Hydrogeochemical assessment of mine water discharges from mining activity. Case of the Haut Beht mine (central Morocco)

1.   Introduction

There has been some work in the study of the relationship between persistent infection with hepatitis B virus and immune responses (see, e.g., ^[2]). Hepatitis B virus (HBV) is a major cause of various liver diseases around the world. Except acute and chronic hepatitis, it causes liver fibrosis and even hepatocellular carcinoma. When an adult gets first infected with the hepatitis B virus during the early period of six months, it is called an acute infection. On the other hand, innate immune responses on persons may drive huge effector immune cells (CD8 T cells, help T cells, B cells) against infection. It is probably due to such immune system, in clinical observations, only 5–10 percent of healthy adults will develop a chronic hepatitis B infection after they get infection. This motivates researchers to investigate the topic that whether antibodies against hepatitis B play a central role in virus clearance (see, e.g., ^[1,2]).

It is practically difficult to obtain experimental results in the study of the antibody response to hepatitis B virus (HBV) infection. Thus, developing suitable mathematical models is an alternative way since it can be used to estimate some crucial factors for the viral infection, and to explore possible mechanisms of protection and viral infection process (see, e.g., ^{[1,2,3,4,5,6,7]} and the references therein). We first mention a model of virus infection in the absence of antibody responses, namely, the following model consists of three compartments of populations, corresponding to target hepatocytes ($T$), infected hepatocytes ($I$), and virus ($V$).

The growth of target cells ($T$) in system (1.1) is described by a logistic term with carrying capacity $T_m$ and and maximal growth rate $r$ (see, e.g., ^[8,9]); target cells ($T$) also get infected at a rate $\beta VT$. Infected cells ($I$) are gained at rate $\beta VT$, and die at rate $\delta$. Infected cells ($I$) produce virus ($V$) at rate $\pi$, and virus clearance rate is denoted by $c$. Further, system (1.1) also assumes that infected class ($I$) can get recovery and move back into the target class at rate $\rho$.

In order to incorporate antibody response, the authors in ^[1] ignore the curing of infected cells by setting $\rho = 0$, and introduce two additional classes, free antibody ($A$) and virus-antibody complexes ($X$), into system (1.1). Then the governing system takes the following form:

The free antibody ($A$) is produced at rate $p_{A}$ proportional to the viral and subviral concentrations, and is degraded at rate $d_{A}$. Without virus, we also introduce a logistic term with maximum growth rate $r_{A}$ and carrying capacity $A_{m}$ for the antibody maintenance. In system (1.2), for simplicity, we have imposed the assumption that the concentration of subviral particles is proportional to the concentration of free virus $V$, and $\theta$ is a constant proportionality. Antigen clearance is caused by the constitution of antigen-antibody complexes. The binding rate with antigen-antibody is $k_{p}$ that causes the free antibody population to descend; $k_{m}$ represents the disassociation rate for antibody reacting to viral particles. The complexes ($X$) are produced by a productive combination rate $k_{p}$ and it decreases at a disassociation rate $k_{m}$ and a degradation rate $c_{AV}$. During infection, free virus ($V$) are gained at a rate $\pi$ and binding rate $k_{m}$ with complexes, and are degraded by a rate $c$ and binding rate $k_{p}$ with antibody.

In ^[1], the authors also mention that it can be a further topic in the investigation of spatial effects in HBV infection. In fact, spatial clustering of infected cells has recently been observed for hepatitis C virus (HCV) infection (see, e.g., ^[10]). The effects of spatial heterogeneity was also added to within-host HIV models, see ^[11,12]. Motivated by those previous works, we intend to consider system (1.2) with spatial variations. For this purpose, we add diffusion terms $D_A\Delta A$, $D_X\Delta X$ and $D_V\Delta V$ into the model, which reflects the spatial variations of free antibody ($A$), virus-antibody complexes ($X$) and free virus ($V$), respectively. Then the modified version of system (1.2) is as follows

Here, we consider a general bounded domain $\Omega \subset \mathbb{R}^3$ where virus and cells stay and interact, and pose zero-flux condition on the boundary of $\Omega$ (i.e., homogeneous Neumann boundary condition). The notation $\frac{\partial }{\partial \nu}$ denotes the differentiation along the outward normal $\nu$ to $\partial\Omega$. The location dependent parameters are continuous and strictly positive functions on $\bar{\Omega}$.

