Abstract
This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number \(\mathfrak {R}_0>1\), there exists a critical wave speed \(c^*>0\), such that for each \(c \ge c^*\) the system admits a nontrivial TWS and for \(c<c^*\) there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behavior of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In a pioneering work, the classical susceptible-infectious-recovered (SIR) epidemic model was introduced by Kermack and McKendrick (1927) in 1927. Since then, epidemic modeling has become one of the most important tools to study spread of the disease, we refer readers to a good survey (Hethcote 2000) on this topic. In order to understand the geographic spread of infectious diseases, the spatial effect would give insights into disease spread and control. Due to this fact, epidemic models with spatial diffusion have been studied for decades. Considering spatial effects, Hosono and Ilyas 1995 proposed and studied the following SIR epidemic model with diffusion:
with initial conditions
where S(x, t) and I(x, t) denote the densities of susceptible and infected individuals at position x and time t, respectively; \(d_i(i=1,2)\) are the diffusion rates of each compartment; \(\beta \) denotes the transmission rate between susceptible and infected individuals; \(\gamma \) is the remove rate. All parameters in system (1.1) are assumed to be positive. It was shown that the existence of traveling wave solutions of system (1.1) with a constant speed. Reaction–diffusion epidemic models have been investigated extensively, regarding the existence problem of traveling wave solutions, see (Bai and Zhang 2015; Ducrot and Magal 2011; Fu 2016; He and Tsai 2019; Lam et al. 2018; Li et al. 2017, 2014; Tian and Yuan 2017; Wang and Ma 2018; Zhang and Xu 2013; Zhao et al. 2017) and references therein.
However, there are relatively few works on epidemic models with discrete spatial structure. In contrast to continuous media, lattice dynamical systems are more realistic in describing the discrete diffusion (for example, patch environment (San and Wang 2019)). Lattice dynamical systems are systems of ordinary differential equations with a discrete spatial structure. Such systems arise from practical backgrounds, such as biology (Fang et al. 2010; Guo and Wu 2012; Han and Kloeden 2019; Weng et al. 2003; Wu et al. 2015; Yang and Zhang 2018), chemical reaction (Erneux and Nicolis 1993; Kapral 1991) and material science (Bates and Chmaj 1999; Brucal-Hallare and Vleck 2011). In a recent paper (Fu et al. 2016), Fu et al. studied the existence of traveling wave solutions for a lattice dynamical system arising in a discrete diffusive epidemic model:
where \(S_n(t)\) and \(I_n(t)\) denote the populations densities of susceptible and infectious individuals at niche n and time t, respectively; 1 and d denote the random migration coefficients for susceptible and infectious individuals, respectively; \(\beta \) is the transmission coefficient between susceptible and infectious individuals; \(\gamma \) is the recovery rate of infectious individuals. Note that system (1.2) is a spatially discrete version of system (1.1). It was proved in Fu et al. (2016) that the conditions of existence and nonexistence of traveling wave solution for system (1.2) are determined by a threshold number and the critical wave speed \(c^*\). If the threshold number is greater than one, then there exists a traveling wave solution for any \(c>c^*\) and there is no traveling wave solutions for \(c<c^*\). Also, the nonexistence of traveling wave solutions for the threshold number less than 1 was derived. Furthermore, Wu 2017 studied the existence of traveling wave solutions with critical speed \(c=c^*\) of system (1.2). Moreover, we refer to Zhang and Wu 2019 and Zhou et al. 2020 for the existence and nonexistence of traveling wave solutions with saturated incidence rate. By introducing the constant recruitment, Chen et al. 2017 studied the traveling wave solutions for the following discrete diffusive epidemic model:
where \(\mu \) is the input rate of the susceptible population; meanwhile, the death rates of susceptible and infectious individuals are also assumed to be \(\mu \). In Chen et al. (2017), the authors showed that the existence of traveling wave solutions connects the disease-free equilibrium to the endemic equilibrium, but they do not prove that the traveling wave solutions converge to the endemic equilibrium at \(+\infty \). As explained in Chen et al. (2017), the main difficulties come from the fact that (1.3) is a system and is nonlocal. In fact, the traveling wave solutions of (1.2) and (1.3) are totally different: For the system like (1.2) without constant recruitment, it can be shown that I tends to 0 as \(\xi \rightarrow \pm \infty \), where \(\xi =n+ct\) is the wave profile, which will be introduced in the next section; however, for the diffusive model with positive constant recruitment, it is more likely to get that \(I(\xi )\rightarrow 0\) as \(\xi \rightarrow -\infty \) and \(I(\xi )\rightarrow I^*\) as \(\xi \rightarrow +\infty \), where \(I^*\) is the positive endemic equilibrium (see Li et al. 2014 for nonlocal diffusive epidemic model; Fu 2016 for random diffusive epidemic model). Therefore, it naturally raises a question: For discrete diffusive systems, does the traveling wave solutions converge to the endemic equilibrium as \(\xi \rightarrow +\infty \)? This constitutes our first motivation of the present paper.
Our second motivation is the nonlinear incidence rate which plays a critical role in the epidemic modeling (Anderson and May 1991). For the discrete diffusive systems with nonlinear incidence rate, will the traveling wave solutions still converge to the endemic equilibrium as \(\xi \rightarrow +\infty \)? Generally, the incidence rate of an infectious disease in most of the literature is assumed to be of mass action form \(\beta SI\) (Anderson and May 1991). Yet the disease transmission process is generally unknown (Korobeinikov and Maini 2005), some nonlinear incidence rates have been introduced and studied, for example, the saturated incidence rate with \(f(I) = \frac{I}{1+\alpha I}\) by Capasso and Serio (1978), the saturated nonlinear incidence rate with \(f(I) = \frac{I}{1+\alpha I^p}(0<p<1)\) by Liu et al. (1986), and so on. For more general cases, Capasso and Serio (1978) considered a more general incidence rate with the form Sf(I), and the general nonlinear incidence rate could bring nontrivial challenges in analysis. Therefore, it is of great significance to study the convergence property of traveling wave solutions of the system with nonlinear incidence rate.
In this paper, we consider a discrete diffusive SIR epidemic model with general nonlinear incidence rate. The main model of this paper is formulated as the following system:
where \(S_n(t)\), \(I_n(t)\) and \(R_n(t)\) denote the densities of susceptible, infectious and removed individuals at niche n and time t, respectively; \(d_i(i=1,2,3)\) is the random migration coefficients for each compartment; \(\Lambda \) is the input rate of susceptible individuals. The biological meaning of other parameters is the same as in model (1.3).
Since \(R_n(t)\) is decoupled from other equations and denote \(\mu _2 = \gamma + \mu _1\), then we only need to study the following system:
Our first goal in this paper is to study the existence and boundedness of the traveling wave solutions of model (1.4). Adopting nonlinear incidence, random and nonlocal diffusive SIR model is studied in Bai and Zhang (2015) and Zhou et al. (2018), respectively. It was shown that the existence of traveling wave solutions for each model by analyzing auxiliary system. Unlike in Bai and Zhang (2015); Zhou et al. (2018) where there is no constant recruitment, here we allow constant recruitment in model (1.4) and it is necessary to consider the boundedness of traveling wave solutions, which is different from Bai and Zhang (2015); Zhou et al. (2018). In a recent paper (Zhang et al. 2018), Zhang et al. studied a time delay nonlocal diffusive SIR model with general incidence, and they established the boundedness of the traveling wave solutions, but with an additional assumption:
(H) \(S_0 f(I_0) - \gamma I_0\le 0\) for some \(I_0>0\),
where \(S_0\) is the disease-free steady state. This assumption has also been used in a recent literature (Zhou et al. 2020). However, we should point out here that (H) cannot be applied for some incidence, such as bilinear incidence. Hence, it comes naturally to consider the boundedness of traveling wave solution for system (1.4) without this assumption. As a result, our model could cover more special cases.
