Transmission dynamics to a spatially diffusive Tuberculosis model subject to age-since-infection

Abstract

Distinct Tuberculosis models with one features of age structure and spatial diffusion have been put forward; however, few Tuberculosis models take all two into account. The main objective of this work is to analyse the recent result (the transmission dynamics) on the long-time behaviour of solutions to the model arising from the spreading of Tuberculosis with the fast and slow progression. Such model is traditionally given by ordinary differential equation system. Here, the local diffusion term, which is used to represent the random walk of the population in a connected domain, and age-since-infection, which is employed to describe the contamination process, are introduced. First, one obtains the well-posedness of the model. Second, one denotes the basic reproduction number through the spectral radius of a compact positive linear operator, which determines the dichotomy of disease persistence and extinction. Third, one proves the global dynamics of the model.

1 Introduction

1.1 Research background and current situation

Tuberculosis is a bacterial disease caused by Mycobacterium tuberculosis and is usually acquired through airborne infection from active Tuberculosis cases [1,2,3]. In [3, 4], Blower et al. formulate and analyse mathematical models describing the transmission dynamics of untreated tuberculosis epidemics. It was assumed in [3, 4] that infected individuals remain non-infectious until they develop disease by one of two pathogenic mechanisms: direct progression or endogenous reactivation. Two types of tuberculosis contribute to the incidence rate of tuberculosis disease: one type of tuberculosis develops through direct progression soon after infection which is named fast progression and the other type of tuberculosis develops through endogenous reactivation in the latently infected individuals which is named slow progression.

As everyone knows, Tuberculosis is a familiar and deadly infectious disease, which has developed into a chronic infectious disease threatening human health all over the world. In 2016, an estimated 10 million 400 thousand cases of Tuberculosis were equivalent to 140 cases per 100 thousand population. At the same time, the proportion of people suffering from Tuberculosis and dying of the disease (fatality rate) is 16%. Hence, Tuberculosis has become a social and public health issue of global concern [5].

Overall, half of those affected by Tuberculosis face catastrophic costs (more than 20% of household income) due to Tuberculosis and are distributed in 16 countries, ranging from 19 to 83%. For drug-resistant Tuberculosis patients, this proportion rose to 80%. In most of the countries surveyed, the poorest households are more than 20% more likely to face catastrophic costs. Tuberculosis is a poor disease. The most at risk are often those who have the most problems with access to health services and are most adversely affected by the high out of pocket costs of health care. If there are no strong mitigation measures, including social protection, a higher proportion of Tuberculosis patients and their families will face a risk of catastrophic costs [6].

Many mathematical models have been become the helpful tools attempting to obtain a better understanding of the spread and control of Tuberculosis. For the past few years, many scholars have done a lot of research on the transmission mechanism and prevention strategy of Tuberculosis, such as [7,8,9,10,11,12,13].

Now, age is one of the most significant and fashionable variables constituting a population. In short, at the individual level, many internal variables inevitably account on age, since different ages mean different reproductive and survival abilities, as well as different behaviours. Therefore, with a deeper understanding of age, some age-dependent Tuberculosis models have been proposed, analysed and studied, such as [14,15,16,17]. Besides, at different age stages, the effects of Tuberculosis transmission are various, which is another important and key factor that needs necessarily to be included in modelling this Tuberculosis transmission process.

Because the population distribute heterogeneously in diverse spatial location in the real life and they will move or diffuse for many reasons, there is increasing testimony that environmental heterogeneity and individual motility have momentous influence on the spread of Tuberculosis.

Owing to the uneven distribution of people in different spatial locations in real life, and the migration or spread of people due to a variety of reasons, more and more evidence in epidemiology shows that environmental heterogeneity and individual initiative have an significant impact on the spread of infectious diseases [18, 19]. In recent years, the global behaviour of the spatial diffusion system for Tuberculosis and other diseases has attracted extensive attention and become one of the research hotspots. Spatial diffusion is an intrinsic properties for studying the roles of spatial heterogeneity on Tuberculosis mechanisms and transmission routes and can lead to rich dynamics. Based on this reality, one generalizes (1.1)–(1.7) by taking account of the case that individuals move or diffuse around on the spatial habitat $x \in \Omega \subset {\mathbb {R}}^n$ with smooth boundary $\partial \Omega$. However, there are relatively few works on Tuberculosis models with both of the infection age and spatial diffusion.

1.2 Mathematical model

Inspired by the above discussions, in this article, one will do with the following spatially diffusive Tuberculosis model version possessing age-since-infection which is generalization of the model investigated in [5] for the first time to allow for individuals moving around on the spatial habitat $x \in \Omega \subset {\mathbb {R}}^n$ with smooth boundary $\partial \Omega$, and is very necessary and reasonable. In order to consider the dynamical structure of Tuberculosis model, the total population is decomposed into four compartments: the susceptible compartment (S) (individuals not infected but capable of infection), the exposed compartment (E) (individuals infected but not yet infectious, i.e. undetected non-symptomatic (latent) carriers), the infectious compartment (I) (individuals capable of transmitting the disease to susceptibles, i.e. symptomatic infectious individuals), and the removed compartment (R) (individuals who have died or who have recovered with permanent immunity). One assumes that the total population is constant and confined to a bounded spatial domain $\Omega$ in $\mathrm {R}^n$ with smooth boundary $\partial \Omega$.

Let i(x, t, a) be the density function of the infected population, where x is a point of space in $\Omega$, t is the time elapsed since the beginning of the Tuberculosis, and a is the infective age (i.e. the time elapsed since acquiring the infection), S(x, t), E(x, t), I(x, t), and R(x, t) be the density functions of the compartment (S), (E), (I), and (R), individually, $\tau$ be the length of the incubation period and $\hat{\tau }$ the length of the disease duration. Evidently,

$$\begin{aligned} E(x, t) = \int \limits _0^{\tau } i(x, t, a) \mathrm {d} a, I(x, t) = \int \limits _{\tau }^{\tau + \hat{\tau }} i(x, t, a) \mathrm {d} a. \end{aligned}$$

Remark 1.1

These formulas are applicable only to the case where the incubation period is constant. Moreover, the age a in (1.3) is different from that in the above formulas, that is the age a in (1.3) stands for the age from which the individual became infected and the incubation period is not included in a.

If one takes time frames that are comparable with the life span of the infectious individuals into consideration, then one can hypothesize that the length of the disease duration will be infinite. Namely,

$$\begin{aligned} I(x, t) = \int \limits _{0}^{+ \infty } i(x, t, a) \mathrm {d} a, \end{aligned}$$

and the total population N(x, t) is given by

$$\begin{aligned} N(x, t) = S(x, t) + E(x, t) + \int \limits _0^{+ \infty } i(x, t, a) \mathrm {d}a + R(x, t). \end{aligned}$$

Obviously , the travel of latent individuals demonstrating no symptoms can spread the Tuberculosis geographically which makes Tuberculosis harder to control.

Fig. 1

The transfer diagram of system (1.1)–(1.7)

The transfer diagram of the model is shown in Fig. 1. The transfer diagram leads to the following system of reaction–diffusion equations.

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + \delta - S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a - \mu S(x, t), \end{aligned}$$

(1.1)

$$\begin{aligned} \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + \overbrace{ (1 - q) S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a }^{\text {slow} ~ \text {progression}} - (\mu + \varepsilon ) E(x, t), \end{aligned}$$

(1.2)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) i (x, t, a)&= D_3 \frac{\partial ^2}{\partial x^2} i(x, t, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (x, t, a), \end{aligned}$$

(1.3)

$$\begin{aligned} i (x, t, 0)&= \underbrace{q S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a}_{\text {fast} ~ \text {progression}} + \varepsilon E(x, t), \end{aligned}$$

(1.4)

$$\begin{aligned} \frac{\partial }{\partial t} R(x, t)&= D_4 \frac{\partial ^2}{\partial x^2} R(x, t) + \int \limits _0^{+ \infty } \gamma (a) i(x, t, a) \mathrm {d}a - \mu R(x, t), \end{aligned}$$

(1.5)

corresponding to initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), i(x, 0, a), R(x, 0) \right] \nonumber \\&\quad = \left[ \varphi _1(x), \varphi _2(x), \varphi _3(x, a), \varphi _4(x) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

(1.6)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} i(x, t, a) = \frac{\partial }{\partial \mathbf {n}} R(x, t) = 0, x \in \partial \Omega . \end{aligned}$$

(1.7)

All the parameters of system (1.1)–(1.7) are positive constants.

• $\vartriangleright$ $D_1$ is the diffusion coefficient of susceptible individuals.

• $\vartriangleright$ $\delta$ is the constant recruitment rate of the population (comprising the birth and immigration).

• $\vartriangleright$ $\beta (a)$ is the transmission coefficient of Tuberculosis which relies upon age-since-infection a, and $\beta (\cdot ) \in L^{\infty } (0, + \infty )$.

• $\vartriangleright$ $\mu$ is the natural death rate independent of age-since-infection a of population.

• $\vartriangleright$ $D_2$ is the diffusion coefficient of undetected non-symptomatic (latent) carriers.

• $\vartriangleright$ q is the proportion of disease by fast progression.

• $\vartriangleright$ $\varepsilon$ is the progression rate from the exposed individuals to the infected individuals.

• $\vartriangleright$ $D_3$ is the diffusion coefficient of symptomatic infectious individuals with age-since-infection a.

• $\vartriangleright$ $\gamma (a)$ is the recovery rate which depends on age-since-infection a, and $\gamma (\cdot ) \in L^{\infty } (0, + \infty )$.

• $\vartriangleright$ $\mu _1(a)$ is the natural death rate which accounts upon age-since-infection a, and $\mu (\cdot ) \in L^{\infty } (0, + \infty )$.

• $\vartriangleright$ d(a) is the additional death rate induced by the Tuberculosis which relies on age-since-infection a, and $d(\cdot ) \in L^{\infty } (0, + \infty )$.

• $\vartriangleright$ i(x, t, 0) is used to reflect the resources fluxing into compartment, because the infection occurs at age 0.

• $\vartriangleright$ $\Omega \in {\mathbb {R}}^n$ is a spatial habitat with smooth boundary $\partial \Omega$.

• $\vartriangleright$ $\mathbf {n}$ is the outward normal to $\partial \Omega$, where $\Omega$ is bounded and connected.

• $\vartriangleright$ $\frac{\partial }{\partial \mathbf {n}}$ is the differentiation along the outward unit normal $\mathbf {n}$.

The homogeneous Neumann boundary conditions indicate that there is no population flux across the boundary $\partial \Omega$.

Since the first four equations of system (1.1)–(1.7) are independent of R(x, t), one only needs to consider the following system:

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + \delta - S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a - \mu S(x, t), \end{aligned}$$

(1.8)

$$\begin{aligned} \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + \overbrace{ (1 - q) S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a }^{\text {slow} ~ \text {progression}} - (\mu + \varepsilon ) E(x, t), \end{aligned}$$

(1.9)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) i (x, t, a)&= D_3 \frac{\partial ^2}{\partial x^2} i(x, t, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (x, t, a), \end{aligned}$$

(1.10)

$$\begin{aligned} i (x, t, 0)&= \underbrace{q S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a}_{\text {fast} ~ \text {progression}} + \varepsilon E(x, t), \end{aligned}$$

(1.11)

associating with initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), i(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _1(x), \varphi _2(x), \varphi _3(x, a)\right] , a \ge 0, x \in \Omega , \end{aligned}$$

(1.12)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} i(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(1.13)

For systems (1.8)–(1.13), there are two main reduction forms as below:

(1) Spatially diffusive Tuberculosis model.