The dynamics of system (1.3) is challenging since there are no diffusion terms in the first two equations, resulting in the loss of compactness of the solution maps. In order to determine the disease-free steady state of system (1.3), we also need to investigate the following system:

It is easy to see that $T = 0$ and $T = T_{m}$ are two steady states of (1.4). However, the global dynamics of system(1.4) is still open to us, due to the loss of compactness of the solution maps. This stops us from using persistence theory in the investigation of the dynamics of system (1.3). Instead, we will focus on the study of the existence of the positive steady states of system (1.3), $(\hat{T}(x), \hat{I}(x), \hat{A}(x), \hat{X}(x), \hat{V}(x))$, which satisfies the following equations:

In view of the first two equations of (1.5), it follows that

Then

Substituting the second equality of (1.7) into the fifth equation of (1.5), we arrive at the following elliptic system

The standard approach in seeking for the positive steady states of system (1.8) is the bifurcation argument. Here, we are going to adopt another approach, using the persistence theory, to study the following parabolic system associated with (1.8):

where

If one can show that system (1.9) is uniformly persistent, then (1.9) must admit a positive steady state (see, e.g., [13,CH1]). We point out that the dynamics of systems (1.3) and (1.9) may be different, but they admit the same positive steady states. Thus, we will focus on the search for positive steady state(s) of system (1.9) via the establishment of uniform persistence of system (1.9).

2.   Persistence of HBV with antibody

Let $\mathbb{Y}: = C(\bar{\Omega}, \mathbb{R}^3)$ be the Banach space with the supremum norm $\parallel \cdot \parallel_{\mathbb{Y}}$. Define $\mathbb{Y}^{+}: = C(\bar{\Omega}, \mathbb{R}_{+}^3)$, then $(\mathbb{Y}, \mathbb{Y}^{+})$ is a strongly ordered space. By the similar arguments in [14,Lemma 2.2] (see also ^[15]), together with [16,Corollary 4] (see also [17,Theorem 7.3.1]), we have the following result:

Lemma 2.1. For every initial value function $\phi\in \mathbb{Y}^{+}$, system (1.9) has a unique mild solution $u(x, t, \phi)$ on $(0, \tau_{\phi})$ with $u(\cdot, 0, \phi) = \phi$, where $\tau_{\phi}\leq \infty$. Furthermore, $u(\cdot, t, \phi)\in \mathbb{Y}^{+}$, $\forall t\in(0, \tau_{\phi})$ and $u(x, t, \phi)$ is a classical solution of (1.9).

Next, we show that solutions of system (1.9) are ultimately bounded, and system (1.9) admits a compact attractor in $\mathbb{Y}^{+}$.

Lemma 2.2. For every initial value function $\phi\in \mathbb{Y}^{+}$, system (1.9) admits a unique solution $u(x, t, \phi)$ on $[0, \infty)$ with $u(\cdot, 0, \phi) = \phi$. Furthermore,

(ⅰ) $u(x, t, \phi)$ is ultimately bounded;

(ⅱ) The semiflow $\Psi(t):\mathbb{Y}^{+}\rightarrow\mathbb{Y}^{+}$ generated by (1.9) is defined by $\Psi(t)\phi = u(\cdot, t, \phi), \ t\geq 0$, which admits a global compact attractor in $\mathbb{Y}^{+}$, $\forall \ t \geq0$.

Proof. In view of (1.10), it is not hard to see that

Thus,

Setting

Then it follows from system (1.9) and (2.1) that

where $c_{\min}: = \min\{c_{AV}, c\}$. Thus, we have

Using (2.2) and the similar arguments to those in the end of [18,Proposition 2.3], we can show that $X(\cdot, t, \phi)$ and $V(\cdot, t, \phi)$ are ultimately bounded. Therefore, there exists $\hat{C} > 0$ and $t_1 > 0$ such that

In view of the first equation of system (1.9) and (2.3), it follows that

Then

where $\hat{A} > 0$ is a constant such that

From the above discussions, we see that $\Psi(t):\mathbb{Y}^{+}\rightarrow\mathbb{Y}^{+}$ is point dissipative. Obviously, $\Psi(t):\mathbb{Y}^{+}\rightarrow\mathbb{Y}^{+}$ is compact, $\forall\ t > 0$. It follows from [19,Theorem 3.4.8] that $\Psi(t):\mathbb{Y}^{+}\rightarrow\mathbb{Y}^{+}$, $t\geq0$, admits a global compact attractor.

