R ×
(0,+∞). For any given 0 ≤ t0 < t and x, y ∈ R satisfying |x − y| ≤ L, it follows
that

w1(x, t)

= T1(t− t0)w1(x, t0) +

∫ t

t0

T1(t− r)[−αw1(x, r) + h(u2(x, r))− h(v2(x, r))]dr



6 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

= T1(t− t0)w1(x, t0) +

∫ t

t0

T1(t− r)[−αw1(x, r) + h′(ξ)w2(x, r)]dr

≥ T1(t− t0)w1(x, t0)− α
∫ t

t0

T1(t− r)w1(x, r)dr.

Let

z(x, t) = e−α(t−t0)T1(t− t0)w1(x, t0), t ≥ t0.

Then z(x, t) is the solution of the nonlocal dispersal equation

∂u

∂t
= d1(J1 ∗ u− u)− αu.

Thus,

z(x, t) = T1(t− t0)z(x, t0)− α
∫ t

t0

T1(t− r)z(x, r)dr, t ≥ t0.

It then follows that w1(x, t) ≥ z(x, t), t ≥ t0, and hence,

w1(x, t) ≥ e−α(t−t0)T1(t− t0)w1(x, t0)

= e−α(t−t0)e−d1(t−t0)
∞∑
m=0

(t− t0)m

m!
am[w1(x, t0)]

= e−(α+d1)(t−t0)
∞∑
m=0

(t− t0)m

m!

∫
R
J1(x− y)am−1[w1(y, t0)]dy

≥ e−(α+d1)(t−t0)(t− t0)

∫ y+1

y

J1(x− z)w1(z, t0)dz

≥ C1(L)e−(α+d1)(t−t0)(t− t0)

∫ y+1

y

w1(z, t0)dz,

where C1(L) = minx∈[−L−1,L+1] J1(x).
Similarly, one has

w2(x, t)

= T2(t− t0)w2(x, t0) +

∫ t

t0

T2(t− r)[−βw2(x, r) + g(u1(x, r))− g(v1(x, r))]dr

= T1(t− t0)w2(x, t0) +

∫ t

t0

T2(t− r)[−βw2(x, r) + g′(ξ)w1(x, r)]dr

≥ T2(t− t0)w2(x, t0)− β
∫ t

t0

T2(t− r)w2(x, r)dr,

and then

w2(x, t) ≥ e−β(t−t0)T2(t− t0)w2(x, t0)

≥ e−(β+d2)(t−t0)(t− t0)

∫
R
J2(x− y)w2(y, t0)dy

≥ C2(L)e−(β+d2)(t−t0)(t− t0)

∫ y+1

y

w2(z, t0)dz,

where C2(L) = minx∈[−L−1,L+1] J2(x). The proof is complete. �



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 7

The second lemma is about the limit behavior of the derivative of wave profiles
at ±∞. For the convenience, in what follows of this paper, we always denote
E+ = (u∗2, v

∗
2) := (k1, k2).

Lemma 2.4. Let φ(ξ) = (φ1(ξ), φ2(ξ)) be any traveling wavefront of (1.1) satisfy-
ing 0 ≤ φi(ξ) ≤ ki. Then lim|ξ|→+∞ φ′(ξ) = 0.

3. Lyapunov stability of traveling wavefronts

In this section, we prove the Lyapunov stability of traveling wavefronts of (1.1).
By (A3)–(A5), we can find sufficiently small constants pi > 0, i = 1, 2, such that

αp1 > ρp2, βp2 > ρp1, (3.1)

where ρ = max{ρ1, ρ2} with

ρ1 = max
{
h′(x) ≥ 0 : x ∈ [0, p2] ∪ [k2 − p2, k2]

}
> 0,

ρ2 = max
{
g′(x) ≥ 0 : x ∈ [0, p1] ∪ [k1 − p1, k1]

}
> 0.

We construct an upper solution and a lower solution of (1.1).

Lemma 3.1. Assume that (A1)–(A5) hold. Let φ(x+ ct) = (φ1(x+ ct), φ2(x+ ct))
be a traveling wavefront of (1.1). Define w±(x, t) = (w±1 (x, t), w±2 (x, t)) by

w+
i (x, t) = min

{
φi(η

+(x, t)) + δpie
−β0t, ki

}
,

w−i (x, t) = max
{
φi(η

−(x, t))− δpie−β0t, 0
}
,

where η±(x, t) = x + ct + ξ0 ± σ0δ(1 − e−β0t), i = 1, 2. Then there exist σ0 > 0,
β0 > 0, δ0 > 0 such that for any δ ∈ (0, δ0] and every ξ0, w+(x, t) and w−(x, t) are
an upper solution and a lower solution of (1.1), respectively.

Proof. We only verify that w+(x, t) is an upper solution of (1.1), since the lower
solution w−(x, t) can be treated similarly. Note that when w+

i (x, t) = ki for i = 1, 2,
it is easy to see that w+

i satisfies (2.4). Hence, in what follows, we consider the case
w+
i (x, t) = φi(η

+(x, t)) + δpie
−β0t.

For simplicity, we denote η+(x, t) by η. Let

µ := min
{αp1 − ρp2

p1
,
βp2 − ρp1

p2

}
> 0.

Fix β0 ∈ (0, µ) and δ∗ ∈ (0, p0) with p0 := max{p1, p2}. Then there exists M =
M(Φ, β0, δ

∗) > 0 large enough such that

φi(η) + δpi ≥ ki − δ∗, ∀δ ∈ (0, δ∗], η ≥M,

φi(η)− δpi ≤ δ∗, ∀δ ∈ (0, δ∗], η ≤ −M, i = 1, 2.

We can directly calculate that

∂w+
i (x, t)

∂t
= cφ′i(η) + β0σ0δe

−β0tφ′i(η)− β0δpie−β0t.

By (2.4), we need to prove that

∂w+
1 (x, t)

∂t
≥ d1D1[w+

1 ]− αw+
1 + h(w+

2 ),

∂w+
2 (x, t)

∂t
≥ d2D2[w+

2 ]− βw+
2 + g(w+

1 ).