If $i(x, t, a) = i(x, t)$ which is not relying upon the age parameter, then systems (1.8)–(1.13) will reduce to the spatially diffusive Tuberculosis model as follows:

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + \delta - \beta S(x, t) i(x, t) - \mu S(x, t), \end{aligned}$$

(1.14)

$$\begin{aligned} \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + (1 - q) \beta S(x, t) i(x, t) - (\mu + \varepsilon ) E(x, t), \end{aligned}$$

(1.15)

$$\begin{aligned} \frac{\partial }{\partial t} i(x, t)&= D_3 \frac{\partial ^2}{\partial x^2} i(x, t) + q \beta S(x, t) i(x, t) + \varepsilon E(x, t) - \left( \gamma + \mu + d \right) i (x, t), \end{aligned}$$

(1.16)

relating to initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), i(x, 0)\right] \nonumber \\&\quad = \left[ \varphi _1(x), \varphi _2(x), \varphi _3(x)\right] , x \in \Omega , \end{aligned}$$

(1.17)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} i(x, t) = 0, x \in \partial \Omega . \end{aligned}$$

(1.18)

(2) Age-dependent Tuberculosis model.

If $i(x, t, a) = i(t, a)$ which is not depending upon the spatial variable, then systems (1.8)–(1.13) will reduce to the age-dependent Tuberculosis model as follows:

$$\begin{aligned} \frac{\partial }{\partial t} S(t)&= \delta - S(t) \int \limits _0^{+ \infty } \beta (a) i(t, a) \mathrm {d}a - \mu S(t), \end{aligned}$$

(1.19)

$$\begin{aligned} \frac{\partial }{\partial t} E(t)&= (1 - q) S(t) \int \limits _0^{+ \infty } \beta (a) i(t, a) \mathrm {d}a - (\mu + \varepsilon ) E(t), \end{aligned}$$

(1.20)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) i (t, a)&= - \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (t, a), \end{aligned}$$

(1.21)

$$\begin{aligned} i (t, 0)&= q S(t) \int \limits _0^{+ \infty } \beta (a) i(t) \mathrm {d}a + \varepsilon E(t), \end{aligned}$$

(1.22)

$$\begin{aligned} S(0)&= S_0 \ge 0, E(0) = E_0 \ge 0, i(t, \cdot ) = i_0 \in L^1 ((0, + \infty ), {\mathbb {R}}). \end{aligned}$$

(1.23)

For systems (1.14)–(1.18) and (1.19)–(1.23), nowdays, there have some good results. Namely, in [20], authors obtain that there exist a traveling wave solutions (TWS) for the model if the threshold number $\Re _0 > 1$ and $c > c^*$, where $c^*$ is the minimum wave speed by using sub-super solution method, Schauders fixed point theorem and Lyapunov functional. In [3], authors show that the transmission dynamics of the disease is fully determined by the basic reproduction number. Besides, authors establish the local stability of a disease-free steady state and an endemic steady state of the model by analysing corresponding characteristic equations. Meanwhile, authors prove the system is uniformly persistent when the basic reproduction number is greater than unity by using the persistence theory for infinite dimensional system. Finally, authors verify that the global dynamics of the system is completely determined by the basic reproduction number by constructing suitable Lyapunov functionals and using LaSalles invariance principle. The major task of this article is to explore the transmission dynamics of system (1.8)–(1.13).

One organizes this manuscript below.

• $\vartriangleright$ One gives the well-posedness of system (1.8)–(1.13) in part 2.

• $\vartriangleright$ One denotes the basic reproduction number of system (1.8)–(1.13) in part 3.

• $\vartriangleright$ One discusses the local dynamics of system (1.8)–(1.13) in part 4.

• $\vartriangleright$ One considers the disease persistence of system (1.8)–(1.13) in part 5.

• $\vartriangleright$ One studies the global dynamics of system (1.8)–(1.13) in part 6.

• $\vartriangleright$ One provides some discussions in part 7.

2 Well-posedness

Consider Banach spaces

which are equipped with the norm

Next, the positive cones of ${\mathbb {X}}_1$, ${\mathbb {X}}_2$ are defined by ${\mathbb {X}}_1^+$, ${\mathbb {X}}_2^+$. Let

$$\begin{aligned} \chi (a) = e^{- \int \limits _0^a \left[ \gamma (s) + \mu _1(s) + d(s) \right] \mathrm {d} s}, \end{aligned}$$

$G: {\mathbb {X}}_1 \rightarrow {\mathbb {X}}_1$ be the $C_0$ semigroup generated by $D_3 \frac{\partial ^2}{\partial x^2}$ subject to the homogeneous Neumann boundary condition. According to [21], one has

$$\begin{aligned} (G(t)[\xi ])(x) = \int \limits _{\Omega } G(y, x, t) \xi (y) \mathrm {d} y, \quad \forall t > 0, \xi \in {\mathbb {X}}_1, \end{aligned}$$

where G(y, x, t) is the Green function. Due to [22], one knows that G is compact and strongly positive for $\forall t > 0$. Via directly solving the equations (1.10) by the way of the characteristic line $t - a = c$ (as shown in Fig. 2), where c is a constant, we have

$$\begin{aligned} i(x, t, a) = {\left\{ \begin{array}{ll} \chi (a) \int \limits _{\Omega } G (y, a, x) i(y, t - a, 0) \mathrm {d} y, &{} t - a > 0, x \in \Omega , \\ \frac{\chi (a)}{\chi (a - t)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a - t) \mathrm {d} y, &{} a - t \ge 0, x \in \Omega . \end{array}\right. } \end{aligned}$$

(2.1)

Fig. 2

The characteristic line

Set

then substituting (2.1) into systems (1.8)–(1.13) derives

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + \delta - \frac{1}{q} j(x, t) + \frac{1}{q} \varepsilon E(x, t) - \mu S(x, t), \end{aligned}$$

(2.2)

$$\begin{aligned} \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + \frac{1 - q}{q} j(x, t) - \left( \mu + \frac{1}{q} \varepsilon \right) E(x, t), \end{aligned}$$

(2.3)

$$\begin{aligned} j(x, t)&= q S(x, t) \left[ h_1(x, t) + h_2(x, t) \right] + \varepsilon E(x, t), \end{aligned}$$

(2.4)

associating with initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), j(x, 0) \right] \nonumber \\&\quad = \left[ \varphi _1(x), \varphi _2(x), q \varphi _1(x) \int \limits _0^{+ \infty } \beta (a) \varphi _3(x, a) \mathrm {d}a + \varepsilon \varphi _2(x) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

(2.5)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} j(x, t) = 0, x \in \partial \Omega \end{aligned}$$

(2.6)

with

$$\begin{aligned} h_1(x, t)&= \int \limits _0^t \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) j(y, t - a) \mathrm {d} y \mathrm {d} a, \\ h_2(x, t)&= \int \limits _t^{+ \infty } \beta (a) \frac{\chi (a)}{\chi (a - t)} \int \limits _{\Omega } G (y, x, t) \varphi _3 (y, a - t) \mathrm {d} y \mathrm {d} a. \end{aligned}$$

Next, one concentrates on the well-posedness of system (2.2)–(2.6).

Theorem 2.1

For $\forall (\varphi _1, \varphi _2, \varphi _3) \in {\mathbb {X}}_1^+ \times {\mathbb {X}}_2^+ \times {\mathbb {X}}_2^+$, system (2.2)–(2.6) exists an only solution (S, E, j) on $\overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty$.

Proof

Set

which is taken the norm

Solving Eqs. (2.2), (2.3) in $(x, t) \in \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty$, one gets

$$\begin{aligned} S(x, t)&= h_3(x, t) + \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \hat{G} (y, x, t - a) \left[ \delta - \frac{1}{q} j(y, a) + \frac{1}{q} \varepsilon E(y, a) \right] \mathrm {d} y \mathrm {d} a, \end{aligned}$$

(2.7)

$$\begin{aligned} E(x, t)&= h_4(x, t) + \frac{1 - q}{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \tilde{G} (y, x, t - a) j(a, y) \mathrm {d} y \mathrm {d} a, \end{aligned}$$

(2.8)

$$\begin{aligned} j(x, t)&= q S(x, t) \left[ h_1(x, t) + h_2(x, t) \right] + \varepsilon E(x, t), \end{aligned}$$

(2.9)

where

$$\begin{aligned} h_3(x, t)&= e^{- \mu t} \int \limits _0^{\pi } \hat{G} (y, x, t) \varphi _1 (y) \mathrm {d} y, \\ h_4(x, t)&= e^{- \left( \mu + \frac{1}{q} \varepsilon \right) t} \int \limits _0^{\pi } \tilde{G} (y, x, t) \varphi _2 (y) \mathrm {d} y \end{aligned}$$

with $\hat{G}$, $\tilde{G}$ are the Green function concerning $D_1 \frac{\partial ^2}{\partial x^2}$, $D_2 \frac{\partial ^2}{\partial x^2}$ subject to the homogeneous Neumann boundary condition. Substituting Eqs. (2.7), (2.8) into (2.9) allows us to denote $H: {\mathbb {X}}_3 \rightarrow {\mathbb {X}}_3$ as:

(2.10)

For convenience, one lets

$$\begin{aligned} h_5(x, t)&= \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \hat{G} (y, x, t - a) \left[ \delta - \frac{1}{q} j(y, a) + \frac{1}{q} \varepsilon E(y, a)\right] \mathrm {d} y \mathrm {d} a, \\ h_6(x, t)&= \frac{1 - q}{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \tilde{G} (y, x, t - a) j(y, a) \mathrm {d} y \mathrm {d} a, \end{aligned}$$

then (2.10) becomes

Via standard procedures, so as to yield a strict contraction mapping H in ${\mathbb {X}}_3$, one puts

Then one has

$$\begin{aligned} \left| H [j_1] - H [j_2] \right|&= \left| q h_3 h_1 \left( \hat{j} \right) + \tilde{h}_5 \left( \hat{j} \right) h_1 (j_2) + h_5 (j_1) h_1 \left( \hat{j} \right) + \varepsilon h_6 \left( \hat{j} \right) \right| \\&\le \left| (q h_3 + h_5) \tilde{h}_1 + \tilde{\tilde{h}}_5 h_1 + \varepsilon \tilde{h}_6 \right| \cdot \sup \limits _{0 \le s \le t} \left| \hat{j} (s, \cdot ) \right| _{{\mathbb {X}}_1}, \end{aligned}$$

where

$$\begin{aligned} \tilde{h}_5 (x, t)&= - \frac{1}{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \hat{G} (y, x, t - a) \hat{j}(y, a) \mathrm {d} y \mathrm {d} a,\\ \tilde{h}_1(x, t)&= \int \limits _0^t \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) \mathrm {d} y \mathrm {d} a,\\ \tilde{\tilde{h}}_5 (x, t)&= - \frac{1}{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \hat{G} (y, x, t - a) \mathrm {d} y \mathrm {d} a,\\ \tilde{h}_6(x, t)&= \frac{1 - q}{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \tilde{G} (y, x, t - a) \mathrm {d} y \mathrm {d} a. \end{aligned}$$

Let

then

$$\begin{aligned} \left| H [j_1] - H [j_2] \right| _{{\mathbb {X}}_3} \le C(t_{\mathrm {max}}) \left| j_1 - j_2 \right| _{{\mathbb {X}}_3}, t_{\mathrm {max}} \le + \infty . \end{aligned}$$

Select $0 < t_{\mathrm {max}} \le 1$ small enough such that $C(t_{\mathrm {max}}) < 1$. Thus, H is a strict contraction in ${\mathbb {X}}_3$. According to the theorem 9.23 (i.e. contraction mapping theorem) of [23], one finishes the proof of this theorem.

