SINGULAR RENORMALIZATION GROUP APPROACH TO SIS PROBLEMS

. In this paper, we consider the boundary value problems of a one-dimensional steady-state SIS epidemic reaction-diﬀusion-advection system in the following two cases: (i) the advection rate is relatively large comparing to the diﬀusion rates of infected and susceptible populations; (ii) the diﬀusion rate of the susceptible population approaches zero. By introducing a singular parameter, the system can be viewed as a singularly perturbed problem. By the renormalization group method, we construct the ﬁrst-order approximate solutions and obtain error estimates.


1.
Introduction. In recent years, it has been commonly recognized that environmental heterogeneity and individual motility are two significant factors that should be considered in studying disease dynamics. In order to capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, there are a lot of works have been devoted into this investigation [1,4,11,16,17,25,26,27,28]. In [1], Allen et al. proposed a frequencydependent Susceptible-Infected-Susceptible (SIS) reaction-diffusion model for a population inhabiting a continuous spatial habitat. Peng and Yi [25,27] considered the effects of large and small diffusion rates of the susceptible and infected individuals on the persistence and extinction of the epidemic disease. However, in some heterogeneous environments, the external environmental forces, such as wind [10] and water flow [19,20,21,22], could be added into the existing reaction-diffusion models as an advection term, which is significant to predict the patterns of disease occurrence and design optimal control strategies. Cui and Lou [9] considered the effects of diffusion and advection on asymptotic profiles of steady states of an epidemic reaction-diffusion-advection model.

NING SUN, SHAOYUN SHI AND WENLEI LI
We consider the following parameterized second order boundary valued problem: where S(x) and I(x) denote the density of susceptible and infected individuals at equilibrium at x ∈ [0, L], respectively. (1) can be, in fact, viewed as the equilibrium equations of the one dimensional SIS reaction-diffusion-advection model [9]: whereS(x, t) andĪ(x, t) respectively denote the density of susceptible and infected individuals at time t and position x in the interval [0, L], the positive constants d S and d I are diffusion rates for the susceptible and infected populations, the positive constant q presents the advection rate of a stream or wind which carries the susceptible and infected populations from the upstream x = 0 to the downstream x = L. The functions β(x) and γ(x) are assumed to be positive Hölder continuous functions on [0, L] which represent the rates of disease transmission and recovery at x, respectively. The no-flux boundary conditions indicate there is no population flux across the upstream and downstream ends, so that both susceptible and infected populations live in a closed environment. Summing two equations of (2) and integrating over (0, L) gives which means the total population size is constant in time, i.e., and hence It is well-known that, from the ecological point of view, only solutions (S(x), I(x)) satisfying S(x) ≥ 0 and I(x) ≥ 0 on [0, L] are of interest. Therefore, we only concern the non-negative solution of (1) and (H) with I(x) > 0 for some x ∈ (0, L), we call it endemic equilibrium(EE). Let Cui, Lam and Lou [8] proved that R 0 > 1 is a necessary and sufficient condition for the existence of EE, furthermore, they studied the qualitative properties of EE of (1) and (H) when diffusion and advection rates d S , d I and q vary in the following two cases: • β(L) > γ(L) and d I q is sufficiently small; • β(L) < γ(L) and d S approaches zero with R 0 > 1.
In the first case, they obtained an exponential decay result by constructing the upper solutions and lower solutions, and showed that two populations persist and concentrate at the downstream end. In the second case, they proved susceptible individuals concentrate at the downstream end and I(x) decays exponentially for positive advection rate. Similar results were also obtained by Kuto, Matsuzawa and Peng [15].
The main purpose of this paper is to provide a new perspective to understand system (1) in the above two cases. In fact, it is easily found that the steady state system (1) can be treated as a singularly perturbed system by introducing a new singular parameter. Therefore, we can study system (1) from the point of singular perturbation. Our main idea is the so called renormalization group (RG) method developed by Chen, Goldenfeld and Oono [5], which has been showed to be effective for singular perturbation systems, and been applied successfully to many fields, such as quantum kinetics [3], invariant manifold [14,6,7], ODEs [30,12,29,18], PDEs [23,24] and SDEs [2,13]. Ziane [30] considered one-order autonomous initial value problems and proved that the approximation results obtained by RG method are valid over long time intervals. Chiba [6,7] treated differential equations on manifolds and proved that RG method could provide approximate vector fields and invariant manifolds as well as approximate solutions. Zhou et al. [29] presented a new formulation of the RG method, as well as applications to second-order boundary layer problems and to boundary layer problems with delay. In [13], the authors investigated a class of stochastic differential equations and proved that the approximate solutions constructed by RG method remain valid with high probability on large time scales. Here, we will firstly present a RG strategy for a kind of integrodifferential perturbed problem, and then use the obtained RG formula to deal with (1) and (H), construct the approximate solution of EE, and give the estimate of the error of the exact EE and the approximate EE and make a comparison with the previous results.
The rest of the paper is organized as follows. In section 2, we give the RG formula for a class of intergo-differential equations. In section 3, we rewrite the system (1) as the form introduced in section 2 by introducing new parameters and variables. Then we apply RG method to construct the first-order approximate solutions and obtain error estimates for the actual solution and the approximate solutions in two cases respectively. The last section is a short conclusion.
2. RG formula. In order to use RG method to study system (1) systematically, we firstly consider the following intergo-differential equations where ε > 0 is a positive small parameter, denotes the derivative to τ , A is a nonsingular n × n matrix, which is assumed for simplicity to be a diagonalizable matrix, andF (u) is a nonlinear vector valued functions on R n . Without loss of generality, we may suppose A is a diagonal matrix.

