A network model of immigration: Enclave formation vs. cultural integration

1.   Introduction

In this paper, we are concerned with the following susceptible-infected-susceptible(SIS) model

Here $ S(x, t) $ and $ I(x, t) $ denote the density of susceptible and infected individuals in a given spatial interval $ (0, L) $, $ d_S $ and $ d_I $ are positive constants which stand for the diffusion coefficients for the susceptible and infected populations, $ a'(x) $ is a smooth nonnegative function which represents the advection speed rate, while $ \beta(x) $ and $ \gamma(x) $ represent the rates of disease transmission and recovery at location x, which are Hölder continuous functions on $ (0, L) $. In addition, $ S_0(x) $ and $ I_0(x) $ are continuous and satisfy

We would like to give the survey of some results on SIS model. In ^[1], Allen et al. investigated a discrete SIS model, in ^[2], they also proposed the SIS model with no advection in a given spatial region $ \Omega $, where they dealt with the existence, uniqueness and asymptotic behaviors of the endemic equilibrium as the diffusion rate of the susceptible individuals approaches to zero. Many authors also considered the SIS reaction–diffusion model, including the global stability of the endemic equilibrium, the effects of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, discuss how the disease vanish or spreading in high-risk or low-risk domain, and so on. For the dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, we can see^[3]. For A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, we can see ^[4]. For Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, we can see^[5], For Concentration profile of endemic equilibrium of a reaction- diffusion-advection SIS epidemic model, we can see ^[6]. For the varying total population enhances disease persistence, we can see ^[7]; For the asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, we can see ^[8]. For the global stability of the steady states of an SIS epidemic reaction-diffusion model, we can see ^[9]. For the asymptotic profile of the positive steady state for an SIS epidemic reaction- diffusion model: effects of epidemic risk and population movement, we can see ^[10]; For reaction-diffusion SIS epidemic model in a time-periodic environment, we can see ^[11]. For the global dynamics and traveling waves for a periodic and diffusive chemostat model with two nutrients and one microorganism, we can see ^[12]. For more information about dynamical systems in population biology, we also can refer to see ^[13] and the references therein. Recently, Cui and Lou studied (1.1) when $ a'(x)\equiv q $ for $ x\in [0, L] $ in ^[14], that is, it is a constant advection. Besides establishing the asymptotic stability of the unique disease-free equilibrium(DFE) when $ \mathcal{R}_0 < 1 $ and the existence of the endemic equilibrium when $ \mathcal{R}_0 > 1 $, they found that the DFE changes its stability at most once as $ d_I $ varies from zero to infinity, which is strong contrast with the case of no advection. Since (1.1) has vary advection, an natural and interesting question is whether we can establish the similar results on (1.1) to those in the case of no advection or not.

Since the functions $ a'(x) $, $ \beta(x) $, $ \gamma(x) $, $ S_0(x) $ and $ I_0(x) $ are continuous in $ (0, L) $, by the standard theory for a system of semilinear parabolic equations, (1.1) is locally wellposedness in $ (0, T_{\max}) $. Noticing (A1), by the maximum principle, $ S(x, t) $ and $ I(x, t) $ are positive and bounded for $ x\in [0, L] $ and $ t\in (0, T_{\max}) $. Hence, by the results in ^[15], $ T_{\max} = \infty $ and (1.1) posses a unique classical solution $ (S(x, t), I(x, t)) $ for all time.

It is easy to verify that

Inspired by ^[2] and ^[14], we say that $ (0, L) $ is a low-risk domain if $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $ and high-risk domain if $ \int_0^L\beta(x)dx > \int_0^L\gamma(x)dx $.

The corresponding equilibrium system of (1.1) is

The half trivial solution $ (\tilde{S}(x), 0) $ of (1.3) is called a disease-free equilibrium(DFE), while the solution $ (\tilde{S}(x), \tilde{I}(x)) $ of (1.3) is called endemic equilibrium(EE) if $ \tilde{I}(x) > 0 $ for some $ x\in (0, L) $.

We also introduce the following basic reproduction number as those in literatures ^[2] and ^[14]. We also can refer to ^[16] and see the definition and the computation of the basic reproduction ratio $ R_0 $ in models for infectious diseases in heterogeneous populations, refer to ^[17] and see reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, see basic reproduction numbers for reaction-diffusion epidemic models ^[18].

Our first result is concerned with the qualitative properties for $ \mathcal{R}_0 $.

Theorem 1.1. Let $ \hat{\mathcal{R}}_0 $ be the basic reproduction number when $ a(x)\equiv 0 $ which was introduced in ^[2]. Then the following conclusions hold.

(1) For any given $ a'(x) > 0 $, $ \mathcal{R}_0\rightarrow \frac{\beta(L)}{\gamma(L)} $ as $ d_I\rightarrow 0 $ and $ \mathcal{R}_0\rightarrow \frac{\int_0^L\beta(x)dx}{\int_0^L\gamma(x)dx} $ as $ d_I\rightarrow +\infty $;

(2) For any given $ d_I > 0 $, $ \mathcal{R}_0\rightarrow \hat{\mathcal{R}}_0 $ as $ \max_{x\in [0, L]}a'(x)\rightarrow 0 $ and $ \mathcal{R}_0\rightarrow \frac{\beta(L)}{\gamma(L)} $ as $ \min_{x\in [0, L]}a'(x)\rightarrow +\infty $;

(3) If $ \beta(x) > (<) \gamma(x) $ on $ [0, L] $, then $ \mathcal{R}_0 > (<1) $ for any given $ d_I > 0 $ and $ a'(x) > 0 $.

Our second result deals with the stability of DFE, which will extend those of ^[2] and ^[14].

Theorem 1.2. The DFE is unstable if $ \mathcal{R}_0 > 1 $ while it is globally asymptotically stable if $ \mathcal{R}_0 < 1 $.

We will analyze (1.1) under the following assumptions on $ \beta(x) $ and $ \gamma(x) $:

(C1) $ \beta(0)-\gamma(0) < 0 < \beta(L)-\gamma(L) $, i.e., $ \beta(x)-\gamma(x) $ changes sign from negative to positive,

or

(C2) $ \beta(0)-\gamma(0) > 0 > \beta(L)-\gamma(L) $, i.e., $ \beta(x)-\gamma(x) $ changes sign from positive to negative.

In the point view of biological,

(C1) all lower-risk sites are located at the upstream and all high-risk sites are at the downstream,

or

(C2) all high-risk sites are distributed at the upstream and lower-risk sites are at the downstream.

To state other results, in convenience, let $ q = \max_{x\in [0, L]}a'(x) $ and denote $ a(x) = q\tilde{a}(x) $ sometimes in the sequels.

We can get further properties of $ \mathcal{R}_0 $ when $ \int_0^L \beta(x)dx > \int_0^L \gamma(x)dx $.

Theorem 1.3. Assume that $ \int_0^L \beta(x)dx > \int_0^L \gamma(x)dx $. Denote $ \mathcal{R}_0 = \mathcal{R}_0(d_I, q) $.

(i) If (C1) holds, then the DFE is unstable for any $ q > \min_{x\in [0, L]}a'(x) > 0 $ and $ d_I > 0 $;

(ii) If (C2) holds, then there exists a unique curve in $ d_I $–$ q $ plane

with the function $ \rho_1 = \rho_1(d_I): (0, +\infty)\rightarrow (0, +\infty) $ satisfying

and such that for every $ d_I > 0 $, the DFE is unstable for $ 0 < \min_{x\in [0, L]}a'(x) < q < \rho_1(d_I) $ and it is globally and asymptotically stable for $ q > \min_{x\in [0, L]}a'(x) > \rho_1(d_I) $.

Here $ \theta_1 $ is the unique positive solution of

Similarly, we can get further properties of $ \mathcal{R}_0 $ when $ \int_0^L \beta(x)dx < \int_0^L \gamma(x)dx $.

Theorem 1.4. Assume that $ \int_0^L \beta(x)dx < \int_0^L \gamma(x)dx $. Let $ d^*_I $ is the unique positive root of the equation $ \hat{\mathcal{R}}_0 = 1 $, where $ \hat{\mathcal{R}}_0 $ was introduced in ^[2].

(1) If (C1) holds, then the DFE is unstable for any $ q > \min_{x\in [0, L]}a'(x) > 0 $ and $ d_I\in (0, d^*_I] $, while for $ d_I\in (d^*_I, +\infty) $ there exists a unique curve in $ d_I $–$ q $ plane

with the monotone function $ \rho_2 = \rho_2(d_I): (d^*_I, +\infty)\rightarrow (0, +\infty) $ satisfying

and such that the DFE is unstable for $ 0 < \min_{x\in [0, L]}a'(x) < q < \rho_2(d_I) $ and it is globally asymptotically stable for $ q > \min_{x\in [0, L]}a'(x) > \rho_2(d_I) $.

Here $ \theta_2 $ is the unique positive solution of

(2) If (C2) holds, then for $ d_I\in (0, d^*_I) $, there exists a unique curve in $ d_I-q $ plane

with the function $ \rho_3 = \rho_3(d_I): (0, d^*_I)\rightarrow (0, +\infty) $ satisfying

and such that the DFE is unstable for $ 0 < \min_{x\in [0, L]}a'(x) < q < \rho_3(d_I) $ and it is globally and asymptotically stable for $ q > \min_{x\in [0, L]}a'(x) > \rho_3(d_I) $, while for $ d_I\in (d^*_I, +\infty) $, the DFE is globally and asymptotically stable for any $ q > \min_{x\in [0, L]}a'(x) > 0 $.

The following theorem deals with the existence of $ EE $.

Theorem 1.5. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $. If $ \mathcal{R}_0 > 1 $, then problem (1.3) possesses at least one EE.