Our second goal in this paper is to study the convergence property of the traveling wave solutions of model (1.4). In proving the convergence property of traveling wave solutions for random diffusive epidemic model, the method combined with Lyapunov functional and Lebesgue dominated convergence theorem played a crucial role, see (Ducrot and Magal 2011; Fu 2016).
In general, there are two ways to construct Lyapunov functionals for the random diffusive model: (i) rewriting the random diffusive model as a system of first-order ODEs and constructing a Lyapunov function for ODE systems (Fu 2016; ii) constructing a Lyapunov functional that contains a first derivative term (Li et al. 2015). However, the above two methods are not applicable to discrete diffusive models (1.4), and it is challenging to construct a suitable Lyapunov functional for the model with lattice structure.
In the present paper, a new Lyapunov functional will be constructed, which contains a specific functional to handle the lattice structure in the corresponding wave profile system. Additionally, the well-posedness for the Lyapunov functional will be also investigated. For the random diffusive epidemic model, the well-posedness of its Lyapunov functional could be achieved via Harnack inequality (for instance, see Li et al. 2015), but for discrete diffusive models, more deeper analysis is needed to verify the permanence of traveling wave solutions, which plays a crucial role in proving the well-posedness for the Lyapunov functional.
We make the following assumptions on function f:
-
(A1)
\(f(I)\ge 0\) and \(f'(I) > 0\) for all \(I\ge 0\), \(f(I) = 0\) if and only if \(I = 0\).
-
(A2)
\(\frac{f(I)}{I}\) is continuous and monotonously nonincreasing for all \(I\ge 0\) and \(\lim \limits _{I\rightarrow 0^+}\frac{f(I)}{I}\) exists.
The conditions of Assumption (A1) and (A2) are satisfied in all the following specific incidence rates:
-
(i)
the bilinear incidence rate with \(f(I)=I\) (see Anderson and May 1991);
-
(ii)
the saturated incidence rate with \(f(I) = \frac{I}{1+\alpha I}\) (see Capasso and Serio 1978);
-
(iii)
the saturated nonlinear incidence rate with \(f(I) = \frac{I}{1+\alpha I^p}\), where \(\alpha >0\) and \(0<p<1\) (a special case in Liu et al. (1986), see also Muroya et al. (2015));
-
(iv)
the nonlinear incidence rate with \(f(I) = \frac{I}{1+kI+\sqrt{1+2kI}}\) (see Heesterbeek and Metz 1993; Thieme 2011);
-
(v)
the nonlinear incidence rate with \(f(I) = \frac{I}{(\epsilon ^\alpha + I^\alpha )^\gamma }\), where \(\epsilon , \alpha , \gamma >0\) and \(\alpha \gamma <1\) (see Thieme 2011);
-
(vi)
the nonlinear incidence rate for pathogen transmission in infection of insects with \(f(I) = k \ln \left( 1+\frac{\nu I}{k}\right) \), which could be described by epidemic model (see Briggs and Godfray 1995).
Hence, system (1.5) covers many models as special cases. Now, we introduce some results on the system (1.5) without migration, which takes the form as:
It is well known that the global dynamics of (1.6) is completely determined by the basic reproduction number \(\mathfrak {R}_{0} = \frac{\beta S_0 f'(0)}{\mu _2}\) (see Korobeinikov 2006): That is, if the number is less than unity, then the disease-free equilibrium \(E_0=(S_0,0)=(\Lambda /\mu _1,0)\) is globally asymptotically stable, while if the number is greater than unity, then a positive endemic equilibrium \(E^*=(S^*,I^*)\) exists and it is globally asymptotically stable, where \(E^*\) satisfy
The organization of this paper is as follows: In Sect. 2, we apply Schauder’s fixed point theorem to construct a family of solutions of the truncated problem. In Sect. 3, we show the existence and boundedness of traveling wave solutions. Further, we use a Lyapunov functional to show that the convergence of traveling wave solutions at \(+\infty \). In Sect. 4, we investigate the nonexistence of traveling wave solutions by using two-sided Laplace transform. At last, there is an application for our general results and a brief discussion.
2 Preliminaries
In this section, since system (1.5) does not enjoy the comparison principle, we will construct a pair upper and lower solutions and apply Schauder’s fixed point theorem to investigate the existence of traveling wave solutions of system (1.5). Consider traveling wave solutions which can be expressed as bounded profiles of continuous variable \(n+ct\) such that
where c denotes the wave speed. Let \(\xi =n+ct\), then we can rewrite system (1.5) as follows:
for all \(\xi \in \mathbb {R}\), where \(J[\phi ](\xi ):= \phi (\xi +1) + \phi (\xi -1) - 2 \phi (\xi )\). We want to find traveling wave solutions with the following asymptotic boundary conditions:
and
where \((S_0,0)\) is the disease-free equilibrium and \((S^*,I^*)\) is the positive endemic equilibrium, which is defined in Section 1. Linearizing the second equation of system (2.2) at disease-free equilibrium \((S_0, 0)\), we have
Letting \(I(\xi )=e^{\lambda \xi }\) and substituting it in (2.5) yield
Denote
By some calculations, we have
for \(\lambda >0\) and \(c>0\). Therefore, we have the following lemma.
Lemma 1
Let \(\mathfrak {R}_{0}>1.\) There exist \(c^*>0\) and \(\lambda ^*>0\) such that
Furthermore,
-
(i)
if \(c=c^*,\) then \(\Delta (\lambda , c)=0\) has only one positive real root \(\lambda ^*;\)
-
(ii)
if \(0<c<c^*,\) then \(\Delta (\lambda , c)>0\) for all \(\lambda \in (0, +\infty )\);
-
(iii)
if \(c>c^*,\) then \(\Delta (\lambda , c)=0\) has two positive real roots \(\lambda _1,\) \(\lambda _2\) with \(\lambda _1<\lambda ^*<\lambda _2\).
From Lemma 1, we have
In the following of this section, we always fix \(c>c^*\) and \(\mathfrak {R}_0>1\).
2.1 Construction of Upper and Lower Solutions
Definition 1
\((S^+(\xi ), I^+(\xi ))\) and \((S^-(\xi ), I^-(\xi ))\) are called a pair upper and lower solutions of (2.2) if \(S^+, I^+, S^-, I^-\) satisfy
Define the following functions:
where \(M_i\) and \(\varepsilon _i(i=1,2)\) are some positive constants to be determined in the following lemmas. Now we show that (2.8) are a pair upper and lower solutions of (2.2).
Lemma 2
The function \( I^+(\xi ) = e^{\lambda _1 \xi }\) satisfies
Lemma 3
The function \(S^+(\xi )=S_0\) satisfies
The proof of the above two lemmas is straightforward, so we omit the details.
Lemma 4
For each sufficiently small \(0<\varepsilon _1<\lambda _1\) and \(M_1>0\) is large enough, the function \(S^-(\xi )=\max \{S_0(1-M_1 e^{\varepsilon _1 \xi }),0\}\) satisfies
with \(\xi \ne \frac{1}{\varepsilon _1}\ln \frac{1}{M_1}:=\mathfrak {X}_1\).
Proof
If \(\xi >\mathfrak {X}_1\), then inequality (2.11) holds since \(S^-(\xi )=S^-(\xi +1)=0\) and \(S^-(\xi -1)\ge 0\). If \(\xi <\mathfrak {X}_1\), then
and
From the concavity of function f(I), we have
thus,
Select \(0<\varepsilon _1<\lambda _1\) small enough such that \(-d_1(2 - e^{\varepsilon _1} - e^{-\varepsilon _1}) - \mu - c \varepsilon _1 < 0\) and note that \(e^{(\lambda _1 - \varepsilon _1)\xi }\le 1\) since \(\xi<\mathfrak {X}_1<0\). Hence, we need to choose
large enough to make sure inequality (2.11) holds. This completes the proof. \(\square \)
Lemma 5
For each sufficiently small \(0<\varepsilon _2<\lambda _1\) and \(M_2>0\) is large enough, the function \( I^-(\xi )=\max \{e^{\lambda _1\xi }(1-M_2e^{\varepsilon _2 \xi }),0\}\) satisfies
with \(\xi \ne \frac{1}{\varepsilon _2}\ln \frac{1}{M_2}:=\mathfrak {X}_2\).
