Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

Abstract

In this paper we propose an age-structured susceptible-infectious-susceptible epidemic model with nonlocal (convolution) diffusion to describe the geographic spread of infectious diseases via long-distance travel. We analyze the well-posedness of the model, investigate the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and study the local and global stability of this nontrivial steady state. Moreover, we discuss the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the nonlocal diffusion rate. The analysis is carried out by using the theory of semigroups and the method of monotone and positive operators. The spectral radius of a positive linear operator associated to the solution flow of the model is identified as a threshold.

1 Introduction

Age structure and spatial movement are two important features on the geographic spread of infectious diseases. The age could be chronological age (the age of the host), infection age (the time elapsed since infection), recovery age (the time elapsed since the last infection), class age (the length of time in the present group), etc. After the early work of Webb (1980, 1982), various age-structured epidemic models with spatial diffusion described by combined hyperbolic-parabolic partial differential equations have been proposed and studied, see Chekroun and Kuniya (2020), Di Blasio (2010), Fitzgibbon et al. (1994, 1996), Kim (2006), Kubo and Langlais (1994), Kuniya et al. (2018), Kuniya and Oizumi (2015), Chekroun and Kuniya (2020), Langlais and Busenberg (1997), and the references cited therein. In these studies, the classical Laplace diffusion was used to describe the random walk of the host in a connected domain. However, in modern time many infectious diseases are spread geographically via long-distance travel such as air travel (Findlater and Bogoch (2018)). It seems that the classical Laplace diffusion is unsuitable to model the spatial spread of infectious diseases via long-distance travel (Blyuss (2005) and Ruan (2007)).

The nonlocal (convolution) diffusion provides a way to describe long-distance dispersal (Yang et al. (2013) and Xu et al. (2020)). The goal of this paper is to propose a new type of epidemic models, namely age-structured susceptible-infectious-susceptible (SIS) epidemic model with nonlocal spatial diffusion. Note that for scalar linear and nonlinear age-structured equations with nonlocal diffusion, recently we (Kang et al. (2020b), Kang and Ruan (2021b) and two forthcoming papers) developed some basic theories including the semigroup of linear operators, asymptotic behavior, spectral theory, asynchronous exponential growth, strong maximum principle, global dynamics, etc. Here we generalize our previous modeling scheme and techniques to study the dynamics of an age-structured SIS epidemic model with nonlocal spatial diffusion.

Let S(t, a, x) and I(t, a, x) be the densities of susceptible and infective individuals of age $a\ge 0$ at time $t\ge 0$ in position $x\in \Omega \subset {\mathbb {R}}^N$, respectively, where $\Omega$ is a bounded subset of ${\mathbb {R}}^N$ and $a<a^+$, the maximum age which is finite. Let $\lambda (t, a, x)$ be the force of infection of infectious individuals to susceptible individuals of age a at time t in position x, and $\gamma (a, x)$ be the recovery rate of infective individuals of age a in position x. We consider the following system of age-structured equations with nonlocal spatial diffusion

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)S(t, a, x)=d[{\mathcal {L}}S](t, a, x)-\lambda (t, a, x)S(t, a, x)-&{}\mu (a, x)S(t, a, x)+\gamma (a, x)I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in {\overline{\Omega }},\\ (\partial _t+\partial _a)I(t, a, x)=d[{\mathcal {L}}I](t, a, x)+\lambda (t, a, x)S(t, a, x)-&{}(\mu (a, x)+\gamma (a, x))I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in {\overline{\Omega }},\\ S(t, 0, x)=P(t, 0, x),\quad I(t, 0, x)=0,&{}t>0, x\in {\overline{\Omega }},\\ S(0, a, x)=S_0(a, x),\quad I(0, a, x)=I_0(a, x),&{}a\in [0, a^+], x\in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(1.1)

where $\mu (a, x)$ denotes the mortality of individuals of age a in position x and the nonlocal operator ${\mathcal {L}}$ is defined by

$$\begin{aligned} {[}{\mathcal {L}}u](t, a, x):=(J*u-u)(t, a, x)=\int _{\Omega }J(x-y)u(t, a, y)dy-u(t, a, x). \end{aligned}$$

(1.2)