(3.2)



8 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

The first inequality of (3.2) holds if and only if

cφ′1(η) + β0σ0δe
−β0tφ′1(η)− β0δp1e−β0t

≥ d1D1[φ1(η) + δp1e
−β0t]− α[φ1(η) + δp1e

−β0t] + h(w+
2 )

= d1
{
J1 ∗ (φ1(η) + δp1e

−β0t)− (φ1(η) + δp1e
−β0t)

}
− α[φ1(η) + δp1e

−β0t] + h(w+
2 )

= d1[J1 ∗ φ1(η)− φ1(η)]− αφ1(η)− αδp1e−β0t + h(w+
2 )

= d1D1[φ1](η)− αφ1(η)− αδp1e−β0t + h(w+
2 ) + h(φ2(η))− h(φ2(η)).

In view of (1.7), we only need to verify

δe−β0t[β0σ0φ
′
1(η)− β0p1] ≥ δe−β0t(−αp1) + [h(w+

2 )− h(φ2(η))]

⇔ β0σ0φ
′
1(η)− β0p1 ≥ −αp1 + δ−1eβ0t[h(w+

2 )− h(φ2(η))]

⇔ β0σ0φ
′
1(η)− β0p1 ≥ −αp1 + δ−1eβ0th′(θ)δp2e

−β0t = −αp1 + h′(θ)p2,

where θ ∈ [φ2(η), w+
2 (x, t)]. For |η| ≥ M , by the choice of M , it suffices to show

that

σ0β0φ
′
1(η)− β0p1 ≥ −αp1 + ρp2.

Obviously, the above inequality holds by φ′1(η) > 0 and the choice of β0. Since
φ′i(η) > 0, i = 1, 2, for |η| ≤M , we let

m0 := min
i=1,2

min{φ′i(η)||η| ≤M} > 0.

In addition, we take

s1 := max{h′(ξ)
∣∣ξ ∈ [0, k2]}, s2 := max{g′(ξ)

∣∣ξ ∈ [0, k1]},

s0 := max{s1, s2}, σ0 :=
p0(β0 + s0)

m0β0
> 0, δ0 := min

{
δ∗,

1

σ0

}
.

Then for |η| ≤M , we obtain

σ0β0φ
′
1(η)− β0p1 + αp1 − h′(θ)p2 ≥ σ0β0m0 − p0(β0 + s0) = 0.

The second inequality of (3.2) holds if and only if

cφ′2(η) + β0σ0δe
−β0tφ′2(η)− β0δp2e−β0t

≥ d2D2[φ2(η) + δp2e
−β0t]− β[φ2(η) + δp2e

−β0t] + g(w+
1 )

= d2
[
J2 ∗ (φ2(η) + δp2e

−β0t)− (φ2(η) + δp2e
−β0t)

]
− β[φ2(η) + δp2e

−β0t] + g(w+
1 )

= d2
{

[J2 ∗ φ2(η)− φ2(η)] + δe−β0t[J2 ∗ p2 − p2]
}
− βφ2(η)− βδp2e−β0t + g(w+

1 )

= d2D2[φ2](η)− βφ2(η)− βδp2e−β0t + g(w+
1 ) + g(φ1(η))− g(φ1(η)).

By (1.7), we only need to verify that

δe−β0t[β0σ0φ
′
2(η)− β0p2] ≥ δe−β0t(−βp2) + [g(w+

1 )− g(φ1(η))]

⇔ β0σ0φ
′
2(η)− β0p2 ≥ −βp2 + δ−1eβ0t[g(w+

1 )− g(φ1(η))]

⇔ β0σ0φ
′
2(η)− β0p2 ≥ −βp2 + δ−1eβ0tg′(θ)δp1e

−β0t = −βp2 + g′(θ)p1,



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 9

where θ ∈ [φ1(η), w+
1 (x, t)]. For |η| ≥ M , by the choice of M , it is equivalent to

show that

σ0β0φ
′
2(η)− β0p2 ≥ −βp2 + ρp1.

It is easy to see that the above inequality holds due to φ′2(η) ≥ 0 and the choice of
β0. For |η| ≤M , since φ′i(η) > 0, i = 1, 2, taking σ0 as above, we obtain

σ0β0φ
′
2(η)− β0p2 + βp2 − g′(θ)p1 ≥ m0σ0β0 − p0[β0 + s0] = 0.

The proof is complete. �

With the help of the upper and lower solution in Lemma 3.1 and the comparison
principle, we can observe the following Lyaponov stability theorem.

Theorem 3.2 (Lyapunov Stability). Assume that (A1)–(A5) hold. Let u(x, t;ϕ)
be the solution of (1.1) with the initial value ϕ. Then the traveling wavefront (φ, c)
of (1.1) is Lyapunov stable in the sense that for any ε > 0, there exists δ > 0 such
that ‖u(·, t;ϕ)− φ(·+ ct)‖ < ε provided that ‖ϕ− φ‖ ≤ δ.

Proof. It is clear that φ′i(ξ) is continuous in ξ ∈ R, i = 1, 2. Then by Lemma 2.4,
we obtain that φi(·) is uniformly continuous on R. Hence, for any ε > 0, there
exists δ1 = δ1(ε) > 0 such that for any |y| ≤ δ1, it holds

|φi(·+ y)− φi(·)| <
ε

4
. (3.3)

Choose δ = δ(ε) ∈ (0,min{ ε
4(1+max{p1,p2}) ,

δ1
σ0
, δ0}), where σ0 and δ0 are given in

Lemma 3.1. Then for every ϕ := (ϕ1, ϕ2) with ‖ϕ− φ‖ < δ, we have

max
{
φi(x)− δpi, 0

}
≤ ϕi(x) ≤ min

{
φi(x) + δpi, ki

}
, ∀x ∈ R, i = 1, 2.

By Lemma 3.1 and the comparison principle, one has

max
{
φi(η

−(x, t))− δpie−β0t, 0
}

≤ ui(x, t;ϕi)

≤ min
{
φi(η

+(x, t)) + δpie
−β0t, ki

}
, i = 1, 2,

(3.4)

where x ∈ R, t ≥ 0, and η±(x, t) := x+ ct± σ0δ(1− e−β0t). It is easy to see that

| ± σ0δ(1− e−β0t)| ≤ σ0δ < δ1, ∀t ≥ 0.