The last theorem will consider the results on (1.1) when $ \beta(x)-\gamma(x) $ changes sign twice in $ (0, L) $.

Theorem 1.6. Assume that $ \beta(x)-\gamma(x) $ changes sign twice in $ (0, L) $.

(1) If $ \int_0^L \beta(x)dx > \int_0^L \gamma(x)dx $ and $ \beta(L) < \gamma(L) $, then there exists some positive constant $ \Lambda $ which is independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $, we can find a positive constant $ Q $ which depends on $ d_I $ such that $ \mathcal{R}_0 > 1 $ when $ 0 < \min_{x\in [0, L]}a'(x) < q < Q $ and $ \mathcal{R}_0 < 1 $ when $ q > Q $.

(2) If $ \int_0^L \beta(x)dx > \int_0^L \gamma(x)dx $ and $ \beta(L) > \gamma(L) $, then there exists some positive constant $ \Lambda $ which is independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $ one of the following conclusions holds:

(i) $ \mathcal{R}_0 > 1 $ for any $ q > \min_{x\in [0, L]}a'(x) > 0 $;

(ii) There exists a positive constant $ \hat{Q} $ which is independent of $ d_I $ and satisfies that $ \mathcal{R}_0 > 1 $ for $ q\neq \hat{Q} $ and $ \mathcal{R}_0 = 1 $ when $ q = \hat{Q} $;

(iii) There exist two positive constants $ Q_2 > Q_1 $ both depending on $ d_I $ such that $ \mathcal{R}_0 > 1 $ when $ q\in (0, Q_1)\cup (Q_2, +\infty) $ while $ \mathcal{R}_0 < 1 $ when $ q\in (Q_1, Q_2) $.

(3) If $ \int_0^L \beta(x)dx < \int_0^L \gamma(x)dx $ and $ \beta(L) > \gamma(L) $, then there exists some positive constant $ \Lambda > d^*_I $ which is independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $, we can find a positive constant $ Q $ which depends on $ d_I $ such that $ \mathcal{R}_0 < 1 $ when $ 0 < \min_{x\in [0, L]}a'(x) < q < Q $ and $ \mathcal{R}_0 > 1 $ when $ q > Q $.

(4) If $ \int_0^L \beta(x)dx < \int_0^L \gamma(x)dx $ and $ \beta(L) < \gamma(L) $, then there exists some positive constant $ \Lambda > d^*_I $ which is independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $ one of the following conclusions holds:

(iv) $ \mathcal{R}_0 < 1 $ for any $ q > \min_{x\in [0, L]}a'(x) > 0 $;

(v) There exists a positive constant $ \hat{Q} $ which is independent of $ d_I $ and satisfies that $ \mathcal{R}_0 < 1 $ for $ q\neq \hat{Q} $ and $ \mathcal{R}_0 = 1 $ when $ q = \hat{Q} $;

(vi) There exist two positive constants $ Q_2 > Q_1 $ both depending on $ d_I $ and satisfy that $ \mathcal{R}_0 < 1 $ when $ q\in (0, Q_1)\cup (Q_2, +\infty) $ while $ \mathcal{R}_0 > 1 $ when $ q\in (Q_1, Q_2) $.

The rest of this paper is organized as follows. In Section 2, we give the proofs of Theorem 1.1 and Theorem 1.2. In Section 3, we will prove Theorem 1.3. In Section 4, we will prove Theorem 1.4. In Section 5, we will prove Theorem 1.5. In Section 6, we will prove Theorem 1.6.

2.   Materials and method

2.1.   The proofs of Theorem 1.1 and Theorem 1.2

In this section, we first give some qualitative properties of $ \mathcal{R}_0 $, then we deal with the stability of DFE, and we can finish the proofs of Theorem 1.1 and Theorem 1.2.

By the definition of $ \mathcal{R}_0 $, there exits some positive function $ \Phi(x)\in C^2([0, L]) $ such that

Letting $ \varphi(x) = e^{-\frac{a(x)}{d_I}}\Phi(x) $, we have

Linearizing (1.1) around $ (\hat{S}, 0) $ and letting $ \bar{\xi}(x, t) = S(x, t)-\hat{S}(x, t) $, $ \bar{\eta}(x, t) = I(x, t) $, we have

For the linear system, seeking for the solution which is separation of variables, i.e., $ \bar{\xi}(x, t) = e^{-\lambda t}\xi(x) $ and $ \bar{\eta}(x, t) = e^{-\lambda t}\eta(x) $, we have

subject to boundary conditions

By the conservation of total population, we need to impose that

Noticing that the second equation of (2.3) is independent of $ \xi $, letting $ \zeta(x) = e^{-\frac{a(x)}{d_I}}\eta(x) $, we only need to consider the following eigenvalue problem

By the results of ^[19], all the eigenvalues are real, the smallest eigenvalue $ \lambda_1(d_I, q) $ is simple, and its corresponding eigenfunction $ \phi_1 $ can be chosen positive.

We will show a fact below.

Lemma 2.1.1. For any $ d_I $ and $ q > \min_{x\in [0, L]}a'(x) > 0 $, $ \lambda_1(d_I, q) < 0 $ if $ \mathcal{R}_0 > 1 $, $ \lambda_1(d_I, q) = 0 $ if $ \mathcal{R}_0 = 1 $ and $ \lambda_1(d_I, q) > 0 $ if $ \mathcal{R}_0 < 1 $.

Proof. Note that $ (\lambda_1(d_I, q), \phi_1) $ satisfies

Multiplying (2.1) by $ e^{\frac{a(x)}{d_I}}\phi_1 $ and (2.7) by $ e^{\frac{a(x)}{d_I}}\Phi $, integrating by parts in $ (0, L) $, and subtracting the resulting equations, we get

Using the mean value theorem of integrating, we have

for some $ 0\leq x_1\leq L $ and $ 0\leq x_2\leq L $. Using $ \beta(x_1)\Phi(x_1)\phi_1(x_1) > 0 $ and $ \Phi(x_2)\phi_1(x_2) > 0 $, we know that

which implies the conclusions are true.

Lemma 2.1.2. If $ \frac{d_I}{q}\rightarrow 0 $ and $ \frac{d_I}{q^2}\rightarrow 0 $, $ \tilde{a}'(x) > \delta > 0 $ for some constant $ \delta $, then $ \mathcal{R}_0\rightarrow \frac{\beta(L)}{\gamma(L)} $.

Proof. Let $ w(x) = e^{-\frac{q}{d_I}A\tilde{a}(x)}\Phi(x) $, where $ \Phi(x) $ is the solution of (2.1), $ A $ is a constant which will be chosen later. It is easy to verify that $ w $ satisfies

First we chose $ A = 1+\frac{C_1d_I}{q^2} $, where $ C_1 $ is a positive constant to be chosen later. Then (2.8) becomes

Assume that $ w(x_*) = \min_{x\in [0, L]} w(x) $. We will show that $ x_* = L $ below. $ w_x(0) < 0 $ implies that $ x_*\neq 0 $. If $ x_*\in (0, L) $, then $ w_{xx}(x_*)\geq 0 $ and $ w_x(x_*) = 0 $, (2.9) means that

Taking $ C_1 = Kq $ with $ K $ large enough, we can get a contradiction. Therefore, $ x_* = L $ and $ w(x)\geq w(L) $ for $ x\in [0, L] $, which implies that

Next, we chose $ A = 1-\frac{C_2d_I}{q^2} $, where $ C_2 $ is a positive constant to be chosen later. Then (2.8) becomes

Assume that $ w(x^*) = \max_{x\in [0, L]} w(x) $. We will show that $ x^* = L $ below. $ w_x(0) > 0 $ implies that $ x^*\neq 0 $. If $ x^*\in (0, L) $, then $ w_{xx}(x^*)\geq 0 $ and $ w_x(x^*) = 0 $, (2.10) means that

Taking $ C_2 = K'q $ with $ K' $ large enough, we can get a contradiction. Therefore, $ x^* = L $ and $ w(x)\leq w(L) $ for $ x\in [0, L] $, which implies that

Dividing (2.1) by $ \Phi(L) $ and integrating the result in $ (0, L) $, we have

Letting $ y = \frac{q[\tilde{a}(L)-\tilde{a}(x)]}{d_I} $, i.e., $ x = \tilde{a}^{-1}[\tilde{a}(L)-\frac{d_Iy}{q}] $, we have

and

Using (2.12), by Lebesgue dominant convergence theorem, then passing to the limit in (2.13), we get

We have the following corollary.

Corollary 2.1.1. The following statements hold.

(i) Given $ d_I > 0 $, $ \mathcal{R}_0\rightarrow \hat{\mathcal{R}}_0 $ as $ q\rightarrow 0 $;

(ii) Given $ d_I > 0 $, $ \mathcal{R}_0\rightarrow \frac{\beta(L)}{\gamma(L)} $ as $ q\rightarrow +\infty $;

(iii) Given $ q > 0 $, $ \mathcal{R}_0\rightarrow \frac{\beta(L)}{\gamma(L)} $ as $ d_I\rightarrow 0 $;

(iv) Given $ q > 0 $, $ \mathcal{R}_0\rightarrow \frac{\int_0^L\beta(x)dx}{\int_0^L\gamma(x)dx} $ as $ d_I\rightarrow +\infty $.

Proof. (i) For any fixed $ \varphi\in H^1((0, L)) $, $ \varphi\neq 0 $, we have

Taking $ \inf_{\varphi\in H^1((0, L)), \varphi\neq 0} $ both sides, we have $ \frac{1}{\mathcal{R}_0}\rightarrow \frac{1}{\hat{\mathcal{R}}_0} $ as $ q\rightarrow 0 $.

(ii) and (iii) are the direct conclusions of Lemma 2.2.

