Volume 2013, Issue 1 921401

Research Article

Open Access

The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Yuan Yuan,

Yuan Yuan

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China wzu.edu.cn

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Hailing Wang,

Hailing Wang

Department of Mathematics, Hubei Minzu University, Enshi, Hubei 445000, China muc.edu.cn

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Weiming Wang,

Corresponding Author

Weiming Wang

[email protected]

orcid.org/0000-0002-7562-8260

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China wzu.edu.cn

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Yuan Yuan,

Yuan Yuan

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China wzu.edu.cn

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Hailing Wang,

Hailing Wang

Department of Mathematics, Hubei Minzu University, Enshi, Hubei 445000, China muc.edu.cn

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Weiming Wang,

Corresponding Author

Weiming Wang

[email protected]

orcid.org/0000-0002-7562-8260

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China wzu.edu.cn

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First published: 22 December 2013

https://doi.org/10.1155/2013/921401

Citations: 1

Academic Editor: Kaifa Wang

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Abstract

We investigate the disease’s dynamics of a reaction-diffusion epidemic model. We first give a priori estimates of upper and lower bounds for positive solutions to model and then give the conditions of the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

1. Introduction

Infectious diseases are the second leading cause of death worldwide, after heart disease, and are responsible for more deaths annually than cancer [1]. Since the pioneer work of Kermark and McKendrick [2], mathematical models have been contributing to improve our understanding of infectious disease dynamics and help us develop preventive measures to control infection spread qualitatively and quantitatively.

Many studies indicate that spatial epidemiology with self-diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission [3]. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics. In particular, the spatial spread of infections has been studied by analyzing traveling wave solutions and calculating spread rates [4–10].

Besides, there has been some research on pattern formation in the spatial epidemic model, starting with Turing’s seminal paper [11]. Turing’s revolutionary idea was that the passive diffusion could interact with chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetric solutions with the result that the system with diffusion has them [12]. In these studies [3, 13–20], via standard linear analysis, the authors obtained the conditions of Turing instability, and, via numerical simulation, they showed the pattern formation induced by self-diffusion or cross-diffusion and found that model dynamics exhibits a diffusion controlled formation growth to stripes, spots, and coexistence or chaos pattern replication.

Recently, the researchers are interested in research on the stationary patterns due to the existence and nonexistence nonconstant solutions of the reaction-diffusion model [21–29]. But the research on the existence and nonexistence nonconstant solutions of reaction-diffusion epidemic model, seems rare [3].

In this paper, we will focus on the disease’s dynamics through studying the existence of the constant and nonconstant steady states of a simple reaction-diffusion epidemic model.

The rest of this paper is organized as follows. In Section 2, we derive a reaction-diffusion epidemic model. In Section 3, we give a priori estimates of upper and lower bounds for positive solutions to model. In Section 4, we give the main results on the existence and nonexistence of positive nonconstant steady states of the model. The paper ends with a brief discussion in Section 5.

2. Basic Model

In [30], Berezovsky and coworkers introduced a simple epidemic model through the incorporation of variable population, disease induced mortality, and emigration into the classic model of Kermark and McKendrick [2]. The total population (N) is divided into two groups susceptible (S) and infectious (I); that is, N = S + I. The model describing the relations between the state variables is

(1)

where the birth process incorporates density dependent effects via a logistic equation with the intrinsic growth rate r and the carrying capacity K; S(t), I(t) represent population densities of susceptible and infected population, respectively; β denotes the transmission rate (the infection rate constant); μ is the natural mortality; d denotes the disease-induced mortality; θ is the per-capita emigration rate of noninfective.

For model (1), the epidemic threshold of basic reproduction number R₀ is then computed as

(2)

The basic demographic reproductive number R_d is given by

(3)

For simplicity, rescalling the model (1) by letting S → S/K, I → I/K, and t → t/(μ + d) leads to the following model:

(4)

where ν = (μ + θ)/(μ + d) defined by the ratio of the average life-span of susceptibles to that of infections and S + I ≤ 1.

See [30] for more details.