Putting $X = V = 0$ into (1.9), we see that

It is easy to see that $A = 0$ is the trivial steady state solution of system (2.4). The stability of the trivial steady state solution $A = 0$ is determined by the following eigenvalue problem:

Assume that $\mu^0$ is the principal eigenvalue of system (2.5). By [20,Proposition 4.4], we see that $\mu^0 > 0$ if the following condition is satisfied

Thus, trivial steady state solution $A = 0$ is unstable for system (2.4) if condition (2.6) holds. If condition (2.6) is true, then one can use [13,Theorem 2.3.2] to show that system (2.4) admits a unique positive steady state $A^*(x)$ which is globally attractive. Thus, two possible steady states of system (1.9) are as follows:

and

Note that $E_0(x)$ always exists, and $E_1(x)$ exists when (2.6) holds. Linearizing system (1.9) around $E_1(x)$, we get the following cooperative system for the infectious compartments:

Substituting $X(x, t) = e^{\lambda t}\psi_{X}(x)$ and $V(x, t) = e^{\lambda t}\psi_{V}(x)$ into (2.7) and we get the associated eigenvalue problem:

It is not hard to see that the linear system (2.7) generates a strongly positive semigroup on $C(\overline{\Omega}, \mathbb{R}_+^2)$ (see, e.g., Section 4 of CH 7 in ^[17]). In addition, the semigroup associated with system (2.7) is compact. By a similar argument as in [17,Theorem 7.6.1], we have the following result which is related to the existence of the principal eigenvalue of (2.8):

Lemma 2.3. The eigenvalue problem (2.8) admits a principal eigenvalue, denoted by $\lambda^{0}$, which corresponds a strongly positive eigenfunction.

Next, we shall adopt the theory developed in [21,Section 3] to define the basic reproduction number for system (1.9). For this purpose, we assume

and

Let $\mathbf{w} = (X, V)^T$, $\mathbf{D} \Delta\mathbf{w} = (D_X\Delta X, D_V\Delta V)^T$, and $\mathbb{S}(t):C(\overline{\Omega}, \mathbb{R}^2)\rightarrow C(\overline{\Omega}, \mathbb{R}^2)$ be the $C_0$-semigroup generated by the following system

Assume that the state variables are near the disease-free steady state $E_1(x)$ and the distribution of initial infection is described by $\varphi\in C(\overline{\Omega}, \mathbb{R}^2)$. Then $\mathbb{S}(t)\varphi(x)$ represents the distribution of those infectious cases as time evolves to time $t$, and hence, the distribution of new infection at time $t$ is $\mathbb{F}(x)\mathbb{S}(t)\varphi(x)$. Let $\mathbb{L}:C(\overline{\Omega}, \mathbb{R}^2)\rightarrow C(\overline{\Omega}, \mathbb{R}^2)$ be defined by

It then follows that $\mathbb{L}(\varphi)(\cdot)$ represents the distribution of accumulated infectious cases during the infection period, and hence, $\mathbb{L}$ is the next generation operator. By the idea of next generation operators (see, e.g., ^[21,22,23]), we define the spectral radius of $\mathbb{L}$ as the basic reproduction number for system (1.9), that is,

From [24,Theorem 3.5] or [21,Theorem 3.1], the following observation holds.

Lemma 2.4. $\mathcal{R}_0-1$ and $\lambda^{0}$ have the same sign.

Next, we are going to find an explicit formula for $\mathcal{R}_0$ when coefficients of system (1.9) are all positive constants. For this special case, we see that $\mathbb{F}(x) = \mathbb{F}$ and $\mathbb{V}(x) = \mathbb{V}$, for all $x\in\bar{\Omega}$, and hence, $\mathcal{R}_0 = r(\mathbb{F}\mathbb{V}^{-1})$ (see e. g., [21,Theorem 3.4]). By direct computations, it follows that

Thus,

In the establishment of the persistence for (1.9), the following results will be necessary.