(iv) By the definition of $ \frac{1}{\mathcal{R}_0} $, for $ \varphi\equiv 1 $, we have

which implies that $ \frac{1}{\mathcal{R}_0} $ is uniformly bounded for $ d_I > 0 $, passing to a subsequence if necessary, it has a finite limit $ \frac{1}{\bar{\mathcal{R}}_0} $ as $ d_I\rightarrow \infty $.

On the other hand, by the standard elliptic regularity and the Sobolev embedding theorem, $ \Phi $ is uniformly bounded for all $ d_I\geq 1 $. Dividing both sides of (2.1) by $ d_I $ and letting $ d_I\rightarrow +\infty $, we have $ \Phi_{xx}\rightarrow 0 $ for $ x\in (0, L) $ and $ \Phi_x(0)\rightarrow 0 $, $ \Phi_x(L)\rightarrow 0 $. Consequently, there exists a positive constant $ \bar{\Phi} $ such that $ \Phi(x)\rightarrow \bar{\Phi} $ as $ d_I\rightarrow +\infty $. Integrating (2.1) by parts over $ (0, L) $, we can get

Letting $ d_I\rightarrow +\infty $, we obtain $ \bar{\mathcal{R}}_0 = \frac{\int_0^L\beta(x)dx}{\int_0^L\gamma(x)dx} $.

Lemma 2.1.3. The following statements hold.

(i) If $ \beta(x) > \gamma(x) $ on $ [0, L] $, then $ \mathcal{R}_0 > 1 $ for any $ d_I > 0 $ and $ q > \min_{x\in [0, L]}a'(x) > 0 $;

(i) If $ \beta(x) < \gamma(x) $ on $ [0, L] $, then $ \mathcal{R}_0 < 1 $ for any $ d_I > 0 $ and $ q > \min_{x\in [0, L]}a'(x) > 0 $.

Proof. (i) If $ \beta(x) > \gamma(x) $ on $ [0, L] $, by the definition of $ \frac{1}{\mathcal{R}_0} $, for $ \varphi\equiv1 $, we have

i.e., $ \mathcal{R}_0 > 1 $.

(ii) Subtracting both sides of (2.2) by $ \beta(x)\varphi $, multiplying by $ e^{\frac{a(x)}{d_I}}\varphi $, we have

Integrating it by parts over $ (0, L) $, using $ \varphi_x(0) = \varphi_x(L) = 0 $, we obtain

Since $ \beta(x) < \gamma(x) $ on $ [0, L] $, the left side of the above equality is positive, and

which implies that $ \mathcal{R}_0 < 1 $.

Proof. Theorem 1.1 is the direct results of Lemma 2.1.2, Corollary 2.1.1 and Lemma 2.1.3.

Next we will consider the stability of DFE.

Lemma 2.1.4. The DFE is stable if $ \mathcal{R}_0 < 1 $, while it is unstable if $ \mathcal{R}_0 > 1 $.

Proof. 1. Assume contradictorily the DFE is unstable if $ \mathcal{R}_0 < 1 $. Then we can find $ (\lambda, \xi, \eta) $ which is a solution of (2.3)–(2.4) subject to (2.5), with at least one of $ \xi $ and $ \eta $ is not identical zero, and $ \Re(\lambda)\leq 0 $. Suppose that $ \eta\equiv 0 $, then $ \xi\not\equiv 0 $ on $ [0, L] $. Using (2.3)–(2.4), we have

It is easy to see that $ \lambda $ is real and nonnegative, and therefore $ \lambda = 0 $. We find that $ \xi = \xi_0e^{\frac{q}{d_I}\tilde{a}(x)} $, where $ \xi_0 $ is some constant to be determined later. By (1.2), we impose that $ \int_0^L[\xi(x)+\eta(x)]dx = 0 $, $ \xi_0 = 0 $, i.e., $ \xi\equiv 0 $ on $ [0, L] $. This is a contradiction. Then we conclude that $ \eta\equiv 0 $ on $ [0, L] $. From (2.6), $ \lambda $ must be real and $ \lambda\leq 0 $. Since $ \lambda_1(d_I, q) $ is the principal eigenvalue, then $ \lambda_1(d_I, q)\leq \lambda\leq 0 $. Lemma 2.1 implies that $ \mathcal{R}_0\geq 1 $, which is a contradiction. Then we conclude that if $ (\lambda, \xi, \eta) $ is a solution of (2.3)–(2.4), with at least one of $ \xi $ and $ \eta $ not identical zero on $ [0, L] $, then $ \Re(\lambda) > 0 $. This proves the linear stability of the DFE.

2. Suppose that $ \mathcal{R}_0 > 1 $. Since $ (\lambda_1(d_I, q), \phi_1) $ is the principal eigen-pair of (2.6), $ (\lambda_1(d_I, q), e^{\frac{a(x)}{d_I}}\phi_1) $ satisfies

By the result of Lemma 2.1.1, $ \lambda_1(d_I, q) < 0 $. On the other hand,

There exists a unique solution $ \xi_1 $ of (2.16). And (2.5) becomes

which implies that (2.3)–(2.4) has a solution $ (\lambda_1(d_I, q), \xi_1, e^{\frac{a(x)}{d_I}}\phi_1(x)) $ satisfying $ \lambda_1(d_I, q) < 0 $ and $ e^{\frac{a(x)}{d_I}}\phi_1(x) > 0 $ in $ (0, L) $. Therefore, the DFE is linearly unstable.

Lemma 2.1.5. If $ \mathcal{R}_0 < 1 $, then $ (S, I)\rightarrow (\hat{S}, 0) $ in $ C([0, L]) $ as $ t\rightarrow +\infty $.

Proof. If $ \mathcal{R}_0 < 1 $, letting $ u(x, t) = Me^{-\lambda_1(d_I, q)t}e^{\frac{a(x)}{d_I}}\phi_1(x) $, then we have

Here $ (\lambda_1(d_I, q), \phi_1) $ is the principal eigen-pair, $ \lambda_1(d_I, q) > 0 $ and $ \phi_1(x) > 0 $ on $ [0, L] $. $ M $ is large enough such that $ I(x, 0)\leq u(x, 0) $ for every $ x\in (0, L) $. Noticing that

By the comparison principle, we have $ I(x, t)\leq u(x, t) $ for every $ x\in (0, L) $ and $ t\geq 0 $. Obviously, $ u(x, t)\rightarrow 0 $ for every $ x\in (0, L) $ as $ t\rightarrow \infty $, which implies that $ I(x, t)\rightarrow 0 $ for every $ x\in (0, L) $ as $ t\rightarrow \infty $.

Now we will show that $ S\rightarrow \hat{S} $ as $ t\rightarrow +\infty $. Since

we have

for $ 0 < x < L $, $ t > 0 $. Noticing that

as $ t\rightarrow +\infty $, we know that there exists a positive function $ \tilde{S}(x) $ such that

Therefore, $ \lim_{t\rightarrow +\infty} S(x, t) = \tilde{S}(x) = \hat{S}(x) $.

Proof. Theorem 1.2 is the direct results of Lemma 2.1.4 and Lemma 2.1.5.

2.2.   Further properties of $ \mathcal{R}_0 $: $ \beta(x)-\gamma(x) $ changing sign once

In this section, we will study further properties of $ \mathcal{R}_0 $ in the case of $ \beta(x)-\gamma(x) $ changing sign once.

Lemma 2.2.1. Assume that $ \phi_1 $ is a positive eigenfunction corresponding to $ \mathcal{R}_0 = 1 $, $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $. If assumption (C1)(or (C2)) holds, then $ (\phi_1)_x > 0 $(or $ (\phi_1)_x < 0 $) in $ (0, L) $.

Proof. If $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and assumption (C1) holds, then there exists some $ x_0\in (0, L) $ such that $ \beta(x)-\gamma(x) < 0 $ in $ (0, x_0) $, $ \beta(x_0) = \gamma(x_0) $ and $ \beta(x)-\gamma(x) > 0 $ in $ (x_0, L) $.

By the definition of $ \phi_1 $, we have

Multiplying (2.17) by $ e^{\frac{a(x)}{d_I}} $, we obtain

Under the assumptions on $ \beta(x) $ and $ \gamma(x) $, we can obtain $ (e^{\frac{a(x)}{d_I}}(\phi_1)_{x})_x > 0 $ in $ (0, x_0) $, $ (e^{\frac{a(x)}{d_I}}(\phi_1)_{x})_x = 0 $ at $ x_0 $ and $ (e^{\frac{a(x)}{d_I}}(\phi_1)_{x})_x < 0 $ in $ (x_0, L) $. That is, $ e^{\frac{a(x)}{d_I}}(\phi_1)_{x} $ is strictly increasing in $ (0, x_0) $ and strictly decreasing in $ (x_0, L) $. Noticing that $ (\phi_1)_x(0) = (\phi_1)_x(L) = 0 $, we can get $ e^{\frac{a(x)}{d_I}}(\phi_1)_{x} > 0 $ in $ (0, L) $. So $ (\phi_1)_x > 0 $ in $ (0, L) $.

Similarly, if $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and assumption (C2) holds, $ (\phi_1)_x < 0 $ in $ (0, L) $. We omit the details here.

Now we prove two general lemmas below.

For any continuous function $ m(x) $ on $ [0, L] $, define

Lemma 2.2.2. Assume that $ m(x)\in C^1([0, L]) $ and $ m(L) > 0 $(or $ m(L) < 0 $). Then there exists some positive constant $ M $ such that $ F(\eta) > 0 $(or $ F(\eta) < 0 $) for any $ \eta > M $.

Proof. Since $ m'(x) $ and $ \tilde{a}'(x) $ is uniformly bounded independent of $ \eta $, we have

Therefore, there exists some positive constant $ M $ such that $ F(\eta) > 0(<0) $ for $ \eta > M $.