$\square$

Theorem 2.2

For $\forall (\varphi _1, \varphi _2, \varphi _3) \in {\mathbb {X}}_1^+ \times {\mathbb {X}}_2^+ \times {\mathbb {X}}_2^+$, the solution (S, E, j) of system (2.2)–(2.6) meets

$$\begin{aligned} S > 0, E \ge 0, j \ge 0 ~ on ~ \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty . \end{aligned}$$

Proof

For $\forall \eta \in {\mathbb {X}}_2$, define

And then, $A: {\mathbb {X}}_2 \rightarrow {\mathbb {X}}_2$ is the positive linear operator in the sense that $A \left( {\mathbb {X}}_2^+ \right) \subset {\mathbb {X}}_2^+$. Noting that $h_1 (x, t)$ can be expressed by $A (\eta )$. Then for $(x, t) \in \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty$, one gains

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t) >&D_1 \frac{\partial ^2}{\partial x^2} S(x, t) - S(x, t) \left[ A(\eta ) + h_2(x, t) + \mu \right] , \\ \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + (1 - q) S(x, t) \left[ A(\eta ) + h_2(x, t) \right] - (\varepsilon + \mu ) E(x, t), \\ j(x, t)&= q S(x, t) \left[ A(\eta ) + h_2(x, t) \right] + \varepsilon E(x, t), \end{aligned}$$

relating with initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), j(x, 0)\right] \\&\quad = \left[ \varphi _1(x), \varphi _2(x), q \varphi _1(x) \int \limits _0^{+ \infty } \beta (a) \varphi _3(x, a) \mathrm {d}a + \varepsilon \varphi _2(x) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} j(x, t) = 0, x \in \partial \Omega . \end{aligned}$$

Owing to the continuity and boundedness of $A(\eta ) + h_2(x, t) + \mu$, one attains

$$\begin{aligned} S(x, t) > 0 ~ \text {on} ~ \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty . \end{aligned}$$

Proceed to the next step, for $(x, t) \in \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty$, one obtains

$$\begin{aligned} \frac{\partial }{\partial t} S(x, t)&> D_1 \frac{\partial ^2}{\partial x^2} S(x, t) - S(x, t) \left[ A(\eta ) + h_2(x, t) + \mu \right] , \\ \frac{\partial }{\partial t} E(x, t)&> D_2 \frac{\partial ^2}{\partial x^2} E(x, t) - (\varepsilon + \mu ) E(x, t), \\ j(x, t)&= q S(x, t) \left[ A(\eta ) + h_2(x, t) \right] + \varepsilon E(x, t), \end{aligned}$$

associating with initial data

$$\begin{aligned}&\left[ S(x, 0), E(x, 0), j(x, 0)\right] \\&\quad = \left[ \varphi _1(x), \varphi _2(x), q \varphi _1(x) \int \limits _0^{+ \infty } \beta (a) \varphi _3(x, a) \mathrm {d}a + \varepsilon \varphi _2(x) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S(x, t) = \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} j(x, t) = 0, x \in \partial \Omega . \end{aligned}$$

Consequently,

$$\begin{aligned} E(x, t) \ge 0, J(x, t) \ge 0 ~ \text {on} ~ \overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty . \end{aligned}$$

One finishes the proof of this theorem. $\square$

Theorem 2.3

For $\forall (\varphi _1, \varphi _2, \varphi _3) \in {\mathbb {X}}_1^+ \times {\mathbb {X}}_2^+ \times {\mathbb {X}}_2^+$, the solution (S, E, j) of system (2.2)–(2.6) is bounded in $\overline{\Omega } \times [0, t_{\mathrm {max}}), t_{\mathrm {max}} \le + \infty$.

Proof

According to system (1.8)–(1.13), one obtains

$$\begin{aligned}&\frac{\partial }{\partial t} S(x, t) + \frac{\partial }{\partial t} E(x, t) + \int \limits _0^{+ \infty } \frac{\partial }{\partial t} i(x, t, a) \mathrm {d} a \\&\quad = D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + \delta - S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a \\&\qquad - \mu S(x, t) + D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + (1 - q) S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a \\&\qquad - (\mu + \varepsilon ) E(x, t) + D_3 \int \limits _0^{+ \infty } \frac{\partial ^2}{\partial x^2} i(x, t, a) \mathrm {d} a \\&\qquad - \int \limits _0^{+ \infty } \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (x, t, a) \mathrm {d} a - \int \limits _0^{+ \infty } \frac{\partial }{\partial a} i(x, t, a) \mathrm {d} a \\&\quad = D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + D_3 \int \limits _0^{+ \infty } \frac{\partial ^2}{\partial x^2} i(x, t, a) \mathrm {d} a \\&\qquad - q S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a + \delta - \mu S(x, t) - (\mu + \varepsilon ) E(x, t) \\&\qquad - \int \limits _0^{+ \infty } \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (x, t, a) \mathrm {d} a - \int \limits _0^{+ \infty } \frac{\partial }{\partial a} i(x, t, a) \mathrm {d} a \\&\quad < D_1 \frac{\partial ^2}{\partial x^2} S(x, t) + D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + D_3 \int \limits _0^{+ \infty } \frac{\partial ^2}{\partial x^2} i(x, t, a) \mathrm {d} a \\&\qquad + \delta - \mu \left[ S(x, t) + E(x, t) + \int \limits _0^{+ \infty } i (x, t, a) \mathrm {d} a \right] . \end{aligned}$$

Noting the homogeneous Neumann boundary conditions of system (1.8)–(1.13) and employing the Gauss formula, one derives

$$\begin{aligned} \int \limits _{\Omega } D_1 \frac{\partial ^2}{\partial x^2} S(x, t) \mathrm {d} x = \int \limits _{\Omega } D_2 \frac{\partial ^2}{\partial x^2} E(x, t) \mathrm {d} x = \int \limits _{\Omega } D_3 \int \limits _0^{+ \infty } \frac{\partial ^2}{\partial x^2} i(x, t, a) \mathrm {d} a \mathrm {d} x = 0. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{\mathrm {d} \tilde{N} (t)}{\mathrm {d} t} =&\int \limits _{\Omega } \left[ \frac{\partial }{\partial t} S(x, t) + \frac{\partial }{\partial t} E(x, t) + \int \limits _0^{+ \infty } \frac{\partial }{\partial t} i(x, t, a) \mathrm {d} a \right] \mathrm {d} x \\&< \int \limits _{\Omega } \left\{ \delta - \mu \left[ S(x, t) + E(x, t) + \int \limits _0^{+ \infty } i (x, t, a) \mathrm {d} a \right] \right\} \mathrm {d} x \\&= \delta \left| \Omega \right| - \mu \tilde{N} (t). \end{aligned}$$

Hence, if

$$\begin{aligned} \tilde{N} (t) > \frac{\delta }{\mu } \left| \Omega \right| , \end{aligned}$$

then

$$\begin{aligned} \frac{\mathrm {d} \tilde{N} (t)}{\mathrm {d} t} < 0. \end{aligned}$$

In addition, one finds the ODE

$$\begin{aligned} \frac{\mathrm {d} \tilde{N} (t)}{\mathrm {d} t} = \delta \left| \Omega \right| - \mu \tilde{N} (t) \end{aligned}$$

with general solution

$$\begin{aligned} \tilde{N} (t) = \frac{\delta }{\mu } \left| \Omega \right| + \left[ \tilde{N} (0) - \frac{\delta }{\mu } \left| \Omega \right| \right] e^{- \mu t}, \end{aligned}$$

where $\tilde{N} (0)$ shows the initial value of total population in region $\Omega$. Via utilizing the standard comparison theorem, one finds out for all $t \ge 0$,

$$\begin{aligned} \tilde{N} (t) \le \frac{\delta }{\mu } \left| \Omega \right| , \end{aligned}$$

if

$$\begin{aligned} \tilde{N} (0) \le \frac{\delta }{\mu } \left| \Omega \right| . \end{aligned}$$

Therefore,

$$\begin{aligned} \Phi = \left\{ (S, E, i) | \tilde{N} \le \frac{\delta }{\mu } \left| \Omega \right| \right\} \end{aligned}$$

is positive invariant for system (2.2)–(2.6), where $\left| \Omega \right|$ is the volume of $\Omega$. One finishes the proof of this theorem. $\square$

3 Basic reproduction number

Apparently, systems (1.8)–(1.13) admit the disease-free equilibrium $P_0 = (S_0, 0, 0)$ with $S_0 = \frac{\delta }{\mu }$. Linearizing system (1.8)–(1.13) near such equilibrium in disease invasion phase, one has

$$\begin{aligned} \frac{\partial }{\partial t} E(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E(x, t) + (1 - q) S(x, t) \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a - (\mu + \varepsilon ) E(x, t), \end{aligned}$$

(3.1)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) i (x, t, a)&= D_3 \frac{\partial ^2}{\partial x^2} i(x, t, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] i (x, t, a), \end{aligned}$$

(3.2)

$$\begin{aligned} i (x, t, 0)&= q \frac{\delta }{\mu } \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d}a + \varepsilon E(x, t), \end{aligned}$$

(3.3)

coresponding with initial data

$$\begin{aligned}&\left[ E(x, 0), i(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _2(x), \varphi _3(x, 0)\right] , a \ge 0, x \in \Omega , \end{aligned}$$

(3.4)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} E(x, t) = \frac{\partial }{\partial \mathbf {n}} i(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(3.5)

According to the above results, one finds

$$\begin{aligned} j(x, t)&= \frac{q \delta }{\mu } \int \limits _0^t \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) j(y, t - a) \mathrm {d} y \mathrm {d} a \nonumber \\&+ \frac{\varepsilon \left( 1 - q \right) }{q} \int \limits _0^t e^{- \mu (t - a)} \int \limits _{\Omega } \tilde{G} (y, x, t - a) j(y, a) \mathrm {d} y \mathrm {d} a. \end{aligned}$$

(3.6)

Making the Laplace transformation to (3.6), one gets

Hence, after multiple interchanging the order of integration, one derives

$$\begin{aligned} L[j]&= \frac{q \delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) e^{- \lambda a} \int \limits _{\Omega } G (y, x, a) \int \limits _0^t j(y, t) \mathrm {d} t \mathrm {d} y \mathrm {d} a \\&+ \frac{\varepsilon \left( 1 - q \right) }{q} \int \limits _0^{+ \infty } e^{\mu a} j(y, a) \int \limits _{\Omega } \tilde{G} (y, x, t - a) \int \limits _0^t e^{- (\lambda + \mu ) t} \mathrm {d} t \mathrm {d} y \mathrm {d} a. \end{aligned}$$

Setting $\lambda = 0$, one figures out

$$\begin{aligned} \int \limits _0^{+ \infty } j(t, x) \mathrm {d} t&= \frac{q \delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \int \limits _{\Omega } G (a, x, y) \int \limits _0^t j(t, y) \mathrm {d} t \mathrm {d} y \mathrm {d} a \\&+ \frac{\varepsilon \left( 1 - q \right) }{q} \int \limits _0^{+ \infty } e^{\mu a} j(a, y) \int \limits _{\Omega } \tilde{G} (t - a, x, y) \int \limits _0^t e^{- \mu t} \mathrm {d} t \mathrm {d} y \mathrm {d} a. \end{aligned}$$

So, due to the discussions of [24], the operator ${\mathcal {Q}}: {\mathbb {X}}_1 \rightarrow {\mathbb {X}}_1$ can be regarded as the next-generation operator,

(3.7)

For operator ${\mathcal {Q}}$, one obtains the following result.

Theorem 3.1

The operator ${\mathcal {Q}}$ is strictly positive and compact.

Proof

The positivity of the operator ${\mathcal {Q}}$ is apparent. In order to yield the compactness of the operator ${\mathcal {Q}}$, one needs the two steps as below.

First step The operator ${\mathcal {Q}}$ is uniformly bounded.

