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Influence of cross-diffusion on the fecally-orally epidemic model with spatial heterogeneity
- Jing Ge^{1, 2},
- Zhigui Lin^{2}Email authorView ORCID ID profile and
- Qunying Zhang^{2}
https://doi.org/10.1186/s13662-017-1422-9
© The Author(s) 2017
Received: 30 August 2017
Accepted: 8 November 2017
Published: 28 November 2017
Abstract
A strongly coupled cooperative parabolic system, which describes fecally-orally epidemic model with cross-diffusion in a heterogeneous environment, was formulated and analyzed. The basic reproduction number \(R_{0} ^{D}\), which serves as a threshold parameter that predicts whether the coexistence will exist or not, is introduced by the next infection operator and the related eigenvalue problems. By applying upper and lower solutions method, we present the sufficient conditions for the existence of the coexistence solution. The true positive solutions can also be obtained by monotone iterative method. Our results imply that the fecally-orally epidemic model with cross-diffusion admits at least one coexistence solution when the basic reproduction number exceeds one and the cross-diffusion coefficient is sufficiently small, while no coexistence exists when the basic reproduction number is smaller than one or the cross-diffusion coefficient is large enough. Finally, some numerical simulations are exhibited to confirm our analytical findings.
Keywords
- fecally-orally epidemic model
- eigenvalue problem
- basic reproduction number
- cross-diffusion
MSC
- 35J47
- 92D30
1 Introduction
To understand the dynamics of solutions to problem (1.1) and its corresponding Cauchy problem, traveling waves solutions were studied in [3, 4], and the entire solutions (that is, solutions defined for all times \(t\in \mathbb {R}\) and for all points \(x\in \mathbb {R}\)) was established in [5, 6] and the spreading fronts of an infective environment was given in [7] by considering a free boundary problem.
On the other hand, in ecology, different concentration levels of species can affect the diffusive direction of another interacting species, which is called cross-diffusion [10]. This is also another hot issue and attracts much attention in recent years; see [11–16] and the references therein. Combined the spatial heterogeneity and cross-diffusion, the corresponding ecosystem can induce more complicated dynamical behaviors [17, 18].
- (H1)
\(g\in C^{1}([0, \infty)), g(0)=0, g'(z)>0, \forall z\geq0\);
- (H2)
\(\frac{g(z)}{z}\) is decreasing and \(\limsup_{z\rightarrow\infty}\frac{g(z)}{z} <\min_{\overline{\Omega}}\{{\frac{a_{11}(x)}{a_{12}(x)}}\}\cdot \min_{\overline{\Omega}}a_{22}(x)\).
The rest of this paper is organized as follows. Section 2 is devoted to introducing the basic reproduction number of epidemic model (1.3) by using the next infection operator and corresponding eigenvalue problem. The sufficient condition for the existence and non-existence of the coexistence solution are presented in Section 2. Finally, some numerical simulations which confirm our analytical findings, as well as a brief discussion, are present in Section 4.
2 The basic reproduction numbers
Theorem 2.1
Proof
Remark 2.1
It is well known that \(\lambda_{0}\) is monotonically decreasing with respect to \(a_{12}(x)\), and we deduce from Theorem 2.1 that \(R^{D}_{0}\) is monotonically increasing with respect to \(a_{12}(x)\), thus \(R^{D}_{0}>1\) if \(a_{12}(x)\) is sufficiently large.
When all coefficients in problem (1.3) are constants, we shall present an explicit formula for \(R_{0}^{D}\), which is in line with the basic reproduction number for the corresponding fecally-orally epidemic model with homogeneous boundary condition in a fixed region [22, 25].
Theorem 2.2
Proof
3 Coexistence
Definition 3.1
Theorem 3.1
Assume that \(R_{0}^{D}>1\), and \(\frac{\beta_{1}}{\gamma_{1}^{2}}\), \(\frac {\beta_{2}}{\gamma_{2}^{2}}\) are sufficiently small, then problem (1.3) admits at least one coexistence solution \((u(x), v(x))\).