Assume that the habitat Ω ⊂ ℝ^m (m ≥ 1) is a bounded domain with smooth boundary ∂Ω (when m > 1), and n is the outward unit normal vector on ∂Ω. We consider the following reaction-diffusion SI epidemic model:

(5)

where the nonnegative constants d_S and d_I are the diffusion coefficients of S and I, respectively. The symbol Δ is the Laplacian operator. The homogeneous Neumann boundary condition implies that the above model is self-contained and there is no infection across the boundary.

The corresponding kinetic model (5) with m = 2 has been investigated by Wang et al. [20].

In this paper, we concentrated on the steady states of model (5) which satisfy

(6)

Throughout this paper, the positive solution (S, I) satisfying model (6) refers to a classical one with S > 0, I > 0 on

. Clearly, model (6) has a unique positive constant solution (endemic equilibrium) E^* = (S^*, I^*) if R_d > (ν + R₀ − 1)/R₀ν and R₀ > 1, where

(7)

3. A Priori Estimates for Positive Solutions to Model (6)

The main purpose of this section is to give a priori upper and lower positive bounds for positive solution of model (6). To this aim, we first cite two known results. The first is due to Lin et al. [31] and the second to Lou and Ni [32]. In the following, let us denote the constants ν, R_d, and R₀ collectively by Λ. The positive constants C, , , C^*, and so forth will depend only on the domains Ω and Λ.

Lemma 1 (Harnack inequality [31]). Let be a positive solution to Δw(x) + c(x)w(x) = 0, where , satisfying the homogeneous Neumann boundary conditions. Then there exists a positive constant C^* = C^*(∥c∥_∞, Ω), such that

(8)

Lemma 2 (maximum principle [32]). Let Ω be a bounded Lipschitz domain in ℝ^m and .

(a)
Assume that and satisfies
(9)
If , then g(x₀, w(x₀)) ≥ 0.
(b)
Assume that and satisfies
(10)
If , then g(x₁, w(x₁)) ≤ 0.

Theorem 3. If R₀ > 1, then the positive solution (S(x), I(x)) of model (6) satisfies

(11)

Proof. Assume that (S(x), I(x)) is a positive solution of model (6). We set

(12)

By applying Lemma 2, we have

(13)

and R₀S(x₂)I(x₂)/(S(x₂) + I(x₂)) ≥ I(x₂). This clearly gives

(14)

Theorem 4. Assume that R_d > 1 and R₀ > 1. Let d and D be fixed positive constants. Then there exists a positive constant such that, if d_S, d_I > d, every positive solution (S(x), I(x)) of model (6) satisfies

(15)

Proof. Let

(16)

In view of Theorem 3, there exists a positive constant C = C(Λ) such that

provided that d_S, d_I > d. As S and I satisfy

(17)

It follows from Lemma 1 that there exists a positive constant C^* = C^*(Λ, d) such that

(18)

for d_S, d_I ≥ d.

Now, on the contrary, suppose that (15) is not true, then there exist sequences , with (d_S,i, d_I,i)∈[d, ∞)×[d, ∞) and the positive solution (S_i, I_i) of model (6) corresponding to (d_S, d_I) = (d_S,i, d_I,i), such that

(19)

It follows from Lemma 1 that

(20)

(S_i, I_i) satisfies

(21)

Integrating by parts, we obtain that, for i = 1,2, …,

(22)

By the regularity theory for elliptic equations [33], we see that there exist a subsequence of

, which we will still denote by

, and two nonnegative functions

, such that

in [C²(Ω)] ² as i → ∞. By (20), we have that

Letting i → ∞ in (22) we obtain that

(23)

Case 1 ( , or , ). Since I_i satisfies the second inequality of (18), I_i > 0 on . Therefore, R₀S_i/(S_i + I_i) − 1 → −1 < 0 on as i → ∞. Hence, ∫_Ω I_i(R₀S_i/(S_i + I_i) − 1)dx < 0 for sufficiently large i which contradicts the second integral identity of (22).

Case 2 ( , ). As above, on . It follows from the first integral identity of (23) that

(24)

This fact combines with

yielding to

, which implies that R₀S_i/(S_i + I_i) → R₀ uniformly on

as i → ∞, since I_i → 0 uniformly on

. As R₀ > 1, this contradicts the second integral identity of (23) and the fact that I_i > 0. This completes the proof.