Hence, combining with (3.3) and (3.4), we obtain

φi(x+ ct)− ε

2
≤ ui(x, t;ϕi) ≤ φi(x+ ct) +

ε

2
, ∀x ∈ R, t ≥ 0, i = 1, 2,

which implies that

‖u(·, t;ϕ)− φ(·+ ct)‖ < ε, ∀t ≥ 0.

The proof is complete. �



10 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

4. Global stability of traveling wavefronts

In this section, we establish the global stability of traveling wavefronts of (1.1)
by applying the squeezing technique, and then show the uniqueness of traveling
wavefronts. To obtain the global stability of traveling wavefronts, we construct
another pair of upper and lower solutions which are different from that in Lemma
3.1. Let ζ(·) ∈ C∞(R) be a fixed function with the following properties:

ζ(s) = 0 ∀s ∈ (−∞, 0]; ζ(s) = 1, ∀s ∈ [4,∞);

0 < ζ ′(s) < 1, |ζ ′′(s)| ≤ 1, ∀s ∈ (0, 4).

We define v±(x, t) = (v±1 (x, t), v±2 (x, t)) by

v+i (x, t) = min
{
ki + δpi − [ki − (1− 2δ)pie

−εt]ζ(ς+ε,C(x, t)), ki
}
,

v−i (x, t) = max
{
− δpi + [ki − (1− 2δ)pie

−εt]ζ(ς−ε,C(x, t)), 0
}
,

where ζ(ς±ε,C(x, t) = ∓ε(x− ξ ± Ct), i = 1, 2.

Lemma 4.1. Assume that (A1)–(A5) hold. Then, for any δ ∈ (0, 1/2], there
exist two positive numbers ε = ε(δ) and C = C(δ) such that, for any ξ ∈ R, the
functions v+(x, t) and v−(x, t) are an upper solution and a lower solution of (1.1),
respectively.

Proof. We only prove that v+(x, t) is an upper solution of (1.1), since the lower
solution v−(x, t) can be showed in a similar way. We define

L1[v+1 ] :=
∂v+1
∂t
− d1D1v

+
1 + αv+1 − h(v+2 ),

L2[v+2 ] :=
∂v+2
∂t
− d2D2v

+
2 + βv+2 − g(v+1 ).

For simplicity, set ν = x− ξ + Ct. It is easy to calculate that

∂v+i
∂t

= εC[ki − (1− 2δ)pie
−εt]ζ ′(−εν)− ε(1− 2δ)pie

−εtζ(−εν)

≥ εC(ki − pi)ζ ′(−εν)− kiε, i = 1, 2.

In view of (3.1), we take ε = ε(δ) > 0 sufficiently small such that

−k1ε− d1k1ε
∫
R
J1(y)|y|dy + δ(αp1 − ρp2) > 0,

−k2ε− d2k2ε
∫
R
J2(y)|y|dy + δ(βp2 − ρp1) > 0.

(4.1)

Since p1, p2 are small enough such that δp1
2k1

+ δp2
2k2

< 1 and ζ ′(s) > 0 for ζ(s) ∈ (0, 1),

we can choose C = C(δ) large enough such that

min
{
εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1ε

∫
R
J1(y)|y|dy + αv+1 − h(v+2 ) :

δp1
2k1
≤ ζ(−εν) ≤ 1− δp2

2k2
, v+1 ∈ [δp1, k1], v+2 ∈ [δp2, k2]

}
> 0

(4.2)



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 11

and

min
{
εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2ε

∫
R
J1(y)|y|dy + βv+2 − g(v+1 ) :

δp2
2k2
≤ ζ(−εν) ≤ 1− δp1

2k1
, v+1 ∈ [δp1, k1], v+2 ∈ [δp2, k2]

}
> 0.

(4.3)

Let

ρ̃1 := max{h′(x) > 0|x ∈ (0, p2]}, ρ̃2 := max{g′(x > 0|x ∈ (0, p1]}.

Then by (A3) and (A5), we have

αk1 − ρ̃1k2 > 0, βk2 − ρ̃2k1 > 0. (4.4)

We prove that L1[v+1 ] ≥ 0 by distinguishing three cases:

Case(i): ζ(−εν) ≤ δp1
2k1

. It is easy to see that v+1 (x, t) = k1, and L1[v+1 ] ≥ 0 by

(A3) and (A5).

Case(ii): ζ(−εν) > 1− δp2
2k2

. In this case, we have δp2 < v+2 < p2− δp2
2 . In view of

(3.1), (4.1) and (4.4), we obtain

L1[v+1 ]

≥ εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1
{
J1 ∗ ζ(−εν)− ζ(−εν)

}
+ d1(1− 2δ)p1e

−εt{J1 ∗ ζ(−εν)− ζ(−εν)
}

+ α
{
k1 + δp1 − [k1 − (1− 2δ)p1e

−εt]ζ(−εν)
}
− h(v+2 )

≥ εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1
∫
R
J1(y)|ζ(−ε(ν − y))− ζ(−εν)|dy

+ α
{
k1 + δp1 − [k1 − (1− 2δ)p1e

−εt]ζ(−εν)
}
− h′(θ)v+2

≥ εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1ε
∫
R
J1(y)|y|dy

+ α
{
k1 + δp1 − [k1 − (1− 2δ)p1e

−εt]ζ(−εν)
}

− ρ̃1
{
k2 + δp2 − [k2 − (1− 2δ)p2e

−εt]ζ(−εν)
}

= εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1ε
∫
R
J1(y)|y|dy + (αk1 − ρ̃1k2)[1− ζ(−εν)]

+ δ(αp1 − ρ̃1p2) + (1− 2δ)(αp1 − ρ̃1p2)e−εtζ(−εν)

≥ −k1ε− d1k1ε
∫
R
J1(y)|y|dy + δ(αp1 − ρp2) > 0,

where θ ∈ [0, v+2 (x, t)].