Choosing a bounded sequence $\left\{ \eta _n\right\} _{n \in {\mathbb {N}}}$ in ${\mathbb {X}}_1$ with $\left\{ \eta _n\right\} _{{\mathbb {X}}_1} \le {\mathbb {K}}$ for some ${\mathbb {K}} > 0$. Denoting a sequence $\left\{ \zeta _n\right\} _{n \in {\mathbb {N}}}$ by

Thus, for all $n \in {\mathbb {N}}$ and $x \in \Omega$, one has

$$\begin{aligned} \zeta _n (x)&\le \frac{q \delta {\mathbb {K}}}{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) \mathrm {d} y \mathrm {d} a \\&+ \frac{\varepsilon \left( 1 - q \right) {\mathbb {K}}}{q} \int \limits _0^{+ \infty } e^{- \mu a} \int \limits _{\Omega } \tilde{G} (y, x, a) \mathrm {d} y \mathrm {d} a. \end{aligned}$$

Go a step further, one finds out that $\left\{ \zeta _n\right\} _{n \in {\mathbb {N}}}$ is uniformly bounded.

Second step The sequence $\left\{ \zeta _n\right\} _{n \in {\mathbb {N}}}$ is equi-continuous.

For $\forall x, \hat{x} \in \Omega$, one yields

$$\begin{aligned} \left| \zeta _n (x) - \zeta _n \left( \hat{x} \right) \right|&= \left| {\mathcal {Q}} \eta _n (x) - {\mathcal {Q}} \eta _n \left( \hat{x} \right) \right| \\&\le \frac{q \delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \int \limits _{\Omega } \left| G (y, x, a) - G (y, \hat{x}, a) \right| \eta _n (y) \mathrm {d} y \mathrm {d} a \\&+ \frac{\varepsilon \left( 1 - q \right) }{q} \int \limits _0^{+ \infty } e^{- \mu a} \int \limits _{\Omega } \left| \tilde{G} (y, x, a) - \tilde{G} (y, \hat{x}, a)\right| \eta _n (y) \mathrm {d} y \mathrm {d} a \\&\le \frac{q \delta {\mathbb {K}}}{\mu } \mathrm {ess.sup}_{a \ge 0} \beta (a) \int \limits _0^{+ \infty } \chi (a) \int \limits _{\Omega } \left| G (y, x, a) - G (y, \hat{x}, a) \right| \mathrm {d} y \mathrm {d} a \\&+ \frac{\varepsilon \left( 1 - q \right) {\mathbb {K}}}{q} \int \limits _0^{+ \infty } e^{- \mu a} \int \limits _{\Omega } \left| \tilde{G} (y, x, a) - \tilde{G} (y, \hat{x}, a)\right| \mathrm {d} y \mathrm {d} a. \end{aligned}$$

Owing to the compactness of the operator $\frac{\partial ^2}{\partial x^2}$ and the uniform continuity of G(y, x, a) and $\tilde{G} (y, x, a)$, there exists $\hat{\delta } > 0$ such that

$$\begin{aligned} \left| G (y, x, a) - G (y, \hat{x}, a) \right|&\le \frac{\mu \hat{\varepsilon }}{2 \delta \mathrm {ess.sup}_{a \ge 0} \beta (a) {\mathbb {K}} \int \limits _0^{+ \infty } \chi (a) \mathrm {d} a \left| \Omega \right| },\\ \left| \tilde{G} (y, x, a) - \tilde{G} (y, \hat{x}, a)\right|&\le \frac{\mu \hat{\varepsilon }}{2 \delta {\mathbb {K}} \int \limits _0^{+ \infty } e^{- \mu a} \mathrm {d} a}, \end{aligned}$$

$\forall \hat{\varepsilon } > 0$, $\left| x - \hat{x} \right| < \hat{\delta }$, $y \in \Omega$. For this $\hat{\delta }$, $\hat{\varepsilon }$,

$$\begin{aligned} \left| \zeta _n (x) - \zeta _n \left( \hat{x} \right) \right|< \hat{\varepsilon }, ~ \text {for} ~ \text {all} \left| x - \hat{x} \right| < \hat{\delta }, \end{aligned}$$

namely, the sequence $\left\{ \zeta _n\right\} _{n \in {\mathbb {N}}}$ is equi-continuous. One finishes the proof of this theorem. $\square$

According to [24], one denotes that the basic reproduction number of (2.2)–(2.6) is

$$\begin{aligned} R_0 = r \left( {\mathcal {Q}} \right) , \end{aligned}$$

where $r \left( {\mathcal {Q}} \right)$ is the spectral radius of ${\mathcal {Q}}$. Theorem 3.1 together with Krein–Rutman theorem [25] indicates that the basic reproduction number $R_0$ is the unique positive eigenvalue of ${\mathcal {Q}}$ associating with a positive eigenvector. Without losing generality, substituting $\eta (x) \equiv 1 > 0$ into (3.7) and applying

$$\begin{aligned} \int \limits _{\Omega } G (y, x, \cdot ) \mathrm {d} y = 1, \int \limits _{\Omega } \tilde{G} (y, x, \cdot ) \mathrm {d} y = 1, \end{aligned}$$

one has

$$\begin{aligned} {\mathcal {Q}} [1]&= \frac{q \delta [1]}{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) \mathrm {d} y \mathrm {d} a \nonumber \\&+ \frac{\varepsilon \left( 1 - q \right) [1]}{q} \int \limits _0^{+ \infty } e^{- \mu a} \int \limits _{\Omega } \tilde{G} (y, x, a) \eta (y) \mathrm {d} y \mathrm {d} a \nonumber \\&= \frac{q \delta [1]}{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a + \frac{\varepsilon \left( 1 - q \right) [1]}{q} \int \limits _0^{+ \infty } e^{- \mu a} \mathrm {d} a. \end{aligned}$$

(3.8)

Thereby, $R_0 = r \left( {\mathcal {Q}} \right)$ can be explicitly expressed by:

$$\begin{aligned} R_0 = \frac{q \delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a + \frac{\varepsilon \left( 1 - q \right) }{ \mu q}. \end{aligned}$$

Nowadays, one can easily figures out that $R_0$ is a threshold value for the existence of a positive space independent endemic steady state $P_* = \left[ S_*, E_*, i_*(a) \right]$ of the original system (1.8)–(1.13), which is a solution of the following equations:

$$\begin{aligned}&0 = \delta - S_* \int \limits _0^{+ \infty } \beta (a) i_*(a) \mathrm {d}a - \mu S_*, \end{aligned}$$

(3.9)

$$\begin{aligned}&0 = (1 - q) S_* \int \limits _0^{+ \infty } \beta (a) i_*(a) \mathrm {d}a - (\mu + \varepsilon ) E_*, \end{aligned}$$

(3.10)

$$\begin{aligned}&\frac{\mathrm {d} i_*(a)}{\mathrm {d}a} = - \left[ \gamma (a) + \mu _1(a) + d(a) \right] i_*(a), \end{aligned}$$

(3.11)

$$\begin{aligned}&i_*(0) = q S_* \int \limits _0^{+ \infty } \beta (a) i_*(a) \mathrm {d}a + \varepsilon E_*. \end{aligned}$$

(3.12)

Theorem 3.2

Assume that $R_0 > 1$, then the original system (1.8)–(1.13) has a space independent endemic steady state $P_* = \left[ S_*, E_*, i_*(a) \right]$, which is a solution of (3.9)–(3.12), where

$$\begin{aligned} S_*&= \frac{\mu + \varepsilon }{(\varepsilon + q \mu ) \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a},\\ E_*&= \frac{ \delta (\varepsilon + q \mu ) \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a - \mu (\mu + \varepsilon ) }{\varepsilon (\mu + \varepsilon )}\\&- \frac{ q \delta (\varepsilon + q \mu ) \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a - \mu (\mu + \varepsilon ) }{\varepsilon (\varepsilon + q \mu )},\\ i_*(a)&= \frac{\chi (a) \left[ \delta (\varepsilon + q \mu ) \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a - \mu (\mu + \varepsilon ) \right] }{\mu + \varepsilon }. \end{aligned}$$

4 Local dynamics

This part will prove that $P_0$ is locally asymptotically stable if $R_0 < 1$ and $P_*$ are locally asymptotically stable if $R_0 > 1$.

Theorem 4.1

(1) The disease-free equilibrium $P_0 = (S_0, 0, 0)$ with $S_0 = \frac{\delta }{\mu }$ is locally asymptotically stable if $R_0 < 1$;

(2) The space independent endemic steady state $P_* = \left[ S_*, E_*, i_*(a) \right]$ is locally asymptotically stable if $R_0 > 1$.

Proof

First step One proves (1). Set

$$\begin{aligned} \tilde{S}(x, t) = S(x, t) - S_0, \tilde{E}(x, t) = E(x, t), \tilde{i}(x, t, a) = {i}(x, t, a), \end{aligned}$$

then the linearized equation of system (1.8)–(1.13) near $P_0 = (S_0, 0, 0)$ reads as:

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{S}(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} \tilde{S}(x, t) - S_0 \int \limits _0^{+ \infty } \beta (a) \tilde{i}(x, t, a) \mathrm {d}a - \mu \tilde{S}(x, t), \end{aligned}$$

(4.1)

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{E}(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} \tilde{E}(x, t) + (1 - q) S_0 \int \limits _0^{+ \infty } \beta (a) \tilde{i}(x, t, a) \mathrm {d}a - (\mu + \varepsilon ) \tilde{E}(x, t), \end{aligned}$$

(4.2)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) \tilde{i} (x, t, a)&= D_3 \frac{\partial ^2}{\partial x^2} \tilde{i} (x, t, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] \tilde{i} (x, t, a), \end{aligned}$$

(4.3)

$$\begin{aligned} \tilde{i} (x, t, 0)&= q S_0 \int \limits _0^{+ \infty } \beta (a) \tilde{i}(x, t, a) \mathrm {d}a + \varepsilon \tilde{E}(x, t), \end{aligned}$$

(4.4)

relating with initial data

$$\begin{aligned}&\left[ \tilde{S}(x, 0), \tilde{E}(x, 0), \tilde{i}(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _1(x) - S_0, \varphi _2(x), \varphi _3(x, a)\right] , a \ge 0, x \in \Omega , \end{aligned}$$

(4.5)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} \tilde{S}(x, t) = \frac{\partial }{\partial \mathbf {n}} \tilde{E}(x, t) = \frac{\partial }{\partial \mathbf {n}} \tilde{i}(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(4.6)

Because the linear system contains $\frac{\partial ^2}{\partial x^2}$, one introduces the related theory from [19]. Define $\theta _k, k = 1, 2, \cdots$ be the eigenvalues of operator $- \frac{\partial ^2}{\partial x^2}$ on a bounded set $\Omega$ with boundary condition (1.13), that is to say,

$$\begin{aligned} \frac{\partial ^2}{\partial x^2} \nu (x) = - \theta _k \nu (x), k = 1, 2, \cdots . \end{aligned}$$

Thus,

$$\begin{aligned} 0 = \theta _0< \theta _1< \theta _2 < \cdots , \end{aligned}$$

associating to which, there is the space of eigenfunctions in $C^1 (\Omega )$, defined by $E \left( \theta _k \right) , k = 1, 2, \cdots$. Define

$$\begin{aligned} \left\{ \omega _{k m} | m = 1, 2, \cdots , \mathrm {dim} E \left( \theta _k \right) , k = 1, 2, \cdots \right\} \end{aligned}$$

be the orthogonal basis of $E \left( \theta _k \right) , k = 1, 2, \cdots$. Make further efforts, put

$$\begin{aligned} {\mathbb {Y}}_{k m} = \left\{ c \omega _{k m} | c \in {\mathbb {R}}^3, m = 1, 2, \cdots , \mathrm {dim} E \left( \theta _k \right) , k = 1, 2, \cdots \right\} , \end{aligned}$$

then

$$\begin{aligned} \tilde{{\mathbb {Y}}} = \bigoplus _{k = 0}^{+ \infty } {\mathbb {Y}}_k, {\mathbb {Y}}_k = \bigoplus _{m = 1}^{\mathrm {dim} E \left( \theta _k \right) } {\mathbb {Y}}_{k m}, m = 1, 2, \cdots , \mathrm {dim} E \left( \theta _k \right) , k = 1, 2, \cdots . \end{aligned}$$

