Complex dynamics of a diffusive epidemic model with strong Allee effect
Introduction
Since the pioneer work of W.C. Allee [1], there is an ongoing interest in the Allee effect on the dynamics of the population models [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The Allee effect is described by the positive relationship between any component of individual fitness and either numbers or density of conspecifics [3], [4]. The Allee effect can be caused by difficulties in finding mating partners at small densities, genetic inbreeding, demographic stochasticity or a reduction in cooperative interactions, see [2].
The impact of disease can be particularly devastating in populations with a strong Allee effect, since any further reduction might tip the population density below the critical threshold and lead to extinction [8], [9], [10]. And the joint interplay of infectious diseases and Allee effects has been studied extensively in epidemiology and lots of important phenomena have been observed [11], [12], [13], [14], [15].
The aim of this paper is to explore the consequences of the Allee effect on the disease transmission as well as on the spatial spread. Based on the previous insightful studies, we make the following assumptions:
- (H1)
In the absence of disease, the total host population grows according to a logistic law with carrying capacity and an intrinsic birth rate constant :
An Allee limit is incorporated in model (1) such that per capita population growth is negative below and the host deterministically goes toward extinction, and when , the per capita growth rate is positive [3], [7]. As becomes largely relative to , the model approaches the standard logistic model [16]: We thus choose to regard such Allee effect as a “strong Allee effect”, as described by Deredec and Courchamp [12] with a critical threshold.
- (H2)
In the presence of disease, we assume that is split into a susceptible part and an infected part :
- (H3)
Following [17], [18], [19], to keep the model simple, we assume that only susceptible is capable of reproducing with logistic law and strong Allee effect (Eq. (2)), and there is no disease related mortality. However the infected still contributes with to population growth toward the carrying capacity. The infected is removed only by death at rate . There is no recovery from the disease.
- (H4)
The disease transmission is assumed to be standard incidence term and no vertical transmission, i.e., the number of contacts between infected and susceptible individuals is constant [20]. The transmission coefficient is .
From the above assumptions, we can establish the following model:
Model (4) is an extension of the model established by Xiao and Chen [18], [19], which describes the transmission of disease for the special case with mass action incidence term and without Allee effect, i.e., .
For model (4), the basic reproduction number is defined as: then it can be rewritten as:
We note that system (5) is not defined at the point (0,0), but both isoclines pass through this point, and in this case, it is a point of particular interest. In the rest of this paper, we assume that , and the analysis of will be shown later. With this assumption, we see that both and are continuous on the positive cone . Also, we assume that the initial conditions for the above model satisfy .
Suppose that the susceptible and the infectious individuals move randomly in the space—described as Brownian random motion [21], and then we propose a simple spatial model corresponding to (5) as follows:
In the above, indicates the size of the system in the directions of and , respectively, is the outward unit normal vector of the boundary and the homogeneous Neumann boundary condition is being considered. The diffusion coefficients and are positive constants, and the initial data are continuous functions. is the Laplacian operator in two-dimensional space, which describes the Brownian random motion. The diffusion model (e.g., model (6)) provides a useful framework to evaluate some spatially related control measures[22].
And the spatial propagation of diseases has been investigated in the literature mainly by way of traveling wave approaches and by approximating the asymptotic rate of spread [8]. In this paper, we mainly focus on the spatial disease spread with the Turing pattern formation in model (6). Turing in his pioneering work in 1952 [23] has shown that under certain conditions two chemical substances, called the morphogens, can generate a stable inhomogeneous pattern if one of the morphogens, the activator, diffuses slowly and the other, the inhibitor, diffuses much faster. Turing’s revolutionary idea was that passive diffusion could interact with the chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetry so that the system with diffusion can have them [24]. Spatial epidemiology with diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission.
The rest of the paper is organized as follows. In Section 2, we present our main result about the stability and bifurcation analysis of model with the strong Allee effect, including boundedness and nonpersistent of solutions, the existence and the stability of disease-free and endemic equilibria for non-spatial and spatial model. In Section 3, we give the conditions of the Turing instability and determine the Turing space, and by performing a series of simulations, we illustrate the emergence of different patterns. Finally, in Section 4, some conclusions and discussion are given.
Section snippets
Some properties of the solutions
The following lemma is due to Xiao and Chen [18], [19].
Lemma 2.1 Assume that initial condition of model(5) satisfies , then either for all and, therefore, as ; or there exists a such that for all .
If , then for all .
Proof Firstly, we consider for all . From model (5) we can get Hence, for all , we have . Since is monotonously decreased
Turing instability and pattern formation
In this section, we mainly focus on the effects of reaction diffusion on Turing instability (or called Turing bifurcation) and pattern formation for model (6).
Mathematically speaking, an equilibrium is a Turing instability meaning that it is an asymptotically stable equilibrium of model (5) but is unstable with respect to the solutions of spatial model (6). Since , then is always true. Hence has an eigenvalue with a positive real part, then it must be a real value and the
Conclusions and remarks
In this paper, we have studied qualitative behavior of a diffusive epidemic model with the strong Allee effect under the zero-flux boundary conditions. The value of this study lies in three-folds. First, it presents the stability of the equilibria with the strong Allee effect, which indicates that the dynamics of the model with Allee effect are rich and complex. Second, it gives an analysis of Turing instability, which determines the Turing space in the spatial domain. Third, it illustrates the
Acknowledgments
The authors would like to thank the anonymous referee for very helpful suggestions and comments which led to improvements of our original manuscript. This research was supported by the National Science Foundation of China (60970065), Zhejiang Provincial Natural Science Foundation (R1110261 & LY12A01014) and the National Basic Research Program of China (2012CB426510).
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