Lemma 2.5. For every initial value function $\phi\in \mathbb{Y}^{+}$, we assume that system (1.9) admits a unique solution $u(x, t, \phi)$ on $[0, \infty)$ with $u(\cdot, 0, \phi) = \phi$.

(ⅰ) If $\phi_{2}(\cdot)\not\equiv0$ and $\phi_{3}(\cdot)\not\equiv0$, then

(ⅱ) Assume that $\phi_{i}(\cdot)\not\equiv0$, for $i = 2, 3$. If there exists a $\sigma_1 > 0$ such that

Then there exists a $\sigma > 0$ such that

Proof. Part (ⅰ). By the positivity of solutions (see Lemma 2.1), it follows that $X(x, t)\geq 0, \ \forall \ x\in \overline{\Omega}, \ t\geq 0$. Suppose, by contradiction, there exists $x_1\in\overline{\Omega}$ and $t_1\in (0, \infty)$ such that $X(x_1, t_1) = 0$. Let $\tau_1 > 0$ be such that $t_1 < \tau_1$. Then $(x_1, t_1)\in\overline{\Omega} \times [0, \tau_1]$ and $X$ attains its minimum on $\overline{\Omega} \times [0, \tau_1]$ at the point $(x_1, t_1)$. In view of the second equation of (1.9), it follows that

In case $x_1\in \partial \Omega$, we apply the Hopf boundary lemma (see, e.g., [25,p. 170,Theorem 3]) and we have $\frac{\partial X(x_1, t_1, \phi)}{\partial\nu} < 0$, which is impossible. In case where $x_1\in\Omega$, then the strong maximum principle (see [25,p. 174,Theorem 7]) implies that

which contradicts the assumption that $\phi_{2}(\cdot)\not\equiv0$. Thus, $X(x, t, \phi) > 0, \ \forall\ x\in\bar{\Omega}, \ t > 0$. Similarly, we see that $V(x, t)\geq 0, \ \forall \ x\in \overline{\Omega}, \ t\geq 0$ (see Lemma 2.1). Suppose, by contradiction, there exists $x_2\in\overline{\Omega}$ and $t_2\in (0, \infty)$ such that $V(x_2, t_2) = 0$. Let $\tau_2 > 0$ be such that $t_2 < \tau_2$. Then $(x_2, t_2)\in\overline{\Omega} \times [0, \tau_2]$ and $V$ attains its minimum on $\overline{\Omega} \times [0, \tau_2]$ at the point $(x_2, t_2)$. Using the third equation of (1.9) and (1.10), it follows that

In case $x_2\in \partial \Omega$, we apply the Hopf boundary lemma (see, e.g., [25,p. 170,Theorem 3]) and we have $\frac{\partial V(x_2, t_2, \phi)}{\partial\nu} < 0$, which is a contradiction. In case where $x_2\in\Omega$, then the strong maximum principle (see [25,p. 174,Theorem 7]) implies that

which contradicts the assumption that $\phi_{3}(\cdot)\not\equiv0$. Thus, $V(x, t, \phi) > 0, \ \forall\ x\in\bar{\Omega}, \ t > 0$.

Claim. $A(x, t, \phi) > 0, \ \forall\ x\in\bar{\Omega}, \ t > 0$.

By Lemma 2.1, it follows that $A(x, t)\geq 0, \ \forall \ x\in \overline{\Omega}, \ t\geq 0$. Suppose, by contradiction, there exists $x_3\in\overline{\Omega}$ and $t_3\in (0, \infty)$ such that $A(x_3, t_3) = 0$. Let $\tau_3 > 0$ be such that $t_3 < \tau_3$. Then $(x_3, t_3)\in\overline{\Omega} \times [0, \tau_3]$ and $A$ attains its minimum on $\overline{\Omega} \times [0, \tau_3]$ at the point $(x_3, t_3)$. By the first equation of (1.9), it follows that

In case $x_3\in \partial \Omega$, we apply the Hopf boundary lemma (see, e.g., [25,p. 170,Theorem 3]) and we have $\frac{\partial A(x_3, t_3, \phi)}{\partial\nu} < 0$, which is a contradiction. In case where $x_3\in\Omega$, then the strong maximum principle (see [25,p. 174,Theorem 7]) implies that

This together with the first equation of (1.9) imply that

which is a contradiction. Thus, $A(x, t, \phi) > 0, \ \forall\ x\in\bar{\Omega}, \ t > 0$.