Note that the parabolic equation

$$\begin{aligned} \frac{\partial }{\partial t} u(x, t)&= \frac{\partial ^2}{\partial x^2} u(x, t),\\ \frac{\partial }{\partial \mathbf {n}} u(x, t)&= 0 \end{aligned}$$

exists the exponential solution

$$\begin{aligned} u(x, t) = e^{\varsigma t} \nu (x), \nu (x) \in {\mathbb {Y}}_k, k = 1, 2, \cdots . \end{aligned}$$

Substituting

$$\begin{aligned} \left[ \tilde{S}(x, t), \tilde{E}(x, t), \tilde{i}(x, t, a) \right] = e^{\varsigma t} \left[ \rho _1^0(x), \rho _2^0(x), \rho _3^0(x, a) \right] \end{aligned}$$

into system (4.1)–(4.6), one has

$$\begin{aligned} \varsigma \rho _1^0(x)&= - D_1 \theta _k \rho _1^0(x) - S_0 \int \limits _0^{+ \infty } \beta (a) \rho _3^0(x, a) \mathrm {d}a - \mu \rho _1^0(x), \end{aligned}$$

(4.7)

$$\begin{aligned} \varsigma \rho _2^0(x)&= - D_2 \theta _k \rho _2^0(x) + (1 - q) S_0 \int \limits _0^{+ \infty } \beta (a) \rho _3^0(x, a) \mathrm {d}a - (\mu + \varepsilon ) \rho _2^0(x), \end{aligned}$$

(4.8)

$$\begin{aligned} \varsigma \rho _3^0(x, a) + \frac{\partial \rho _3^0(x, a)}{\partial a}&= - D_3 \theta _k \rho _3^0(x, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] \rho _3^0(x, a), \end{aligned}$$

(4.9)

$$\begin{aligned} \rho _3^0(x, 0)&= q S_0 \int \limits _0^{+ \infty } \beta (a) \rho _3^0(x, a) \mathrm {d}a + \varepsilon \rho _2^0(x), \end{aligned}$$

(4.10)

associating with initial data

$$\begin{aligned}&\left[ \tilde{S}(x, 0), \tilde{E}(x, 0), \tilde{i}(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _1(x) - S_0, \varphi _2(x), \varphi _3(x, a)\right] , a \ge 0, x \in \Omega , \end{aligned}$$

(4.11)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} \tilde{S}(x, t) = \frac{\partial }{\partial \mathbf {n}} \tilde{E}(x, t) = \frac{\partial }{\partial \mathbf {n}} \tilde{i}(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(4.12)

Solving equation (4.9), one yields

$$\begin{aligned} \rho _3^0(x, a) = \rho _3^0(x, 0) \tilde{\chi }(a) e^{- \varsigma a}, \tilde{\chi }(a) = \chi (a) e^{- D_3 \theta _k a}, k = 1, 2, \cdots . \end{aligned}$$

(4.13)

Substituting (4.13), (4.8) into (4.10), one derives

(4.14)

Clearly,

$$\begin{aligned} {\mathcal {A}}'(\varsigma )&= - \varsigma S_0 \int \limits _0^{+ \infty } \beta (a) \rho _3^0(x, 0) \tilde{\chi }(a) e^{- \varsigma a} \mathrm {d}a\\ <&0. \end{aligned}$$

If (4.14) has a unique real positive root $\varsigma$, one gets

$$\begin{aligned} 1&= \frac{\delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \tilde{\chi }(a) e^{- \varsigma a} \mathrm {d}a \\ <&\frac{q \delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a + \frac{\varepsilon \left( 1 - q \right) }{ \mu q}\\&= R_0, \end{aligned}$$

which leads to a contradiction. Thus, all roots of (4.14) are negative. If (4.14) has complex roots in the form of $\varsigma = x' \pm y' i$ with $x' > 0$, one obtains

$$\begin{aligned} 1&= \frac{\delta }{\mu } \int \limits _0^{+ \infty } \beta (a) \tilde{\chi }(a) e^{- x' a} \cos (y' a) \mathrm {d}a \\&\le R_0, \end{aligned}$$

which creats a contradiction. Thereform, the disease-free equilibrium $P_0 = (S_0, 0, 0)$ with $S_0 = \frac{\delta }{\mu }$ is locally asymptotically stable if $R_0 < 1$.

Second step One proves (2). Let

$$\begin{aligned} \widehat{S}(x, t) = S(x, t) - S_*, \widehat{E}(x, t) = E(x, t) - E_*, \widehat{i}(x, t, a) = {i}(x, t, a) - i_*(a), \end{aligned}$$

then the linearized equation of system (1.8)–(1.13) around $P_* = \left[ S_*, E_*, i_*(a) \right]$ reads as:

$$\begin{aligned} \frac{\partial }{\partial t} \widehat{S}(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} \widehat{S}(x, t) - S_* \int \limits _0^{+ \infty } \beta (a) \widehat{i}(x, t, a) \mathrm {d}a - \mu \widehat{S}(x, t), \end{aligned}$$

(4.15)

$$\begin{aligned} \frac{\partial }{\partial t} \widehat{E}(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} \widehat{E}(x, t) + (1 - q) S_* \int \limits _0^{+ \infty } \beta (a) \widehat{i}(x, t, a) \mathrm {d}a - (\mu + \varepsilon ) \widehat{E}(x, t), \end{aligned}$$

(4.16)

$$\begin{aligned} \left( \frac{\partial }{\partial t} + \frac{\partial }{\partial a} \right) \widehat{i} (x, t, a)&= D_3 \frac{\partial ^2}{\partial x^2} \widehat{i} (x, t, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] \widehat{i} (x, t, a),\end{aligned}$$

(4.17)

$$\begin{aligned} \widehat{i} (x, t, 0)&= q S_* \int \limits _0^{+ \infty } \beta (a) \widehat{i}(x, t, a) \mathrm {d}a + \varepsilon \widehat{E}(x, t), \end{aligned}$$

(4.18)

conresponding with initial data

$$\begin{aligned}&\left[ \widehat{S}(x, 0), \widehat{E}(x, 0), \widehat{i}(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _1(x) - S_*, \varphi _2(x) - E_*, \varphi _3(x, a) - i_*(a) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

(4.19)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} \widehat{S}(x, t) = \frac{\partial }{\partial \mathbf {n}} \widehat{E}(x, t) = \frac{\partial }{\partial \mathbf {n}} \widehat{i}(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(4.20)

Homoplastically, substituting

$$\begin{aligned} \left[ \widehat{S}(x, t), \widehat{E}(x, t), \widehat{i}(x, t, a) \right] = e^{\varsigma t} \left[ \rho _1(x), \rho _2(x), \rho _3(x, a) \right] \end{aligned}$$

into system (4.15)–(4.20), one gets

$$\begin{aligned} \varsigma \rho _1(x)&= - D_1 \theta _k \rho _1(x) - S_* \int \limits _0^{+ \infty } \beta (a) \rho _3(x, a) \mathrm {d}a - \mu \rho _1(x), \end{aligned}$$

(4.21)

$$\begin{aligned} \varsigma \rho _2(x)&= - D_2 \theta _k \rho _2(x) + (1 - q) S_* \int \limits _0^{+ \infty } \beta (a) \rho _3(x, a) \mathrm {d}a - (\mu + \varepsilon ) \rho _2(x), \end{aligned}$$

(4.22)

$$\begin{aligned} \varsigma \rho _3(x, a) + \frac{\partial \rho _3(x, a)}{\partial a}&= - D_3 \theta _k \rho _3(x, a) - \left[ \gamma (a) + \mu _1(a) + d(a) \right] \rho _3^0(x, a), \end{aligned}$$

(4.23)

$$\begin{aligned} \rho _3(x, 0)&= q S_* \int \limits _0^{+ \infty } \beta (a) \rho _3(x, a) \mathrm {d}a + \varepsilon \rho _2(x), \end{aligned}$$

(4.24)

associating with initial data

$$\begin{aligned}&\left[ \widehat{S}(x, 0), \widehat{E}(x, 0), \widehat{i}(x, 0, a)\right] \nonumber \\&\quad = \left[ \varphi _1(x) - S_*, \varphi _2(x) - E_*, \varphi _3(x, a) - i_*(a)\right] , a \ge 0, x \in \Omega , \end{aligned}$$

(4.25)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} \widehat{S}(x, t) = \frac{\partial }{\partial \mathbf {n}} \widehat{E}(x, t) = \frac{\partial }{\partial \mathbf {n}} \widehat{i}(x, t, a) = 0, x \in \partial \Omega . \end{aligned}$$

(4.26)

Solving the equation (4.23), one yields

$$\begin{aligned} \rho _3(x, a) = \rho _3(x, 0) \tilde{\chi }(a) e^{- \varsigma a}, \tilde{\chi }(a) = \chi (a) e^{- D_3 \theta _k a}, k = 1, 2, \cdots . \end{aligned}$$

(4.27)

Plugging (4.27) into systems (4.21)–(4.26), one gets that the characteristic equation of system (1.8)–(1.13) at $P_* = \left[ S_*, E_*, i_*(a) \right]$ is

$$\begin{aligned} \mu \left( R_0 - 1 \right) \Xi (\varsigma ) - \left( D_1 \theta _k + \mu R_0 + \varsigma \right) \left[ \Xi (\varsigma ) - 1 \right] = 0, k = 1, 2, \cdots , \end{aligned}$$

where

$$\begin{aligned} \Xi (\varsigma ) = S_* \int \limits _0^{+ \infty } \beta (a) \tilde{\chi }(a) e^{- \varsigma a} \mathrm {d}a. \end{aligned}$$

Hence,

$$\begin{aligned} \left( \varsigma + D_1 \theta _k + \mu \right) \Xi (\varsigma ) - \left( \varsigma + D_1 \theta _k + \mu R_0 \right) = 0, k = 1, 2, \cdots . \end{aligned}$$

(4.28)

Assume that (4.28) has a positive real root $\varsigma$, due to $R_0 > 1$, then one yields

$$\begin{aligned} \Xi (\varsigma )&= \frac{\varsigma + D_1 \theta _k + \mu R_0}{\varsigma + D_1 \theta _k + \mu }\nonumber \\ >&1, k = 1, 2, \cdots . \end{aligned}$$

(4.29)

Evidently, $\Xi '(\varsigma ) < 0$. This manifests

$$\begin{aligned} \Xi (\varsigma ) <&\Xi (0) \\&= S_* \int \limits _0^{+ \infty } \beta (a) \tilde{\chi }(a) e^{- \varsigma a} \mathrm {d}a\\&= 1, \end{aligned}$$

which creates a contradiction with (4.29). Therefore, the total real roots of (4.28) are negative.

Suppose that (4.28) has complex roots $\varsigma = x{''} \pm y{''} i$ with $x{''} > 0$, then

$$\begin{aligned} \left( x{''} \pm y{''} i + D_1 \theta _k + \mu \right) \Xi \left( x{''} \pm y{''} i \right) - \left( x{''} \pm y{''} i + D_1 \theta _k + \mu R_0 \right) = 0, k = 1, 2, \cdots . \end{aligned}$$

Furthermore, one obtains

$$\begin{aligned} \mathrm {Re} \left[ \Xi \left( x{''} \pm y{''} i \right) \right]&= \frac{\left( x{''} + D_1 \theta _k + \mu R_0 \right) \left( x{''} + D_1 \theta _k + \mu \right) + {y{''}}^2}{\left( x{''} + D_1 \theta _k + \mu \right) ^2 + {y{''}}^2}\nonumber \\ >&1, k = 1, 2, \cdots . \end{aligned}$$

(4.30)

On the other side,

$$\begin{aligned} \mathrm {Re} \left[ \Xi \left( x{''} \pm y{''} i \right) \right]&\le \Xi (0)\\ <&1, \end{aligned}$$

which leads to a contradiction with (4.30). Consequently, the space independent endemic steady state $P_* = \left[ S_*, E_*, i_*(a) \right]$ is locally asymptotically stable if $R_0 > 1$. One finishes the proof of this theorem. $\square$

5 Disease persistence

In this part, one will demonstrate the disease persistence when $R_0 > 1$. First, one rewrites

$$\begin{aligned} i(x, t, a) = {\left\{ \begin{array}{ll} \frac{\chi (a)}{\chi (a - t)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a - t) \mathrm {d} y, &{} a - t \ge 0, x \in \Omega , \\ \chi (a) \int \limits _{\Omega } G (y, a, x) j(y, t - a, 0) \mathrm {d} y, &{} t - a > 0, x \in \Omega . \end{array}\right. } \end{aligned}$$

(5.1)

Next, one gives the first result of this portion.