NING SUN, SHAOYUN SHI AND WENLEI LI
Now we construct an approximate solution of (4) with initial condition u(0).
Step 1. The naive expansion. Substituting the naive perturbation expansion into equations (4) yields here F (u 0 ) = A −1F (u 0 ). Solve equations (5) as u 0 (τ ) = e −τ A a, where a is an arbitrary constant vector. Taking u 0 (τ ) into equations (6) and solving it with u 1 (0) = 0, we obtain AF (e −tA a)dsdt, which becomes into F (e −tA a)dt, by using integration by parts. Thus, the naive expansion to first order is F (e −tA a)dt.
Step 2. The introduction of a free parameter. The solution u 1 (τ ) can be split into two parts. The first is the resonant part, generated in our case by the time-constant part of the integrand e tA F (e −tA a) and F (e −tA a), which is called secular term. The secular term is proportional to τ , causing the asymptotic property of (7) to be lost on time scales of O( 1 ε ). The second is the non-resonant part corresponding to the time-dependent part of the integrand e tA F (e −tA a) and F (e −tA a). Thus, we decompose Taking (8) into (7) leads tõ In order to remove the secular term to get the effective asymptotic solution, the key step of RG method is to introduce a free parameter ζ to split τ as τ − ζ + ζ, i.e., Differentiating the renormalized expansion (10) with respect to ζ, we derive the following renormalization group equations up to O(ε) Suppose now that we can solve RG equation (11) as (10), we obtain the approximate solution of (4)ũ 3. Main results. In this section, we construct the approximation for EE of boundary value problem(BVP) (1) for two different parameter settings and investigate the asymptotic profiles of EE respectively. For simplicity of presentation, we rewrite BVP (1) as: where f (x, S, I) = β(x) SI S+I − γ(x)I. It was shown in [8] that if R 0 > 1, there exists at least one EE. Moreover, if d S = d I , the EE is unique.
In this paper, we define for matrix A = (a ij ) n×m , and denote the n-order diagonal matrix with diagonal elements λ 1 , · · · , λ n by diag(λ 1 , · · · , λ n ). And for ease of notations, we set σ = max 3.1. The asymptotic profiles of EE when d I q 1 and β(L) > γ(L). In this subsection, we investigate the asymptotic profiles of EE when d I q is sufficiently small and the downstream site is high risk, i.e., d I q 1 and β(L) > γ(L), which ensures the existence of EE by the above analysis.
There exists a positive constant K 0 such that if ε qη is sufficiently small, then Furthermore, when ε → 0, u(0) = (S(0), I(0)) T satisfies Case 1. 0 < η < 1 Following the same RG procedure mentioned in Section 2, we obtain the naive expansion of (17) up to order O(ε) where a = (a 1 , a 2 ) T is a constant vector with a 1 , a 2 > 0 (since we only focus on the positive solutions of (17)), and As in Section 2, we obtain the following ansatz approximate EE of (17) where V (τ ) = e εRτ u(0). A straightforward calculation gives where DG(τ, V (τ )) denotes the Jacobian matrix of G(τ, V (τ )) with respect to V . To obtain the error between the exact EE and the approximate EE (24), we need the following lemma.
In the rest of the article, we will use the same notations as in Case 1 for simplicity.
Case 2. η > 1 Adopting the same procedure as in Case 1, we obtain the ansatz approximate solutionũ where V (τ ) = e εRτ u(0), R = − 1 q diag(β(L), γ(L)) and , Using the same method as in the proof of Proposition 1 and Proposition 2 yields the following results.
Theorem 3.1. Assume that β(L) > γ(L), d I q is sufficiently small and d S σ q 2 < 1, then any EE (S e (x), I e (x)) of (14)- (16) with (H) satisfies for 0 < d I d S < 1, (S,Ĩ) T is given by (24) and K = ; for d I d S = 1, (S,Ĩ) T is given by (41) and K = d I q 2 K 5 , and δ = min{1, d I d S }. Remark 3. In this subsection, we proved the error of the exact EE and the approximate EE is exponentially small, which is equivalent to the Theorem 1.2 in [8]. Moreover, the approximate EE is up to and including O(ε), which extends the results of [8].

The asymptotic profiles of EE when d S
1, β(L) < γ(L) and R 0 > 1. In this subsection, we study the asymptotic profiles of EE when d S approaches zero and the downstream site is low risk, i.e., d S 1 and β(L) < γ(L), under the assumption R 0 > 1 which ensures the existence of EE.
The following lemma can be found in [8].
Taking (44) and (H) into consideration yields the following result.
Remark 5. By Remark 3.4, we can obtain that Theorem 3.2 leads to I(x) ≤ K 7 e − κ ε for some appropriate positive constant K 7 . Furthermore, since we have constructed the approximate solutionS(x) and obtained the error estimate between S(x) andS(x), then we can adopt the same method as in the proof of (64) to obtain that for x ∈ [0, L], 4. Conclusion. In this paper, we studied the SIS reaction-diffusion-advection model in a new perspective of singular perturbation theory by using RG method, which has not been done in previous studies to the best of our knowledge. In order to deal with system (1) and (H) with RG method, we introduced an intergo-differential equations using one boundary value condition, and applied RG method to construct an approximate solution of the intergo-differential equations. Then we proved that the approximate solution we constructed satisfying another boundary value condition is the approximate solution of (1) and (H) under some conditions. Our results extend the applicability of RG method, which is of great significance to study other BVPs. Furthermore, we gave the error of the exact solution and the approximate solution we constructed, and compared our results with those in [8], which illustrates the effectiveness of our results.