Case(iii): δp1
2k1
≤ ζ(−εν) ≤ 1− δp2

2k2
. By (4.2), one has

L1[v+1 ]

≥ εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1ε
∫
R
J1(y)|y|dy + αv+1 − h(v+2 )

≥ min
{
εC(k1 − p1)ζ ′(−εν)− k1ε− d1k1ε

∫
R
J1(y)|y|dy + αv+1 − h(v+2 )

}
> 0.

Next, we verify that L2[v+2 ] ≥ 0 by considering the following three cases.



12 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

Case(i): ζ(−εν) ≤ δp2
2k2

. It is clear that v+2 (x, t) = k2, and hence, L2[v+2 ] ≥ 0.

Case(ii): ζ(−εν) > 1− δp1
2k1

. In this case, we have δp1 < v+1 < p1− δp1
2 . In view of

(3.1), (4.1) and (4.4), we obtain

L2[v+2 ]

≥ εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2
{
J2 ∗ ζ(−εν)− ζ(−εν)

}
+ d2(1− 2δ)p2e

−εt{J2 ∗ ζ(−εν)− ζ(−εν)
}

+ β
{
k2 + δp2 − [k2 − (1− 2δ)p2e

−εt]ζ(−εν)
}
− g(v+1 )

≥ εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2
∫
R
J2(y)|ζ(−ε(ν − y))− ζ(−εν)|dy

+ β
{
k2 + δp2 − [k2 − (1− 2δ)p2e

−εt]ζ(−εν)
}
− g′(θ)v+1

≥ εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2ε
∫
R
J2(y)|y|dy

+ β
{
k2 + δp2 − [k2 − (1− 2δ)p2e

−εt]ζ(−εν)
}

− ρ̃2
{
k1 + δp1 − [k1 − (1− 2δ)p1e

−εt]ζ(−εν)
}

= εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2ε
∫
R
J2(y)|y|dy + (βk2 − ρ̃2k1)[1− ζ(−εν)]

+ δ(βp2 − ρ̃2p1) + (1− 2δ)(βp2 − ρ̃2p1)e−εtζ(−εν)

≥ −k2ε− d2k2ε
∫
R
J2(y)|y|dy + δ(βp2 − ρp1) > 0,

where θ ∈ [0, v+1 (x, t)].

Case(iii): δp2
2k2
≤ ζ(−εν) ≤ 1− δp1

2k1
. By (4.3), we derive

L2[v+2 ]

≥ εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2ε
∫
R
J2(y)|y|dy + βv+2 − g(v+1 )

≥ min
{
εC(k2 − p2)ζ ′(−εν)− k2ε− d2k2ε

∫
R
J1(y)|y|dy + βv+2 − g(v+1 )

}
> 0.

The proof is complete. �

Remark 4.2. Clearly, the functions v±i (x, t), i = 1, 2 in Lemma 4.1 have the
following properties:

v+i (x, 0) = ki, ∀x ∈ [ξ,∞),

v+i (x, 0) ≥ (1− δ)pi, ∀x ∈ (−∞,∞),

v+i (x, t) ≤ δpi + (1− 2δ)pie
−εt, ∀(x, t) ∈ (−∞, ξ − Ct− 4ε−1]× R+,

v−i (x, 0) = 0, ∀x ∈ (−∞, ξ],
v−i (x, 0) ≤ ki − (1− δ)pi, ∀x ∈ (−∞,∞),

v−i (x, t) ≥ ki − δpi − (1− 2δ)pie
−εt, ∀(x, t) ∈ [ξ + Ct+ 4ε−1,∞)× R+.

By Lemma 3.1, we define w±(x, t, ξ0, δ) = (w±1 (x, t, ξ0, δ), w
±
2 (x, t, ξ0, δ)) where

w+
i (x, t, ξ0, δ) := min

{
φi(x+ ct+ ξ0 + σ0δ(1− e−β0t)) + δpie

−β0t, ki
}
,

w−i (x, t, ξ0, δ) := max
{
φi(x+ ct+ ξ0 − σ0δ(1− e−β0t))− δpie−β0t, 0

}
.



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 13

Lemma 4.3. Assume that (A1)–(A5) hold. Let φ(x+ ct) = (φ1(x+ ct), φ2(x+ ct))
be a traveling wavefront of (1.1). Then there exists ε∗ > 0 such that, if u(x, t) =
(u1(x, t), u2(x, t)) is a solution of (1.1) on [0,∞) with the initial data u(x, 0) sat-
isfying 0 ≤ ui(x, 0) ≤ ki for all x ∈ R, i = 1, 2, and the following is true:

w−(x, 0, cT + ξ, δ) ≤ u(x, T ) ≤ w+(x, 0, cT + ξ + h, δ)

on R provided that for some ξ ∈ R, T ≥ 0, h > 0 and δ ∈ (0,min{ δ02 ,
1
σ0
}), then

for every t ≥ T + 1, there exist ξ̂(t), δ̂(t) and ĥ(t) satisfying

w−(x, 0, ct+ ξ̂(t), δ̂(t)) ≤ u(x, t) ≤ w+(x, 0, ct+ ξ̂(t) + ĥ(t), δ̂(t)), (4.5)

where ξ̂(t), δ̂(t) and ĥ(t) are defined as follows

ξ̂(t) ∈ [ξ − σ0δ, ξ + h+ σ0δ],

ĥ(t) ∈ [0, h− σ0ε∗min{h, 1}+ 2σ0δ],

δ̂(t)) ∈ (δe−β0 + ε∗min{h, 1})e−β0(t−(T+1)).