Part (ⅱ). From Lemma 2.2, we see that $V(x, t)$ is ultimately bounded. This together with assumption (2.13) imply that there exists $t_4 > 0$ and $C > 0$ such that

From the above inequalities and the first equation of (1.9), it follows that

Let $\underline{A} > 0$ satisfy the following inequality

By (2.16) and the standard parabolic comparison theorem (see, e.g., [17,Theorem 7.3.4]), we deduce that

Let $\sigma: = \min\{\sigma_1, \underline{A}\}$. Then (2.14) holds.

We show that $\mathcal{R}_0$ is an important index for the persistence of HBV in system (1.9).

Theorem 2.1. Assume that (2.6) holds. For every initial value function $u^0(\cdot) = (A^0, X^0, V^0)(\cdot)\in \mathbb{Y}^+$, we assume that system (1.9) admits a unique solution

on $[0, \infty)$ with $u(\cdot, 0, u^0) = u^0$. If $\mathcal{R}_0 > 1$, then system (1.9) admits at least one (componentwise) positive steady state $\hat{u}(x)$ and there exists a $\sigma > 0$ such that for any $u^0(\cdot)\in \mathbb{Y}^+$ with $X^0(\cdot)\not\equiv0$ and $V^0(\cdot)\not\equiv0$, we have

uniformly for $x\in\overline{\Omega}$.

Proof. Let

and

Recall that the semiflow $\Psi(t):\mathbb{Y}^{+}\rightarrow\mathbb{Y}^{+}$ generated by (1.9) is defined in Lemma 2.2. By Lemma 2.5 (ⅰ), it follows that for any $u^0(\cdot)\in \mathbb{W}_{0}$, we have

In other words, $\Psi(t)\mathbb{W}_{0}\subseteq \mathbb{W}_{0}, \ \forall \ t\geq 0.$ Let

and $\omega(u^0(\cdot))$ be the omega limit set of the orbit $O^{+}(u^0(\cdot)): = \{\Psi(t)u^0(\cdot):t\geq 0\}$.

Claim 1. $\omega(v^0(\cdot))\subseteq\{E_0(x)\}\cup \{E_1(x)\}, \ \forall \ v^0(\cdot)\in M_{\partial}$.

Since $v^0(\cdot)\in M_{\partial}$, we have $\Psi(t)v^0(\cdot)\in M_{\partial}, \ \forall \ t\geq0$, that is, $X(\cdot, t, v^0(\cdot))\equiv 0$ or $V(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t\geq 0.$

In case where $V(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t\geq 0$. Then it follows from the third equation in system (1.9) that $X(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t\geq 0$. Thus, $X(x, t, v^0(\cdot))$ satisfies system (2.4), and hence,

Thus,

In case where $V(\cdot, \hat{t}_0, v^0(\cdot))\not\equiv 0$, for some $\hat{t}_0\geq 0$. Then we can use similar arguments in Lemma 2.5 to show that $V(x, t, v^0) > 0$, for all $x\in\bar{\Omega}$ and $t > \hat{t}_0$, and hence, $X(\cdot, t, v^0)\equiv0$, for all $t > \hat{t}_0$. Then it follows from the second equation in system (1.9) that $A(\cdot, t, v^0(\cdot))V(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t > \hat{t}_0$. From the above discussions, it follows that $A(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t > \hat{t}_0$. Thanks to the first equation in system (1.9), it follows that $V(\cdot, t, v^0(\cdot))\equiv 0, \ \forall \ t > \hat{t}_0$. This is a contradiction, and hence, we cannot allow the possibility that $V(\cdot, \hat{t}_0, v^0(\cdot))\not\equiv 0$, for some $\hat{t}_0\geq 0$. Therefore, we complete the proof of Claim 1.