P(t, 0, x) is the boundary condition given by

$$\begin{aligned} P(t, 0, x)=\int _{0}^{a^+}\int _{\Omega }\beta (a, x, y)P(t, a, y)dyda, \; t>0, \; x\in {\overline{\Omega }}, \end{aligned}$$

in which $\beta (a, x, y)$ denotes the birth rate at which individuals of age a in position y give to newborns in position x and $P(t, a, x)=S(t, a, x)+I(t, a, x)$ represents the total population. $S_0(a, x)$ and $I_0(a, x)$ are positive initial value functions. We mention that the nonlocal diffusion operator in (1.1) corresponds to an elliptic operator with zero Dirichlet boundary condition. The convolution $\int _{\Omega }J(x-y)u(t, a, y)dy$ is the rate at which individuals are arriving at position x from other places while $\int _{\Omega }J(y-x)u(t, a, x)dy$ is the rate at which individuals are leaving location x for other locations. Moreover, J is a diffusion kernel which is a $C^0$ nonnegative function with unit integral representing the spatial dispersal and $J(0)>0$; i.e.,

$$\begin{aligned} \int _{{\mathbb {R}}^N}J(x)dx=1,\quad J(x)\ge 0,\quad \forall x\in {\mathbb {R}}^N. \end{aligned}$$

In addition, we assume that the force of infection $\lambda (t, a, x)$ is given by

$$\begin{aligned} \lambda (t, a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I(t, \sigma , y)dyd\sigma , \end{aligned}$$

(1.3)

where $k(a, \sigma , x, y)$ denotes the rate of disease transmission from infective individuals of age $\sigma$ in position y to susceptible individuals of age a in position x.

(a) Age-structured SIS epidemic models. If $S(t, a, x)=S(t, a)$ and $I(t, a, x)=I(t, a)$ do not depend on the spatial variable, then model (1.1) reduces to the following age-structured SIS epidemic model provided $\int _{\Omega }J(x-y)dy=1$ for all $x\in \Omega$:

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)S(t, a)=-S(t, a) \int _{0}^{a^+}k(a, \sigma )I(t, \sigma )d\sigma -\mu (a)S(t, a)+\gamma (a)I(t, a), \; t>0, a\in (0, a^+],\\ (\partial _t+\partial _a)I(t, a)=S(t, a) \int _{0}^{a^+}k(a, \sigma )I(t, \sigma )d\sigma -(\mu (a)+\gamma (a))I(t, a), \; t>0, a\in (0, a^+],\\ S(t, 0)=P(t, 0),\quad I(t, 0)=0, \; t>0, \\ S(0, a)=S_0(a),\quad I(0, a)=I_0(a), \; a\in [0, a^+]. \end{array}\right. } \end{aligned}$$

(1.4)

Such age-structured epidemic models have been studied extensively in the literature, see Busenberg et al. (1991), Iannelli (1995), Iannelli et al. (1999), Iannelli et al. (1992), Iannelli and Milner (2017), Inaba (1990, 2014, 2017), Kang et al. (2020a), Kuniya and Inaba (2013), Kuniya (2014), Kuniya and Iannelli (2014), Kuniya et al. (2016, 2018), Thieme (1991), and Webb (1984). Since the model is independent of the location, most studies focused on the temporal dynamics of these models.

(b) Nonlocal SIS epidemic models. If $S(t, a, x)=S(t, x)$ and $I(t, a, x)=I(t, x)$ do not depend on the age variable, then model (1.1) becomes the following nonlocal SIS epidemic model:

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \partial _t S(t, x) = d[\int _{\Omega }J(x-y)S(t, y)dy-S(t, x)] - S(t, x) \int _{\Omega }k(x, y)I(t, y)dy \\ &{}\qquad \quad \quad \quad \quad -\mu (x)S(t, x)+\gamma (x)I(t, x), \; t>0, x\in {\overline{\Omega }},\\ &{} \partial _t I(t, x) \,= d[\int _{\Omega }J(x-y)I(t, y)dy-I(t, x)]+S(t, x) \int _{\Omega }k(x, y)I(t, y)dy \\ &{}\qquad \quad \quad \quad \quad -(\mu (x)+\gamma (x))I(t, x),\; t>0, x\in {\overline{\Omega }},\\ &{} S(0, x)~~\,=S_0(x),\quad I(0, x)=I_0(a, x), \; x\in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(1.5)