Proof. By Lemma 3.1, we can see that w+(x, t, cT +ξ+h, δ) and w−(x, t, cT +ξ, δ),
respectively, are upper and lower solutions of (1.1). It is clear that ũ(x, t) =
u(x, T + t), t ≥ 0 is a solution of (1.1) with the initial value ũ(x, 0) = u(x, T )
for x ∈ R. Then by the comparison principle, we have

w−(x, t, cT + ξ, δ) ≤ u(x, T + t) ≤ w+(x, t, cT + ξ + h, δ), ∀(x, t) ∈ R× [0,+∞),

i.e.,

max
{
φi(η

−(x, t, T ))− δpie−β0t, 0
}

≤ ui(x, T + t)

≤ min
{
φi(η

+(x, t, T ) + h) + δpie
−β0t, ki

}
, ∀(x, t) ∈ R× [0,+∞),

(4.6)

where i = 1, 2, η±(x, t) = x + c(T + t) + ξ ± σ0δ(1 − e−β0t). Take y = −cT − ξ.
Then by the comparison principle, we have that for every nonnegative constant L,
any x ∈ R satisfying |x− y| ≤ L and every t > 0, i = 1, 2,

ui(x, T + t)− w−i (x, t, cT + ξ, δ)

≥ Ni(L, t)
∫ y+1

y

(ui(z, T )− w−i (z, 0, cT + ξ, δ))dz.
(4.7)

Note that lim|x|→+∞ φ′i(x) = 0 in Lemma 2.4, i = 1, 2. We can choose M > 0 large

enough such that φ′i(x) ≤ min{p1,p2}
2σ0

for all |x| ≥M . Let

L = M + |c|+ 1, h̄ = min{h, 1}, ε1 =
1

2
min{φ′1(x), φ′2(x)

∣∣|x| ≤ 2} > 0.

Since

w−i (z, 0,−y, δ) < φi(z − y), w+
i (z, 0,−y + h̄, δ) > φi(z − y + h̄), i = 1, 2,

we obtain ∫ y+1

y

[w+
i (z, 0, cT + ξ + h̄, δ)− w−i (z, 0, cT + ξ, δ)]dz

≥
∫ y+1

y

[φi(z + cT + ξ + h̄)− φi(z + cT + ξ)]dz



14 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

=

∫ y+1

y

[φi(z + h̄)− φi(z)]dz

=

∫ y+1

y

φ′i(ν)h̄dz ≥ 2ε1h̄.

Then at least one of the following two assertions is true:

(i)
∫ y+1

y
[ui(z, T )− w−i (z, 0, cT + ξ, δ)]dz ≥ ε1h̄;

(ii)
∫ y+1

y
[w+
i (z, 0, cT + ξ + h̄, δ)− ui(z, T )]dz ≥ ε1h̄;

Subsequently, we consider only the case (i). The case (ii) is similar and thus omitted.
For any |x− y| ≤ L, letting t = 1 in (4.7), it holds

ui(x, T + 1) ≥ w−i (x, 1, cT + ξ, δ) +Ni(L, 1)ε1h̄

≥ φi(x− y + c− σ0δ(1− e−β0))− δpie−β0 +N0(L)ε1h̄, i = 1, 2,

where N0(L) = min{N1(L, 1),N2(L, 1)}. Let

L1 = L+ |c|+ 2, ε∗ = min
1≤i≤2

{
min
|x|≤L1

N0(L)ε1
2σ0φ′i(x)

,
1

2σ0
,
δ0
2

}
.

Then for |x− y| ≤ L, by the mean value theorem, we have

φi(x− y + c+ 2σ0ε
∗h̄− σ0δ(1− e−β0))− φi(x− y + c− σ0δ(1− e−β0))

= φ′i(x− y + c+ 2θiσ0ε
∗h̄− σ0δ(1− e−β0))2σ0ε

∗h̄

≤ N0(L)ε1h̄, i = 1, 2, θi ∈ (0, 1).

Thus,

ui(x, T + 1) ≥ φi(η−(x, 1, T ) + 2σ0ε
∗h̄)− δpie−β0 , i = 1, 2. (4.8)

From the mean value theorem and the definitions of M and L, we obtain that for
|x− y| ≥ L,

φi(η
−(x, 1, T ))− φi(η−(x, 1, T ) + 2σ0ε

∗h̄)

= φ′i(η
−(x, 1, T )− 2θiσ0ε

∗h̄)(−2σ0ε
∗h̄)

≥ −ε∗h̄pi, i = 1, 2, θi ∈ (0, 1).

That is, for all |x− y| ≥ L,

φi(η
−(x, 1, T )) ≥ φi(η−(x, 1, T ) + 2σ0ε

∗h̄))− ε∗h̄pi, i = 1, 2,

and therefore, by (4.6) with t = 1, it holds

ui(x, T + 1) ≥ max
{
φi(η

−(x, 1, T ) + 2σ0ε
∗h̄)− (δe−β0 + ε∗h̄)pi, 0

}
(4.9)

for all |x−y| ≥ L, i = 1, 2. By (4.8) and (4.9), we obtain that for all x ∈ R, i = 1, 2,

ui(x, T + 1) ≥ max
{
φi(η

−(x, 1, T ) + 2σ0ε
∗h̄)− (δe−β0 + ε∗h̄)pi, 0

}
= max

{
φi(x+ ι)− (δe−β0 + ε∗h̄)pi, 0

}
,

where

ι = c(T + 1) + 2σ0ε
∗h̄+ ξ + ξ̄, ξ̄ = σ0δ(e

−β0 − 1). (4.10)

Then

u(x, T + 1) ≥ w−(x, 0, ι, µ̄), ∀x ∈ R,
where µ̄ = δe−β0 + ε∗h̄ ≤ δ0.



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 15

In view of the comparison principle and by the choice of ε∗, one has

w−(x, t̃, ι, µ̄) ≤ u(x, T + 1 + t̃), ∀t̃ ≥ 0. (4.11)

Then for all t ≥ T + 1, setting t̃ = t− (T + 1) in (4.11), we have

ui(x, t) ≥ w−i (x, t− (T + 1), ι, µ̄)

= φi[x+ ct− c(T + 1) + ι− σ0µ̄(1− e−β0(t−(T+1)))]− µ̄pie−β0(t−(T+1))

≥ φi[x+ ct− c(T + 1) + ι− σ0µ̄]− δ̂(t)pi, i = 1, 2,

where δ̂(t) = µ̄e−β0(t−(T+1)). Since φi(·) is monotone increasing, together with the
choice of η and (4.10), it holds

ui(x, t) ≥ w−i (x, 0, ct+ ξ̂, δ̂(t)), ∀x ∈ R, i = 1, 2, (4.12)

where

ξ̂ = 2σ0ε
∗h̄+ ξ − σ0δ(1− e−β0)− σ0µ̄ = σ0ε

∗h̄+ ξ − σ0δ.
By simple computations, we obtain the following estimate of ξ̂,

ξ − σ0δ ≤ ξ̂(t) ≤ ξ + σ0ε
∗h̄ ≤ ξ + h+ σ0δ.