Lemma 2.2.3. Assume that $ m(x) $ changes sign once in $ (0, L) $. Then

(i) If $ m(L) > 0 $ and $ \int_0^L \tilde{a}'(x)m(x)dx > 0 $, then $ F(\eta) > 0 $ for any $ \eta > 0 $;

(ii) If $ m(L) < 0 $ and $ \int_0^L \tilde{a}'(x)m(x)dx < 0 $, then $ F(\eta) < 0 $ for any $ \eta > 0 $;

(iii) If $ m(L) > 0 $ and $ \int_0^L \tilde{a}'(x)m(x)dx < 0 $, then there exists a unique $ \eta_1\in (0, +\infty) $ such that $ F(\eta_1) = 0 $ and $ F'(\eta_1) > 0 $;

(iv) If $ m(L) < 0 $ and $ \int_0^L \tilde{a}'(x)m(x)dx > 0 $, then there exists a unique $ \eta_1\in (0, +\infty) $ such that $ F(\eta_1) = 0 $ and $ F'(\eta_1) < 0 $.

Proof. We only prove part (i) and part (iii). The proofs of part (ii) and part (iv) are similar.

(i) If $ m(L) > 0 $ and $ m(x) $ changes sign once in $ (0, L) $, then there exists $ x_1\in (0, L) $ such that $ m(x) < 0 $ for $ x\in (0, x_1) $ and $ m(x) > 0 $ for $ x\in (x_1, L) $. Since $ \tilde{a}(x) $ is increasing, we have $ m(x)[\tilde{a}(x)-\tilde{a}(x_1)] > 0 $ for $ x\in (0, L) $ and $ x\neq x_1 $. And

which implies that $ e^{-\tilde{a}(x_1)\eta}F(\eta) $ is strictly increasing in $ \eta\in (0, \infty) $, $ e^{-\tilde{a}(x_1)\eta}F(\eta) > F(0) = \int_0^L\tilde{a}'(x)m(x)dx > 0 $. Consequently, $ F(\eta) > 0 $ for any $ \eta > 0 $. Here the prime notation denotes differentiation by $ \eta $. Part (i) is proved.

(iii) $ \int_0^L\tilde{a}'(x)m(x)dx < 0 $ means that $ F(0) < 0 $, while, by the result of Lemma 2.2.2, $ m(L) > 0 $ means that $ F(\eta) > 0 $ for $ \eta > M $ with $ M $ large enough. By continuity, there at least exists a positive root for $ F(\eta) = 0 $. But $ e^{-\tilde{a}(x_1)\eta}F(\eta) $ is increasing in $ \eta\in (0, \infty) $, so $ F(\eta) = 0 $ only has a unique positive root $ \eta_1 $. By (2.18), we have $ F'(\eta_1) > a(x_1)F(\eta_1) = 0 $. Part (iii) is proved.

2.3.   The stability of DFE

In this section, we consider the stability of DFE. First we have

Lemma 2.3.1. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and $ \int_0^L\beta(x)dx > \int_0^L\gamma(x)dx $.

(i) If $ \beta(x) $ and $ \gamma(x) $ satisfy (C1), then $ \mathcal{R}_0 > 1 $ for $ d_I > 0 $ and $ q > \min_{x\in [0, L]}a'(x) > 0 $;

(ii) If $ \beta(x) $ and $ \gamma(x) $ satisfy (C2), then for every $ d_I > 0 $, there exists a unique $ \bar{q} = \bar{q}(d_I) $ such that $ \mathcal{R}_0 > 1 $ for $ 0 < \min_{x\in [0, L]}a'(x) < q < \bar{q} $, $ \mathcal{R}_0 = 1 $ for $ q = \bar{q} $ and $ \mathcal{R}_0 < 1 $ for $ q > \bar{q} $.

Proof. (i) Subtracting both sides of (2.2) by $ \beta(x)\varphi $, multiplying by $ \frac{e^{\frac{a(x)}{d_I}}}{\varphi} $, we have

Integrating it by parts over $ (0, L) $, using $ \varphi_x(0) = \varphi_x(L) = 0 $, we obtain

Using Lemma 2.2.3(i) with $ m(x) = \frac{[\beta(x)-\gamma(x)]}{\tilde{a}'(x)} $, $ \int_0^L[\beta(x)-\gamma(x)]e^{\frac{a(x)}{d_I}}dx > 0 $, and

which implies that $ \mathcal{R}_0 > 1 $.

(ii) Differentiating both sides of (2.2) with respect to $ q $, denoting the differentiation with respect to $ q $ by the dot notation, we obtain

Multiplying (2.19) by $ e^{\frac{a(x)}{d_I}}\varphi $ and integrating the resulting equation in $ (0, L) $, we have

Multiplying (2.2) by $ e^{\frac{a(x)}{d_I}}\dot{\varphi} $ and integrating the resulting equation in $ (0, L) $, we get

Subtracting (2.20) and (2.21), we obtain

By the result of Corollary 2.1.1, we know that

Meanwhile, we have

for any $ d_I $. Then there must exist at least some $ \bar{q} $ such that $ \mathcal{R}_0(\bar{q}) = 1 $. By Lemma 2.1.1, for any $ \bar{q} > 0 $ satisfying $ \mathcal{R}_0(\bar{q}) = 1 $, $ (\phi_1)_x < 0 $ in $ (0, L) $. Recalling (2.22), we have

which implies that $ \bar{q} $ is the unique point satisfying $ \mathcal{R}_0(\bar{q}) = 1 $.

The following lemma will tell us that there exists a function $ q = \rho_1(d_I) $ such that $ \mathcal{R}_0(d_I, \rho_1(d_I)) = 1 $ and give the asymptotic profile of $ \rho_1(d_I) $ if $ \int_0^L\beta(x)dx > \int_0^L\gamma(x)dx $.

Lemma 2.3.2. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $, $ \int_0^L\beta(x)dx > \int_0^L\gamma(x)dx $, and $ \theta_1 $ is the unique solution of

Suppose that $ \beta(x) $ and $ \gamma(x) $ satisfy (C2). Then there exists a function $ \rho_1:(0, \infty)\rightarrow (0, \infty) $ such that $ \mathcal{R}_0(d_I, \rho_1(d_I)) = 1 $. And $ \rho_1 $ satisfies

Proof. 1. Let's first consider the limit of $ \frac{\rho_1(d_I)}{d_I} $ as $ d_I\rightarrow \infty $. Assume that $ \frac{\rho_1(d_I)}{d_I}\rightarrow \infty $ as $ d_I\rightarrow \infty $. Under the assumption (C2), by Lemma 2.1.4, we have

which is a contradiction to $ \mathcal{R}_0(d_I, \rho_1(d_I)) = 1 $.

Next, we will prove that $ \frac{\rho_1(d_I)}{d_I}\rightarrow \theta_1 $ as $ d_I\rightarrow \infty $. Here $ \theta_1 $ is the unique positive root of $ \int_0^L[\beta(x)-\gamma(x)]e^{\theta_1\tilde{a}(x)}dx = 0 $. By the discussions above, we know that $ \frac{\rho_1(d_I)}{d_I} $ is bounded for large $ d_I $. Passing to a subsequence if necessary, we suppose that $ \frac{\rho_1(d_I)}{d_I}\rightarrow \theta_* $ for some nonnegative number $ \theta_* $ as $ d_I\rightarrow \infty $. Let $ \tilde{\varphi} $ be the unique normalized eigenfunction of the eigenvalue $ \mathcal{R}_0(d_I, \rho_1(d_I)) = 1 $. Then

Integrating (2.23) in $ (0, L) $, we get

Recalling that, up to a subsequence if necessary, $ \tilde{\varphi}\rightarrow 1 $ in $ C([0, 1]) $ as $ d_I\rightarrow \infty $. Letting $ d_I\rightarrow \infty $ in (2.24), we have

By Lemma 2.2.3 with $ m(x) = \frac{[\beta(x)-\gamma(x)]}{\tilde{a}'(x)} $, $ F(\eta) $ has a unique positive root, i.e., $ \theta_* = \theta_1 $.

2. Contradictorily, assume that $ q = \rho_1(d_I)\rightarrow q^* > 0 $ or $ q = \rho_1(d_I)\rightarrow \infty $ as $ d_I\rightarrow 0 $. By Lemma 2.1.4, we know that

or

which is a contradiction to $ \mathcal{R}_0(d_I, \rho_1(d_I)) = 1 $. Therefore, we have $ \lim_{d_I\rightarrow 0}\rho_1(d_I) = 0 $.

To study the properties of $ \mathcal{R}_0 $ when $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $, we need the following results which were stated in ^[2]:

Proposition 2.3.1. Assume that $ \beta(x)-\gamma(x) $ changes sign in $ (0, L) $.

(i) $ \hat{\mathcal{R}}_0 $ is a monotone decreasing function of $ d_I $ with $ \hat{\mathcal{R}}_0\rightarrow \max\{\beta(x)/\gamma(x):x\in[0, L]\} $ as $ d_I\rightarrow 0 $ and $ \hat{\mathcal{R}}_0\rightarrow \int_0^L\beta(x)dx/\int_0^L\gamma(x)dx $ as $ d_I\rightarrow +\infty $;

(ii) $ \hat{\mathcal{R}}_0 > 1 $ for all $ d_I > 0 $ if $ \int_0^L\beta(x)dx\geq \int_0^L\gamma(x)dx $;

(iii) There exists a threshold value $ d^*_I\in (0, +\infty) $ such that $ \hat{\mathcal{R}}_0 > 1 $ for $ d_I < d^*_I $ and $ \hat{\mathcal{R}}_0 < 1 $ for $ d_I > d^*_I $ if $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $.

Lemma 2.3.3. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $. Then there exists some constant $ d^*_I > 0 $ such that $ d^*_I $ is the unique positive root of the equation $ \hat{\mathcal{R}_0}(d_I) = 1 $ and the following statements hold.