Nonlocal dispersal epidemic models (without age structure) including (1.5) have also been studied by some researchers, see Xu et al. (2020), Yang and Li (2017), and Yang et al. (2013, 2019). These studies were mainly concerned with the existence of traveling wave solutions in such nonlocal dispersal models. For existence and stability of steady states in such models, see Yang et al. (2019) and Zhao and Ruan (2018). Notice that model (1.5) is more general than those considered in Xu et al. (2020), Yang and Li (2017), and Yang et al. (2013, 2019), since it includes not only nonlocal dispersal terms but also nonlocal interaction terms.

(c) Age-structured SIS epidemic models with Laplace diffusion. If the spatial domain $\Omega \subset {\mathbb {R}}^N$ is connected and the dispersal of individuals follows is a random walk, we obtain the following age-structured SIS epidemic model with Laplace diffusion:

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)S(t, a, x)=d\Delta S(t, a, x)-\lambda (t, a, x)S(t, a, x)-&{}\mu (a, x)S(t, a, x)+\gamma (a, x)I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in \Omega ,\\ (\partial _t+\partial _a)I(t, a, x)=d \Delta I(t, a, x)+\lambda (t, a, x)S(t, a, x)-&{}(\mu (a, x)+\gamma (a, x))I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in \Omega ,\\ S(t, 0, x)=P(t, 0, x),\quad I(t, 0, x)=0,&{}t>0, x\in \Omega ,\\ S(0, a, x)=S_0(a, x),\quad I(0, a, x)=I_0(a, x),&{}a\in [0, a^+], x\in \Omega , \end{array}\right. } \end{aligned}$$

(1.6)

where $\lambda (t, a, x)$ is given by (1.3). There could be appropriate boundary conditions on $\partial \Omega$ in (1.6), but we omit them. Various age-structured epidemic models with Laplace diffusion related to model (1.6) have been proposed and studied, see Di Blasio (2010), Kim (2006), Kubo and Langlais (1994), Kuniya et al. (2018), Chekroun and Kuniya (2020), Kuniya and Oizumi (2015), Langlais and Busenberg (1997), and the references cited therein. However, due to their complex structures, the spatial-temporal dynamics of this kind of epidemic models are not well-understood.

Here we provide a comprehensive analysis on the dynamics of age-structured SIS epidemic model (1.1) with nonlocal diffusion. We first analyze the well-posedness of the model, then investigate the existence and uniqueness of the nontrivial steady state, and finally study the local and global stability of this nontrivial steady state. Moreover, we consider the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the diffusion rate. We would like to mention that the analysis carried out in this paper is also valid for the age-structured SIS epidemic model (1.6) with random diffusion. We also refer to our recent study in Kang and Ruan (2021a) on the approximation of random diffusion by nonlocal diffusion with properly rescaled kernels in age-structured models.

The paper is organized as follows. In Sect. 2, we define the function spaces, introduce a key lemma and two theorems, and make some assumptions. In Sect. 3, we study the well-posedness of the model and provide an important monotone property of the semigroup generated by the solution flow. In Sect. 4, we investigate the existence of the nontrivial steady state. In Sect. 5 we define the basic reproduction number. A result on the uniqueness of the nontrivial steady state is obtained in Sect. 6. Then, we study local stability of the nontrivial steady state and asymptotic properties of the principal eigenvalue and nontrivial steady state in terms of the diffusion rate d in Sect. 7. In Sect. 8, we analyze global stability of the nontrivial steady state. We end up the paper with a discussion in Sect. 9.

2 Preliminaries

Define the function spaces as follows:

$$\begin{aligned} Z:= L^1(\Omega ),\quad X:=L^1((0, a^+), Z)=L^1((0, a^+)\times \Omega ) \end{aligned}$$

with positive cones

$$\begin{aligned}&Z_+:=L^1_+(\Omega ):=\{f\in Z: f\ge 0,\,\text {a.e.}\},\\&X_+:=L^1_+((0, a^+), Z) :=\{f\in X: f\ge 0, \,\text {a.e.}\}, \end{aligned}$$

respectively, Then we give a lemma based on Guo and Chan (1994), and its proof can be found in (Kang et al. 2020b, Lemma 3.1).