Recall that $\mu^0$ is the principal eigenvalue of the eigenvalue problem (2.5), and $\mu^0 > 0$ since (2.6) holds. By continuity, there is a $\delta_0 > 0$ such that $\mu_{\delta_0} > 0$, where $\mu_{\delta_0} > 0$ is the principal eigenvalue of the following eigenvalue problem:

Claim 2. $E_0(x)$ is a uniform weak repeller for $\mathbb{W}_{0}$ in the sense that

Suppose, by contradiction, that there exists $u^0(\cdot) \in \mathbb{W}_{0}$ such that

Then there exists $t_0 > 0$ such that

From the first equation of (1.9), we see that

Assume that $\varphi_{\delta_0}(x)$ is the positive eigenfunction corresponding to $\mu_{\delta_0}$, and there exists a $C_0 > 0$ such that

where we have used the fact that $A(x, t_0) > 0, \ \forall \ x\in\bar{\Omega}$ (see Lemma 2.5). The comparison principle and the inequality (2.19) imply that

Since $\mu_{\delta_0} > 0$, it follows that $A(x, t)$ is unbounded. This contradiction proves the Claim 2.

Since $\mathcal{R}_0 > 1$, it follows from Lemma 2.4 that $\lambda^{0} > 0$. By continuity of the principal eigenvalue, we can find an $\epsilon_1 > 0$ such that $\lambda_{\epsilon_1} > 0$, where $\lambda_{\epsilon_1}$ is the principal eigenvalue of the following eigenvalue problem:

By continuity of $f(V)$, we can choose a $\delta_1$ with $0 < \delta_1\leq\epsilon_1$ such that

Claim 3. $E_1(x)$ is a uniform weak repeller for $\mathbb{W}_{0}$ in the sense that

Suppose, by contradiction, there exists $u^0(\cdot) \in \mathbb{W}_{0}$ such that

Then there exists $t_1 > 0$ such that

and

From the second and third equations in system (1.9), it follows that

Assume that $(\psi_{X}^{\epsilon_1}(x), \psi_{V}^{\epsilon_1}(x))$ is the positive eigenfunction corresponding to $\lambda_{\epsilon_1}$, and there exists a $C_1 > 0$ such that

where we have used the fact that $X(x, t_1) > 0, \ V(x, t_1) > 0, \ \forall \ x\in\bar{\Omega}$ (see Lemma 2.5). The comparison principle and the inequality (2.22) imply that

Since $\lambda_{\epsilon_1} > 0$, it follows that $(X(x, t), V(x, t))$ is unbounded. This contradiction proves Claim 3.

Define a continuous function $\mathbb{P}:\mathbb{Y}^+ \rightarrow [0, \infty)$ by

By Lemma 2.5 (ⅰ), it follows that $\mathbb{P}^{-1}(0, \infty)\subseteq \mathbb{W}_{0}$ and $\mathbb{P}$ has the property that if $\mathbb{P}(u^0(\cdot)) > 0$ or $u^0(\cdot)\in \mathbb{W}_{0}$ with $\mathbb{P}(u^0(\cdot)) = 0$, then $\mathbb{P}(\Psi(t)u^0(\cdot)) > 0, \ \forall \ t > 0.$ That is, $\mathbb{P}$ is a generalized distance function for the semiflow $\Psi(t):\mathbb{Y}^+ \rightarrow \mathbb{Y}^+$ (see, e.g., ^[26]).

From the above claims, it follows that any forward orbit of $\Psi(t)$ in $M_{\partial}$ converges to $\{E_0(x)\}\cup \{E_1(x)\}$. For $i = 0, 1$, $\{E_i(x)\}$ is isolated in $\mathbb{Y}^+$ and $W^{s}(\{E_i(x)\})\cap \mathbb{W}_{0} = \emptyset$, where $W^{s}(\{E_i(x)\})$ is the stable set of $\{E_i(x)\}$ (see ^[26]). It is obvious that no subset of $\{E_0(x)\}\cup \{E_1(x)\}$ forms a cycle in $\partial \mathbb{W}_{0}$. By Lemma 2.2, the semiflow $\Psi(t):\mathbb{Y}^+\rightarrow\mathbb{Y}^+$ has a global compact attractor in $\mathbb{Y}^+$, $\forall \ t \geq0$. Then it follows from [26,Theorem 3] that there exists a $\sigma_1 > 0$ such that