1. If $ \beta(x) $ and $ \gamma(x) $ satisfy (C1), then

(i) for $ d_I\in (0, d^*_I] $, $ \mathcal{R}_0 > 1 $ for any $ q > \min_{x\in [0, L]}a'(x) > 0 $;

(ii) for $ d_I\in (d^*_I, \infty) $, there exists a unique $ \bar{q} = \bar{q}(d_I) $ such that $ \mathcal{R}_0 < 1 $ for any $ 0 < \min_{x\in [0, L]}a'(x) < q < \bar{q} $ and $ \mathcal{R}_0 > 1 $ for any $ q > \bar{q} $.

2. If $ \beta(x) $ and $ \gamma(x) $ satisfy (C2), then

(iii) for $ d_I\in (0, d^*_I] $, there exists a unique $ \bar{q} = \bar{q}(d_I) $ such that $ \mathcal{R}_0 > 1 $ for any $ 0 < \min_{x\in [0, L]}a'(x) < q < \bar{q} $ and $ \mathcal{R}_0 < 1 $ for any $ q > \bar{q} $;

(iv) for $ d_I\in (d^*_I, \infty) $, $ \mathcal{R}_0 < 1 $ for any $ q > \min_{x\in [0, L]}a'(x) > 0 $.

Proof. (i) Noticing that $ \beta(x) $ and $ \gamma(x) $ satisfy (C1), similar to the proof of (ii) in Lemma 2.1.4, we can prove that there exists a unique $ \bar{q} > 0 $ satisfying $ \mathcal{R}_0(\bar{q}) = 1 $ and $ \mathcal{R}'_0(\bar{q}) > 0 $. Hence, the conclusion is true for $ d_I\in (d^*_I, +\infty) $.

For $ d_I\in (0, d^*_I] $, by the results of Proposition 2.3.1, we have $ \lim_{q\rightarrow 0}\mathcal{R}_0 = \hat{\mathcal{R}}_0\geq 1 $. By the results of Corollary 2.1.1, $ \lim_{q\rightarrow +\infty}\mathcal{R}_0 = \beta(L)/\gamma(L) > 1 $ under the condition (C1). Hence $ \mathcal{R}_0 > 1 $ for any $ q > 0 $.

(ii) The proof of Lemma 2.3.3 under the condition (C2) is similar to that of Lemma 2.1.4, we omit the details here.

Lemma 2.3.4. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $. Then there exists a constant $ d^*_I > 0 $ such that $ d^*_I $ is the unique positive root of the equation $ \hat{\mathcal{R}_0}(d_I) = 1 $ and the following statements hold.

1. If $ \beta(x) $ and $ \gamma(x) $ satisfy (C1), then there exists a function $ \rho_2:(d^*_I, \infty)\rightarrow (0, \infty) $ such that $ \rho_2 $ is a monotone increasing function of $ d_I $ and $ \mathcal{R}_0(d_I, \rho_2(d_I)) = 1 $. Let $ \theta_2 $ be the unique solution of

Then

2. If $ \beta(x) $ and $ \gamma(x) $ satisfy (C2), then there exists a function $ \rho_3:(0, d^*_I)\rightarrow (0, \infty) $ such that $ \mathcal{R}_0(d_I, \rho_3(d_I)) = 1 $ and

Proof. 1. If we can prove that $ \rho'_2(d_I) > 0 $ for $ d_I\in (d^*_I, \infty) $, then $ \rho_2(d_I) $ is a monotone increasing function of $ d_I $. Here the prime notation denotes differentiation by $ d_I $. Since $ \mathcal{R}_0(d_I, \rho_2(d_I)) = 1 $, we can get

By Lemma 2.3.1, $ \frac{\partial \mathcal{R}_0}{\partial q} > 0 $ for $ \mathcal{R}_0(d_I, \rho_2(d_I)) = 1 $. So we need to prove that $ \frac{\partial \mathcal{R}_0}{\partial d_I} < 0 $.

Differentiating both sides of (2.2) with respect to $ d_I $, denoting the differentiation with respect to $ d_I $ by the dot notation, we obtain

Multiplying (2.26) by $ e^{\frac{a(x)}{d_I}}\varphi $ and integrating the resulting equation in $ (0, L) $, we obtain

Multiplying (2.2) by $ e^{\frac{a(x)}{d_I}}\dot{\varphi} $ and integrating the resulting equation in $ (0, L) $, we get

Subtracting (2.27) and (2.28), we have

By Lemma 2.2.1, for any $ d_I $ satisfying $ \mathcal{R}_0(d_I, q) = 1 $, $ (\phi_1)_x > 0 $, we can get

(2.25) and (2.30) imply that $ \rho'_2(d_I) > 0 $ for $ d_I\in (d^*_I, \infty) $.

The proof of $ \lim_{d_I\rightarrow \infty} \frac{\rho_2(d_I)}{d_I} = \theta_2 $($ \theta_2 $ is the unique solution of $ \int_0^L[\beta(x)-\gamma(x)]e^{\theta_2 a(x)}dx = 0 $) is similar to the proof of Lemma 2.3.2, we omit the details here.

Now we will prove that $ \lim_{d_I\rightarrow d^*_I+}\rho_2(d_I) = 0 $. Assume that there exists $ q^* $ such that $ q = \rho_2(d_I)\rightarrow q^* $ as $ d_I\rightarrow d_I^*+ $. Then there exists a positive function $ \phi^*(x)\in C^2([0, L]) $ such that

Noticing that $ d^*_I $ is the unique positive root of $ \hat{\mathcal{R}}_0 = 1 $ and the definition of $ \hat{\mathcal{R}}_0 $ implies $ q = 0 $, there exists a positive function $ \hat{\phi}(x)\in C^2([0, L]) $ such that

Multiplying (2.31) by $ \hat{\phi} $, (2.32) by $ \phi^* $, subtracting the two resulting equations, then integrating by parts over $ (0, L) $, we get

Since $ \phi^*_x $ is positive(by Lemma 2.2.1), we have $ q^* = 0 $. Therefore, $ \lim_{d_I\rightarrow d^*_I+}\rho_2(d_I) = 0 $.

2. Using the arguments above, similar to the proof of Lemma 2.3.2, we can obtain the conclusions.

2.4.   The endemic equilibrium

In this section, we will show that: If the disease-free equilibrium is unstable, then we can use the bifurcation analysis and degree theory to study the existence of endemic equilibrium.

Letting $ \tilde{S} = e^{\frac{a(x)}{d_S}}\bar{S} $, $ \tilde{I} = e^{\frac{a(x)}{d_I}}\bar{I} $, we have

Since the structure of the solution set of (2.33) is the same as that of (1.3), we study (2.33) instead of (1.3). Denote the unique disease-free equilibrium of (2.33) by $ (\hat{\bar{S}}, 0) = (\frac{N}{\int_0^L e^{\frac{a(x)}{d_S}}}, 0) $. We will consider a branch of positive solutions of (2.33) bifurcating from the branch of semi-trivial solutions given by

through using the local and global bifurcation theorems. For fixed $ d_S $, $ d_I > 0 $, we take $ q $ as the bifurcation parameter. Let

for $ p > 1 $ and the set of positive solution of (2.33) to be

Lemma 2.4.1 Assume that $ d_S $, $ d_I > 0 $ and $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $. Then

1. $ q_* > 0 $ is a bifurcation point for the positive solutions of (2.33) from the semi-trivial branch $ \Gamma_S $ if and only if $ q_* $ satisfies $ R_0(d_I, q_*) = 1 $. That is,

(I) If $ \int^L_0 \beta(x)dx > \int^L_0\gamma(x)dx $, then such $ q_* $ exists uniquely for any $ d_I > 0 $ if and only if assumption (C2) holds;

(II) If $ \int^L_0 \beta(x)dx < \int^L_0\gamma(x)dx $, let $ d^*_I $ be the unique positive root of $ \hat{R}_0 = 1 $, then such $ q_* $ exists uniquely for any $ d_I > 0 $ if and only if either $ \beta(x) $ and $ \gamma(x) $ satisfy condition (C1) and $ d > d^*_I $ or they satisfy condition (C2) and $ 0 < d < d^*_I $.

2. There exits some $ \delta > 0 $ such that all positive solutions of (2.33) near $ (q_*, (\hat{\bar{S}}, 0)))\in\mathbb{R}\times X\times X $ can be parameterized as

where $ (q(\tau), (\hat{\bar{S}}+\bar{S}_1(\tau), I_1(\tau))) $ is a smooth curve with respect to $ \tau $ and satisfies $ q(0) = q_* $, $ \hat{S}_1(0) = I_1(0) = 0 $.

3. There exists a connected component $ \Sigma $ of $ \bar{O} $ satisfying $ \Gamma\subseteq \Sigma $, and $ \Sigma $ possesses some properties as follows.

Case (I) Assume that $ \int^L_0\beta(x)dx > \int^L_0\gamma(x)dx $ and (C2) holds. Then there exists some endemic equilibrium $ (\hat{S}_*, \hat{I}_*) $ of (2.33) when $ q = 0 $ such that for $ \Sigma $, the projection of $ \Sigma $ to the $ q $-axis satisfies $ Proj_q \Sigma = [0, q_*] $ and the connected component $ \Sigma $ connects to $ (0, (\hat{S}_*, \hat{I}_*)) $.

Case (II) Assume that $ \int^L_0\beta(x)dx < \int^L_0\gamma(x)dx $. Then

(i) If (C1) holds and $ d_I > d^*_I $, then (2.33) has no positive solution for $ 0 < \min_{x\in [0, L]}a'(x) < q < q_* $ and for $ \Sigma $, the projection of $ \Sigma $ to the $ q $-axis satisfies $ Proj_q \Sigma = [q_*, \infty) $.