Theorem 5.1

For $\forall (\varphi _1, \varphi _2, \varphi _3) \in {\mathbb {X}}_1^+ \times {\mathbb {X}}_2^+ \times {\mathbb {X}}_2^+$, system (2.2)–(2.6) denotes a continuous semiflow

Proof

For $\forall \pi \ge 0$, $t \ge 0$, $a \ge 0$, $x \in \Omega$, set

$$\begin{aligned} S_{\pi } (x, t) = S(x, \pi + t), j_{\pi } (x, t)= j(x, \pi + t), i_{\pi } (x, t, a) = i(x, \pi + t, a), \end{aligned}$$

then

$$\begin{aligned} \frac{\partial }{\partial t} S_{\pi }(x, t)&= D_1 \frac{\partial ^2}{\partial x^2} S_{\pi }(x, t) + \delta - \frac{1}{q} j_{\pi }(x, t) + \frac{1}{q} \varepsilon E_{\pi }(x, t) - \mu S_{\pi }(x, t), \end{aligned}$$

(5.2)

$$\begin{aligned} \frac{\partial }{\partial t} E_{\pi }(x, t)&= D_2 \frac{\partial ^2}{\partial x^2} E_{\pi }(x, t) + \frac{1 - q}{q} j_{\pi }(x, t) - \left( \mu + \frac{1}{q} \varepsilon \right) E_{\pi }(x, t), \end{aligned}$$

(5.3)

$$\begin{aligned} j_{\pi }(x, t)&= q S_{\pi }(x, t) \int \limits _0^{+ \infty } \beta (a) i_{\pi } (x, t, a) \mathrm {d} a + \varepsilon E_{\pi }(x, t), \end{aligned}$$

(5.4)

associating with initial data

$$\begin{aligned}&\left[ S_{\pi }(x, 0), E_{\pi }(x, 0), j_{\pi }(x, 0) \right] \nonumber \\&\quad = \left[ \varphi _{1 \pi }(x), \varphi _{2 \pi }(x), q \varphi _{1 \pi }(x) \int \limits _0^{+ \infty } \beta (a) \varphi _{3 \pi }(x, a) \mathrm {d}a + \varepsilon \varphi _{2 \pi }(x) \right] , a \ge 0, x \in \Omega , \end{aligned}$$

(5.5)

and the homogeneous Neumann boundary condition

$$\begin{aligned} \frac{\partial }{\partial \mathbf {n}} S_{\pi }(x, t) = \frac{\partial }{\partial \mathbf {n}} E_{\pi }(x, t) = \frac{\partial }{\partial \mathbf {n}} j_{\pi }(x, t) = 0, x \in \partial \Omega . \end{aligned}$$

(5.6)

And then, (5.1) becomes

$$\begin{aligned} i_{\pi }(x, t, a) = {\left\{ \begin{array}{ll} \frac{\chi (a)}{\chi (a - t - \pi )} \int \limits _{\Omega } G (y, \pi + t, x) \varphi _3 (y, a - t - \pi ) \mathrm {d} y, &{} a - t \ge \pi , x \in \Omega , \\ \chi (a) \int \limits _{\Omega } G (y, a, x) j_{\pi }(y, t - a, 0) \mathrm {d} y, &{} t - a > \pi , x \in \Omega . \end{array}\right. } \end{aligned}$$

(5.7)

In addition, for $\forall \pi \ge 0$, $a > t \ge 0$, $x \in \Omega$, one gets

$$\begin{aligned} i_{\pi }(x, 0, a - t) = {\left\{ \begin{array}{ll} \frac{\chi (a - t)}{\chi (a - t - \pi )} \int \limits _{\Omega } G (y, \pi , x) \varphi _3 (y, a - t - \pi ) \mathrm {d} y, &{} a - t \ge \pi , x \in \Omega , \\ \chi (a - t) \int \limits _{\Omega } G (y, a - t, x) j_{\pi }(y, t - a, 0) \mathrm {d} y, &{} \pi > a - t \ge 0, x \in \Omega . \end{array}\right. } \end{aligned}$$

(5.8)

According to the property of G, one derives

$$\begin{aligned}&\frac{\chi (a)}{\chi (a - t)} \int \limits _{\Omega } G (y, t, x) i_{\pi } (y, a - t, 0) \mathrm {d} y \nonumber \\&\quad = {\left\{ \begin{array}{ll} \frac{\chi (a)}{\chi (a - t - \pi )} \int \limits _{\Omega } G (y, \pi + t, x) \varphi _3 (y, a - t - \pi ) \mathrm {d} y, &{} a - t \ge \pi , x \in \Omega , \\ \chi (a) \int \limits _{\Omega } G (y, a, x) j_{\pi }(y, t - a, 0) \mathrm {d} y, &{} \pi > a - t \ge 0, x \in \Omega . \end{array}\right. } \end{aligned}$$

(5.9)

Combining with (5.7) and (5.8), one can yield

$$\begin{aligned} i_{\pi }(x, t, a) = {\left\{ \begin{array}{ll} \frac{\chi (a)}{\chi (a - t)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a - t, 0) \mathrm {d} y, &{} a - t \ge 0, x \in \Omega , \\ \chi (a) \int \limits _{\Omega } G (y, a, x) j_{\pi }(y, t - a) \mathrm {d} y, &{} t - a > 0, x \in \Omega . \end{array}\right. } \end{aligned}$$

(5.10)

Thus, from (5.2)–(5.6) and (5.10), one has

$$\begin{aligned} \Phi \left[ S(\cdot , \pi ), E(\cdot , \pi ), i(\cdot , \cdot , \pi ), t \right]&= \left[ S_{\pi }(t), E_{\pi }(t), i_{\pi }(\cdot , \cdot , t) \right] \\&= \Phi \left( \varphi _1, \varphi _2, \varphi _3, \pi + t \right) , \forall \pi \ge 0, t \ge 0. \end{aligned}$$

Therefore, the time continuity of $\Phi$ follows from theorem 2.1–2.3. One finishes the proof of this theorem.

$\square$

Theorem 5.2

If $\varphi \in \Psi$, $R_0 > 1$, then there exists $\omega > 0$, such that

$$\begin{aligned} \limsup \limits _{t \rightarrow + \infty } |j(\cdot , t)|_{{\mathbb {X}}_1} > \omega , \end{aligned}$$

where

Proof

According to the expression of $R_0$, opting $\omega > 0$, such that

$$\begin{aligned} \frac{q \delta - \omega }{\mu } \int \limits _0^{+ \infty } \beta (a) \chi (a) \mathrm {d} a + \frac{\varepsilon \left( 1 - q \right) }{ \mu q} > 1. \end{aligned}$$

(5.11)

Presume

$$\begin{aligned} j(x, t) \le \omega , \forall x \in \Omega , t \ge t_1 > 0, \end{aligned}$$

then due to (5.11), one obtains that there exist enough small $\lambda > 0$ and sufficiently large $t_2 > t_1$, such that

(5.12)

Therefore, one has

$$\begin{aligned}&\frac{\partial }{\partial t} \left[ S(x, t) + E(x, t) \right] \nonumber \\&\quad \ge D_1 \frac{\partial ^2}{\partial x^2} \left[ S(x, t) + E(x, t) \right] + q \delta - \omega - \mu \left[ S(x, t) + E(x, t) \right] ~ \text {on} ~ [t_2, + \infty ) \times \Omega . \end{aligned}$$

(5.13)

Solving (5.13) and employing comparison principle, one obtains

$$\begin{aligned} \left[ S(x, t) + E(x, t) \right] \ge \frac{q \delta - \omega }{\mu } \left[ 1 - e^{- \mu \left( t_2 - t_1 \right) } \right] ~ \text {on} ~ [t_2, + \infty ) \times \Omega . \end{aligned}$$

(5.14)

Combining theorem 5.1 and (5.14), and letting $S(x, t_2)$, $E(x, t_2)$, $j(x, t_2)$ with $t_2 = 0$ be initial conditions, we have

$$\begin{aligned}&j(x,t) \nonumber \\&\quad \ge \frac{q \delta - \omega }{\mu } \left[ 1 - e^{- \mu \left( t_2 - t_1 \right) } \right] \int \limits _0^t \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) j(y, t - a) \mathrm {d} y \mathrm {d} a ~ \text {on} ~ [t_2, + \infty ) \times \Omega . \end{aligned}$$

(5.15)

Clearly,

Denote

According to (5.15), one gets

$$\begin{aligned}&L[j] \left( \hat{x} \right) \\&\quad \ge \frac{q \delta - \omega }{\mu } \left[ 1 - e^{- \mu \left( t_2 - t_1 \right) } \right] \int \limits _0^{+ \infty } e^{- \lambda t} \int \limits _0^t \beta (a) \chi (a) \int \limits _{\Omega } G (y, x, a) j(y, t - a) \mathrm {d} y \mathrm {d} a \mathrm {d} t ~ \text {on} ~ [t_2, + \infty ) \times \Omega . \end{aligned}$$

Thus, after multiple interchanging the order of integration, one obtains

$$\begin{aligned} L[j] \left( \hat{x} \right) \ge&\frac{q \delta - \omega }{\mu } \left[ 1 - e^{- \mu \left( t_2 - t_1 \right) } \right] \int \limits _0^{+ \infty } \beta (a) \chi (a) e^{- \lambda a} \int \limits _{\Omega } G (y, x, a) \int \limits _0^{+ \infty } e^{- \lambda t} j(y, t) \mathrm {d} t \mathrm {d} y \mathrm {d} a \\&\ge \tilde{R} L[j] \left( \hat{x} \right) ~ \text {on} ~ [t_2, + \infty ) \times \Omega , \end{aligned}$$

which creates a contradiction with (5.12). The second claim directly follows from (5.14). One finishes the proof of this theorem. $\square$

According to theorem 5.2, one will prove the strong $|\cdot |_{{\mathbb {X}}_1}$-persistence.