Proof
If \(\xi > \mathfrak {X}_2\), then inequality (2.12) holds since \(I^-(\xi )=I^-(\xi -1)=0\) and \(I^-(\xi +1)\ge 0\). If \(\xi < \mathfrak {X}_2\), then
and
Inequality (2.12) is equivalent to the following inequality:
Note that \(\frac{f(I)}{I}\) is nonincreasing on \((0,\infty )\) in Assumption (A2). Then, for any \(\epsilon \in (0,f'(0))\), there exists a small positive number \(\delta >0\) such that
For \(0< I<\delta \), we have
Recall that \(\xi <\mathfrak {X}_2\), we can choose \(M_2\) large enough such that
Since (2.14) is valid for any \(\epsilon \), we have
Furthermore, the right hand of (2.13) satisfies
Using the definition of \(\Delta (\lambda ,c)\) and \([ I^-(\xi )]^2\le e^{2\lambda _1\xi }\), noticing that \(\Delta (\lambda _1+\varepsilon _2,c)<0\) for small \(\varepsilon _2>0\) by (2.7), then it is suffice to show that
The above inequality holds for \(M_2\) large enough, since the left-hand side vanishes and the right-hand side tends to infinity as \(M_2\rightarrow +\infty \). This ends the proof. \(\square \)
Hence, functions (2.8) are a pair upper and lower solutions of (2.2) by Definition 1.
2.2 Truncated Problem
Let \(X>\max \{-\mathfrak {X}_1,-\mathfrak {X}_2\}\). Define the following set:
It is clear that \(\Gamma _X\) is a nonempty bounded closed convex set in \(C([-X,X],\mathbb {R}^2)\). For any \((\phi ,\psi )\in C([-X,X],\mathbb {R}^2)\), extend it as
Denote
Consider the following truncated initial problem:
where \((\phi ,\psi )\in \Gamma _X\) and \(\alpha \) is a constant large enough such that \(H_1(\phi ,\psi )\) is nondecreasing on \(\phi (\xi )\). By the ordinary differential equation theory, system (2.15) has a unique solution \((S_X(\xi ),I_X(\xi ))\) satisfying \((S_X(\xi ),I_X(\xi ))\in C^1([-X,X],\mathbb {R}^2)\). Then, we define an operator
by
Next we show the operator \(\mathcal {A}=(\mathcal {A}_1,\mathcal {A}_2)\) has a fixed point in \(\Gamma _X\) by Schauder’s fixed point theorem (see Chang 2005, Corollary 2.3.10).
Lemma 6
The operator \(\mathcal {A}=(\mathcal {A}_1,\mathcal {A}_2)\) maps \(\Gamma _X\) into itself.
Proof
Firstly, we show that \(S^-(\xi )\le S_X(\xi )\) for any \(\xi \in [-X,X].\) If \(\xi \in (\mathfrak {X}_1, X),\) it follows that \(S^-(\xi )=0\) and it is a lower solution of the first equation of (2.15). If \(\xi \in (-X,\mathfrak {X}_1),\) then \(S^-(\xi )=S_0(1-M_1 e^{\varepsilon _1 \xi })\), from Lemma 4, we have
which implies that \(S^-(\xi )=S_0(1-M_1 e^{\varepsilon _1 \xi })\) is a lower solution of the first equation of (2.15). Thus, \(S^-(\xi )\le S_X(\xi )\) for any \(\xi \in [-X,X].\)
Secondly, we show that \(S_X(\xi )\le S^+(\xi ) = S_0\) for any \(\xi \in [-X,X].\) In fact,
thus, \(S^+(\xi )=S_0\) is an upper solution to the first equation of (2.15), which gives us \(S_X(\xi )\le S_0\) for any \(\xi \in [-X,X].\)
Similarly, we can show that \(I^-(\xi )\le I_X(\xi )\le I^+(\xi )\) for any \(\xi \in [-X,X].\) This completes the proof. \(\square \)
Lemma 7
The operator \(\mathcal {A}{:\Gamma _X\rightarrow \Gamma _X}\) is completely continuous.
Proof
Suppose \((\phi _i(\xi ),\psi _i(\xi ))\in \Gamma _X,\ i=1,2.\) Denote
We show that the operator \(\mathcal {A}\) is continuous. By direct calculation, we have
and
where \(H_i(\phi ,\psi )(i=1,2)\) are defined in (2.15). For any \((\phi _i, \psi _i)\in \Gamma _X\), \(i=1,2\), we have
Since \(S_X\) and \(I_X\) are class of \(C^1([-X,X])\), note that
Same arguments on \(I_X\). Thus, it is easy to see that the operator \(\mathcal {A}\) is continuous. Next, it follows from (2.15), we can obtain that \(S_X'\) and \(I_X'\) are bounded. Hence, the operator \(\mathcal {A}\) is compact and is completely continuous. This ends the proof. \(\square \)
Applying Schauder’s fixed point theorem, we have the following lemma.
Lemma 8
There exists \((S_X,I_X)\in \Gamma _X\) such that
for \(\xi \in [-X,X]\).
In the following, we show some prior estimates for \((S_X,I_X)\). Define
with the norm
Lemma 9
There exists a constant \(C(Y)>0\) such that
for \(0<Y<X\) and \(X>\max \{-\mathfrak {X}_1,-\mathfrak {X}_2\}\).
Proof
Recall that \((S_X,I_X)\) is the fixed point of the operator \(\mathcal {A},\) then
where
Since \(0\le S_X(\xi )\le S_0\) and \(0\le I_X(\xi )\le e^{\lambda _1 Y}\) for all \(\xi \in [-Y,Y]\), from (2.16) we have
and
Thus, there exists some constant \(C_1(Y) > 0\) such that
For any \(\xi ,\eta \in [-Y,Y]\), it follows from Zhang and Wu (2019) that
Then, \(|\hat{S}_X(\xi +1)-\hat{S}_X(\eta +1)|\le C_1(Y)|\xi -\eta |\) for all \(\xi ,\eta \in [-Y,Y]\). Similarly, we have
for all \(\xi ,\eta \in [-Y,Y]\). Furthermore,
for all \(\xi ,\eta \in [-Y,Y]\). Hence, there exists some constant \(C(Y) > 0\) such that
Similarly,
for any \(Y<X\). This completes the proof. \(\square \)
3 Existence of Traveling Wave Solutions
We first state the main results of this section as follows.
Theorem 1
For any wave speed \(c>c^*\), system (1.5) admits a nontrivial traveling wave solution \((S(\xi ),I(\xi ))\) satisfying
Furthermore,
The proof of Theorem 1 is divided into the following several steps.
Step 1 We show that system (1.5) admits a nontrivial traveling wave solution \((S(\xi ),I(\xi ))\) in \(\mathbb {R}\) and satisfying \(\lim _{\xi \rightarrow -\infty }(S(\xi ), I(\xi ))=(S_0, 0)\).