For every t ≥ T , in view of (4.6), it follows that

ui(x, t) ≤ min
{
φi(η

+(x, t− T, T ) + h) + δpie
−β0(t−T ), ki

}
≤ min

{
φi(x+ ct+ ξ + h+ σ0δ) + δ̂(t)pi, ki

}
, ∀x ∈ R, i = 1, 2.

Hence, for any t ≥ T + 1, one has

ui(x, t) ≤ w+
i (x, 0, ct+ ξ̂(t) + ĥ(t), δ̂(t)), ∀x ∈ R, i = 1, 2,

that is, for x ∈ R,

u(x, t) ≤ w+(x, 0, ct+ ξ̂(t) + ĥ(t), δ̂(t)), (4.13)

where

ĥ(t) = ξ + h+ σ0δ − ξ̂(t) = h− σ0ε∗h̄+ 2σ0δ.

From the definition of ε∗, it holds h− σ0ε∗h̄ ≥ h− σ0ε∗h > 0, and thus,

ĥ(t) ∈ (0, h− σ0ε∗h̄+ 2σ0δ].

Therefore, (4.12) and (4.13) imply that (4.5) holds. The proof is complete. �

Lemma 4.4. Assume that (A1)–(A5) hold. Let φ(x+ ct) = (φ1(x+ ct), φ2(x+ ct))
be a traveling wavefront of (1.1) and ϕ(x) = (ϕ1(x), ϕ2(x)) with ϕi ∈ [0, ki] be such
that

lim
x→+∞

ϕi(x) > ki − pi, lim
x→−∞

ϕi(x) < pi, i = 1, 2.

Then, for every δ > 0, there exist T = T (ϕ, δ) > 0, ξ = ξ(ϕ, δ) ∈ R and h =
h(ϕ, δ) > 0 such that

w−(x, 0, cT + ξ, δ) ≤ u(x, T ;ϕ) ≤ w+(x, 0, cT + ξ + h, δ), ∀x ∈ R.

Proof. By the comparison principle, u(x, t;ϕ) = (u1(x, t;ϕ1), u2(x, t;ϕ2)) exists on
R+ and 0 ≤ ui(x, t;ϕi) ≤ ki for any (x, t) ∈ R× R+, i = 1, 2. For every δ > 0, one
can take δ1 = δ1(δ, ϕ) ∈ (0,min{δ, δ0}) satisfying

lim
x→+∞

ϕi(x) > ki − (1− δ1)pi, lim
x→−∞

ϕi(x) < (1− δ1)pi, i = 1, 2.



16 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

Thus, there exists a constant M = M(ϕ, δ1) > 0 such that, for i = 1, 2,

ϕi(x) ≤ (1− δ1)pi, ∀x ≤ −M ; ϕi(x) ≥ ki − (1− δ1)pi, ∀x ≥M. (4.14)

Let ε = ε(δ1), C = C(δ1) and v±(x, t) be described by Lemma 4.1 with δ replaced
by δ1 and ξ = ξ±, where ξ± = ∓M . Together with (4.14) and Remark 4.2, we
obtain that for i = 1, 2,

ϕi(x) ≤ (1− δ1)pi ≤ v+i (x, 0), ∀x ≤ −M,

ϕi(x) ≤ ki = v+i (x, 0), ∀x ≥ −M,

and

ϕi(x) ≥ ki − (1− δ1)pi ≥ v−i (x, 0), ∀x ≥M,

ϕi(x) ≥ 0 = v−i (x, 0), ∀x ≤M.

Thus,

v−(x, 0) ≤ ϕ(x) ≤ v+(x, 0), ∀x ∈ R.
By Lemma 4.1 and the comparison principle, one has

v−(x, t) ≤ u(x, t;ϕ) ≤ v+(x, t), ∀(x, t) ∈ R× R+.

Since δ1 < δ, we can take T > 0 sufficiently large such that, for any t ≥ T ,

δ1pi + (1− 2δ1)pie
−εt < δpi, ki − δ1pi + (1− 2δ1)pie

−εt > ki − δpi, i = 1, 2.

and hence, by Remark 4.2 again, for i = 1, 2,

ui(x, t;ϕi) ≤ v+i (x, t) < δpi, ∀x ≤ x−(t),

ui(x, t;ϕi) ≥ v−i (x, t) > ki − δpi, ∀x ≥ x+(t),
(4.15)

where x±(t) = ξ∓ ± Ct± 4ε−. By (4.15), we have

ui(x, T ;ϕi) < δpi, ∀x ≤ x−(T ),

ui(x, T ;ϕi) > ki − δpi, ∀x ≥ x+(T ), i = 1, 2.
(4.16)

Since limx→−∞ φi(x) = 0 and limx→+∞ φi(x) = ki, i = 1, 2, we can choose H > 0
large enough such that H

2 > x+(T ), −H2 < x−(T ), and

φi(x) + δpi > ki, ∀x ≥ H

2
,

φi(x)− δpi < 0, ∀x ≤ −H
2
.

(4.17)

Since 0 ≤ φi(x) ≤ ki and 0 ≤ ui(x, t;ϕi) ≤ ki for every (x, t) ∈ R × R+, and
together with (4.16) and (4.17), we have that for i = 1, 2,

max{φi(−H + x)− δpi, 0} ≤ ui(x, T ;ϕi) ≤ min{φi(H + x) + δpi, ki}, (4.18)

for all x ∈ R.
Let −H = ξ0 + cT , h0 = 2H > 0. It is clear that (4.18) implies that for i = 1, 2,

max
{
φi(x+ cT + ξ0)− δpi, 0

}
≤ ui(x, T ;ϕi)

≤ min{φi(x+ cT + ξ0 + h0) + δpi, ki}, ∀x ∈ R.
(4.19)

Let ξ = ξ0 and h = h0 > 0. It then follows from (4.19) that

ui(x, T ;ϕi) ≥ w−i (x, 0, cT + ξ0, δ) = w−i (x, 0, cT + ξ, δ),



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 17

ui(x, T ;ϕi) ≤ w+
i (x, 0, cT + ξ0 + h0, δ) = w+

i (x, 0, cT + ξ + h, δ), i = 1, 2.