(ii) If (C2) holds and $ 0 < d_I < d^*_I $, then there exists some endemic equilibrium $ (\hat{S}_*, \hat{I}_*) $ of (2.33) when $ q = 0 $ such that for $ \Sigma $, the projection of $ \Sigma $ to the $ q $-axis satisfies $ Proj_q \Sigma = [0, q_*] $ and the connected component $ \Sigma $ connects to $ (0, (\hat{S}_*, \hat{I}_*)) $.

Proof. 1. Let $ F:\mathbb{R}^+\times X\times X\rightarrow Y\times Y\times \mathbb{R} $ be the mapping as follows.

It is to verify that the pair $ (\bar{S}, \bar{I}) $ is a solution of (2.33) if only if $ F(q, (\bar{S}, \bar{I})) = 0 $. Obviously, $ F(q, (\hat{\bar{S}}, 0)) = 0 $ for any $ q > \min_{x\in [0, L]}a'(x) > 0 $. The Fr$ \acute{e} $chet derivatives of $ F $ at $ (\hat{\bar{S}}, 0) $ are given by

If $ (\Phi_1, \Psi_1) $ is a nontrivial solution of the following problem

then $ (q_*, (\hat{\bar{S}}, 0))) $ is degenerate solution of (2.33). The second equation of (2.33) has a positive solution $ \Psi_1 $ only if $ q = q_* $ satisfies $ \mathcal{R}_0(d_I, q_*) = 1 $. And $ \Phi_1 $ satisfies

Obviously, $ \Phi_1 $ is uniquely determined by $ \Psi_1 $ in (2.36). Therefore, $ q = q_* $ is the only possible bifurcation point along $ \Gamma_S $ where positive solutions of (2.33) bifurcates and such $ q_* $ exists if and only if $ \mathcal{R}_0 = 1 $. We can obtain the necessary and sufficient conditions for the occurrence of bifurcation by Lemma 2.3.1 and Lemma 2.3.3.

2. At $ (q, (\bar{S}, \bar{I})) = (q_*, (\hat{\bar{S}}, 0)) $, the kernel

where $ (\Phi_1, \Psi_1) $ is the solution of (2.35) with $ q = q_* $. Up to a multiple of constant, $ (\Phi_1, \Psi_1) $ is unique. And the range of $ D_{(\bar{S}, \bar{I})}F(q_*, (\hat{\bar{S}}, 0)) $ is given by

and it is co-dimension one. By the result of Lemma 2.1.1, $ (\Psi_1)_x $ keeps one sign in $ (0, L) $ and $ \int_0^L(\Psi_1)_x\Psi_1e^{\frac{a(x)}{d_I}}dx\neq 0 $, which implies that

Therefore, using the local bifurcation theorem in ^[20] to $ F(q, (\bar{S}, \bar{I})) $ at $ (q_*, (\hat{\bar{S}}, 0)) $, we know that the set of positive solutions of (2.33) is a smooth curve

satisfying $ q(0) = q_* $, $ \bar{S}_1(\tau) = \tau \hat{\bar{S}}+o(|\tau|) $ and $ I_1(\tau) = o(|\tau|) $. Similar to the procedure in ^[21] and ^[22], (also see ^[23]), we can compute

Here $ l $ is the linear functional on $ Y\times Y\times \mathbb{R} $ defined by $ < l, [f, g, k] > = \int_0^Lg\Psi_1e^{\frac{a(x)}{d_I}}dx $.

3. By the global bifurcation theorem in ^[23] and ^[24], we can get the existence of the connected component $ \Sigma $. Moreover, $ \Sigma $ is either unbounded, or connects to another $ (q, (\hat{\bar{S}}, 0)) $, or $ \Sigma $ connects to another point on the boundary of $ O $.

Case (I) Assume that $ \int^L_0\beta(x)dx > \int^L_0\gamma(x)dx $ and (C2) holds. By Lemma 2.2.1 and the proof of part 2, we see that there exits a unique $ q_* $ such that the local bifurcation occurs at $ (q_*, (\hat{\bar{S}}, 0)) $ and $ q'(0) < 0 $, which means that the bifurcation direction is subcritical. Therefore, there exists some small $ \delta > 0 $ such that (2.33) has a positive solution if $ q_*-\delta < q < q_* $. By Lemma 2.1.4, $ \mathcal{R}_0 > 1 $ if $ q_*-\delta < q < q_* $ for $ \delta > 0 $ small enough. By Lemma 2.1.5, (2.33) has no positive solution if $ \mathcal{R}_0 < 1 $, which implies that (2.33) has no positive solution if $ q > q_* $. Consequently, the projection of $ \Sigma $ to the $ q $-axis $ Proj_q\Sigma \subset [0, q_*] $. And $ \Sigma $ must be bounded in $ \bar{O} $ because the positive solutions are uniformly bounded in $ L^{\infty} $ for $ 0\leq q\leq q_* $. So the third option must happen here. Hence $ \Sigma $ must connect to $ (0, (\bar{S}_*, \bar{I}_*)) $, so $ 0\in Proj_q\Sigma $. Here $ (\bar{S}_*, \bar{I}_*) $ is the unique endemic equilibrium of (2.33) when $ q = 0 $.

Case (II) Assume that $ \int^L_0\beta(x)dx < \int^L_0\gamma(x)dx $.

(i) If (C1) holds and $ d_I > d^*_I $, by Lemma 2.2.1 and the bifurcation analysis above, there exists unique bifurcation point $ q_* $ satisfying $ q'(0) > 0 $, which means the bifurcation direction is supercritical. Then there exists some small $ \delta > 0 $ such that (2.33) has a positive solution if $ q_* < q < q_*+\delta $. By Lemma 2.3.3, $ \mathcal{R}_0 > 1 $ if $ q_* < q < q_*+\delta $ for some $ \delta > 0 $ small enough. By Lemma 2.1.5, (2.33) has no positive solution if $ \mathcal{R}_0 < 1 $, which implies that (2.33) has no positive solution if $ 0 < q < q_* $. So the first option must happen here. If there exists some finite $ q^* > q_* $ such that $ Proj_q\Sigma = [q_*, q^*) $, then it contradicts to the fact that all positive solutions are uniformly bounded in $ L^{\infty} $ for $ q = q^* $. Consequently, the projection of $ \Sigma $ to the $ q $-axis $ Proj_q\Sigma = [q_*, \infty) $.

(ii) If (C2) holds and $ 0 < d_I < d^*_I $, the proof is similar to that of Case (I), we omit the details here.

We will give the Leray-Schauder degree argument.

Lemma 2.4.2. For any $ \epsilon > 0 $, there exist two constants $ \underline{C} $ and $ \bar{C} $ which depend on $ d_I $, $ \epsilon $, $ \|\beta\|_{\infty} $, $ \|\gamma\|_{\infty} $ and $ N $ such that if $ \mathcal{R}_0\neq 1 $, then for any positive solution of (2.33),

for any $ \epsilon\leq d_S\leq \frac{1}{\epsilon} $ and $ 0\leq q\leq \frac{1}{\epsilon} $.

Proof. $ \int_0^L[e^{\frac{a(x)}{d_S}}\bar{S}+e^{\frac{a(x)}{d_I}}\bar{I}]dx = N $ means that $ \bar{S}(x) $ and $ \bar{I}(x) $ are bounded in $ L^1 $ space. Using the standard theory of elliptic equation, it is easy to see that $ \bar{S} $ and $ \bar{I} $ have the upper bound $ \bar{C} $ depending on $ d_I $, $ \epsilon $, $ \|\beta\|_{\infty} $, $ \|\gamma\|_{\infty} $ and $ N $.

Therefore, we just need to prove that $ \bar{S} $ and $ \bar{I} $ have lower bounds.

Suppose contradictorily that there exist a sequence of $ \{(d_{S, i}, q_i)\}_{i = 1}^{\infty} $ satisfies $ \epsilon\leq d_{S, i}\leq \frac{1}{\epsilon} $ and $ 0\leq q_i\leq \frac{1}{\epsilon} $ and $ \mathcal{R}_0\neq 1 $, and $ \{(\bar{S}_i(x), \bar{I}_i(x))\}_{i = 1}^{\infty} $ are the corresponding positive solutions of (2.33) satisfying

and $ (\bar{S}_i(x), \bar{I}_i(x)) $ satisfies

Up to a subsequence, we assume that $ d_{S, i}\rightarrow d_S > 0 $ and $ q_i\rightarrow q\geq 0 $. Note that $ \|\bar{I}_i\|_{\infty} $ are uniformly bounded. Letting $ \tilde{\bar{I}}_i = \frac{\bar{I}_i}{\|\bar{I}_i\|_{\infty}} $, we have

By standard regularity and Sobolev embedding theorem in ^[25], up to a subsequence, $ \bar{I}_i\rightarrow 0 $ in $ C^1([0, L]) $ and there exists $ I^* > 0 $ such that $ \tilde{\bar{I}}_i\rightarrow I^* $ in $ C^1([0, L]) $ and $ \|I^*\|_{\infty} = 1 $. Since $ \bar{I}_i\rightarrow 0 $ in $ C^1([0, L]) $ and $ \int_0^L[e^{\frac{q_i}{d_{S, i}}\tilde{a}(x)}\bar{S}_i+e^{\frac{q_i}{d_I}\tilde{a}(x)}\bar{I}_i]dx = N $ implies that $ \bar{S}_i $ is bounded in $ L^1([0, L]) $, using the equation of $ \bar{S}_i $, we get $ \bar{S}_i\rightarrow \hat{\bar{S}} > 0 $ in $ C^1([0, L]) $. Letting $ i\rightarrow \infty $ in the equation of $ \bar{I}_i $, we have

Since $ I^* > 0 $, (2.39) means that $ 0 $ is the principle eigenvalue, which is a contradiction of the assumption of $ \mathcal{R}_0\neq 1 $ for any $ d_I > 0 $ and $ 0\leq q \leq \frac{1}{\epsilon} $. Therefore, there must exist some positive constant $ \underline{C} $ such that $ \max_{x\in [0, L]}I(x)\geq \underline{C} $. Similar to the argument in ^[26], by Harnack inequality, we have

for some constant $ C^* $ depending on $ d_I $, $ \epsilon $, $ \|\beta\|_{\infty} $, $ \|\gamma\|_{\infty} $ and $ N $, which implies that $ \bar{I}(x) $ has uniformly positive lower bound.