Let \(\{X_n\}_{n=1}^{+\infty }\) be an increasing sequence such that \(X_n>-\mathfrak {X}_2\), \(X_n>Y\) and \(X_n\rightarrow +\infty \) as \(n\rightarrow +\infty \) for all \(n\in \mathbb {N}\), where Y is from Lemma 9. Denote \((S_n, I_n)\in \Gamma _{X_n}\) be the solution of system (2.15). For any \(N\in \mathbb {N}\), since the function \( I^+(\xi )\) is bounded in \([-X_N,X_N]\), then the sequences
are uniformly bounded in \([-X_N,X_N]\). Then, by (2.15), we can obtain that
are also uniformly bounded in \([-X_N,X_N]\). Again with (2.15), we can express \(S''_n(\xi )\) and \( I''_n(\xi )\) in terms of \(S_n(\xi )\), \( I_n(\xi )\), \(S_n(\xi \pm 1)\), \( I_n(\xi \pm 1)\), \(S_n(\xi \pm 2)\) and \( I_n(\xi \pm 2)\), which give us
are uniformly bounded in \([-X_N+2,X_N-2]\). By the Arzela–Ascoli theorem (see Rudin 1991, Theorem A5), we can use a diagonal process to extract a subsequence, denoted by \(\{S_{n_k}\}_{k\in \mathbb {N}}\) and \(\{ I_{n_k}\}_{k\in \mathbb {N}}\) such that
uniformly in any compact subinterval of \(\mathbb {R}\), for some functions S and I in \(C^1(\mathbb {R})\). Thus, \((S(\xi ), I(\xi ))\) is a solution of system (2.2) with
Furthermore, by the definition of (2.8), it follows that
Step 2 We claim that the functions \(S(\xi )\) and \(I(\xi )\) satisfy \(0<S(\xi )<S_0\ \ \mathrm{and}\ \ I(\xi )>0\ \ \mathrm{in}\ \ \mathbb {R}\).
We first show that \(S(\xi )>0\) for all \(\xi \in \mathbb {R}\). Assume reversely, that is, assume that if there exists some real number \(\xi _0\) such that \(S(\xi _0) = 0\), then \(S'(\xi _0) = 0\) and \(J[S](\xi _0)\ge 0\). By the first equation of (2.2), we have
which is a contradiction. Thus, \(S(\xi )>0\) for all \(\xi \in \mathbb {R}\).
Next, we show that \(I(\xi )>0\) for all \(\xi \in \mathbb {R}\). By way of contradiction, we assume that if there exists \(\xi _1\) such that \(I(\xi _1) = 0\) and \(I(\xi )>0\) for all \(\xi <\xi _1\). From the second equation of (2.2), we have
Consequently, \( I(\xi _1+1) = I(\xi _1-1) = 0\) since \(I(\xi )\ge 0\) in \(\mathbb {R}\), which is a contradiction to the definition of \(\xi _1\).
To show that \(S(\xi )<S_0\) for all \(\xi \in \mathbb {R}\), we assume that if there exists \(\xi _2\) such that \(S(\xi _2) = S_0\), it is easy to obtain
This contradiction leads to \(S(\xi )<S_0\) for all \(\xi \in \mathbb {R}\).
Step 3 Boundedness of traveling wave solutions \(S(\xi )\) and \(I(\xi )\) in \(\mathbb {R}\).
We need to consider two cases of the nonlinear incidence function f(x). In fact, the function f(x) satisfying Assumptions (A1) and (A2) has two possibilities: (i) \(\lim \limits _{x\rightarrow +\infty }f(x)\) exists; (ii)\(\lim \limits _{x\rightarrow +\infty }f(x) = +\infty \). For example, the saturated incidence with \(f(x)=\frac{bx}{1+cx}\) satisfies (i) since \(\lim \limits _{x\rightarrow +\infty }\frac{bx}{1+cx} = \frac{b}{c}\) and the bilinear incidence with \(f(x)=bx\) satisfies (ii).
Case 1. \(\lim \limits _{x\rightarrow +\infty }f(x)\) exists. Without losing generality, we assume that \(\lim \limits _{x\rightarrow +\infty }f(x) = \bar{f} < +\infty \), then it is easy to verify that \(\frac{\Lambda }{\mu _1+\beta \bar{f}}\) is a lower solution of \(S(\xi )\) and \(\frac{\beta S_0\bar{f}}{\mu _2}\) is an upper solution of \(I(\xi )\). Then, we obtain
Case 2. \(\lim \limits _{x\rightarrow +\infty }f(x) = +\infty \). In this case, we have the following lemmas.
Lemma 10
The functions \(\frac{ I(\xi \pm 1)}{ I(\xi )}\) and \(\frac{ I'(\xi )}{ I(\xi )}\) are bounded in \(\mathbb {R}\).
Proof
From the second equation of (2.2), one has that
Denote \(U(\xi ) = e^{\nu \xi } I(\xi )\), where \(\nu = (2d_2 + \mu _2)/c\). It follows that
thus \(U(\xi )\) is increasing on \(\xi \). Then, \(U(\xi -1) < U(\xi )\), that is,
Note that
Integrating (3.2) over \([\xi ,\xi +1]\) and using the fact that \(e^{\nu \xi } I(\xi )\) is increasing, we have
By (3.2), we obtain
Integrating (3.3) over \([\xi -\frac{1}{2},\xi ]\) yields
that is
Similarly, integrating (3.3) over \([\xi , \xi +\frac{1}{2}]\), we have
Thus,
By the second equation of (2.2), it follows that
which gives us \(\frac{ I'(\xi )}{ I(\xi )}\) is bounded in \(\mathbb {R}\). The proof is end. \(\square \)
Lemma 11
Let \(\{c_k,S_k, I_k\}\) be a sequence of traveling wave solutions of (1.5) with speed \(\{c_k\}\) in a compact subinterval of \((0,\infty )\). If there is a sequence \(\{\xi _k\}\) such that \( I(\xi _k)\rightarrow +\infty \) as \(k\rightarrow +\infty \), then \(S(\xi _k)\rightarrow 0\) as \(k\rightarrow +\infty \).
Proof
Assume that there is a subsequence of \(\{\xi _k\}_{k\in \mathbb {N}}\) again denoted by \(\xi _k\), such that \( I_k(\xi _k)\rightarrow +\infty \) as \(k\rightarrow +\infty \) and \(S_k(\xi _k)\ge \varepsilon \) in \(\mathbb {R}\) for all \(k\in \mathbb {N}\) with some positive constant \(\varepsilon \). From the first equation of (2.2), we have
where \(\tilde{c}\) is a positive lower bound of \(\{c_k\}\). It follows that
for all \(k\in \mathbb {N}\), where \(\delta = \frac{\varepsilon }{\delta _0}\). By Lemma 10, we can assume that \(\bigg |\frac{ I'_k(\xi )}{ I_k(\xi )}\bigg |<C_0\) for some \(C_0>0\). Then,
for all \(k\in \mathbb {N}\). Thus,
which give us
since \( I_k(\xi _k)\rightarrow +\infty \) as \(k\rightarrow +\infty \). Recalling we assumed that \(\lim _{x\rightarrow +\infty }f(x)=+\infty \) in this case, one has that
where \(\varpi _k:=\min \limits _{\xi \in [\xi _k-\delta , \xi _k]} I_k(\xi )\). Moreover, there exists some \(K>0\) such that
Note that \(S_k<S_0\) in \(\mathbb {R}\) for each \(k\in \mathbb {N}\). Hence, \(S_k(\xi _k)\le -S_0\) for all \(k\ge K\), which reduces to a contradiction since \(S_k(\xi _k)\ge \varepsilon \) in \(\mathbb {R}\) for all \(k\in \mathbb {N}\) with some positive constant \(\varepsilon \). This completes the proof. \(\square \)
Lemma 12
If \(\limsup \limits _{\xi \rightarrow +\infty } I(\xi )=+\infty \), then \(\lim \limits _{\xi \rightarrow +\infty } I(\xi )=+\infty \).
The proof of Lemma 12 is similar to that of (Chen et al. 2017, Lemma 3.4), so we omit the details.
Lemma 13
The function \( I(\xi )\) is bounded in \(\mathbb {R}\).
