Onset and termination of oscillation of disease spread through contaminated environment.

We consider a reaction diffusion equation with a delayed nonlocal nonlinearity and subject to Dirichlet boundary condition. The model equation is motivated by infection dynamics of disease spread (avian influenza, for example) through environment contamination, and the nonlinearity takes into account of distribution of limited resources for rapid and slow interventions to clean contaminated environment. We determine conditions under which an equilibrium with positive value in the interior of the domain (disease equilibrium) emerges and determine conditions under which Hope bifurcation occurs. For a fixed pair of rapid and slow response delay, we show that nonlinear oscillations can be avoided by distributing resources for both fast or slow interventions.

1. Introduction. We consider the spread of a disease carried by a biological species and transmitted through contaminated environment. We assume the diseased individuals move randomly in a spatial domain Ω (a smooth open bounded set in a finite dimensional space) following the standard diffusion, and subject to the Dirichlet condition on the boundary ∂Ω as the boundary is not suitable for the diseased individuals to survive (due to disease prevention and control, or due to the natural environmental constraints). We model the situation where the growth of the infection in the biological population is proportional to the number of diseased individuals as the amount of pathogen loads released to the environment is proportional to this number of diseased cases. We further consider the case where a certain amount of resources is available to clean the environment, a portion of the sources can be used to respond to the contamination relatively faster (with a delay given by τ 1 ) and the rest can be used for slower response characterized by another average delay τ 2 > τ 1 . This yields the following model ∂u ∂t = d∆u + ru[1 − a 1 diseased populations. The total environment available for the pathogen contamination is normalized to 1. In the first nonlocal delayed integration, u(y, t − τ 1 ) is the pathogen loads released by the infected individuals at time t − τ 1 and spatial location y and P 1 (x, y) is the probability of the pathogen moved from the spatial location y to current location x. A certain biosafety intervention measure is implemented, in proportion to the pathogen loads Ω P 1 (x, y)u(y, t − τ 1 )dy, but with a time lag τ 1 . Similar interpretations apply to the second integration, but with a longer delay τ 2 . The constants a 1 and a 2 satisfy a 1 + a 2 = 1, where a 1 ∈ [0, 1] represents the allocation of resources to be allocated to implement the intervention measure for either rapid or slow response to protect the environment from being used to be contaminated to spread the disease back to the biological species under consideration. The kernel function are relevant to the mobility of the virus and this can be derived in a similar fashion as in [14]. Note that we assume the time for the biosafety intervention is much slower than the virus spread in the environment, and hence the delay in the spread process is ignored. ∆ stands for the Laplacian operator, with following Dirichlet boundary condition u(x, t) = 0, x ∈ ∂Ω and t ∈ (0, +∞) which implies that the exterior environment is hostile and the species cannot move across the boundary of environment, and initial condition satisfies where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω, τ = max(τ 1 , τ 2 ), η ∈ C := C([−τ, 0], Y ) and Y = L 2 (Ω). This study is motivated by the spread of avian influenza, an infectious disease of birds that is caused by influenza virus type A strains. The involvement of different bird species and their interactions with environments together lead to complex transmission pathways which include birds to birds, birds to mammals, birds to human, birds to insects, human to human, and environment to birds/mammals/human and vice-versa [9]. How to model the interplay of different transmission pathways and its impact on the spread of avian influenza imposes significant challenge [1][11] [16]. In the study of Wang et al. [17], a system of reaction diffusion equations on unbounded domains was proposed to establish the existence and nonexistence of traveling wave solutions of a reaction-convection epidemic model for the spatial spread of avian influenza involving a wide range of bird species and environmental contamination. In the earlier studies of Gourley et al. [6], the role of migrating birds were examined using partial differential equations and their reduction to delay differential systems. Here we focus on the spread of avian influenza among the wild birds, where the virus is shredded into the environment, through which the virus further spreads and infects other wild birds coming to contact with contaminated environment. The parameter r represents the intrinsic susceptibility and transmissibility of the environment, which can be reduced through biosafety intervention so the nonlinearity in the kinetic equation for the infected individuals resembles the classical delayed non-local logistic equations. Note that a large portion of the environment for the virus spread and contamination involves water, the kernel functions P 1 and P 2 for the virus spread in the environment can involve both diffusion and convercation.
Our goal in this paper is to 1). Determine whether there is a critical value of r above which the disease will persist in the population in the form of a nonnegative non-trivial equilibrium (note this is necessarily spatially varying due to the Dirichlet condition); 2). Identify critical value of the rapid response delay where the nontrivial equilibrium remains locally stable when all resources are committed for the rapid biosafety intervention; 3). Identify critical value of the slow response delay where the nontrivial equilibrium loses its locally stability even when all resources are committed for the slow biosafety intervention; 4). Identify the critical resource allocation parameter α when a Hopf bifurcation takes place from the nontrivial equilibrium, in this case we examine the patterns of bifurcated periodic solutions to examine impact of parameters on the peak and frequency of the spatiotemporally varying stable patterns.
We use r * to denote the principal eigenvalue of the following one-dimensional eigenvalue problem and φ is the corresponding eigenfunction of r * with φ(x) > 0 for x ∈ Ω. The following notations are needed. Let L p (Ω) (p ≥ 1) be the space consisting of measurable functions on Ω that are p-integrable, and H k (Ω) (k ≥ 0) be the space consisting of functions whose k-th order weak derivatives belong to L 2 (Ω). Denote the s- For any real-valued vector space Z, we also denote the complexification of Z to be Z C := Z ⊕ iZ = {x 1 + ix 2 |x 1 , x 2 ∈ Z}. For a linear operator L : Z 1 → Z 2 , we denote the domain of L by D(L), the null space by N (L) and the range of L by R(L). For the complex-valued Hilbert space Y C , the standard inner product is < u, v >= Ωū (x)v(x)dx. In what follows, we assume the comparability condition between the kernel functions P i (x, y), i = 1, 2 and the eigenfunction φ. Namely, we assume Ω Ω (P 1 (x, y) + P 2 (x, y))φ(y)φ(x)dydx = 0. This is because one of the kernel functions can be zero.

