Complex dynamics of a diffusive epidemic model with strong Allee effect

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Abstract

In this paper, we investigate the dynamics of a diffusive epidemic model with strong Allee effect in the susceptible population. We show some properties of solutions of the model, the asymptotic stability of the equilibria. Especially, we show that there exists a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spot–stripe mixtures and stripes patterns.

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 N(t) grows according to a logistic law with carrying capacity K(K>0) and an intrinsic birth rate constant r(r>0): dNdt=rN(1NK).

    An Allee limit (m) is incorporated in model (1) such that per capita population growth is negative below m and the host N deterministically goes toward extinction, and when 0<m<N<K, the per capita growth rate is positive [3], [7]. As N becomes largely relative to m, the model approaches the standard logistic model [16]: dNdt=rN(1NK)(1mN). 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 N is split into a susceptible part S and an infected part I: N(t)=S(t)+I(t).

  • (H3)

    Following [17], [18], [19], to keep the model simple, we assume that only susceptible S is capable of reproducing with logistic law and strong Allee effect (Eq. (2)), and there is no disease related mortality. However the infected I still contributes with S to population growth toward the carrying capacity. The infected I is removed only by death at rate α. There is no recovery from the disease.

  • (H4)

    The disease transmission is assumed to be standard incidence term βSIS+I and no vertical transmission, i.e., the number of contacts between infected and susceptible individuals is constant [20]. The transmission coefficient is β>0.

From the above assumptions, we can establish the following model: {dSdt=rS(1S+IK)(1mS+I)βSIS+I,dIdt=βSIS+IαI.

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 βSI and without Allee effect, i.e., m=0.

For model (4), the basic reproduction number is defined as: R0=βα, then it can be rewritten as: {dSdt=rS(1S+IK)(1mS+I)αR0SIS+If1(S,I),dIdt=αI(R0SS+I1)f2(S,I).

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 f1(0,0)=f2(0,0)=0, and the analysis of E0=(0,0) will be shown later. With this assumption, we see that both f1(S,I) and f2(S,I) are continuous on the positive cone R+2={(S,I):S0,I0}. Also, we assume that the initial conditions for the above model satisfy S(0)>0,I(0)>0.

Suppose that the susceptible (S) and the infectious individuals (I) move randomly in the space—described as Brownian random motion [21], and then we propose a simple spatial model corresponding to (5) as follows: {Sdt=rS(1S+IK)(1mS+I)αR0SIS+I+d12S,Idt=α(R0SIS+II)+d22I,Sn=In=0,r=(x,y)Ω,S(r,0)=S0(r)>0,I(r,0)=I0(r)>0,rΩ=[0,L]×[0,L].

In the above, L indicates the size of the system in the directions of x and y, respectively, n is the outward unit normal vector of the boundary Ω and the homogeneous Neumann boundary condition is being considered. The diffusion coefficients d1 and d2 are positive constants, and the initial data S0(r),I0(r) are continuous functions. 2=2x2+2y2 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

  • (i)

    Assume that initial condition of model(5) satisfies S(0)+I(0)K, then either

    • (a)

      S(t)+I(t)K for all t0 and, therefore, (S(t),I(t))(K,0) as t ;

      or

    • (b)

      there exists a t1>0 such that S(t)+I(t)K for all t>t1.

  • (ii)

    If S(0)+I(0)K, then S(t)+I(t)<K for all t0.

Proof

Firstly, we consider S(t)+I(t)K for all t0. From model (5) we can get d(S(t)+I(t))dt=rS(1S+IK)(1mS+I)αI. Hence, for all t0, we have d(S(t)+I(t))dt0. Since S(t)+I(t) 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 tr(JEe2)<0, then tr(Ai)<0 is always true. Hence Ai 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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