Proof
Assume that \(\limsup \limits _{\xi \rightarrow +\infty } I(\xi )=+\infty \), then we have \(\lim \limits _{\xi \rightarrow +\infty }S(\xi )=0\) by Lemma 11 and Lemma 12. Set \(\theta (\xi )=\frac{ I'(\xi )}{ I(\xi )}\), from the second equation of (2.2), we have
where
It is easy to have that \(\lim \limits _{\xi \rightarrow +\infty }B(\xi ) = 0\) since \(\frac{f( I(\xi ))}{I(\xi )}\le f'(0)\) and \(\lim \limits _{\xi \rightarrow +\infty }S(\xi )=0\). By using (Chen and Guo 2003, Lemma 3.4), \(\theta (\xi )\) has a finite limit \(\omega \) at \(+\infty \) and satisfies the following equation:
By some calculations, we obtain
Thus, there exists a unique positive real root \(\omega _0\) of \(h(\omega ,c) = 0\). Recall that \(\lambda _1\) is the smaller real root of (2.6) and \(\lambda _2\) is the larger real root of (2.6). From Lemma 1, one has
thus, we have \(\lambda _2<\omega _0\). Since \(\lim \limits _{\xi \rightarrow +\infty }\theta (\xi ) = \omega _0\), there exists some \(\tilde{\xi }\) large enough such that
with some constant C. This is a contradiction since \( I(\xi )\le e^{\lambda _1\xi }\) in \(\mathbb {R}\) and \(\lambda _1<\omega _0\). This ends the proof. \(\square \)
By the above lemmas, we know that \(f(I(\xi ))\) is bounded from the above since \(I(\xi )\) is bounded, then Proposition (3.1) follows. Hence, we obtained that \(S(\xi )\) and \(I(\xi )\) are bounded from the above and \(S(\xi )\) has a strictly positive lower bound in \(\mathbb {R}\). In the following, we will show \(I(\xi )\) could not approach zero.
Lemma 14
Let \(0<c_1\le c_2\) be given and \((S(\xi ),I(\xi ))\) be a solution of system (2.2) with speed \(c\in [c_1,c_2]\) satisfying \(0<S(\xi )<S_0\ \ \mathrm{and}\ \ I(\xi )>0\ \ \mathrm{in}\ \ \mathbb {R}\). Then, there exists some small enough constant \(\varepsilon _0>0\), such that \(I'(\xi )>0\) provided that \(I(\xi )\le \varepsilon _0\) for all \(\xi \in \mathbb {R}\).
Proof
Assume by way of contradiction that there is no such \(\varepsilon _0\), that is there exists some sequence \(\{\xi _k\}_{k\in \mathbb {N}}\) with speed \(c_k\in (\underline{c},\overline{c})\) such that \( I(\xi _k)\rightarrow 0\) as \(k\rightarrow +\infty \) and \(I'(\xi _k)\le 0\), where \(\underline{c}\) and \(\overline{c}\) are two given positive real numbers. Denote
Thus, we have \( I_k(0)\rightarrow 0\) as \(k\rightarrow +\infty \) and \( I_k(\xi )\rightarrow 0\) locally uniformly in \(\mathbb {R}\) as \(k\rightarrow +\infty \). As a consequence, there also holds that \( I_k'(\xi )\rightarrow 0\) locally uniformly in \(\mathbb {R}\) as \(k\rightarrow +\infty \) by the second equation of (2.2). From the proof of (Chen et al. 2017, Lemma 3.8), we can obtain that \(S_\infty = S_0\). Let \(\Psi _k(\xi ):=\frac{ I_k(\xi )}{ I_k(0)}\). In the view of
we have that \(\Psi _k(\xi )\) and \(\Psi _k'(\xi )\) are also locally uniformly in \(\mathbb {R}\) as \(k\rightarrow +\infty \). Letting \(k\rightarrow +\infty \), thus
We claim that \(\Psi _\infty (\xi )>0\) in \(\mathbb {R}\). In fact, if there exists some \(\xi _0\) such that \(\Psi _\infty (\xi _0)=0\), we also have \({\Psi }'_\infty (\xi _0)=0\) since \(\Psi _\infty (\xi )\ge 0\), then
Thus, \(\Psi _\infty (\xi _0+1) = \Psi _\infty (\xi _0-1) = 0\), and it follows that \(\Psi _\infty (\xi _0+m) = 0\) for all \(m\in \mathbb {Z}\). Recall that \(c_\infty \Psi _\infty '(\xi ) \ge - \mu _2 \Psi _\infty (\xi )\), then the map \(\xi \mapsto \Psi _\infty (\xi ) e^{\frac{\mu _2\xi }{c_\infty }}\) is nondecreasing. Since it vanishes at \(\xi _0+m\) for all \(m\in \mathbb {Z}\) and \(e^{\frac{\mu _2\xi }{c_\infty }}\) is increasing, one can concluded that \(\Psi _\infty = 0\) in \(\mathbb {R}\), which is a contradicts with \(\Psi _\infty (0) = 1\).
Denoting \(\mathcal {Z}(\xi ):=\frac{\Psi _\infty '(\xi )}{\Psi _\infty (\xi )}\), it is easy to verify that \(\mathcal {Z}(\xi )\) satisfies
Thanks to (Chen and Guo 2003, Lemma 3.4), \(\mathcal {Z}(\xi )\) has finite limits \(\omega _{\pm }\) as \(\xi \rightarrow \pm \infty \), where \(\omega _\pm \) are roots of
By the analogous arguments in Lemma 1, we have \(\omega _\pm >0\). Thus, \(\Psi _\infty '(\xi )\) is positive at \(\pm \infty \) by the definition of \(\mathcal {Z}(\xi )\). Moreover, \(\Psi _\infty '(\xi )>0\) for all \(\xi \in \mathbb {R}\). Indeed, if there exists some \(\xi ^*\) such that \(\mathcal {Z}(\xi ^*) = \inf _{\mathbb {R}}\mathcal {Z}(\xi )\), then \(\mathcal {Z}(\xi ^*) = 0\). Differentiating (3.4) gives us
it follows that
Hence, \(\mathcal {Z}(\xi ^*) = \mathcal {Z}(\xi ^*+m)\) for all \(m\in \mathbb {Z}\). Then, there is
So \(\Psi _\infty '(\xi )>0\). From the definition of \(\Psi _\infty (\xi )\), we have
Thus, \( I'(\xi _k) = I_k'(0)>0\), which is a contradiction. This completes the proof. \(\square \)
Step 4 Convergence of the traveling wave solutions as \(\xi \rightarrow +\infty \). The key point is to construct a suitable Lyapunov functional.
Let \(g(x)=x-1-\ln x\) for \(x>0\), it is easy to check \(g(x)\ge 0\) since g(x) has the global minimum value 0 only at \(x = 1\). Define the following Lyapunov functional:
where
and
Thanks to the boundedness of \(S(\xi )\) and \(I(\xi )\) (see Step 3), we have \(W_1(S,I)(\xi )\) and \(W_2(S,I)(\xi )\) are well defined and bounded from below. Since \(\lim _{\xi \rightarrow -\infty }I(\xi )=0\), we need to consider the process of \(\xi \) approaching negative infinity for \(W_3(S,I)(\xi )\). For the \(\varepsilon _0\) in Lemma 14, define \(\xi ^* = \min \{\xi \in \mathbb {R}|I(\xi ) = \varepsilon _0\}\), then \(I(\xi )\) is increasing in \((-\infty , \xi ^*]\). By the properties of function g, we have \(W_3(S,I)(\xi )\ge 0\) for \(\xi \in (-\infty , \xi ^*]\). Thus, the Lyapunov functional \(L(S,I)(\xi )\) is well defined and bounded from below.
