Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}$$\end{document} and the local basic reproduction number R¯0(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{R}}_{0}(x)$$\end{document} are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}$$\end{document} and R¯0(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{R}}_{0}(x)$$\end{document} as well as the asymptotic properties of R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}$$\end{document} when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}. Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R0<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}<1$$\end{document}, while the disease is uniformly persistent when R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}>1$$\end{document}. Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle’s invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R0≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}\le 1$$\end{document}, and the endemic equilibrium is globally asymptotically stable if R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}>1$$\end{document} and an additional condition is satisfied.


Introduction
It is common knowledge that infectious diseases remain to threaten the survival and development of mankind.Therefore, one of the most significant and frequently discussed subjects nowadays is the prevention and control of disease transmission.In recent years, the convenient transportation provides the possibility of spatiotemporal transmission of diseases, and dynamic modeling has played a key role in describing spatiotemporal feature on diseases.Review these studies, the important and valuable results have been obtained and applied in realistic prevention and control of disease [1][2][3][4].
Vaccination is important to control measures in preventing the transmission of diseases.Thus, many countries provide routine vaccination against some the diseases, such as the avian influenza vaccine, the spondylitis vaccine, and so on.Over time, however, the vaccine loses its protection.Hence, developing mathematical models describing and understanding this phenomenon are particularly important [2,[5][6][7][8][9].Kribs-Zaleta and Velasco-Hernández [5] investigated a model of an SIS epidemic with vaccination.Li et al. [6] demonstrated that vaccine efficacy is crucial in preventing and controlling diseases.In particular, Liu et al. [2] established and studied the following model to characterize vaccination strategy: Here, S(t), V (t), I(t) and R(t) denote the densities of susceptible, vaccinated, infected and recovered individuals, respectively.For model (1.1), the authors showed that there exhibits rigorous threshold dynamics and vaccination contributes to disease control by reducing the basic reproduction number.Noting that model (1.1) does not take into account incubation periods.But before the host becomes contagious, many diseases have an incubation period, and the duration of this time varies from disease to disease [10].Additionally, we are aware that disease spread is not only time-dependent but also spacedependent.Webby [11] pointed out that infectious cases can be discovered in one place and then spread to other locations.As a result, studying epidemic models with spatial diffusion is intriguing.It has been confirmed by numerous studies in various models [12][13][14][15][16].
On the other hand, incidence rate is also an important substance in modeling the dynamics of epidemic systems.However, incidence functions in many infectious disease models are bilinear or standard incidence.From the perspective of the development mechanism of infectious diseases, the bilinear incidence is generally used for small-scale susceptible group and exposure.As the population size increases, the bilinear incidence will become infinite and lose its practical significance.At this time, the standard incidence is adopted, which is applicable to a large number of people.More and more nonlinear incidence has been mentioned many times (See [17][18][19][20]), such as nonlinear incidences βSg(I), βf (S)g(I) and βf (S, I).At this time, the nonlinear incidence is more realistic and achieves more exact results.Therefore, it is very interesting and natural to investigate epidemic model with nonlinear incidences.