Lemma 2.1

Assume that $f\in L^\infty _+((0, a^+)\times \Omega )$, then there exists a unique mild solution $u(a, x), \; 0\le \tau \le a\le a^+$, to the evolution equation on X for any initial data $\phi \in Z$

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _au(a, x)=[-f(a, x)+d{\mathcal {L}}]u(a, x),&{}a\in (0, a^+), x\in {\overline{\Omega }},\\ u(\tau , x)=\phi (x),&{}x\in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(2.1)

Here a mild solution means that $u\in C([0, a^+], Z)$. Define the solution operator of the initial value problem (2.1) by

$$\begin{aligned} \left( {\mathcal {F}}(\tau , a)\phi \right) (x)=u(a, x),\quad \forall \phi \in Z. \end{aligned}$$

(2.2)

Then $\{{\mathcal {F}}(\tau , a)\}_{0\le \tau \le a\le a^+}$ is a family of linear uniformly bounded positive operators on Z and is strongly continuous in $\tau$ and a. Furthermore,

$$\begin{aligned} e^{-\int _{\tau }^{a}{\overline{f}}(\rho )d\rho }e^{d{\mathcal {L}}(a-\tau )}\le {\mathcal {F}}(\tau , a)\le e^{-\int _{\tau }^{a}{\underline{f}}(\rho )d\rho }e^{d{\mathcal {L}}(a-\tau )}, \end{aligned}$$

(2.3)

where ${\underline{f}}(a):=\inf _{x\in \Omega }f(a, x)$ and ${\overline{f}}(a):=\sup _{x\in \Omega }f(a, x)$.

We make the following assumptions on each parameter.

Assumption 2.2

 

• (i) $\gamma$ and $\mu$ are belonging to $L^\infty _+((0, a^+)\times \Omega )$ with $\gamma ^+:=\Vert \gamma \Vert _{L^\infty _+((0, a^+)\times \Omega )}$ and $\mu ^+:=\Vert \mu \Vert _{L^\infty _+((0, a^+)\times \Omega )}$;

• (ii) $k(a, \sigma , x, y)=0$ for $x, y\in {\mathbb {R}}^N\setminus \Omega$ or for $a, \sigma <0$ and $a, \sigma >a^+$;

• (iii) $k(a, \sigma , x, y)$ satisfies

  $$\begin{aligned} \lim _{\Vert (\kappa , h)\Vert \rightarrow 0}\int _{0}^{a^+}\int _{\Omega }|k(a+\kappa , \sigma , x+h, y)-k(a, \sigma , x, y)|dxda=0 \end{aligned}$$

  uniformly for $\sigma \in [0, a^+]$ and $y\in \Omega$, where $\kappa \in {\mathbb {R}}, h\in {\mathbb {R}}^N$ and $\Vert \cdot \Vert$ denotes the usual Euclidean norm in ${\mathbb {R}}^{N+1}$;

• (iv) There exists two positive functions $k_1\in L^1_+((0, a^+)\times \Omega )$ and $k_2\in L_+^\infty ((0, a^+)\times \Omega )$ respectively such that

  $$\begin{aligned} \epsilon k_1(a, x)k_2(\sigma , y)\le k(a, \sigma , x, y)\le k_1(a, x)k_2(\sigma , y) \end{aligned}$$

  for all $(a, x), (\sigma , y)\in {\mathbb {R}}_+\times {\overline{\Omega }}$, where $\epsilon \in (0, 1)$;

• (v) $k_1(a, x)$ has positive upper and lower bounds $k^+$ and $k^-$ such that $k^+\ge k_1(a, x)\ge k^->0$ for all $(a, x)\in [0, a^+]\times {\overline{\Omega }}$.

The lower bound in Assumption 2.2-(v) means that the transmission can occur between every pair of susceptible and infectious individuals from an epidemiological point of view.