Next we show that the map \(\xi \mapsto L(S, I)(\xi )\) is nonincreasing. The derivative of \(W_1(S, I)(\xi )\) along the solution of (2.2) is calculated as follows:
where
Noticing that the endemic equilibrium \((S^*,I^*)\) of system (1.5) satisfying (1.7) and \(\mu _1 = \mu + \alpha \). By some calculations, we obtain that
For \(W_2(S, I)(\xi )\), one has that
Similarly,
It can be shown that
and
Thus,
Since the arithmetic mean of nonnegative real numbers is greater than or equal to the geometric mean of the same list, then we have
From Assumption (A2), we can conclude that
Then, we have
Here, we use \(\frac{I f(I^*)}{I^*f(I)} - 1 - \ln \left( \frac{I f(I^*)}{I^*f(I)}\right) \ge 0\). Hence, the map \(\xi \mapsto L(S, I)(\xi )\) is nonincreasing. Consider an increasing sequence \(\{\xi _k\}_{k\ge 0}\) with \(\xi _k>0\) such that \(\xi _k\rightarrow +\infty \) when \(k\rightarrow +\infty \) and denote
Since the functions S and I are bounded, the system (2.2) give us that the functions S and I have bounded derivatives. Then, by Arzela–Ascoli theorem, the functions \(\{S_k(\xi )\}\) and \(\{I_k(\xi )\}\) converge in \(C_{loc}^{\infty }(\mathbb {R})\) as \(k\rightarrow +\infty \), and up to extraction of a subsequence, one may assume that the sequences of \(\{S_k(\xi )\}\) and \(\{I_k(\xi )\}\) convergence toward some nonnegative \(C^\infty \) functions \(S_\infty \) and \( I_\infty .\) Furthermore, since \(L(S, I)(\xi )\) is nonincreasing on \(\xi \) and bounded from below, there exists a constant \(C_0\) and large k such that
Therefore, there exists some \(\delta \in \mathbb {R}\) such that \(\lim \limits _{k\rightarrow \infty } L(S_k, I_k)(\xi )=\delta \) for any \(\xi \in \mathbb {R}\). By Lebesgue’s dominated convergence theorem (see (Rudin 1976, Theorem 11.32)), we have
Thus,
Note that \(\frac{\mathrm{d}L}{\mathrm{d}\xi }=0\) if and only if \(S(\xi )\equiv S^*\) and \( I(\xi )\equiv I^*\), it follows that
Hence, we complete the proof of Theorem 1. Next, we give the following theorem on the critical traveling wave solution.
Theorem 2
For the wave speed \(c=c^*\), system (1.5) admits a nontrivial traveling wave solution \((S(\xi ),I(\xi ))\) satisfying
where \(\xi =n+c^*t\). Furthermore,
Proof
This theorem could be obtained by an approximation technique used in (Chen et al. 2017, Section 4), and we will give key sketch for the sake of completeness. Consider a sequence \(\{c_k\}\) such that \(c_k\in (c^*,c^*+1]\) for each \(k\in \mathbb {N}\), and \(c_k\rightarrow c^*\) as \(k\rightarrow +\infty \). Let \((S_k,I_k)\) be a traveling wave solution with wave speed \(c_k\).
We first show that \(\liminf \limits _{k\rightarrow +\infty }\Vert I_k\Vert _{L^\infty (\mathbb {R})}>0\). On the contrary, up to extraction of a subsequence, assuming \(\Vert I_k\Vert _{L^\infty (\mathbb {R})}\rightarrow 0\) as \(k\rightarrow +\infty \). Since \(I_k\) is bounded, then \(I_k(+\infty )\) exists. Thanks to (Chen et al. 2017, Lemma 3.9), we can obtain that \(I_k(+\infty )=I^*>0\), which is a contradiction.
Secondly, we prove that \(\limsup \limits _{k\rightarrow +\infty }\Vert I_k\Vert _{L^\infty (\mathbb {R})}<+\infty \). By way of contradiction, up to extraction of a subsequence, we assume that \(\Vert I_k\Vert _{L^\infty (\mathbb {R})}\rightarrow +\infty \) as \(k\rightarrow +\infty \). There exists \(\xi _k\in \mathbb {R}\) such that
It follows from the second equation of (2.2) that
for all \(\xi \in \mathbb {R}\) and \(k\in \mathbb {N}\). Hence, \(\frac{I_k(\xi \pm 1)}{I_k(\xi )}\) is bounded by Lemma 10. Define
By Lemma 11, we can conclude that \(\Phi _k(\xi )\rightarrow 0\) as \(k\rightarrow +\infty \) locally uniformly in \(\mathbb {R}\) since \(I_k(\xi +\xi _k)\rightarrow +\infty \) as \(k\rightarrow +\infty \). According to Chen et al. (2017), we can obtain that \(\Psi _k(\xi )\) is locally bounded and \(\Psi _k\) converges in \(C_{loc}^1(\mathbb {R})\) to \(\Psi _\infty \). Moreover, \(\Psi _\infty \) satisfies
in \(\mathbb {R}\). The proof of Lemma 14 give us \(\Psi _\infty \) is nonnegative and \(\Psi _\infty (0)=\lim \limits _{k\rightarrow +\infty }\Psi _k(0)=1\). It follows from Chen et al. (2017) that \({\Psi _\infty }'(0)=0\) and \(J[\Psi _\infty ](0)=0\). Hence, there is a contradiction with Eq. (3.5) since \(\mu _2>0\).
Finally, with the help of the above priori bound, passing the limit as \(k\rightarrow +\infty \) (see (Chen et al. 2017, page 2350-2351), we can obtain the existence of critical traveling wave solution for \(c=c^*\), which satisfying asymptotic boundary conditions (2.3). Recalling that the Lyapunov functional is independent on c, we can also have that \((S(n+c^*t),I(n+c^*t))\) satisfying the asymptotic boundary conditions (2.4). The proof is completed. \(\square \)
4 Nonexistence of Traveling Wave Solutions
In this section, we study the nonexistence of traveling wave solutions. Firstly, we show that \(c>0\) if there exists a nontrivial positive solution \((S(\xi ), I(\xi ))\) of system (2.2) satisfying the asymptotic boundary conditions (2.3) and (2.4).
Lemma 15
Assume that \(\mathfrak {R}_0>1\) and there exists a nontrivial solution \((S(\xi ), I(\xi ))\) of system (1.5) satisfying the asymptotic boundary conditions (2.3) and (2.4). Then, \(c>0\), where c is defined in (2.1).
Proof
Assume that \(c\le 0\). Since \(S(\xi )\rightarrow S_0\ \ \mathrm{and}\ \ I(\xi )\rightarrow 0\ \ \mathrm{as}\ \ \xi \rightarrow -\infty \), there exists a \(\xi ^*<0\) such that
here, we used the condition \(\mathfrak {R}_0>1\). Note that inequality (4.1) is valid for any \(\epsilon \in (0,f'(0))\), and then, for \(\xi <\xi ^*\), we have
Denote \(\omega = \frac{\beta S_0 f'(0)-\mu _2}{2}\) and \(Q(\xi ) = \int _{-\infty }^\xi I(y)\mathrm{d}y\) for \(\xi \in \mathbb {R}\), note that \(\omega >0\) since \(\mathfrak {R}_0>1\). Integrating inequality (4.1) from \(-\infty \) to \(\xi \) and using \(I(-\infty )=0\), one has that
Again, integrating inequality (4.1) from \(-\infty \) to \(\xi \) yields
Since \(Q(\xi )\) is strictly increasing in \(\mathbb {R}\) and \(c\le 0\), we can conclude that
which is a contradiction. Hence, \(c>0\). The proof is finished. \(\square \)
Now, we are in position to show the nonexistence of traveling wave solutions, and we will use two-sided Laplace to prove it (see Bai and Zhang 2015; Yang et al. 2013; Zhou et al. 2020)
Theorem 3
Assume that \(\mathfrak {R}_0>1\) and \(c<c^*\). Then, there is no nontrivial solution \((S(\xi ), I(\xi ))\) of system (1.5) satisfying the asymptotic boundary conditions (2.3) and (2.4).