Theorem 5.3

If $\varphi \in \Psi$, $R_0 > 1$, then there exists $\omega ' > 0$, such that

$$\begin{aligned} \limsup \limits _{t \rightarrow + \infty } |j(\cdot , t)|_{{\mathbb {X}}_1} > \omega ', \end{aligned}$$

where

Proof

Hypothesize that

$$\begin{aligned} \limsup \limits _{t \rightarrow + \infty } |j(\cdot , t)|_{{\mathbb {X}}_1} \le \omega ', ~ \text {for some} ~ \omega ' > 0. \end{aligned}$$

And then, owing to theorem 5.2, one indicates that there exist increasing sequences $\left\{ t_{1 l}\right\} _{l = 1}^{+ \infty }$, $\left\{ t_{2 l}\right\} _{l = 1}^{+ \infty }$, $\left\{ t_{3 l}\right\} _{l = 1}^{+ \infty }$ with $t_{1 l}> t_{1 2} > t_{1 3}$, $l = 1, 2, \cdots , + \infty$ and decreasing sequence $\left\{ t_{4 l}\right\} _{l = 1}^{+ \infty }$ with $\lim \limits _{l \rightarrow + \infty } t_{4 l} = 0$. And they also meet

$$\begin{aligned} \left| j\left( \cdot , t_{3 l} \right) \right| _{{\mathbb {X}}_1}&> \omega , t = t_{3 l}, l = 1, 2, \cdots , + \infty , \end{aligned}$$

(5.16)

$$\begin{aligned} \left| j\left( \cdot , t_{2 l} \right) \right| _{{\mathbb {X}}_1}&= \omega , t = t_{2 l}, l = 1, 2, \cdots , + \infty , \end{aligned}$$

(5.17)

$$\begin{aligned} \left| j\left( \cdot , t_{1 l} \right) \right| _{{\mathbb {X}}_1}&< t_{4 l} < \omega , t = t_{2 l}, l = 1, 2, \cdots , + \infty , \end{aligned}$$

(5.18)

$$\begin{aligned} \left| j\left( \cdot , t \right) \right| _{{\mathbb {X}}_1}&< \omega , t \in \left( t_{2 l}, t_{1 l} \right) , l = 1, 2, \cdots , + \infty , \end{aligned}$$

(5.19)

and set $\left\{ S_{l}\right\} _{l = 1}^{+ \infty }$, $\left\{ E_{ l}\right\} _{l = 1}^{+ \infty }$, $\left\{ j_{ l}\right\} _{l = 1}^{+ \infty }$ denote

Due to (2.7)–(2.9), (2.10) and using the Arzela–Ascoli theorem, one gets that there exists $\left( S_*, E_*, j_* \right) \in {\mathbb {X}}_1^+ \times {\mathbb {X}}_1^+ \times {\mathbb {X}}_1^+$, such that

$$\begin{aligned} \liminf \limits _{l \rightarrow + \infty } S_l = S_*, \liminf \limits _{l \rightarrow + \infty } E_l = E_*, \liminf \limits _{l \rightarrow + \infty } j_l = j_*. \end{aligned}$$

Put $\left( \widehat{S}, \widehat{E}, \widehat{j} \right)$ be a solution of (2.2)–(2.6) with

$$\begin{aligned} \varphi _1(x) = S_*(x), \varphi _2(x) = E_*(x), \varphi _3(x, a) = \chi (a) \int \limits _{\Omega } G (y, x, a) j_*(y) \mathrm {d} y, \forall a \ge 0, x \in \Omega . \end{aligned}$$

Because $\varphi _3$ accounts upon (5.1) and due to theorem 5.2, one gets that there exists $t' > 0$, $\omega '' > 0$, such that

$$\begin{aligned} \left| \hat{j} \left( \cdot , t' \right) \right| _{{\mathbb {X}}_1} >&\omega , t = t',\end{aligned}$$

(5.20)

$$\begin{aligned} \left| \hat{j} \left( \cdot , t \right) \right| _{{\mathbb {X}}_1} >&\omega '', 0< t < t'. \end{aligned}$$

(5.21)

Set

then it follows from the semiflow property that

$$\begin{aligned} \left| \hat{j}_{1 l} \left( \cdot , t' \right) \right| _{{\mathbb {X}}_1} >&\omega , t = t', \end{aligned}$$

(5.22)

$$\begin{aligned} \left| \hat{j}_{1 l} \left( \cdot , t \right) \right| _{{\mathbb {X}}_1}>&\omega '' > t_{4 l}, 0< t < t', \end{aligned}$$

(5.23)

for enough large l. In contrast, for

it follows from (5.16)–(5.19) that

$$\begin{aligned} \left| \hat{j}_{1 l} \left( \cdot , t' \right) \right| _{{\mathbb {X}}_1} >&\omega , t = t', \end{aligned}$$

(5.24)

$$\begin{aligned} \left| \hat{j}_{1 l} \left( \cdot , t \right) \right| _{{\mathbb {X}}_1}>&\omega '' > t_{4 l}, 0< t < t', l = 1, 2, \cdots , + \infty . \end{aligned}$$

(5.25)

Obviously, it leads to a contradiction between (5.22)–(5.23) and (5.24)–(5.25). Here, one completes the proof of $\limsup \limits _{t \rightarrow + \infty } |j(\cdot , t)|_{{\mathbb {X}}_1} > \omega '$, for some constant $\omega ' > 0$. One finishes the proof of this theorem. $\square$

6 Global dynamics

The main aim of this portion is to prove that $P_0$ is globally asymptotically stable if $R_0 \le 1$ and $P_*$ is globally asymptotically stable if $R_0 > 1$.

Theorem 6.1

(1) The disease-free equilibrium $P_0 = (S_0, 0, 0)$ with $S_0 = \frac{\delta }{\mu }$ is globally asymptotically stable if $R_0 \le 1$;

(2) The space-independent endemic steady state $P_* = \left[ S_*, E_*, i_*(a) \right]$ is globally asymptotically stable if $R_0 > 1$.

Proof

First step One proves (1). Denote the following Lyapunov functional

with

clearly, $\Gamma (a)$ meets

$$\begin{aligned}&S_0 \beta (a) + \frac{\mathrm {d} \Gamma (a)}{\mathrm {d} a} - \left[ \gamma (a) + \mu _1(a) + d(a) \right] \Gamma (a) = 0, \\&\quad \Gamma (0) = R_0, \end{aligned}$$

then

$$\begin{aligned} \frac{\partial }{\partial t} V_S(x, t)&= D_1 \frac{S(x, t) - S_0}{S(x, t)} \frac{\partial ^2}{\partial x^2} S(x, t) - \frac{\mu \left[ S(x, t) - S_0 \right] ^2}{S(x, t)} \\&\quad - \frac{1}{q} j(x, t) + \frac{1}{q} \varepsilon \frac{S(x, t) - S_0}{S(x, t)} E(x, t) + \frac{1}{q} S_0 \int \limits _0^{+ \infty } \beta (a) i(x, t, a) \mathrm {d} a. \end{aligned}$$

Moreover,

$$\begin{aligned} V_i(x, t)&= \int \limits _0^{+ \infty } \Gamma (t + a) \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a) \mathrm {d} y \mathrm {d} a \nonumber \\&\quad + \int \limits _0^{t} \Gamma (t - a) \chi (t - a) \int \limits _{\Omega } G (y, t - a, x) j (y, a) \mathrm {d} y \mathrm {d} a. \end{aligned}$$

(6.1)

Thus,

$$\begin{aligned} \frac{\partial }{\partial t} V_i(x, t)&= \Gamma (0) \int \limits _{\Omega } G (y, 0, x) j (y, t) \mathrm {d} y\\&\quad + \int \limits _0^t \frac{\mathrm {d} \Gamma (t - a)}{\mathrm {d} t} \chi (t - a) \int \limits _{\Omega } G (y, t - a, x) j (y, a) \mathrm {d} y \mathrm {d} a \\&\quad + \int \limits _0^t \Gamma (t - a) \chi (t - a) \int \limits _{\Omega } \frac{\partial G(y, t - a, x)}{\partial t} j (y, a) \mathrm {d} y \mathrm {d} a \\&\quad - \int \limits _0^t \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] \Gamma (t - a) \chi (t - a) \int \limits _{\Omega } G (y, t - a, x) j (y, a) \mathrm {d} y \mathrm {d} a \\&\quad + \int \limits _0^{+ \infty } \frac{\mathrm {d} \Gamma (t + a)}{\mathrm {d} t} \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a) \mathrm {d} y \mathrm {d} a \\&\quad + \int \limits _0^{+ \infty } \Gamma (t + a) \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } \frac{\partial G(y, t, x)}{\partial t} \varphi _3 (y, a) \mathrm {d} y \mathrm {d} a \\&\quad - \int \limits _0^{+ \infty } \left[ \gamma (t + a) + \mu _1(t + a) + d(t + a) \right] \Gamma (t + a) \frac{\chi (t + a)}{\chi ( a)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a) \mathrm {d} y \mathrm {d} a \\&= \Gamma (0) j(x, t) + \int \limits _0^{+ \infty } \left\{ \frac{\mathrm {d} \Gamma ( a)}{\mathrm {d} a} - \left[ \gamma (a) + \mu _1(a) + d(a) - D_3 \frac{\partial ^2}{\partial x^2} \right] \Gamma (a) \right\} i(x, t, a) \mathrm {d} a. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\mathrm {d} V_{P_0} (t)}{\mathrm {d} t}&= - D_1 \int \limits _{\Omega } \frac{|\nabla S(x, t)|^2}{S(x, t)^2} \mathrm {d} x - \int \limits _{\Omega } \frac{\mu }{S(x, t)} \left[ S(x, t) - S_0 \right] ^2 \mathrm {d} x - \int \limits _{\Omega } \left[ 1 - \Gamma (0) \right] j(x, t) \mathrm {d} x \nonumber \\&+ \int \limits _{\Omega } \int \limits _{0}^{+ \infty } \left\{ S_0 \beta (a) + \frac{\mathrm {d} \Gamma (a)}{\mathrm {d} a} - \left[ \gamma (a) + \mu _1(a) + d(a) - D_3 \frac{\partial ^2}{\partial x^2} \right] \Gamma (a) \right\} i(x, t ,a) \mathrm {d} a \mathrm {d} x. \end{aligned}$$

(6.2)

Accordingly, one has

$$\begin{aligned} \frac{\mathrm {d} V_{P_0} (t)}{\mathrm {d} t}&= - D_1 \int \limits _{\Omega } \frac{|\nabla S(x, t)|^2}{S(x, t)^2} \mathrm {d} x - \int \limits _{\Omega } \frac{\mu }{S(x, t)} \left[ S(x, t) - S_0 \right] ^2 \mathrm {d} x - \int \limits _{\Omega } \left( 1 - R_0 \right) j(x, t) \mathrm {d} x\\&\le 0, \end{aligned}$$

if $R_0 \le 1$. In summary, $\left\{ P_0 \right\}$ is the largest invariant set such that $\frac{\mathrm {d} V_{P_0} (t)}{\mathrm {d} t} = 0$, and according to the invariance principle in [26], one knows that $P_0$ is globally attractive.

Second step One proves (2). Define the Lyapunov functional as follows

with

then

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}_S(x, t)&= D_1 \frac{S(x, t) - S_*}{S(x, t)} \frac{\partial ^2}{\partial x^2} S(x, t) - \frac{\mu \left[ S(x, t) - S_* \right] ^2}{S(x, t)} \nonumber \\&\quad + \frac{1}{q} \varepsilon \frac{S(x, t) - S_*}{S(x, t)} E(x, t) + \frac{i(x, t, 0) S_*}{q S(x, t)}\nonumber \\&\quad + \frac{i_*(0) }{q} - \frac{i(x, t, 0)}{q} - \frac{i_*(0) S_*}{q S(x, t)}. \end{aligned}$$

(6.3)

Besides,

$$\begin{aligned} \tilde{V}_i(x, t) = \int \limits _0^{t} \Xi (t - a) \Delta \left[ u_1, v_1 \right] \mathrm {d} y \mathrm {d} a + \int \limits _0^{+ \infty } \Xi (t + a) \Delta \left[ u_2, v_2 \right] \mathrm {d} y \mathrm {d} a \end{aligned}$$

with

$$\begin{aligned} u_1&= \chi (t - a) \int \limits _{\Omega } G (y, t - a, x) i (y, a, 0) \mathrm {d} y, v_1 = i_*(t - a), \\ u_2&= \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } G (y, t, x) \varphi _3 (y, a) \mathrm {d} y, v_2 = i_*(t + a). \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}_i(x, t)&= \Xi (0) \Delta \left[ \int \limits _{\Omega } G (y, 0, x) i (y, a, 0) \mathrm {d} y, i_*(0) \right] + \int \limits _0^t \frac{\mathrm {d} \Xi (t - a)}{\mathrm {d} t} \Delta \left[ u_1, v_1 \right] \mathrm {d} a \\&\quad + \int \limits _0^{+ \infty } \frac{\mathrm {d} \Xi (t + a)}{\mathrm {d} t} \Delta \left[ u_2, v_2 \right] \mathrm {d} a \\&\quad + \int \limits _0^t \Xi (t - a) \chi (t - a) \int \limits _{\Omega } \frac{\partial G(y, t - a, x)}{\partial t} i (y, a, 0) \mathrm {d} y \frac{\partial \Delta \left[ u_1, v_1 \right] }{\partial u_1} \mathrm {d} a \\&\quad - \int \limits _0^t \Xi (t - a) \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] u_1 \frac{\partial \Delta \left[ u_1, v_1 \right] }{\partial u_1} \mathrm {d} a \\&\quad - \int \limits _0^t \Xi (t - a) \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] i_*(t - a) \frac{\partial \Delta \left[ u_1, v_1 \right] }{\partial v_1} \mathrm {d} a \\&\quad + \int \limits _0^{+ \infty } \Xi (t + a) \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } \frac{\partial G(y, t, x)}{\partial t} \varphi _3 (y, a) \mathrm {d} y \frac{\partial \Delta \left[ u_2, v_2 \right] }{\partial u_2} \mathrm {d} a \\&\quad - \int \limits _0^{+ \infty } \Xi (t + a) \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] u_2 \frac{\partial \Delta \left[ u_2, v_2 \right] }{\partial u_2} \mathrm {d} a \\&\quad - \int \limits _0^{+ \infty } \Xi (t + a) \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] i_*(t + a) \frac{\partial \Delta \left[ u_2, v_2 \right] }{\partial v_2} \mathrm {d} a. \end{aligned}$$