Proof
To verify the existence of a positive solution to problem (1.4), it suffices to find a pair of upper and lower solutions to problem (1.4). We seek such as in the form \((\tilde{u}, \tilde{v})=(M_{1}, M_{2})\), \((\hat{u}, \hat{v})= (g_{1}(\delta(d_{1}+\frac{\beta_{1}}{\gamma_{1}})\phi, \delta(d_{2}+\frac {\beta_{2}}{\gamma_{2}})\psi), g_{2}(\delta(d_{1}+\frac{\beta _{1}}{\gamma_{1}})\phi, \delta(d_{2}+\frac{\beta_{2}}{\gamma_{2}})\psi ))\), where \(M_{i}\) \((i=1,2)\) and δ are some positive constants with δ small enough, \((\phi, \psi)\equiv(\phi(x),\psi(x))\) is (normalized) positive eigenfunction corresponding to \(\lambda_{0}\), and \(\lambda_{0}\) is the principal eigenvalue of the linear eigenvalue problem (2.5).
On one hand, due to \(\limsup_{z\rightarrow\infty}\frac{g(z)}{z} <\min_{\overline{\Omega}}\{{\frac{a_{11}(x)}{a_{12}(x)}}\}\cdot\min_{\overline{\Omega}}\{a_{22}(x)\}\), there exists constant \(M_{0}\) such that \(\frac{g(z)}{z}<\min_{\overline {\Omega}}\{{\frac{a_{11}(x)}{a_{12}(x)}}\}\cdot\min_{\overline{\Omega }}\{a_{22}(x)\}\) for \(z\geq M_{0}\). As a result, the first two inequalities in (3.5) will hold if we set \((\tilde{u}, \tilde {v})=(M_{1},M_{2})\), where \(M_{1}=\max\{M_{0}, \max_{\overline{\Omega }}u(x,0), \max_{\overline{\Omega}}v(x,0)\}\), \(M_{2}=M_{1}\cdot\max_{\overline{\Omega}} \{{\frac{a_{12}(x)}{a_{11}(x)}}\}\).
Consequently, the pair \((\tilde{u},\tilde{v})=(M_{1},M_{2})\), \((\hat{u},\hat{v})= (g_{1}(\delta(d_{1}+\frac{\beta_{1}}{\gamma_{1}})\phi, \delta (d_{2}+\frac{\beta_{2}}{\gamma_{2}})\psi), g_{2}(\delta(d_{1}+\frac{\beta _{1}}{\gamma_{1}})\phi, \delta(d_{2}+\frac{\beta_{2}}{\gamma_{2}})\psi ))\) are ordered upper and lower solutions to problem (1.4), respectively. Applying Theorem 2.1 of [16] leads to the existence of the coexistence solution to problem (1.4). □
In what follows, we will present the non-existence result of any coexistence steady state.
Theorem 3.2
If \(R_{0}^{D}<1\), problem (1.3) has no positive steady-state solution.
Proof
Remark 3.1
Assume that all coefficients of (1.3) are spatially independent. \(R^{D}_{0}\) is represented by (2.7). If the cross-diffusion coefficients, i.e., \(\frac{\beta_{1}}{\gamma_{1}}\) or \(\frac{\beta_{2}}{\gamma_{2}}\) is large enough, then no coexistence solution to problem (1.3) exists.
The above derivations lead to the following theorem.
Theorem 3.3
4 Numerical simulation and discussion
In present paper, we proposed and studied a cross-diffusion fecally-orally epidemic model where we had considered the coexistence solution to problem (1.3) in a spatially heterogeneous environment. Firstly, we introduced the basic reproduction number \(R_{0}^{D}\) via the next generation operator and associated linear eigenvalue problem, and we further proved that \(R_{0}^{D}\) served as a threshold parameter which predicts whether the coexistence exists or not. To be more precise, when all coefficients are constants, we provided an explicit formula for \(R_{0}^{D}\). The coexistence of problem (1.3) is investigated by combining the eigenvalue problem and monotone iterative schemes when \(R_{0}^{D}>1\), while if \(R_{0}^{D}<1\), problem (1.3) has no coexistence solution. We proved that large cross-diffusion will result in non-existence of the coexistence (Theorem 3.2), whereas coexistence is possible if the cross-diffusion coefficients are small (Theorem 3.1). As far as we know, on the propagation of the species, large cross-diffusion will result in more complex dynamic behavior in ecology, for example, large cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system [27]; see also [19, 28] and the references therein. Our results show that cross-diffusion has also a significant impact on the coexistence of the epidemic model.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant Nos.11771381, 11701206 and 11501494).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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