Let Ω ⊂ R n be a bounded domain with smooth boundary ∂Ω, and n be the outward unit normal vector on ∂Ω.Based on the above discussion, we take into account latency, temporal and spatial factors, and nonlinear incidences in model (1.1).As a result, we propose the diffusive SVEIR model with nonlinear incidences shown below: − (μ 3 (x) + σ(x))E, t ≥ 0, x ∈ Ω, ∂I(t, x) ∂t =∇ • (d 4 (x)∇I) + σ(x)E − (μ 4 (x) + δ(x) + γ(x))I, t ≥ 0, x ∈ Ω, and the initial conditions S(0, x) =ψ 1 (x) ≥ 0, V (0, x) = ψ 2 (x) ≥ 0, E(0, x) = ψ 3 (x) ≥ 0, Here, the densities of susceptible, vaccinated, latent, infected and recovered individuals at time t and spatial location x are denoted by S(t, x), V (t, x), E(t, x), I(t, x) and R(t, x), respectively.Λ(x) is the input rate of population; ξ(x) is the vaccination rate of susceptible individuals; α(x) is the rate of vaccinees obtain immunity; σ(x) represents the transition rate from E to I; δ(x) is the disease-induced death rate; γ(x) is the recovery rate after infection; β 1 (x) and β 2 (x) are the infection rate of S and V infected by I in spatial location x, respectively; μ k (x) (1 ≤ k ≤ 5) denote the natural death rates of susceptible, vaccinated, latent, infected and recovered individuals, respectively; d k (x) (1 ≤ k ≤ 5) are the diffusion rate of susceptible, vaccinated, latent, infected and recovered individuals, respectively; and f 1 (S), f 2 (V ) and g(I) denote the nonlinear incidence rates.For the purposes of this article, we assume that all parameters in model (1.2) are continuous, nonnegative and bounded defined on Ω.
The main purpose in this paper is to investigate the basic reproduction number and the dynamics of model (2.1) in the spatially heterogeneous and homogeneous cases, respectively.The model proposed in this paper can more reasonably characterize the spread of epidemics.The main contribution and innovations are summarized as follows: (1) The well-posedness of solutions for model (2.1) is established.It is proved that for any nonnegative initial value, model (2.1) has a unique nonnegative and ultimately bounded solution by using the comparison principle, C 0 -semigroup and the integral expression of model (2.1).
(2) The expression of the basic reproduction number R 0 for model (2.1) is calculated, which depends diffusion rates d i (x)(1 ≤ i ≤ 4).Furthermore, we investigate the asymptotic properties of R 0 when the diffusive rates tend to infinity or zero.
(3) We establish the threshold criteria on the extinction and persistence of solutions for model (2.1) in the spatially heterogeneous case by using the theory of monotone dynamical systems and the persistence theory of dynamical systems.
(4) We establish the complete results for the global stability of model (2.1) in the spatially homogeneous case.Specially, the disease-free equilibrium P 0 of model (5.1) is also globally asymptotically stable when R 0 = 1.
The organization of this paper is as follows.In Sect.2, we prove the existence and ultimate boundedness of solutions for model (2.1).In Sect.3, we calculate the basic reproduction number R 0 of model (2.1) and local basic reproduction number R 0 (x) by using the next-generation operator method.And then, two compact linear operators L 1 and L 2 are introduced to investigate the relationship between R 0 and R 0 (x) and the asymptotic properties of R 0 when the diffusive rates are constants and tend to infinity or zero.In Sect.4, the threshold dynamics of model (2.1) are discussed.It is demonstrated that disease-free equilibrium P 0 (x) is globally stable and that the solutions persist uniformly.In Sect.5, we discuss the global asymptotic stability of the model in the homogeneous space.