Owing to

$$\begin{aligned} u \frac{\partial \Delta \left[ u, v \right] }{\partial u} + v \frac{\partial \Delta \left[ u, v \right] }{\partial v} = \Delta \left[ u, v \right] , \end{aligned}$$

one has

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}_i(x, t)&= \Xi (0) \Delta \left[ \int \limits _{\Omega } G (y, 0, x) i (y, a, 0) \mathrm {d} y, i_*(0) \right] \\&+ \int \limits _0^{+ \infty } \left\{ \frac{\mathrm {d} \Xi (a)}{\mathrm {d} a} - \left[ \gamma (t - a) + \mu _1(t - a) + d(t - a) \right] \Xi (a) \right\} \Delta \left[ i(x, t, a), i_*(a) \right] \mathrm {d} a \\&+ \int \limits _0^t \Xi (t - a) \chi (t - a) \int \limits _{\Omega } \frac{\partial G(y, t - a, x)}{\partial t} i (y, a, 0) \mathrm {d} y \frac{\partial \Delta \left[ u_1, v_1 \right] }{\partial u_1} \mathrm {d} a \\&+ \int \limits _0^{+ \infty } \Xi (t + a) \frac{\chi (t + a)}{\chi (a)} \int \limits _{\Omega } \frac{\partial G(y, t, x)}{\partial t} \varphi _3 (y, a) \mathrm {d} y \frac{\partial \Delta \left[ u_2, v_2 \right] }{\partial v_2} \mathrm {d} a. \end{aligned}$$

Denote the unit semigroup

then

$$\begin{aligned} \frac{\partial G}{\partial t}&= D_2 \frac{\partial ^2}{\partial x^2} G, \\ \frac{\partial \Delta \left[ u, v \right] }{\partial u}&= 1 - \frac{v}{u}. \end{aligned}$$

Furthermore, one gets

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}_i(x, t)&= \Xi (0) \Delta \left[ i (x, t, 0), i_*(0) \right] + \int \limits _0^{+ \infty } \Xi (a) D_2 \frac{\partial ^2}{\partial x^2} i(x, t, a) \left[ 1 - \frac{i_*(a)}{i(x, t, a)} \right] \mathrm {d} a \\&+ \int \limits _0^{+ \infty } \left\{ \frac{\mathrm {d} \Xi (a)}{\mathrm {d} a} - \left[ \gamma (a) + \mu _1( a) + d( a) \right] \Xi (a) \right\} \Delta \left[ i(x, t, a), i_*(a) \right] \mathrm {d} a. \end{aligned}$$

Then, it follows that

$$\begin{aligned}&\left[ \gamma (a) + \mu _1( a) + d( a) \right] \Xi (a) - \frac{\mathrm {d} \Xi (a)}{\mathrm {d} a} = S_* \beta (a), \\&\quad \Xi (0) = \int \limits _0^{+ \infty } S_* \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \\&\quad = S_* \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi . \end{aligned}$$

Consequently, one derives

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}_i(x, t)&= S_* \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \Delta \left[ i (x, t, 0), i_*(0) \right] \nonumber \\&\quad + \int \limits _0^{+ \infty } \Xi (a) D_2 \frac{\partial ^2}{\partial x^2} i(x, t, a) \left[ 1 - \frac{i_*(a)}{i(x, t, a)} \right] \mathrm {d} a \nonumber \\&\quad - \int \limits _0^{+ \infty } S_* \beta (a) \Delta \left[ i(x, t, a), i_*(a) \right] \mathrm {d} a. \end{aligned}$$

(6.4)

Put

$$\begin{aligned} \tilde{V}(x, t) = \tilde{V}_S(x, t) + \tilde{V}_i(x, t), \end{aligned}$$

then uniting with (6.3) and (6.4), one yields

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}(x, t)&= D_1 \frac{S(x, t) - S_*}{S(x, t)} \frac{\partial ^2}{\partial x^2} S(x, t) - \frac{\mu \left[ S(x, t) - S_* \right] ^2}{S(x, t)}\\&\quad + \frac{i(x, t, 0) S_*}{q S(x, t)} - \frac{i_*(0) S_*}{q S(x, t)} + i_*(0) - i(x, t, 0)\\&\quad + S_* \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \Delta \left[ i (x, t, 0), i_*(0) \right] \\&\quad + \int \limits _0^{+ \infty } \Xi (a) D_2 \frac{\partial ^2}{\partial x^2} i(x, t, a) \left[ 1 - \frac{i_*(a)}{i(x, t, a)} \right] \mathrm {d} a \\&\quad - \int \limits _0^{+ \infty } S_* \beta (a) \Delta \left[ i(x, t, a), i_*(a) \right] \mathrm {d} a. \end{aligned}$$

Proceed to the next step, one has

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}(x, t)&= D_1 \frac{S(x, t) - S_*}{S(x, t)} \frac{\partial ^2}{\partial x^2} S(x, t) - \frac{\mu \left[ S(x, t) - S_* \right] ^2}{S(x, t)} + \frac{i(x, t, 0) S_*}{q S(x, t)} \nonumber \\&\quad - \frac{i_*(0) S_*}{q S(x, t)} + i_*(0) - i(x, t, 0) \nonumber \\&\quad + S_* \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \left[ i (x, t, 0) - i_*(0) - i_*(0) \ln \frac{i (x, t, 0)}{i_*(0)} \right] \nonumber \\&\quad + \int \limits _0^{+ \infty } \Xi (a) D_2 \frac{\partial ^2}{\partial x^2} i(x, t, a) \left[ 1 - \frac{i_*(a)}{i(x, t, a)} \right] \mathrm {d} a \nonumber \\&\quad - \int \limits _0^{+ \infty } S_* \beta (a) \left[ i(x, t, a) - i_*(a) - i_*(a) \ln \frac{i(x, t, a)}{i_*(a)} \right] \mathrm {d} a. \end{aligned}$$

(6.5)

By computations, one can get

$$\begin{aligned} S_* \int \limits _0^{+ \infty } \beta (a) i_*(a) \left[ 1 - \frac{S i_*(0) i(x, t, a)}{S_* i(x, t, 0) i_*(a)} \right] \mathrm {d} a = 0. \end{aligned}$$

And then, (6.5) can be transformed into

$$\begin{aligned} \frac{\partial }{\partial t} \tilde{V}(x, t)&= D_1 \frac{S(x, t) - S_*}{S(x, t)} \frac{\partial ^2}{\partial x^2} S(x, t) - \frac{\mu \left[ S(x, t) - S_* \right] ^2}{S(x, t)} \nonumber \\&\quad + \frac{i(x, t, 0) S_*}{q S(x, t)} - \frac{i_*(0) S_*}{q S(x, t)} + i_*(0) - i(x, t, 0) \nonumber \\&\quad + S_* \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \left[ i (x, t, 0) - i_*(0) - i_*(0) \ln \frac{i (x, t, 0)}{i_*(0)} \right] \nonumber \\&\quad + \int \limits _0^{+ \infty } \Xi (a) D_2 \frac{\partial ^2}{\partial x^2} i(x, t, a) \left[ 1 - \frac{i_*(a)}{i(x, t, a)} \right] \mathrm {d} a \nonumber \\&\quad - \int \limits _0^{+ \infty } S_* \beta (a) \left[ 2 - \frac{S_*}{S(x, t)} + \ln \frac{S_*}{S(x, t)} + \frac{S i_*(0) i(x, t, a)}{S_* i(x, t, 0) i_*(a)} + \ln \frac{S i_*(0) i(x, t, a)}{S_* i(x, t, 0) i_*(a)} \right] \mathrm {d} a. \end{aligned}$$

(6.6)

Therefore, integrating (6.6) over $\Omega$, one has

$$\begin{aligned} \frac{\mathrm {d} V_{P_*} (t)}{\mathrm {d} t}&= - D_1 S_* \int \limits _{\Omega } \frac{|\nabla S(x, t)|^2}{S^2(x, t)} \mathrm {d} x - D_3 \int \limits _{\Omega } \int \limits _0^{+ \infty } \Xi (a) i_*(a) \frac{|\nabla i(x, t, a)|^2}{i^2(x, t, a)} \mathrm {d} a \mathrm {d} x \\&\quad - \int \limits _{\Omega } \frac{\mu }{S(x, t)} \left[ S(x, t) - S_* \right] ^2 \mathrm {d} x \\&\quad + S_* \int \limits _{\Omega } \int \limits _0^{+ \infty } \beta (\pi ) \chi (\pi ) \mathrm {d} \pi \left[ i (x, t, 0) - i_*(0) - i_*(0) \ln \frac{i (x, t, 0)}{i_*(0)} \right] \mathrm {d} x \\&\quad - \int \limits _{\Omega } \int \limits _0^{+ \infty } S_* \beta (a) \left\{ h \left[ \frac{S_*}{S(x, t)} \right] + h \left[ \frac{S i_*(0) i(x, t, a)}{S_* i(x, t, 0) i_*(a)} \right] \right\} \mathrm {d} a \mathrm {d} x\\&\le 0, \end{aligned}$$

where $h[s] = 1 - s + \ln s, s \in {\mathbb {R}}_+$ has the properties that $h(s) \le 0$ while $s > 0$ and h(s) reaches the global minimum 0 at $s = 1$. In summary, $\left\{ P_* \right\}$ is the largest invariant set such that $\frac{\mathrm {d} V_{P_*} (t)}{\mathrm {d} t} = 0$, and due to the invariance principle in [26], one gets that $P_*$ is globally attractive. One finishes the proof of this theorem. $\square$

7 Discussions

In this study, one considers the Tuberculosis model with the fast and slow progression, where age structure and spatial diffusion are introduced. Although the basic reproduction number is independent of the diffusion coefficients $D_1$, $D_2$ and $D_3$ and the dynamics of this model is similar to the model (1.19)–(1.23) without spatial diffusions, the process and method of solving and proving are different from models (1.19)–(1.23) without spatial diffusions. The difficulty in obtaining the dynamics of this model is the construction of Lyapunov functional when proving global asymptotic stability. It is a pity that this spatial diffusion is the Laplace (local) diffusion describing the random walk of the population in a connected domain. Nevertheless, in modern time many infectious diseases are spread geographically by long-distance travel such as air travel [27]. It seems that the Laplace (local) diffusion is unsuitable to model the spatial spread of infectious diseases through this phnomenon [28, 29]. Hence, some scholars propose convolution (nonlocal) diffusion to describe the long-distance dispersal [30, 31]. And then, one discusses the new type of Tuberculosis model, namely age-dependent Tuberculosis model with convolution (nonlocal) diffusion. One puts these in the future.