Next we recall some basic concepts of nonsupporting operators and the fixed point theorem from Inaba (1990) (see also (Inaba 2014, Proposition 7.7)). Let B(Y) be the set of bounded linear operators from Y to Y, where Y is a Banach space with a positive cone $Y_+$. $L\in B(Y)$ is said to be positive if $L(Y_+)\subset Y_+$. $L\in B(Y)$ is said to be strongly positive if $\langle f, L\psi \rangle >0$ for every pair $\psi \in Y_+\setminus \{0\}, f\in Y_+^*\setminus \{0\}$, where $Y_+^*$ is the space of positive linear functionals on Y. For $L, V\in B(Y)$, we say $L\ge V$ if $(L-V)(Y_+)\subset Y_+$. A positive operator $L\in B(Y)$ is said to be nonsupporting if for every pair $\psi \in Y_+\setminus \{0\}$ and $f\in Y^*_+\setminus \{0\}$, there exists a positive integer $p=p(\psi , f)$ such that $\langle f, L^n\psi \rangle >0$ for all $n\ge p$. r(L) denotes the spectral radius of $L\in B(Y)$, $\sigma (L)$ denotes the spectrum of L, and $\sigma _P(L)$ denotes the point spectrum of L.

Theorem 2.3

(Inaba (1990)) Let Y be a real Banach space and $Y_+$ be its positive cone. Let $\Psi$ be a positive operator from $E_+$ to itself and $L:=\Psi '[0]$ be its Fréchet derivative at 0. If

• (i) $\Psi (0)=0$;

• (ii) $\Psi$ is compact and bounded;

• (iii) L has a positive eigenvector $v_0\in Y_+\setminus \{0\}$ associated with an eigenvalue $\lambda _0>1$;

• (iv) L has no eigenvector in $Y_+$ associated with the eigenvalue 1,

then $\Psi$ has at least one nontrivial fixed point in $Y_+$.

In the case where L is a majorant of $\Psi$ (that is, L is a linear operator such that $\Psi (\phi )\le L\phi$ for any $\phi \in Y_+$), the following theorem also holds (see (Inaba 2014, Proposition 7.8)).

Theorem 2.4

(Inaba (2014)) Let Y be a real Banach space and $Y_+$ be its positive cone. Let $\Psi$ be a positive operator from $Y_+$ to itself and L be its compact and nonsupporting majorant. Then, $\Psi$ has no trivial fixed point in $Y_+$ provided $r(L)\le 1$.

3 Well-posedness

3.1 McKendrick equation with nonlocal diffusion

Let P(t, a, x) be the total population density at time $t\ge 0$ with age $a\ge 0$ in position $x\in \Omega$. Then the total population P is governed by the McKendrick equation with nonlocal diffusion (see Kuniya and Oizumi (2015)):

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)P(t, a, x)=d[{\mathcal {L}}P](t, a, x)-\mu (a, x)P(t, a, x),&{} t{>}0, a{\in }(0, a^+], x{\in }{\overline{\Omega }},\\ P(t, 0, x)=\int _{0}^{a^+}\int _{\Omega }\beta (a, x, y)P(t, a, y)dyda,&{} t>0, x\in {\overline{\Omega }},\\ P(0, a, x)=P_0(a, x)\in X_+,&{} a\in [0, a^+], x\in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(3.1)

where $\mu (a, x)$ represents the mortality of an individual of age a in position x, $\beta (a, x, y)$ denotes the birth rate at which an individual of age a in position y gives to a newborn in position x. We make the following assumptions on these parameters.

Assumption 3.1

 

• (i) $\mu$ is belonging to $L^\infty _+((0, a^+)\times \Omega )$ with $\mu ^+:=\Vert \mu \Vert _{L^\infty _+((0, a^+)\times \Omega )}$;

• (ii) $\beta (a, \cdot , \cdot )=0$ for $a<0$ and $a>a^+$ with

  $$\begin{aligned} \lim \limits _{\Vert h\Vert \rightarrow 0}\int _{\Omega }|\beta (a, x+h, y)-\beta (a, x, y)|dx=0 \end{aligned}$$

  uniformly in $a\in (0, a^+), y\in {\overline{\Omega }}$, where $h\in {\mathbb {R}}^N$ and $\Vert \cdot \Vert$ denotes the usual Euclidean norm in ${\mathbb {R}}^N$. Moreover,

  $$\begin{aligned} \beta _1(x)\beta _2(a, y)\ge \beta (a, x, y)\ge \epsilon \beta _3(a, y)>0 \end{aligned}$$

  for all $a\in [0, a^+]$ and $x, y\in {\overline{\Omega }}$, where $\beta _1\in Z_+$ and $\beta _2, \beta _3\in L^\infty _+((0, a^+)\times \Omega )$ and $\epsilon \in (0, 1)$.