Now we prove that $ S(x) $ has a uniform positive lower bound. Let $ S(x_0) = \min_{x\in [0, L]}S(x) $. Using the minimum principle in ^[27], we have

Consequently,

and

which completes the proof.

Lemma 2.4.3. Assume that $ \beta(x)-\gamma(x) $ changes sign once in $ (0, L) $ and one of the following conditions holds:

(i) $ d_I > 0 $, $ q > \min_{x\in [0, L]}a'(x) > 0 $, $ \int_0^L\beta(x)dx > \int_0^L\gamma(x)dx $ and (C2) holds;

(ii) $ 0 < d_I < d^*_I $, $ q > \min_{x\in [0, L]}a'(x) > 0 $, $ \int_0^L\beta(x)dx < \int_0^L\gamma(x)dx $ and (C1) holds.

Then (2.33) has at least an endemic equilibrium.

Proof. Note that we can extend the ranges of $ f $ and $ g $ properly for any nonnegative pair $ (f, g)\in C([0, L])\times C([0, L]) $ such that the function $ \frac{fg}{e^{\frac{\tau a(x)}{d_S}}f+e^{\frac{\tau a(x)}{d_I}}g} $ is Lipschitz continuous for $ f, g\in \mathbb{R} $ and $ \tau\in [0, 1] $. Therefore we define the following compact operator family from $ C([0, L])\times C([0, L]) $ to $ C([0, L])\times C([0, L]) $:

Since the operator $ d_I\frac{d^2}{dx^2}+\tau a'(x)\frac{d}{dx}-\gamma(x) $ is invertible, then for any $ \tau\in [0, 1] $ and $ (f, g)\in C([0, L])\times C([0, L]) $, by the second equation of (2.40), $ v $ is uniquely determined. Substituting this $ v $ into the first and last equations of (2.40), $ u $ is also uniquely determined. Therefore, we can define $ \mathcal{G}_{\tau}(f, g): = (u, v) $.

Under conditions (i) and (ii), $ \mathcal{R}_{0, \tau} > 1 $ for any $ \tau\in [0, 1] $. Here

By the result of Lemma 2.4.2, for any $ \tau\in [0, 1] $, there exist two positive constant $ \bar{C} $ and $ \underline{C} $ depending on $ d_S $, $ d_I $, $ q $, $ \|\beta\|_{\infty} $, $ \|\gamma\|_{\infty} $ and $ N $ such that $ \underline{C}\leq u, v\leq \bar{C} $ for any solution of (2.40).

Let

Then $ (\bar{S}, \bar{I})\neq \mathcal{G}(\tau, (\bar{S}, \bar{I})) $ for any $ \tau\in [0, 1] $ and $ (\bar{S}, \bar{I})\in \partial D $, which implies that Leray-Schauder degree $ deg({\bf{I}}-\mathcal{G}(\tau, (\cdot, \cdot)), D, 0) $ is well defined, and it is independent of $ \tau $. Here $ {\bf{I}} $ is the identity map. Moreover, $ (\bar{S}, \bar{I}) $ is a solution of (2.33) if and only if $ (\bar{S}, \bar{I}) $ satisfies $ (\bar{S}, \bar{I}) = \mathcal{G}(1, (\bar{S}, \bar{I})) $. If $ (\bar{S}, \bar{I})\in D $ and $ ({\bf{I}}-\mathcal{G}(0, (\cdot, \cdot)))(\bar{S}, \bar{I}) = 0 $, then $ (\bar{S}, \bar{I}) $ is a positive solution of

By the result of ^[2], (2.41) has a unique positive solution $ (S_*, I_*) $ satisfying $ S_*+I_* = \frac{N}{L} $ if the basic reproduction number $ \hat{\mathcal{R}}_0 > 1 $. Linearizing (2.41) around $ (S_*, I_*) $, we get

Adding the first two equations of (2.42) and using the boundary condition $ \Phi_x = \Psi_x = 0 $, $ x = 0, L $, we get

Solving it, we have $ \Phi = -\Psi $. Substituting this relation into the first equation of (2.42), we obtain

Since $ I_* $ is a positive solution of (2.40), we know that $ -d_I\frac{d^2}{dx^2}+\frac{2L}{N}\beta(x)I_*+\gamma(x)-\beta(x) $ is a positive operator, so $ \mu > 0 $. Hence the unique positive solution $ (S_*, I_*) $ is linearly stable. Using Leray-Schauder degree index (see Theorem 1.2.8.1 in ^[28]), we obtain

Consequently, using the homotopy invariance of Leray-Schauder degree, we have

for $ (d_I, q)\in \Omega_{hh}^U\cup \Omega_{lh}^{U_1} $. By the properties of degree, $ \mathcal{G}(1, (\cdot, \cdot) $ has a fixed point in $ D $ if $ (d_I, q)\in \Omega_{hh}^U\cup \Omega_{lh}^{U_1} $, which implies that (2.33) has at least one positive solution.

2.5.   Properties of $ \mathcal{R}_0 $ when $ \beta(x)-\gamma(x) $ changes sign twice

In this section, we consider the properties of $ \mathcal{R}_0 $ when $ \beta(x)-\gamma(x) $ changes sign twice. We also need the results on the positive roots of $ F(\eta) $ which is defined as

for any given continuous function $ m(x) $ on $ [0, L] $.

Lemma 2.5.1. Assume that there exists $ 0 < x_1 < x_2 < L $ such that $ m(x_1) = m(x_2) = 0 $, i.e., $ m(x) $ change sign twice for $ x\in [0, L] $. Then

(i) If $ m(L) < 0 $ and $ \int_0^L\tilde{a}'(x)m(x)dx > 0 $, then $ F(\eta) $ has a unique positive root $ \eta_1 $ for $ \eta\in (0, +\infty) $ satisfying $ F'(\eta_1) < 0 $;

(ii) If $ m(L) > 0 $ and $ \int_0^L\tilde{a}'(x)m(x)dx < 0 $, then $ F(\eta) $ has a unique positive root $ \eta_1 $ for $ \eta\in (0, +\infty) $ satisfying $ F'(\eta_1) > 0 $;

(iii) If $ m(L) > 0 $ and $ \int_0^L\tilde{a}'(x)m(x)dx > 0 $, then $ F(\eta) $ has at most two positive roots for $ \eta\in (0, +\infty) $;

(iv) If $ m(L) < 0 $ and $ \int_0^L\tilde{a}'(x)m(x)dx < 0 $, then $ F(\eta) $ has at most two positive roots for $ \eta\in (0, +\infty) $.

Proof. We only prove part (i) and part (iii). The proofs of part (ii) and part (iv) are similar.

(i). Let $ G_1(\eta): = e^{-\tilde{a}(x_2)\eta}[\tilde{a}(x_1)F(\eta)-F'(\eta)] $ and the prime notation denote differentiation with respect to $ \eta $. Since $ m(L) < 0 $ and $ m(x) $ changes sign twice, it is easy to see that $ m(x) < 0 $ for $ x\in (0, x_1)\cup (x_2, L) $ and $ m(x) > 0 $ for $ x\in (x_1, x_2) $. Note that $ \tilde{a}(x) $ is increasing. We know that

for $ x\in (0, L) $ and $ x\neq x_i(i = 1, 2) $. As a result, for any $ \eta > 0 $, we have

which implies that $ G'_1(\eta) $ is a strictly increasing function for $ \eta\in (0, \infty) $. By Lemma 2.2.2 and $ m(L) < 0 $, $ F(\eta) < 0 $ for $ \eta > M $ if $ M $ is large enough. But $ F(0) = \int_0^L\tilde{a}'(x)m(x)dx > 0 $, so there exits at least a positive root of $ F(\eta) $. Let $ \eta_1 $ be the smallest positive one, then $ F'(\eta_1)\leq 0 $.

If $ F'(\eta_1) = 0 $, since

then

That is, $ \eta_1 $ is a strict local maximum value point of $ F(\eta) $, which is a contradiction. So $ F'(\eta_1) < 0 $. Now we will prove that $ \eta_1 $ is the unique positive root of $ F(\eta) $. Assume contradictorily that $ \eta_2 > \eta_1 $ is the first number such that $ F(\eta_2) = 0 $. Since $ F(\eta_1) = 0 $ and $ F'(\eta_1) < 0 $, then $ F(\eta) < 0 $ in $ (\eta_1, \eta_2) $, which implies that $ F'(\eta_2)\geq 0 $. By the definition of $ G_1(\eta) $, and noticing that $ F(\eta_1) = F(\eta_2) = 0 $, we have $ G_1(\eta_1) = -\tilde{a}(x_1)e^{\tilde{a}(x_2)\eta_1}F'(\eta_1) > 0 $ and $ G_1(\eta_2) = -\tilde{a}(x_1)e^{\tilde{a}(x_2)\eta_2}F'(\eta_2)\leq 0 $, which is a contradiction to the fact that $ G_1(\eta) $ is strictly increasing.