2.
Existence of steady state solution. The positive steady state solutions of (1) satisfy the following equation: Let N (d∆ + r * ) and R(d∆ + r * ) be the null space and the range of the operator d∆ + r * , then N (d∆ + r * ) = span{φ}, Then we have the following decompositions: Then we have the following result on positive steady state solution of model (1).
In what follows, we always assume that r ∈ (r * , r * ] and r * − r * 1. 3. Eigenvalue analysis. It is easy to see that the linearized equation of the model (1) at the steady state solution u r can be written as where η ∈ C. Define a operator A r : From [13], A r is an infinitesimal generator of a strong continuous semigroup and A r is also self-adjoint. Then the study of the stability of u r is transferred to the analysis of the following eigenvalue problem where ψ ∈ X C \{0}, i.e., the study of the following spectral set where A τ1τ2,r is the infinitesimal generator of the semigroup induced by the solutions of equation (3) with A τ1τ2,r ψ =ψ, and is solvable for some ω > 0 and ψ ∈ X C \{0}.
Next, we discuss the effects of two nonlocal delays on the stability at the positive steady state solution u r in four different cases. Case 1. τ 1 = 0 and τ 2 > 0.
Case 4. τ 1 > 0 and τ 2 ∈ (0, τ 20 ). Assume that there exit iω τ2τ1 (ω τ2τ1 > 0) and ψ τ2τ1 ∈ X C \{0} such that By the similar analysis as in the Case 3, the following result can be obtained: r−r * is uniformly bounded for r ∈ (r * , r * ]. The equivalent equation to (5) is G τ2τ1 = (g 1 (z τ2τ1 , β τ2τ1 , k τ2τ1 , r), g 2 (z τ2τ1 , β τ2τ1 , k τ2τ1 , r)) = 0, which has the same form as G τ1τ2 . Moreover, G τ2τ1 = 0 at r = r * if the following equations are satisfied Since stability analysis is similar for the above four cases, we will only discuss Case 3. For other cases, we omit them in this paper.
7. Discussion. Here we interpreted the classical logistic model with two non-local delayed terms in the framework of avian influenza spread between wild birds and the environment-the environment is contaminated by infected birds and the contaminated environment then pass on the pathogen to other susceptible birds. Due to the random movement of the infected birds and pathogens, the disease spreads in the geographical domain and pathogen loads in any given spatial location are not just the consequence of local contamination. Here we consider the case where resources are available for cleaning the environment. These resources can be used to launch either rapid or slow environment cleaning interventions, but the resources are limited so optimal allocations will be needed. Our study shows that disease outbreak in the form of a nontrivial equilibrium is possible assuming the intrinsic reproduction number is sufficiently large, and nonlinear oscillations around this nontrivial equilibrium can take place. Our analysis and simulations show that to prevent this oscillation, the resources should be distributed for both rapid and slow responses, focusing on either rapid or slow response will require the slow response to be also very rapid. For example, in Figure 3, if we normalized the delay so that the rapid response takes place with τ 1 = 1, then the critical value for nonlinear oscillation (τ 20 ) to take place can be large, and close to 5 when α is close to 0.5.
In recent years, reaction-diffusion equations with time delay have been investigated extensively. Su et al. [15] studied a diffusive logistic equation with mixed delayed and instantaneous density dependence, with some interesting results on global continuation of Hopf bifurcation branches. Hu and Yuan [10] proposed a coupled system of reaction-diffusion system with distributed delay and studied stability of the positive steady state solution and the occurrence of Hopf bifurcation. The Hopf bifurcation was also considered in Ma [12] for a coupled reaction-diffusion systems involving three interacting species. The earlier work introducing nonlocal terms into