Proof
By way of contradiction, assume that there exists a nontrivial positive solution \((S(\xi ), I(\xi ))\) of system (1.5) satisfying the asymptotic boundary condition (2.3) and (2.4). Then, \(c>0\) by Lemma 15 and
Let \(\omega = \frac{\beta S_0 f'(0)-\mu _2}{2}\) and \(Q(\xi ) = \int _{-\infty }^\xi I(y)\mathrm{d}y\) for \(\xi \in \mathbb {R}\). It follows from the proof of Lemma 15, there exists a \(\xi ^*<0\) such that
Recalling that \(Q(\xi )\) is strictly increasing in \(\mathbb {R}\), one has that
Thus,
Hence, there exists some constant \(\delta >0\) such that
Moreover, there exists a \(\nu >0\) is large enough and \(\epsilon _0\in (0,1)\) such that
Set
We have
which implies that \(V(\xi )\) is bounded as \(\xi \rightarrow -\infty \). Since \(\int _{-\infty }^\infty I(\xi )\mathrm{d}\xi <\infty \), we obtain that
From the second equation of (2.2), we have
integrating over \((-\infty ,\xi )\) gives us
Hence, we can obtain that
For \(\lambda \in \mathbb {C}\) with \(0<\text {Re}\lambda <\mu _0\), define the following two-sided Laplace transform of \(I(\cdot )\) by
Note that
and
From the second equation of (2.2), we have
Taking two-sided Laplace transform on (4.8) gives us
It follows from the proof in Lemma 5, as \(\xi \rightarrow -\infty \), we have
Thus, we can obtain that
By the property of Laplace transform (Widder 1941), either there exist a real number \(\mu _0\) such that \(\mathcal {L}(\lambda )\) is analytic for \(\lambda \in \mathbb {C}\) with \(0<\text {Re} \lambda < \mu _0\) and \(\lambda =\mu _0\) is singular point of \(\mathcal {L}(\lambda )\), or \(\mathcal {L}(\lambda )\) is well defined for \(\lambda \in \mathbb {C}\) with \(\text {Re} \lambda >0\). Furthermore, the two Laplace integrals can be analytically continued to the whole right half line; otherwise, the integral on the left of (4.9) has singularity at \(\lambda = \mu _0\) and it is analytic for all \(\lambda <\mu _0\). However, it follows from (4.10) that the integral on the right of (4.9) is actually analytic for all \(\lambda \le 2 \mu _0\), a contradiction. Thus, (4.9) holds for all \(\text {Re}\lambda >0\). From Lemma 1, \(\Delta (\lambda ,c)>0\) for all \(\lambda >0\) and by the definition of \(\Delta (\lambda ,c)\) in (2.6), we know that \(\Delta (\lambda ,c)\rightarrow \infty \) as \(\lambda \rightarrow \infty \), which is a contradiction with Eq. (4.9). This ends the proof. \(\square \)
5 Application and Discussion
As an application, we consider the following two discrete diffusive epidemic models. The first one is a model with saturated incidence rate which has been wildly used in epidemic modeling (see, for example, (Li et al. 2014; Zhang and Wu 2019; Xu and Ma 2009; Zhang et al. 2018; Xu and Guo 2019)).
Example 1
Discrete diffusive epidemic model with saturated incidence rate:
where \(\beta I_n(t)\) is the force of infection and \(\frac{1}{1+\alpha I_n(t)}\) measures the inhibition effect which is dependent on the infected individuals.
Setting \(f(I_n(t)) = \frac{\beta I_n(t)}{1+\alpha I_n(t)}\) in the original system (1.5), we can easily see that (5.1) is a special case of (1.5). In fact, it is obvious that \(f(I_n(t))\) satisfies Assumptions (A1) and (A2). The disease-free equilibrium of system (5.1) is similar to the original one, which is \(\tilde{E}_{0}=(S_0,0)\). Moreover, we obtain the basic reproduction number of system (5.1) as \(\mathfrak {R}_{1} = \frac{\beta S_0}{\gamma +\mu _1}\) and there exists a positive equilibrium \(\tilde{E}^{*}=(\tilde{S}^{*},\tilde{I}^{*})\) if \(\mathfrak {R}_{1}>1\), where
Hence, from Theorems 1, 2 and 3, we obtain the following corollary.
Corollary 1
Assume that \(\mathfrak {R}_1>1\). Then, there exists some \(c^*>0\) such that for any \(c\ge c^*\), system (5.1) admits a traveling wave solution \((S(\xi ),I(\xi ))\) satisfying
Furthermore, system (5.1) admits no traveling wave solutions satisfying (5.2) when \(c<c^*\).
The next example was studied in Chen et al. (2017), and our results will solve the open problem proposed in Chen et al. (2017), which is the traveling wave solutions converging to the endemic equilibrium as \(\xi \rightarrow +\infty \) for discrete diffusive system (1.3).
Example 2
Discrete diffusive epidemic model with mass action infection mechanism:
Setting \(f(I_n(t)) = \beta I_n(t)\) in the original system (1.5), we can easily see that (5.3) is a special case of (1.5) and this model has been studied in Chen et al. (2017). The disease-free equilibrium of system (5.3) is \(\bar{E}_{0}=(S_0,0)\). Moreover, we obtain the basic reproduction number of system (5.3) the same with (5.1) as \(\mathfrak {R}_{1} = \frac{\beta S_0}{\gamma +\mu _1}\), and there exists a positive equilibrium \(\bar{E}^{*}=(\bar{S}^{*},\bar{I}^{*})\) if \(\mathfrak {R}_{1}>1\), where
Then, from Theorems 1, 2 and 3, we obtain the following corollary.
Corollary 2
Assume that \(\mathfrak {R}_1>1\). Then, there exists some \(c^*>0\) such that for any \(c\ge c^*\), system (5.3) admits a traveling wave solution \((S(\xi ),I(\xi ))\) satisfying
Furthermore, system (5.3) admits no traveling wave solutions satisfying (5.4) when \(c<c^*\).
Note that Corollary 2 could answer the open problem proposed in Chen et al. (2017), that is, the traveling wave solutions for system (1.4) converges to the endemic equilibrium at \(+\infty \).
Next, we show that how the parameters affect wave speed. Suppose \((\hat{\lambda },\hat{c})\) be a zero root of \(\Delta (\lambda ,c)\) which defined in (2.6), a direct calculation yields
that is, \(\hat{c}\) is an increasing function on \(\beta \), \(d_2\) and \(\mathfrak {R}_0\). Biologically, this means that the diffusion and infection ability of infected individuals can accelerate the speed of disease spreading.
Now, we are in a position to make the following summary:
In this paper, a discrete diffusive epidemic model with nonlinear incidence rate has been investigated. When the basic reproduction number \(\mathfrak {R}_0>1\), we proved that there exists a critical wave speed \(c^*>0\), such that for each \(c \ge c^*\) the system (1.5) admits a nontrivial traveling wave solution. Moreover, we used a Lyapunov functional to establish the convergence of traveling wave solutions at \(+\infty \). We also showed the nonexistence nontrivial traveling wave solutions when \(\mathfrak {R}_0>1\) and \(c<c^*\). As special example of the model (1.5), we considered two different discrete diffusive epidemic model and apply our general results to show the conditions of existence and nonexistence of traveling wave solutions for the model (5.1). One of the example is studied in Chen et al. (2017), and our result solved the open problem proposed in Chen et al. (2017), which is the traveling wave solutions converge to the endemic equilibrium as \(\xi \rightarrow +\infty \) for discrete diffusive system (1.3).
Here, we mention some functions f(I) considered in the literature that do not satisfy Assumptions (A1) and (A2). For example, the incidence rates with media impact \(f(I) = I e^{-mI}\) in Cui et al. (2008); the specific incidence rate \(f(I) = \frac{kI}{1 + \alpha I^2}\) in Xiao and Ruan (2007); and the nonmonotone incidence rate \(f(I) = \frac{kI}{1 + \beta I + \alpha I^2}\) in Xiao and Zhou (2006). In a recent paper, Shu et al. (2019) studied a SIR model with nonmonotone incidence rates and without constant recruitment, and they investigated the existence and nonexistence of traveling wave solutions. What is the condition of existence and nonexistence of traveling wave solution for our model (1.5) with nonmonotone incidence rates will be an interesting question, and we leave this for future work.
