Wave propagation for a discrete diffusive vaccination epidemic model with bilinear incidence

The aim of the current paper is to study the existence of traveling wave solutions (TWS) for a vaccination epidemic model with bilinear incidence. The existence result is determined by the basic reproduction number $\Re_0$. More specifically, the system admits a nontrivial TWS when $\Re_0>1$ and $c \geq c^*$, where $c^*$ is the critical wave speed. We also found that the TWS is connecting two different equilibria by constructing Lyapunov functional. Lastly, we give some biological explanations from the perspective of epidemiology.


Introduction
Vaccination is critical for the prevention and control of infectious diseases, there are more than 20 life-threatening diseases could be prevented by vaccines up to now.Vaccinators can achieve immunity by having the immune system recognize foreign substances, antibodies are then screened and generated to produce antibodies against the pathogen or similar pathogen, and then giving the injected individual a high level of disease resistance.In [1], Liu et al. proposed the following system with continuous vaccination strategy: where S(t), V (t), I(t) and R(t) are the densities of susceptible, vaccinated, infective and removed individuals at time t, respectively.The parameters of model (1.1) are biologically explained as in Table 1.In [1], the authors shown that the disease-free equilibrium for model ( asymptotically stable (GAS) if the basic reproduction number is less than one, while if the number is greater than one, then a positive endemic equilibrium exists which is GAS.Since then, the epidemic models with vaccination have attracted the attention of many scholars.Kuniya [2] extended the study in [1] to a multi-group case, and then studied the global stability by using the graph-theoretic approach and Lyapunov method.Considering the effect of age, three vaccination epidemic models with age structure are proposed in [3][4][5], and the global stabilities are studied.
For more recent studies on the vaccination epidemic models, we refer to [6,7] and the references therein.
With the increasing trend of globalization and mobility of people, the spatial structure of human density and location has a significant impact on the spread of diseases.It is necessary to investigate the role of diffusion in the epidemic modeling.Mathematically, Laplacian operator in the reaction-diffusion systems could can be well used to study the infectious disease model with diffusion, since it could describe the random diffusion of each individual in the adjacent space.On the other hand, nonlocal operator could describe the long range diffusion on the whole habitat [8].In the study of local and nonlocal diffusive epidemic models, there is a solution called traveling wave solution (TWS).Viewing from infectious diseases perspective, the existence of TWS for epidemic model implies that the disease can be invaded [9].Up to now, there have been many studies on the TWS for local and nonlocal diffusive epidemic models (see, for example, [10][11][12][13][14][15]). By considering both vaccination and spatial diffusion, Xu et al. [16] studied a local diffusive SVIR model, where the global dynamics on bounded domain and TWS on unbounded domain for the model were studied.Meanwhile, the problem of TWS for two different SVIR models with nonlocal diffusion were investigated in [17,18].
Unlike local and nonlocal diffusive, there is another diffusion in infectious disease modeling, which is discrete diffusion.In fact, epidemic model with discrete diffusion can be regarded as lattice system, such system is better to describe the epidemic model with patch structure [19].Recently, Chen et al. [20] proposed a lattice SIR epidemic model: where n ∈ Z. S n and I n denote densities of susceptible and infectious individuals at time t and niche n. β is the disease transmission rate. 1 (normalized) and d denote the random migration parameters for each compartments.Chen et al. shown that system (1.2) admits TWS when ℜ 0 > 1 and c ≥ c * .More recently, the TWS for (1.2) was proved to be converged to the endemic equilibrium by Zhang et al [21,22].Model (1.2) is an SIR model with constant recruitment (i.e. the constant Λ), and the existence of TWS for the discrete diffusive epidemic model without constant recruitment was studied in [23][24][25].However, to our best knowledge, there are only a few studies focus on the problem of TWS for discrete diffusive epidemic models, especially for the model with constant recruitment.Based on the above facts, in order to study the role of vaccination and patch structure in the disease modeling, we consider a discrete diffusive vaccination model as follows where S n , V n , I n and R n denote susceptible, vaccinated, infectious and removed individuals.d is the spatial motility of infectious individuals and the diffusive rate of other compartments are normalized to be 1.The biological significance of the parameters of (1.3) are the same as those in (1.1).
The current paper devotes to study the existence of TWS for system (1.3) with bilinear incidence.In fact, there are very few studies on TWS for the epidemic model with bilinear incidence and the main difficulty is the boundedness of TWS [19].On the other hand, introducing the constant recruitment (i.e.Λ in model (1.3)) will bring much more complexity in mathematical analysis than the system without constant recruitment.Moreover, it is difficult to obtain the behaviour of TWS at +∞ for such model (see, for example, [20]).One motivation of this paper is to show the convergence of TWS for lattice epidemic model (1.3).To gain this purpose, we will construct an appropriate Lyapunov functional for the wave form equations corresponding to lattice dynamical system (1.3).To do this, we prove the persistence of TWS, which is crucial to guarantee the Lyapunov functional has a lower bound.We should be point out that, for different models, the construction of Lyapunov functional is also different and requires technique.Biologically, since the vaccination has an effect of decreasing the basic reproduction number in [1], we want to study how vaccination affects the speed of TWS.
The organization of this paper is as follows.In section 2, we give some preliminaries results.Section 3 devote to study the existence of TWS of system (1.3) by applying Schauder's fixed point theorem.In Section 4, we show the boundedness of TWS.Furthermore, we show the convergence of TWS in Section 5. Finally, there is a brief discussion and some explanations from the perspective of epidemiology will be given in Section 6.