The proof is complete. �

Theorem 4.5 (Global stability). Assume that (A1)–(A5) hold. Let (φ, c) be a
traveling wavefront of (1.1) with wave speed c ∈ R. Then φ(x + ct) is globally
exponentially stable with shift in the sense that there exists a positive constant µ
such that for every ϕi ∈ [0, ki] satisfying

lim
x→+∞

ϕi(x) > ki − pi, lim
x→−∞

ϕi(x) < pi, i = 1, 2,

the solution u(x, t;ϕ) = (u1(x, t;ϕ1), u2(x, t;ϕ2)) of (1.1) satisfies

‖u(·, t;ϕ)− φ(·+ ct+ ξ0)‖ ≤ Ke−µt, t ≥ 0

for some K = K(ϕ) > 0 and ξ0 = ξ0(ϕ) ∈ R, ϕ(x) = (ϕ1(x), ϕ2(x)), where ‖ · ‖ is
the general super norm in R2.

Proof. Let β0, σ0, δ0 be as in Lemma 3.1 and let ε∗ be as in Lemma 4.3 with ε∗

chosen so that σ0ε
∗ < 1. We choose 0 < δ∗ < min{ δ02 ,

1
2σ0
} such that

0 < k∗ := σ0ε
∗ − 2σ0δ

∗ < 1,

and then fix a t∗ ≥ 1 satisfying e−β0(t
∗−1)(1 + ε∗

δ∗ ) < 1 − k∗. The proof needs the
following two claims.

Claim (i). There exist T ∗ = T ∗(ϕ) > 0 and ξ∗ = ξ∗(ϕ) ∈ R such that

w−(x, 0, cT ∗ + ξ∗, δ∗) ≤ u(x, T ∗;ϕ) ≤ w+(x, 0, cT ∗ + ξ∗ + 1, δ∗), ∀x ∈ R. (4.20)

In fact, by Lemma 4.4, there exist T = T (ϕ) > 0, ξ = ξ(ϕ) > 0 and h = h(ϕ) > 0
such that

w−(x, 0, cT + ξ, δ∗) ≤ u(x, T ;ϕ) ≤ w+(x, 0, cT + ξ + h, δ∗), ∀x ∈ R. (4.21)

If h ≤ 1, then by the monotonicity of φi(·), i = 1, 2, Claim (i) holds. If h > 1,
we can choose an integer N := max{m|m ∈ Z+,mk∗ < h}. Since k∗ ∈ (0, 1) and
h > 1, then N ≥ 1, h ∈ (Nk∗, (N + 1)k∗], furthermore, 0 < h − Nk∗ ≤ k∗ < 1.
Note that h̄ := min{1, h} = 1. By (4.21), Lemma 4.3 and the choice of k∗ and t∗,
we obtain

w−(x, 0, c(T + t∗) + ξ̂(T + t∗), δ̂(T + t∗))

≤ u(x, T + t∗;ϕ)

≤ w+(x, 0, c(T + t∗) + ξ̂(T + t∗) + ĥ(T + t∗), δ̂(T + t∗)), ∀x ∈ R,

(4.22)

where

ξ̂(T + t∗) ∈ [ξ − σ0δ∗, ξ + h+ σ0δ
∗],

0 ≤ ĥ(T + t∗) ≤ h− σ0ε∗ + 2σ0δ
∗,

δ̂(T + t∗) = (δ∗e−β0 + ε∗)e−β0(t
∗−1) ≤ (1− k∗)δ∗ < δ∗.

Applying the similar argument N times, we conclude that (4.22), with T + t∗

replaced by T+Nt∗, holds for some ξ∗ = ξ̂ ∈ R, δ̂ ∈ (0, δ∗] and 0 ≤ ĥ ≤ h−Nk∗ < 1.
Then by the monotonicity of φ(·), (4.20) holds.



18 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

Claim (ii). Define p = 2σ0δ
∗ + 1, Tn = T ∗ + nt∗, δ∗n = (1 − k∗)nδ∗ and hn =

(1− k∗)n < 1, n ≥ 0. Then there exist a sequence {ξn}∞n=0 ⊂ R with ξ0 = ξ∗ such
that

|ξn+1 − ξn| ≤ phn, ∀n ≥ 0, (4.23)

w−(x, 0, cTn + ξn, δ
∗
n) ≤ u(x, Tn;ϕ) ≤ w+(x, 0, cTn + ξn + hn, δ

∗
n), (4.24)

for all n ≥ 0 and x ∈ R. Indeed, we use a mathematical induction to show that
(4.24) holds for every n ≥ 0. Clearly, Claim (i) implies that (4.24) holds when
n = 0. Suppose that (4.24) holds for some n = m > 0. By Lemma 4.3 with
T = Tm, ξ = ξm, h = hm, δ = δ∗m, and t = Tm + t∗ = Tm+1 (since t ≥ 1), it follows
that

w−(x, 0, cTm+1 + ξ̂, δ̂) ≤ u(x, Tm+1;ϕ) ≤ w+(x, 0, cTm+1 + ξ̂ + ĥ, δ̂), ∀x ∈ R,

where ξ̂ ∈ [ξm − σ0δ∗m, ξm + hm + σ0δ
∗
m],

δ̂ = (δ∗me
−β0 + ε∗hm)e−β0(Tm+1−Tm−1)

≤ (1− k∗)mδ∗
[
(1 +

ε∗

δ∗
)e−β0(t

∗−1)]
≤ (1− k∗)mδ∗(1− k∗)
= (1− k∗)m+1δ∗ = δ∗m+1,

and

ĥ ≤ hm − σ0ε∗hm + 2σ0δ
∗
m = (1− k∗)m[1− σ0ε∗ + 2σ0δ

∗] = hm+1.

Taking ξm+1 = ξ̂, we have

|ξm+1 − ξm| ≤ |ξm + hm + σ0δ
∗
m − (ξm − σ0δ∗m)|

= |2σ0δ∗m + hm|
= |2σ0(1− k∗)mδ∗ + (1− k∗)m|
= |(1− k∗)m(2σ0δ

∗ + 1)| = phm.