4. Existence and Nonexistence of Positive Nonconstant Steady States

In this section, we provide some sufficient conditions for the existence and nonexistence of nonconstant positive solution of model (6) by using the Leray-Schauder degree theory [34]. From now on, we denote by

(25)

the eigenvalues of the operator −Δ on Ω with the zero-flux boundary conditions.

4.1. Nonexistence for Positive Nonconstant Steady States to Model (6)

This section is devoted to the consideration of the nonexistence for the nonconstant positive solutions of model (6), and, in the following results, the diffusion coefficients do play a significant role.

Theorem 5. Assume that R₀ > 1. Let D₂ be a fixed positive constant with D₂ > (R₀ − 1)/μ₁. Then there exists a positive constant D₁(Λ, D₂) such that model (6) has no positive nonconstant solution provided that d_S ≥ D₁ and d_I ≥ D₂.

Proof. Let (S(x), I(x)) be any positive solution of model (6) and denote . Then, multiplying the first equation of model (6) by , integrating over Ω, by virtue of Theorem 3, we have that

(26)

where C₁, C₂ depend only on Λ. In a similar manner, we multiply the second equation in model (6) by

to have

(27)

It follows from (26), (27) and the ε-Young inequality that

(28)

where C = C₂ + R₀. It follows from the well-known Poincaré inequality that

(29)

Since d_Iμ₁ > R₀ − 1 from the assumption, we can find a sufficiently small ε₀ such that d_Iμ₁ ≥ R₀ − 1 + εC/2. Finally, by taking D₁ : = (1/μ₁)(C₁ + C/2ε₀) one can conclude that

and

, which asserts our results.

4.2. Existence for Positive Nonconstant Steady States to Model (6)

In this section, we discuss the global existence of nonconstant positive classical solutions to model (6), which guarantees the existence of the stationary patterns [21, 24, 26, 27].

Unless otherwise specified, in this section, we always require that R_d > (ν + R₀ − 1)/R₀ν and R₀ > 1, which guarantees that model (6) has one positive constant steady state E^*. From now on, let us denote

(30)

Let X = {u ∈ [C²(Ω)] ²∣∂u/∂n = 0, x ∈ ∂Ω} and

. Then we write model (6) in the form

(31)

where

(32)

Define a compact operator ℱ : X⁺ → X⁺ by

(33)

where (I − Δ) ⁻¹ is the inverse operator of I − Δ subject to the zero-flux boundary condition. Then u is a positive solution of model (31) if and only if u satisfies

(34)

To apply the index theory, we investigate the eigenvalue of the problem

(35)

where Ψ = (Ψ₁, Ψ₂) ^T and ℱ_u(u^*) = (I − Δ) ⁻¹(I + A) with

(36)

λ is an eigenvalue of (35) if and only if λ is an eigenvalue of the matrix (μ_i + 1) ⁻¹(μ_iI − A) for any i ≥ 0. Therefore, I − ℱ_u(u^*) is invertible if and only if, for any i ≥ 0, the matrix

(37)

is invertible. A straightforward computation yields

(38)

where ρ = (1/R₀)(νR₀R_d − R₀ + 1 − ν)(R₀ − 1) > 0. For the sake of convenience, we denote

(39)

Then H(d_S, d_I, μ) = d_Sd₂det (M_i).

If (d_Sa₄ − d_Ia₁) ² > 4d_Sd₂ρ, then H(d_S, d_I, μ) = 0 has two real roots μ^± given by

(40)

Set B : = B(d_S, d_I) = {μ : μ ≥ 0, μ⁻(d_S, d_I) < μ < μ⁺(d_S, d_I)}, S_p = {μ₀, μ₁, μ₂, …}, and m(μ_i) the multiplicity of μ_i.

To compute index (I − ℱ, u^*), we can assert the following conclusion by Pang and Wang [22].

Lemma 6 (see [22].)Suppose H(d_S, d_I, μ_i) ≠ 0 for all μ_i ∈ S_p. Then

(41)

where

(42)

In particular, if H(d_S, d_I, μ) > 0 for all μ ≥ 0, then σ = 0.