Preliminaries
Firstly, the corresponding ODE system for (1.3) is where R equation is decoupled from other equations.Clearly, system (2.1) has a disease-free equilibrium as the basic reproduction number.The well known results for (2.1) is 1) has a globally asymptotically stable positive equilibrium Now, we state our purpose of the current paper.Letting ς = n + ct in system (1.3),where c is wave speed, we arrive at We want to find TWS satisfying: and lim

Eigenvalue problem
Linearizing the third equation of (2.4) at the E 0 yields Let I(ς) = e rς , we have (2.9) By some calculations, for r > 0 and c > 0, we have There exist c * > 0 and r * > 0 such that ∂∆(r, c) ∂r Furthermore,

Sub-and super-solutions
Fix c > c * and ℜ 0 > 1, we show the following lemma.
Lemma 2.2.For ε i > 0 small enough and M i > 0 (i = 1, 2, 3) large enough, we define the following six functions: Then they satisfy and Proof.The proof of (2.10) are trivial, so we omit the details.Now, we focus on the proof of inequalities (2.11) ), and small enough and ς < X 1 < 0. Thus, we need to choose a sufficiently large Then (2.11a) holds.
As for (2.11c), we choose Using the definition of ∆(r, c) and noticing that ∆(r 1 + ε 3 , c) < 0, then it suffices to show that which holds for M 3 is large enough.

Existence of traveling wave solutions
) where ρ 1 is large enough such that ρ 1 φ − β 1 φψ is nondecreasing on φ and ρ 2 is large enough such that ρ 2 ϕ − β 2 ϕψ is nondecreasing on ϕ.Clearly, system (3.1) has a unique solution Lemma 3.1.The operator A maps Γ B into itself and it is completely continuous.
Proof.Firstly, it is easy to show A maps Γ B into Γ B by Lemma 2.2, so we omit the details.Next, we focus on the second part of Lemma 3.1.For i = 1, 2, suppose that (φ i (ς), and Direct calculation yields (τ +B) H 3 (φ, ϕ, ψ)(τ )dτ, For i = 1, 2 and any (φ i , ϕ i , ψ i ) ∈ Γ B , we have Hence, Hence, the operator A is continuous by some similar arguments with V B and I B .Moreover, S ′ B , V ′ B and I ′ B are bounded by (3.1).Thus, the operator A is completely continuous.
By using Schauder's fixed point theorem, there exists (S B , V B , I B ) ∈ Γ B such that . Next, we give some prior estimate for (S B , V B , I B ). Define Lemma 3.2.There exists constant C(X ) > 0 such that Proof.Since (S B , V B , I B ) is the fixed point of A, one has Hence, for some constant C 1 (X ) > 0. It follows from [24] that for any X < B.
With the help of Lemma 3.2 and following from the standard arguments in [22], we can conclude that (S, V, I) is solution for system (2.4) with

Boundedness of traveling wave solution
In the following, we first show the boundedness of (S, V, I).

Now
Integrating (4.2) from ς − 1 2 to ς yields Similarly, integrating (4.2) over [ς, ς + 1 2 ], we have Claim II.I ′ (ς) I(ς) is bounded in R.This claim is true because Claim I and the third equation of (2.4).Choose a sequence {c k , S k , V k , I k } of the TWS for (1.3) in a compact subinterval of (0, ∞), we have the following claim.
The following lemma is to show that I(ς) cannot approach 0. Proof.We only need to show that if I(ς) ≤ ε 0 for ε 0 > 0 is small enough, then I ′ (ς) > 0 for all ς ∈ R. If not, we can choose a sequence {ς k } k∈N with c k ∈ (a, b) so that I(ς k ) → 0 as k → +∞ and I ′ (ς k ) ≤ 0, where a and b are two positive constants.Let then, I k (0) → 0, I k (ς) → 0 and I ′ k (ς) → 0 locally uniformly in R as k → +∞, and we can obtain that S ∞ = S 0 and V ∞ = V 0 by a similar proof in [20,Lemma 3.8].

Convergence of the traveling wave solution
In this section, we show the convergence of traveling wave solutions.
Hence, there exists some δ ∈ R such that lim k→∞ V(S k , V k , I k )(ς) = δ, ∀ς ∈ R. Using Lebegue dominated convergence theorem, one has that Recall that dV dς = 0 if and only if S ≡ S * , V ≡ V * and I ≡ I * , which finishes the proof.Remark 5.1.For the case c = c * , we can obtain the existence of TWS by using a similar approximation technique used in [20,Section 4].The TWS for c = c * is also satisfy (2.5) and (2.6) since the Lyapunov functional is independent of c.

Discussion
In this paper, we proposed a discrete diffusive vaccination epidemic model (i.e., system (1.3)), which seems to be more realistic than the non-delayed model (1.2).Employing Schauder's fixed point theorem and Lyapunov functional, we obtain the existence of nontrivial positive TWS, which is connecting two different equilibrium.Our research examines the conditions (i.e.basic reproduction number) under which an infectious disease can spread, even this disease has a vaccine.Now we finish this section with some explanations from the perspective of epidemiology.Assume that (r, ĉ) is a root of ∆(r, c) = 0, by some calculations, we obtain dĉ dγ 1 < 0, dĉ dd > 0, dĉ dβ 1 > 0, dĉ dβ 2 > 0 and dĉ dℜ 0 > 0.
Mathematically, ĉ is a decreasing on γ 1 , while ĉ is an increasing function on d, β 1 and β 2 .From the biological point of view, this indicates the following three scenarios:

Table 1 .
Biological meaning of parameters in model (1.1).