It follows that (4.23) holds for n = m, and (4.24) holds for n = m+1. By induction,
we obtain that (4.23) and (4.24) hold for all n ≥ 0.

By (4.24) and the comparison principle, for every n ≥ 0, all t ≥ Tn and x ∈ R,

max
{
φi(η

−
n (x, t))− δ∗npie−β0(t−Tn), 0

}
≤ u(x, t, ψi)

≤ min
{
φi(η

+
n (x, t) + hn) + δ∗npie

−β0(t−Tn), ki
}
, i = 1, 2,

(4.25)

where η±n (x, t) = x+ ct+ ξn ± σ0δ∗n(1− e−β0(t−Tn)). For t ≥ T ∗, let n = [ t−T
∗

t∗ ] be

the largest integer not greater than t−T∗

t∗ , and define δ(t) = δ∗n, ξ(t) = ξn − σ0δ∗n,
and h(t) = hn + 2σ0δ

∗
n. Then, Tn = T + nt∗ ≤ t ≤ T ∗ + (n+ 1)t∗ = Tn+1, in other

words, t ∈ [Tn, Tn+1). In view of (4.25), we obtain that for t ≥ T ∗ and x ∈ R,
i = 1, 2,

φi(x+ ct+ ξ(t))− piδ(t) ≤ ui(x, t;ϕi) ≤ φi(x+ ct+ ξ(t) + h(t)) + piδ(t). (4.26)

Set µ := − 1
t∗ ln(1 − k∗) > 0 and q(t) := e(

t−T∗
t∗ −1) ln(1−k∗). Since 0 ≤ n ≤ t−T∗

t∗ <
n+ 1, we have

(1− k∗)n < (1− k∗)
t−T∗

t∗ −1 = q(t).



EJDE-2022/49 BISTABLE TRAVELING WAVEFRONTS IN AN EPIDEMIC SYSTEM 19

Hence,

δ(t) = δ∗n ≤ δ∗q(t), ∀t ≥ T ∗, (4.27)

h(t) = (2σ0δ
∗ + 1)(1− k∗)n ≤ (2σ0δ

∗ + 1)q(t), ∀t ≥ T ∗. (4.28)

By (4.23) and (4.27), it follows that for τ ≥ t ≥ T ∗,
|ξ(τ)− ξ(t)| = |ξm − σ0δ∗m − (ξn − σ0δ∗n)|

≤
m−1∑
s=n

|ξs+1 − ξs|+ 2σ0δ
∗
n

≤
m−1∑
s=n

phs + 2σ0δ
∗
n

≤ phn
1− (1− k∗)

+ 2σ0δ
∗
n =

phn
k∗

+ 2σ0δ
∗
n

≤
( p

k∗δ∗
+ 2σ0

)
δ(t),

(4.29)

where m = [ τ−T
∗

t∗ ] ≥ n = [ t−T
∗

t∗ ]. From (4.29), the limit ξ0 := limt→+∞ ξ(t) exists,
and

|ξ0 − ξ(t)| ≤
( p

k∗δ∗
+ 2σ0

)
δ(t), ∀t ≥ T ∗. (4.30)

Therefore, by combining (4.26)-(4.28) and (4.30), we obtain the assertion of the
theorem. The proof is complete. �

As a corollary of Theorem 4.5, we can obtain the uniqueness of traveling wave-
front of (1.1).

Corollary 4.6. Assume that (A1)–(A5) hold. Let φ̃(x+c̃t) = (φ̃1(x+c̃t), φ̃2(x+c̃t))

be a traveling wavefront with 0 ≤ φ̃i(x+ c̃t) ≤ ki. Then there exists s̃ ∈ R such that

φ̃(·) = φ(· + s̃) and c̃ = c, where φ(x + ct) = (φ1(x + ct), φ2(x + ct)) be traveling
wavefront of (1.1).

Proof. It is clear that

lim
ξ→+∞

φ̃i(ξ) > ki − pi, lim
ξ→−∞

φ̃i(ξ) < pi, i = 1, 2.

By Theorem 4.5, there exist K > 0 and s̃ ∈ R such that

‖φ̃(·+ c̃t)− φ(·+ ct+ s̃)‖ ≤ Ke−µt, ∀t ≥ 0. (4.31)

We choose ξ0 ∈ R such that 0 < φ̃i(ξ0) < ki, i = 1, 2, and define a set

M(ξ0) := {(x, t) ∈ R× [0,+∞)|x+ ct = ξ0}.
It then follows from (4.31) that for any (x, t) ∈M(ξ0),

φi(η)−Ke−µt ≤ φ̃i(ξ0) ≤ φi(η) +Ke−µt, i = 1, 2, (4.32)

where η := s̃+ ξ0 +(c− c̃)t. Note that limξ→+∞ φi(ξ) = ki and limξ→−∞ φi(ξ) = 0,
i = 1, 2. Let t→ +∞ in (4.32), we obtain that c̃ ≥ c and c̃ ≤ c by the left-hand and
right-hand side inequalities, respectively. Then, c̃ = c. Substituting it into (4.31),
we obtain that for any (x, t) ∈M(ξ0),

‖φ̃(·)− φ(·+ s̃)‖ ≤ Ke−µt. (4.33)

Hence, it follows from (4.33) that φ̃(·) = φ(· + s̃) as t → +∞. The proof is
complete. �



20 Y.-C. HAO, G.-B. ZHANG EJDE-2022/49

Acknowledgments. This research was supported by the NSF of China (11861056),
by the NSF of Gansu Province (21JR7RA121), by the Department of Education
of Gansu Province: Youth Doctoral Fund Project (2021QB-018), and by the Out-
standing Youth Science Fund of Gansu Province (20JR10RA111). The authors are
grateful to the anonymous referees for their careful reading of this paper and for
their valuable suggestions.

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Yu-Cai Hao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu
730070, China

Email address: 626526834@qq.com

Guo-Bao Zhang (corresponding author)
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu

730070, China
Email address: zhanggb2011@nwnu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Lyapunov stability of traveling wavefronts
	4. Global stability of traveling wavefronts
	Acknowledgments

	References