Well-posedness of solutions
In this section, we focus on the well-posedness of solutions of model (2.1).Because R(t, x) does not appear in the first four equations of model (1.2), we only discuss the following subsystem of (1.2) + ) be the positive cone of X.
(A 2 ) f 1 (S), f 2 (V ) and g(I) are nondecreasing for all S > 0, V > 0 and I > 0, respectively, and g(I) I is nonincreasing for all I > 0.
Consider the following scalar reaction-diffusion equation: where d(x), μ(x) and β(x) are positive, continuous and bounded functions defined on Ω.The following conclusions are held in light of Lemma 1 in [21].
Next, the following results are established regarding the existence and ultimate boundedness of the global solution of model (2.1).

Basic reproduction number
In this section, we want to calculate the basic reproduction number of model (2.1) and find its display expression and explore its some properties.When (E(t, x), I(t, x)) ≡ (0, 0) in model (2.1), we get the reaction-diffusion equation shown below.
It follows from Lemma 2.1 that the following system admits a unique positive steady state S 0 (x), satisfying the equation with ∂S0(x) ∂n = 0 for x ∈ ∂Ω, which is globally asymptotically stable in C(Ω, R + ).Then, from the second equation of system (3.1),we have the limit system By Lemma 2.1 and Corollary 4.3 in [32], system (3.3)exists a unique positive steady state V 0 (x) satisfying the equation with ∂V0(x) ∂n = 0 for x ∈ ∂Ω, which is globally asymptotically stable in C(Ω, R + ).Thus, we obtain that model (2.1) has disease-free equilibrium P 0 (x) = (S 0 (x), V 0 (x), 0, 0).
At equilibrium P 0 (x), we linearize the last two equations of model (2.1), and then, we obtain the following linearized subsystem We define the matrix operators where w = (w 1 , w 2 ) T .Assume that the illness is introduced at time t = 0 and the initial infection distribution is represented by the expression ψ(x) = (ψ 3 (x), ψ 4 (x)) T .As a result, the distribution of new infection becomes F (x)T (t)ψ(x) at time t as time evolves.Therefore, We call operator L by the next-generation operator.Obviously, L is a continuous and positive operator which maps the initial infection distribution ψ(x) to the distribution of the total infective people produced during the infection period.According to [33], the spectral radius of L is defined as the basic reproduction number R 0 of model (2.1).Namely, R 0 = r(L).
We can obtain by calculating . Thus, we can obtain Namely, R 0 is defined as the spectral radius of operator b 21 (x)f 12 (x).This tells that R 0 is the principal eigenvalue of the following eigenvalue problem: Thus, there exists a strictly positive eigenfunction That is and ∂φ * (x) ∂n = 0 for x ∈ ∂Ω.Multiplying by φ * in both sides of (3.5) and then integrating on Ω, then we have On the other hand, we consider the following eigenvalue problem: From Krein-Rutman theorem, there is a principal eigenvalue λ = λ * and a corresponding strictly positive Then, we get Multiplying by ψ * in both sides of (3.5), we have Then, from (3.6) we also obtain Integrating on Ω, we further have .
When diffusion coefficients d i (x) = 0 (i = 1, 2, 3, 4), then model (2.1) is transformed into the corresponding ordinary differential equation model as follows: (3.9) Theorem 3.1.For model (3.9), in each position x ∈ Ω, the local basic reproduction number R 0 (x) is expressed as (3.11) Theorem 3.1 can be easily proved by using the next-generation matrix method.We here omit it.Based on the above analysis, for the relation between R 0 and R 0 (x), we can have the following obvious observation.
Now, we define two linear operators L 1 and L 2 : C(Ω, R) → C(Ω, R) as follows (3.12) and let the operators L i (i = 1, 2) subject to the Neumann boundary condition.Then, it is easy to see that r(L where R i (x) (i = 1, 2, 3) defined in (3.11) are additive operators on C(Ω, R).
Proof.Recall the calculation of R 0 , we know According to the elliptic estimates and maximum principles, we know that L 1 and L 2 are strongly positive compact linear operators on C(Ω, R).This completes the proof.
By the comparison principle, we have Proof.Because L 1 and L 2 are strongly positive compact linear operators on C(Ω, R), so is L 1 L 2 .By the general result on the Krein-Rutman theorem (Theorem 2.5 [34]), we know that r(L 1 ), r(L 2 ) and r(L 1 L 2 ) are simple positive eigenvalues of operators L 1 , L 2 and L 1 L 2 associated with positive eigenvectors, respectively.Moreover, there is no other such eigenvalue for L 1 , L 2 and L 1 L 2 , respectively.Owing to This completes the proof.By (3) of Theorem 3.6 in [34], we can have the following result.
Lemma 3.5.The following inequalities hold: Similar proof for the opposite part can take.This completes the proof.