Hence,

From Lemma 5 (ⅱ), there exists a $\sigma > 0$ such that (2.17) is valid. Hence, the uniform persistence stated in the conclusion (ⅱ) hold. By [27,Theorem 3.7 and Remark 3.10], it follows that $\Psi(t):\mathbb{W}_{0} \rightarrow \mathbb{W}_{0}$ has a global attractor $\mathcal{A}_0$. Using [27,Theorem 4.7], we deduce that $\Psi(t)$ admits a steady-state $\hat{u}(\cdot)\in \mathbb{W}_{0}$. By Lemma 2.5 (ⅰ), we can further conclude that $\hat{u}(\cdot)$ is a positive steady state of (1.9). The proof of Part (ⅱ) is finished.

3.   Elimination of HBV with antibody

In this section, we focus on the study of elimination of HBV with antibody. Due to technical reasons, we only consider a special case where we assume $k_{m} = 0$ in system (1.9), and the coefficients in (1.9) are all positive constants. Then the equation of $X$ in system (1.9) is decoupled from the other equations, and hence, it suffices to investigate the following system:

We see that two possible steady states of system (3.1) are as follows:

and

where $A^*: = A_{m}(1-\frac{d_A}{r_{A}}) > 0$, provided that $r_{A} > d_A$.

Linearizing system (3.1) around $\mathcal{E}_1$, we get the following scalar system

Substituting $V(x, t) = e^{\Lambda t}\psi(x)$ into (3.2), and we get the associated eigenvalue problem:

By the same argument in [17,Theorem 7.6.1], we can show that the eigenvalue problem (3.3) admits a principal eigenvalue, denoted by $\Lambda^{0}$, which corresponds a strongly positive eigenfunction $\psi^0(x)$. In fact, one can show that $\Lambda^0 = \pi f(0)-c -k_{p}A^*$ and the associated eigenfunction $\psi(\cdot)\equiv 1$. Note that one can also adopt the theory developed in [21,Section 3] to define the basic reproduction number, $\mathcal{R}_0^0$, for system (3.1). For this purpose, we assume $\mathbf{F} = \pi f(0)$ and $\mathbf{V} = c+k_{p}A^*$. By [21,Theorem 3.4], it follows that

Putting $k_{m} = 0$ in (2.12), and it is easy to see that $\mathcal{R}_0^0 = \mathcal{R}_0$ when $k_{m} = 0$. This is the reason why the reproduction number in this section is denoted by $\mathcal{R}_0^0$. Further, it is easy to observe that

We impose the following condition:

Let

Theorem 3.1. Assume that (3.5) holds. For any $(A^0(\cdot), V^0(\cdot))\in \mathcal{Y}_{P}$ with $A^0(\cdot)\not\equiv 0$, let $(A(\cdot, t), V(\cdot, t))$ be the solution of (3.1) with $(A(\cdot, 0), V(\cdot, 0)) = (A^0(\cdot), V^0(\cdot))$. If $\mathcal{R}_0^0 < 1$, then we have

Proof. Assume $\mathcal{R}_0^0 < 1$, that is, $\Lambda^{0} < 0$ (see (3.4)). Then there exists $\xi_0 > 0$ such that $\Lambda_{\xi_0} < 0$, where $\Lambda_{\xi_0}$ is the principal eigenvalue of the following eigenvalue problem:

The first equation of (3.1) can be rewritten as follows

From (3.5), we see that

Then it is not hard to show that $\mathcal{Y}_{P}$ is a positively invariant set for system (3.1). Thus,

In view of the first equation of (3.1), we see that

and hence,

Therefore, we may choose $t_1 > 0$ such that

In view of the second equation of (3.1), we see that

where we have used the fact that $f(V)\leq f(0), \ \forall \ V \geq 0$. Assume that $\psi_{\xi_0}(x)$ is a strongly positive eigenfunction corresponding to $\Lambda_{\xi_0}$, and there exists $\hat{C} > 0$ such that $V(x, t_1)\leq\hat{C}\psi_{\xi_0}(x), \ \forall \ x \in \overline{\Omega}$. From (3.8), the comparison principle implies that

Since $\Lambda_{\xi_0} < 0$, it follows that

Then $A(x, t)$ in (3.1) is asymptotic to system (2.4). Using $A^0(\cdot)\not\equiv 0$ and the theory for asymptotically autonomous semiflows (see, e.g., [28,Corollary 4.3]), we have

The proof is complete.