References
Anderson, R.M., May, R.M.: Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford (1991)
Bai, Z., Zhang, S.: Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay. Commun. Nonlinear Sci. Numer. Simulat. 22, 1370–1381 (2015)
Bates, P.W., Chmaj, A.: A discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150, 281–305 (1999)
Briggs, C.J., Godfray, H.C.J.: The dynamics of insect-pathogen interactions in stage-structured populations. Am. Nat. 145, 855–887 (1995)
Brucal-Hallare, M., Vleck, E.V.: Traveling wavefronts in an antidiffusion lattice Nagumo model. SIAM J. Appl. Dyn. Syst. 10, 921–959 (2011)
Capasso, V., Serio, G.: A generalization of the Kermack-Mackendric deterministic model. Math. Biosci. 42, 43–61 (1978)
Chang, K.-C.: Methods in Nonlinear Analysis. Springer Monographs in Mathematics. Springer, Berlin (2005)
Chen, X., Guo, J.-S.: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326, 123–146 (2003)
Chen, Y.-Y., Guo, J.-S., Hamel, F.: Traveling waves for a lattice dynamical system arising in a diffusive endemic model. Nonlinearity 30, 2334–2359 (2017)
Cui, J., Sun, Y., Zhu, H.: The impact of media on the control of infectious diseases. J. Dyn. Differ. Equ. 20, 31–53 (2008)
Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24, 2891–2911 (2011)
Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993)
Fang, J., Wei, J., Zhao, X.-Q.: Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. A-Math. Phys. Eng. Sci. 466, 1919–1934 (2010)
Fu, S.-C., Guo, J.-S., Wu, C.-C.: Traveling wave solutions for a discrete diffusive epidemic model. J. Nonlinear Convex Anal. 17, 1739–1751 (2016)
Fu, S.-C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435, 20–37 (2016)
Guo, J.-S., Wu, C.-H.: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 252, 4357–4391 (2012)
Han, X., Kloeden, P.E.: Lattice dynamical systems in the biological sciences. In: Yin, G., Zhang, Q. (eds.) Modeling, Stochastic Control, Optimization, and Applications. Springer, Cham (2019)
He, J., Tsai, J.-C.: Traveling waves in the Kermark–McKendrick epidemic model with latent period. Z. Angew. Math. Phys. 70, 2722 (2019)
Heesterbeek, J.A.P., Metz, J.A.J.: The saturating contact rate in marriage and epidemic models. J. Math. Biol. 31, 529–539 (1993)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)
Hosono, Y., Ilyas, B.: Traveling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci. 5, 935–966 (1995)
Kapral, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991)
Kermack, W., McKendrick, A.: A contribution to mathematical theory of epidemics. Proc. R. Soc. A-Math. Phys. Eng. Sci. 115, 700–721 (1927)
Korobeinikov, A., Maini, P.K.: Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113–128 (2005)
Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615–626 (2006)
Lam, K.-Y., Wang, X., Zhang, T.: Traveling waves for a class of diffusive disease-transmission models with network structures. SIAM J. Math. Anal. 50, 5719–5748 (2018)
Li, W.-T., Xu, W.-B., Zhang, L.: Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discret. Contin. Dyn. Syst. 37, 2483–2512 (2017)
Li, Y., Li, W.-T., Lin, G.: Traveling waves of a delayed diffusive SIR epidemic model. Commun. Pur. Appl. Anal. 14, 1001–1022 (2015)
Li, Y., Li, W.-T., Yang, F.-Y.: Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl. Math. Comput. 247, 723–740 (2014)
Liu, W.M., Levin, S.A., Iwasa, X.: Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986)
Muroya, Y., Kuniya, T., Enatsu, Y.: Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discret. Contin. Dyn. Syst. Ser. B 20, 3057–3091 (2015)
Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill, New York (1976)
Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, New York (1991)
San, X.F., Wang, Z.-C.: Traveling waves for a two-group epidemic model with latent period in a patchy environment. J. Math. Anal. Appl. 475, 1502–1531 (2019)
Shu, H., Pan, X., Wang, X.-S., Wu, J.: Traveling waves in epidemic models: non-monotone diffusive systems with non-monotone incidence rates. J. Dyn. Differ. Equ. 31, 883–901 (2019)
Thieme, H.R.: Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators. J. Differ. Equ. 250, 3772–3801 (2011)
Tian, B., Yuan, R.: Traveling waves for a diffusive SEIR epidemic model with standard incidences. Sci. China Math. 60, 813–832 (2017)
Wang, W., Ma, W.: Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discret. Contin. Dyn. Syst. Ser. B 23, 3213–3235 (2018)
Weng, P., Huang, H., Wu, J.: Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68, 409–439 (2003)
Widder, D.V.: The Laplace Transform. Princeton Mathematical Series 6. Princeton University Press, Princeton (1941)
Wu, C.-C.: Existence of traveling waves with the critical speed for a discrete diffusive epidemic model. J. Differ. Equ. 262, 272–282 (2017)
Wu, S., Weng, P., Ruan, S.: Spatial dynamics of a lattice population model with two age classes and maturation delay. Eur. J. Appl. Math. 26, 61–91 (2015)
Xiao, D., Ruan, S.: Global analysis of an epidemic model with a nonlinear incidence rate. Math. Biosci. 208, 419–429 (2007)
Xiao, D., Zhou, Y.: Qualitative analysis of an epidemic model. Can. Appl. Math. Q 14, 469–492 (2006)
Xu, R., Ma, Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal. Real World Appl. 10, 3175–3189 (2009)
Xu, Z., Guo, T.: Traveling waves in a diffusive epidemic model with criss-cross mechanism. Math. Meth. Appl. Sci. 42, 2892–2908 (2019)
Yang, F.-Y., Li, Y., Li, W.-T., Wang, Z.-C.: Traveling waves in a nonlocal dispersal Kermack–McKendrick epidemic model equation with monostable convolution type nonlinearity. Discret. Contin. Dyn. Syst. Ser. B 18, 1969–1993 (2013)
Yang, Z., Zhang, G.: Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Sci. China Math. 61, 1789–1806 (2018)
Zhang, Q., Wu, S.-L.: Wave propagation of a discrete SIR epidemic model with a saturated incidence rate. Int. J. Biomath. 12, 1950029 (2019)
Zhang, S., Xu, R.: Travelling waves and global attractivity of an SIRS disease model with spatial diffusion and temporary immunity. Appl. Math. Comput. 224, 635–651 (2013)
Zhang, Y., Li, Y., Zhang, Q., Li, A.: Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules. Physica A 501, 178–187 (2018)
Zhang, S.-P., Yang, Y.-R., Zhou, Y.-H.: Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence. J. Math. Phys. 59, 011513 (2018)
Zhao, L., Wang, Z.-C., Ruan, S.: Traveling wave solutions in a two-group epidemic model with latent period. Nonlinearity 30, 1287–1325 (2017)
Zhou, J., Song, L., Wei, J.: Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay. J. Differ. Equ. 268, 4491–4524 (2020)
Zhou, J., Xu, J., Wei, J., Xu, H.: Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate. Nonlinear Anal. Real World Appl. 41, 204–231 (2018)
Zhou, J., Yang, Y., Hsu, C.-H.: Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discret. Contin. Dyn. Syst. Ser. B 25, 1469–1495 (2020)
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions which led to a significant improvement of this work. R Zhang and S Liu were supported by Natural Science Foundation of China (11871179; 11771374), J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (nos. LC2018002, LH2019A021) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mary Pugh.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, R., Wang, J. & Liu, S. Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model. J Nonlinear Sci 31, 10 (2021). https://doi.org/10.1007/s00332-020-09656-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-020-09656-3
Keywords
- Lattice dynamical system
- Schauder’s fixed point theorem
- Traveling wave solutions
- Diffusive epidemic model
- Lyapunov functional