The large diffusion rates
In this subsection, we investigate R 0 quantitatively when the diffusion rates increase to infinity.For this purpose, assume d 3 (x) ≡ d 3 and d 4 (x) ≡ d 4 are constants.We firstly give a useful lemma before we will do this.
Lemma 3.6.(see [34]) Let W be an ordered Banach space with positive cone W + such that W + has nonempty interior.Let T n (n ≥ 1), and T be strongly positive compact linear operators on W . Suppose T n SOT − −− → T (strong operator topology) which means T n (u) → T (u) for any u ∈ W .If ∪ n≥1 T n (B) is precompact, where B is the closed unit ball of W , and r(T n ) ≥ r 0 for some r 0 > 0, then r(T n ) → r(T ).
The proof of this lemma can be found in Theorem 4.1 in [34].
Lemma 3.7.Let R 0 be defined in (3.7), and d i (x) ≡ d i (i = 1, 2, 3, 4) be constants.For any given Then, R 0 is decreasing function with respect to d 3 and d 4 .
Proof.Firstly, we prove R 0 is decreasing in d 3 .Let k = 1 R0 .According to the Krein-Rutman theory, there is an eigenvalue k and a corresponding eigenfunction ψ with ψ 2 = 1, which is strictly positive and satisfies Therefore, we have Differentiating both sides of (3.13) with respect to d 3 , we get (3.14) Multiplying (3.14) by ψ and (3.13) by ψ d3 , and integrating their difference over Ω, we have and By divergence theorem and conditions of Lemma 3.7, we have Combine (3.15) and (3.16), we obtain Using a similar approach, we can also prove that R 0 is decreasing in d 4 .This completes the proof.
Proof.For any given u ∈ C(Ω, R), we just need to show that Since the operator L 1 subjects to the Neumann boundary condition, we have that Q d3 is the solution of the following problem Under the comparison principle, we know Hence by the L p estimate, {Q d3 } d3>1 is uniformly bounded in W 2,p (Ω) for any p > 1.By the embedding theorems, we see that Under the maximum principle, we know that Q must be constant.Then integrating both sides of the first equation of (3.17) and taking Proof.For any given u ∈ C(Ω, R), we just need to show that For similar discussions as in Lemma 3.8, we can conclude that This completes the proof.
(1) For fixed . By Lemmas 3.8-3.9,we We obverse that the operators and H 2,∞ are strongly positive compact operators on C(Ω, R).For similar discussion of Theorem 4.5 in [34] and Theorem 3.2, we have that ) when d 4 → +∞.Finally, we can know that the eigenfunctions of H 1,∞ and H 2,∞ must be constants, and This completes the proof.
Proof.By Theorem 3.3, we have The proof of (ii) is similar, and we omit it.This completes the proof.

The small diffusion rates
Now, we investigate the asymptotic properties of basic reproduction number R 0 when the diffusion rates decrease to zero.We first have the result as follows.
(i) For fixed d 4 > 0, when are strongly positive compact operators in C(Ω), by Lemma 3.6, we have The proof of (ii) is similar to (i), and we omit it.This completes the proof.
, where k d4 is the principal eigenvalue of the following system ⎧ ⎨ ⎩ (3.21)By (3.21), we have We just need to prove lim sup d4→0 k d4 ≤ 1 R * 0 . Assume to the contrary, i.e., lim sup Then, there exists ε 0 > 0 and a sequence {d 4,n } with 2 in B(x 0 , δ) for x 0 ∈ Ω and δ > 0. Assume that system (3.21) has a principal eigenvalue k d4,n and a corresponding eigenfunction v d4,n , which is positive.Then in B(x 0 , δ), we obtain .