(iii) By Lemma 2.2.2 and $ m(L) > 0 $, we see that $ F(\eta) > 0 $ for $ \eta > M $ if $ M $ is large enough. Then either $ F(\eta) > 0 $ for any $ \eta > 0 $ or $ F(\eta) $ has positive roots in $ (0, \infty) $. Let $ G_2(\eta) = e^{-\tilde{a}(x_2)\eta}[F'(\eta)-\tilde{a}(x_1)F(\eta)] $ and $ \eta_1 $ be the first positive root of $ F(\eta) = 0 $. Similar to the proof of part (i), it is easy to prove that $ G_2(\eta) $ is strictly monotone increasing in $ (0, +\infty) $ and $ F'(\eta_1)\leq 0 $. We discuss in two cases.

Case 1: $ F'(\eta_1) = 0 $. We will show that $ \eta_1 $ is the unique positive root of $ F(\eta) $. Since

then $ F''(\eta_1)-[\tilde{a}(x_1)+\tilde{a}(x_2)]F'(\eta_1)+\tilde{a}(x_1)\tilde{a}(x_2)F(\eta_1) = F''(\eta_1) > 0 $. That is, $ F(\eta) $ attains a strict local minimum at $ \eta_1 $. Now we will prove that $ \eta_1 $ is the unique positive root of $ F(\eta) $. Assume contradictorily that $ \eta_2 > \eta_1 $ is the first number such that $ F(\eta_2) = 0 $. Since $ \eta_1 $ is a strict local minimum value point, we have $ F(\eta) > 0 $ in $ (\eta_1, \eta_2) $, which implies that $ F'(\eta_2)\leq 0 $. By the definition of $ G_2(\eta) $, and noticing that $ F(\eta_1) = F(\eta_2) = 0 $, we have $ G_2(\eta_1) = 0 $ and $ G_2(\eta_2) = e^{a(x_2)\eta_2}F'(\eta_2)\leq 0 $, which is a contradiction to the fact that $ G_2(\eta) $ is strictly increasing. So $ F(\eta) $ only has a unique positive root $ \eta_1 $ in this case.

Case 2. $ F'(\eta_1) < 0 $. Since $ F(\eta_1) = 0 $, so $ F(\eta) < 0 $ if $ \eta > \eta_1 $ and $ \eta $ close to $ \eta_1 $ enough. By Lemma 3.2 and $ m(L) > 0 $, $ F(\eta) > 0 $ for $ \eta > M $ if $ M $ is large enough. Therefore, there exists at least a root of $ F(\eta) = 0 $ in $ (\eta_1, \infty) $. Assume that $ \eta_2 $ is the first root of $ F(\eta) = 0 $ in $ (\eta_1, \infty) $. Then $ F(\eta) < 0 $ in $ (\eta_1, \eta_2) $ and $ F'(\eta_2)\geq 0 $. If $ F'(\eta_2) = 0 $, then

And $ F(\eta) $ attains a strict local minimum at $ \eta_2 $, which is a contradiction. Hence $ F'(\eta_2) > 0 $.

We need to show that there is no positive root of $ F(\eta) = ) $ for $ \eta > \eta_2 $. Assume contradictorily that there exists $ \eta_3 > \eta_2 $ such that $ F(\eta_3) = 0 $ and $ F(\eta) > 0 $ in $ (\eta_2, \eta_3) $. Then $ F'(\eta_3) < 0 $. And $ G_2(\eta_2) = e^{\tilde{a}(x_2)\eta_2}F'(\eta_2) > 0 $ and $ G_2(\eta_3) = e^{a(x_2)\eta_3}F'(\eta_3) < 0 $, which contradicts the fact that $ G_2(\eta) $ is strictly increasing. Therefore we have proved that there exists a unique $ \eta_2 > \eta_1 $ such that $ F(\eta_2) = 0 $ and $ F'(\eta_2) > 0 $.

Now we give the proof of Theorem 1.6 below.

Proof. We only prove part(i) and (iii). The proofs of (ii) and (iv) are similar.

Part (i): Similar to the proofs of Lemma 2.3.2 and 2.3.3, it is easy to prove that there exists some positive constant $ \Lambda $ which is independent of $ d_I $ and $ q $ and for each $ d_I > \Lambda $, there exists some $ \tilde{q} = \tilde{q}(d_I) $ which satisfies $ \mathcal{R}_0(d_I, \tilde{q}) = 1 $ and $ \frac{\tilde{q}}{d_I}\rightarrow \eta_0 $ as $ d_I\rightarrow \infty $. Here $ \eta_0 $ is the unique positive root of $ F(\eta) = 0 $.

Next, we will prove that

for any $ \tilde{q} $ satisfying $ \mathcal{R}_0(d_I, \tilde{q}) = 1 $ if $ d_I $ is large enough.

Let $ \tilde{\varphi} $ be the unique normalized eigenfunction of the eigenvalue $ \mathcal{R}_0(d_I, \tilde{q}) = 1 $, i.e., $ \max_{[0, L]}\tilde{\varphi} = 1 $ and

By (2.22), we have

Multiplying (2.43) by $ \int_0^x\tilde{\varphi}(s)ds $ and integrating it over $ (0, L) $, we get

Substitute it into (2.44), we obtain

As $ d_I\rightarrow \infty $, $ \frac{\tilde{q}}{d_I}\rightarrow \eta_0 $ and $ \tilde{\varphi}\rightarrow 1 $, we have

By Lemma 2.5.1(i),

Hence, there exists some constant $ Q > 0 $(dependent on $ d_I $) such that $ \mathcal{R}_0 > 1 $ for $ 0 < q < Q $ and $ \mathcal{R}_0 < 1 $ for $ q > Q $.

Part (iii). According to the results of Lemma 2.5.1(iii), we divide into three cases to prove it.

Case 1. $ F(\eta) > 0 $ for any $ \eta > 0 $. It is easy to show that there exists some positive constant $ \Lambda $ independent of $ d_I $ and $ q $ such that $ \mathcal{R}_0 > 1 $ for every $ d_I > \Lambda $ and any $ q > 0 $.

Case 2. $ F(\eta) $ has a unique positive root $ \eta_1 $ for $ \eta\in(0, +\infty) $ and $ F'(\eta_1) = 0 $. Similar to the proof of part (i), we can prove that there exists some positive constant $ \Lambda $ independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $, there exists some $ \tilde{q} = \tilde{q}(d_I) $ such that $ \mathcal{R}_0(d_I, \tilde{q}) = 1 $ and $ \frac{\tilde{q}}{d_I}\rightarrow \eta_0 $ as $ d_I\rightarrow \infty $, where $ \eta_0 $ is the unique positive root of $ F(\eta) = 0 $. Moreover, $ \frac{\partial \mathcal{R}_0}{\partial q}(d_I, \tilde{q}) = 0 $. Therefore there exists some positive constant $ \Lambda $ which is independent of $ d_I $ and $ q $ such that for every $ d_I > \Lambda $, there exists a constant $ Q > 0 $ dependent on $ d_I $ satisfying $ \mathcal{R}_0 = 1 $ for $ q = Q $ and $ \mathcal{R}_0 > 1 $ for $ q\in (0, Q)\cup (Q, \infty) $.

Case 3. $ F(\eta) $ has two positive roots $ \eta_1 $ and $ \eta_2 $($ \eta_1 < \eta_2 $) for $ \eta\in (0, +\infty) $ and $ F'(\eta_1) < 0 $, $ F'(\eta_2) > 0 $. Similar to the discussion of part (i), for each $ d_I > 0 $, there exist $ \tilde{q}_1 = \tilde{q}_1(d_I) $ and $ \tilde{q}_2 = \tilde{q}_2(d_I) $ such that $ \mathcal{R}_0(d_I, \tilde{q}_i) = 1(i = 1, 2) $ and $ \frac{\tilde{q}_1}{d_I}\rightarrow \eta_1 $, $ \frac{\tilde{q}_2}{d_I}\rightarrow \eta_2 $ as $ d_I\rightarrow \infty $. And

Consequently, there exist two constants $ Q_2 > Q_1 > 0 $ which depend on $ d_I $ and satisfy that $ \mathcal{R}_0 > 1 $ for $ q\in(0, Q_1)\cup (Q_2, \infty) $, $ \mathcal{R}_0 < 1 $ for $ q\in(Q_1, Q_2) $.

3.   Results

In this section, we will summarize the main results of this paper.

Theorem 1.1 gives some properties for the basic reproduction number $ \mathcal{R}_0 $ and Theorem 1.2 says that $ \mathcal{R}_0 = 1 $ is the watershed for judging whether the DFE is stable or not. Theorem 1.3 and Theorem 1.4 deal with the stable and unstable regions of the DFE. Theorem 1.5 establishes the existence of EE. Theorem 1.6 considers the results on (1.1) when $ \beta(x)-\gamma(x) $ changes sign twice in $ (0, L) $.

4.   Discussion

We only establish the results on (1.1) under the assumption of $ a'(x) > 0 $ in this paper. However, it is much more difficult to obtain the results on (1.1) if there exists some $ x_0\in (0, L) $ satisfying $ a'(x_0) = 0 $.

5.   Conclusion

Biologically, the influence of advection is from the upstream to the downstream, small diffusion or large advection tends to force the individuals to concentrate at the downstream end. Therefore, the disease persists for arbitrary advection rate if the habitat is a high-risk domain and the downstream end is a high-risk site. While the advection transports the individuals to a favorable location and thus it can help eliminate the disease if the downstream end is a low-risk site. In conclusion, when advection is strong or the diffusion is small, the disease will be eliminated if the downstream end is a low–risk site, while the disease will persist if the downstream end is a high–risk site.

Acknowledgments

The authors thank the anonymous referees for their helpful suggestions.

Xiaowei An was supported by Natural Science Foundation of China People's Police University(No.ZKJJPY201723).

Conflict of interest

All authors declare no conflicts of interest in this paper.