4.   Discussion

This study presents a reaction-diffusion system (1.3) modeling HBV infection, which consists of five compartments of populations, namely, target cells ($T$), infected cells ($I$), free virus ($V$), free antibody ($A$), and virus-antibody complexes ($X$). In system (1.3), we assume that only free virus ($V$), free antibody ($A$), and virus-antibody complexes ($X$) can diffuse, and the host cells (target and infected cells) do not have the ability to move. Thus, the governing equations are coupled by ODEs and PDEs. Due to the lack of diffusion terms of target cells ($T$) and infected cells ($I$) in (1.3), the steady-state solutions involved $T$ and $I$ can be explicitly expressed by free virus ($V$). Thus, investigating the existence of steady-state solutions of (1.3) is equivalent to the study of steady-state solutions of system (1.9).

The standard approach in seeking for positive steady-state solutions of system (1.9) is applying theory of bifurcation to the associated elliptic equations of (1.9). Instead, we adopt dynamical approach in the analysis of (1.9) in the current paper. We define an reproduction number, $\mathcal{R}_0$, for system (1.9), and we show that system (1.9) is uniformly persistent and it admits at least one (componentwise) positive steady state when $\mathcal{R}_0 > 1$ (see Theorem 2.1). Mathematically, it is more difficult to investigate the elimination of HBV in system (1.9). Putting $k_{m} = 0$ in system (1.9), the equation of $X$ in (1.9) is decoupled from the other equations, and we directly study the system (3.1) for the extinction case of HBV. Imposing the assumption (3.5), we can show that HBV will die out for (3.1) if the associated reproduction number $\mathcal{R}_0^0$ is less than one (Theorem 3.1). Here, we also raise some challenging problems related to system (1.9), which can be future research directions:

● The impact of the diffusion coefficients $D_X$ and $D_V$ on the basic reproduction number $\mathcal{R}_0$;

● The dynamics of system (1.9) for the critical case when $\mathcal{R}_0 = 1$;

● The uniqueness and the global attractiveness of the positive steady state of system (1.9) if it exists;

● The asymptotic profile of positive steady state of system (1.9) when the diffusion rates $D_X$ and $D_V$ both tend to zero.

In order to simplify the modeling in system (1.3), we have ignored two compartments of populations, namely, free subviral particles ($S$) and subviral particles-antibody complexes ($X_{s}$) in ^[1] by assuming that subviral particles $S$ (resp. subviral particles-antibody complexes $X_{s}$) is proportional to the concentration of free virus $V$ (resp. virus-antibody complexes $X$) with a constant proportionality $\theta$. The authors in ^[1] developed another more complete model about HBV infection with antibody, which includes the interactions of target cells ($T$), infected cells($I$), free subviral particles ($S$), free antibody ($A$), virus-antibody complexes ($X$), subviral particles-antibody complexes ($X_{s}$), and free virus ($V$). After we add spatial variations into such system, we shall investigate the following more realistic and challenging case in the future:

The meanings of the parameters in system (4.1) were collected in [1,Table 1].

Acknowledgments

We are grateful to three anonymous referees for their careful reading and helpful suggestions which led to significant improvements of our original manuscript. Research of FBW is supported in part by Ministry of Science and Technology, Taiwan; and National Center for Theoretical Sciences (NCTS), National Taiwan University; and Chang Gung Memorial Hospital (BMRPD18, NMRPD5J0201 and CLRPG2H0041). YCS is partially supported by Chang Gung Memorial Hospital (CLRPG2H0041). CLL is partially supported by Chang Gung Memorial Hospital (CRRPG2B0185, CRRPG2H0041, CRRPG2H0081, CLRPG2H0041).

Conflict of interest

The authors declare there is no conflicts of interest.