Threshold dynamics
In this section, we discuss the global stability of disease-free equilibrium P 0 (x) and the uniform persistence of model (2.1).Firstly, the following conclusions are established.
Because (E(t, x), I(t, x)) tend to (0, 0) uniformly for x ∈ Ω as t → ∞, from the first equation of model (2.1), we get the following limit equation From the theory of asymptotically autonomous semiflows (see [32]) and Lemma 2.1, we further acquire that S(t, x) → S 0 (x) uniformly for x ∈ Ω as t → ∞.Similarly, we also obtain that V (t, x) → V 0 (x) uniformly for x ∈ Ω as t → ∞.Thus, by Lemma 3.2, we know that disease-free equilibrium P 0 (x) is globally asymptotically stable.This completes the proof.
As a result of Theorem 4.2, we can obtain Remark 4.1.We here have proved the existence of endemic equilibrium, but its uniqueness and stability are still an open question.In the following section, we will prove the global asymptotic stability of endemic equilibrium in the spatial homogeneous environment.
Thus, by LaSalle's invariable principle, we finally get that the equilibrium P 0 is globally asymptotically stable when R 0 ≤ 1.This completes the proof.Theorem 5.2.Assume that R 0 > 1, and the following inequalities hold Proof.Define a Lyapunov function as follows: Calculating the time derivative of L 1 , we have Due to the divergence theorem, the Neumann boundary conditions and (5.2), we can obtain Thus, we know that dL1 dt ≤ 0. Furthermore, we know that dL1 dt = 0 if and only if S = S * , V = V * , E = E * and I = I * .Thus, by LaSalle's invariable principle, it clear that endemic equilibrium P * is globally asymptotically stable.This completes the proof.Remark 5.1.In Theorem 5.2, an additional condition (5.4) is added.We easily see that when f 1 (S) = S and f 2 (V ) = V , then condition (5.4) holds.However, if we take f 1 (S) = S, f 2 (V ) = V and g(I) =

Conclusions
In this paper, we propose a diffusive SVEIR epidemic model with nonlinear incidences.We mainly concern to the dynamical behaviors and some properties of the basic reproduction number.On nonlinear incidence rate function f 1 (S), f 2 (V ) and g(I), we have introduced assumptions (A 1 ) − (A 2 ).Firstly, the wellposedness of solutions is established.Secondly, we calculate the expression of the basic reproduction number R 0 and the local basic reproduction number R 0 (x), which are characterized as the spectral radius of the next-generation operator.Following, we investigate the relation of R 0 and R 0 (x) by defined compact linear operators L 1 and L 2 with spectral radius one and additive operators R i (x) (i = 1, 2, 3).Furthermore, when the diffusion coefficients d i (x) ≡ d i (i = 3, 4) are constant, we establish quantitative connection of R 0 and R 0 (x), that is, Thirdly, we show the disease-free equilibrium P 0 (x) is globally asymptotically stable if R 0 < 1, while the disease is persistent if R 0 > 1, and model (2.1) has at least one endemic equilibrium P * (x) = (S * (x), V * (x), E * (x), I * (x)).Furthermore, applying Lyapunov functions, we established the global stability of the equilibria in the spatially homogeneous model.That is, when R 0 ≤ 1, then disease-free equilibrium P 0 of model (5.1) is globally asymptotically stable; when R 0 > 1 and the conditions of (5.4) are satisfied, then endemic equilibrium P * of model (5.1) is globally asymptotically stable.It is covered and improved some existing global dynamical results (see [38]).There are still some open problems for model (2.1) that we need to look into in the future.For example, one open problem is when the basic reproduction number R 0 > 1 is the uniqueness and stability of the endemic equilibrium in the spatially heterogeneous environments for model (2.1).The other open problem is that when R 0 = 1, the threshold dynamics of model (2.1) are not establish in this paper.In addition, in the spatial homogeneous environment whether we also only need condition R 0 > 1 to exactly prove the global asymptotic stability of endemic equilibrium P * as in Theorem 5.2.As well as, the asymptotic profiles of endemic equilibrium for model (2.1) as the diffusive coefficients tend to infinity or zero also are not investigated.

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Dynamical analysis for a diffusive SVEIR Page 13 of 27 173

For
any positive a, d 3 and d 4 , let b = b(a, d 3 , d 4 ) be the principal eigenvalue of the following eigenvalue problem: AΦ + aCΦ = bΦ, (3.24)Then, we have b(k, d 3 , d 4 ) = 0.According to Theorem 1.4 in [36], we have lim (d3,d4)→(0,0) b = max x∈Ω b(D a (x)), where b(D a (x)) is an eigenvalue of the matrix D a (x) with a greater real part for each x ∈ Ω and
denote the spectral radius of L i .Next we can have the following result. 19))