From Lemma 6, we see that to calculate the index of index(I − ℱ, u^*), the key step is to determine the range of μ for which H(d_S, d_I, μ) < 0.

Theorem 7. Assume that R_d > max {1, (ν + R₀ − 1)/R₀ν}. If , for some j ≥ 1, and is odd, then there exists a positive constant d^* such that model (6) has at least one nonconstant solution if d_I > d^*.

Proof. Since , equivalently, a₁ > 0, it follows that if d_I is large enough, then (d_Sa₄ − d_Ia₁) ² > 4d_Sd_Iρ and 0 < μ⁻(d_S, d_I) < μ⁺(d_S, d_I). Furthermore,

(43)

Since a₁/d_S ∈ (μ_j, μ_j+1) for some j ≥ 1, there exists d₀ ≫ 1 such that

(44)

By Theorem 5, we know that there exists d > d₀ such that model (6) with diffusion coefficients d_S = d and d_I ≥ d has no nonconstant solutions. Moreover, we can choose d so large that a₁/d < μ₁. It follows that there exists d^* > d such that

(45)

We shall prove that, for any d_I ≥ d^*, model (6) has at least one nonconstant positive solution. On the contrary, suppose that this assertion is not true for some

. In the following, we will derive a contradiction by using a homotopy argument.

By virtue of Theorems 3 and 4, there exists a positive constant such that the positive solution (S(x), I(x)) of model (6) satisfies C⁻¹ < S, I < C.

Set

(46)

and define

(47)

(48)

where

(49)

It is clear that finding the positive solution of model (31) is equivalent to finding the fixed point of Φ(u, 1) in ℳ. Further, by virtue of the definition of ℳ, we have that Φ(u, θ) = 0 has no fixed point in ∂ℳ for all 0 ≤ θ ≤ 1.

Since Φ(u, t) is compact, the Leray-Schauder topological degree deg (I − Φ(u, θ), ℳ, 0) is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce

(50)

In view of μ⁻ ∈ (μ_i, μ_i+1) and μ⁺ ∈ (μ_j, μ_j+1), we have B(d_S, d_I)∩S_j = {μ_i+1, μ_i+2, …, μ_p}. Clearly, I − Φ(u, 1) = I − ℱ. Thus, if model (6) has no other solutions except the constant one u^*, then Lemma 6 shows that

(51)

On the contrary, by the choice of d and d^*, we have that and u^* is the only fixed point of Φ(u, 0). It therefore follows from Lemma 6 that

(52)

From (50)–(52), we get a contradiction. Therefore, there exists a nonconstant solution of model (6). The proof is completed.

5. Discussion

In this paper, we investigate the disease’s dynamics through studying the existence and nonexistence positive constant steady states of a reaction-diffusion epidemic model. We give a priori estimates for positive solutions to model and show that the nonconstant positive steady states exist due to the emergence of diffusion, which demonstrates that stationary patterns can be found as a result of diffusion. The numerical results about the stationary patterns for model (5) can be found in [20].

On the other hand, there are plenty of papers which focus on the pattern formation of reaction-diffusion population models via standard linear analysis method and numerical simulations. But there is little literature analytically concerning the existence of a stationary patterns via theory and methods of partial differential equations infrequently. The methods and results in the present paper may enrich the research of pattern formation in the spatial epidemic model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous referee for very helpful suggestions and comments which led to improvement of our original paper. This research was supported by the National Science Foundation of China (61373005) and Zhejiang Provincial Natural Science Foundation (LY12A01014).

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The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Abstract

1. Introduction

2. Basic Model

3. A Priori Estimates for Positive Solutions to Model (6)

4. Existence and Nonexistence of Positive Nonconstant Steady States

4.1. Nonexistence for Positive Nonconstant Steady States to Model (6)

4.2. Existence for Positive Nonconstant Steady States to Model (6)

5. Discussion

Conflict of Interests

Acknowledgments

References

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The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Abstract

1. Introduction

2. Basic Model

3. A Priori Estimates for Positive Solutions to Model (6)

4. Existence and Nonexistence of Positive Nonconstant Steady States

4.1. Nonexistence for Positive Nonconstant Steady States to Model (6)

4.2. Existence for Positive Nonconstant Steady States to Model (6)

5. Discussion

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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