Let $P^*(a, x)$ be the steady state solution of system (3.1). Then it satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _aP^*(a, x)=d[{\mathcal {L}}P^*](a, x)-\mu (a, x)P^*(a, x),&{}a\in (0, a^+], x\in {\overline{\Omega }},\\ P^*(0, x)=\int _{0}^{a^+}\int _{\Omega }\beta (a, x, y)P^*(a, y)dyda,&{} x\in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(3.2)

By Lemma 2.1, we obtain $P^*(a, x)=\left( {\mathcal {F}}(0, a)P^*(0)\right) (x)$, where ${\mathcal {F}}$ is the solution operator of (2.1) with f being replaced by $\mu$. Substituting it into the second equation of (3.2), one obtains the following integral equation:

$$\begin{aligned} P^*(0, x)=\int _{0}^{a^+}\int _{\Omega }\beta (a, x, y)\left( {\mathcal {F}}(0, a)P^*(0)\right) (y)dyda. \end{aligned}$$

(3.3)

In order to investigate the existence of a nontrivial solution of (3.3), we define a linear operator K on $L^1(\Omega )$ by

$$\begin{aligned} {[}K\psi ](x):=\int _{0}^{a^+}\int _{\Omega }\beta (a, x, y)\left( {\mathcal {F}}(0, a)\psi \right) (y)dyda, \quad \psi \in Z, \end{aligned}$$

(3.4)

and investigate the existence of a nontrivial fixed point $\psi ^*$ of K. We have the following proposition. We mention that its proof is very similar to those of Lemma 4.1 and Proposition 4.2 in the later sections, thus here we only provide a sketch proof.

Proposition 3.2

Let Assumption 3.1 be satisfied and K be defined by (3.4). If $r(K)=1$, then K has a positive fixed point $\psi ^*\in Z_+\setminus \{0\}:=L^1_+(\Omega )\setminus \{0\}$. Here, $r(\cdot )$ denotes the spectral radius of an operator.

Proof

Under Assumption 3.1, we can see from the well-known compactness criterion in $L^1$ (see Brezis (2010, Theorem 4.26)) that K is a compact operator (see (Kang and Ruan 2021b, Lemma 3.3)). Moreover, K is nonsupporting due to the positive lower bound of $\beta$, see Assumption 3.1-(ii). Then using the Krein-Rutman theorem (see Sawashima (1964) or Marek (1970)), we see that r(K) is an eigenvalue of K and there exists a corresponding positive nonzero eigenvector $\psi ^*\in L^1_+(\Omega )\setminus \{0\}$. Thus we have $K\psi ^*=r(K)\psi ^*$. Since $r(K)=1$, the proof is complete. $\square$

The spectral radius r(K) is thought to be the demographic reproduction number and the condition $r(K)=1$ corresponds to the situation where the growth rate is 0 in the well-known Lotka’s characteristic equation. In fact, if the birth rate is separable such that $\beta (a, x, y)=\beta _1(x)\beta _2(a, y)$ with $\beta _1\in L^1_+(\Omega )\setminus \{0\}$ and $\beta _2\in L^\infty _+((0, a^+)\times \Omega )$, then (3.4) can be rewritten as

$$\begin{aligned} {[}K\psi ](x)=\beta _1(x)\int _{0}^{a^+}\int _{\Omega }\beta _2(a, y)\left( {\mathcal {F}}(0, a)\psi \right) (y)dyda, \quad \psi \in Z, \end{aligned}$$

and the range of K becomes one-dimensional. Therefore, $\psi (x)=\beta _1(x)$ is a positive nonzero eigenvector of K such that

$$\begin{aligned} {[}K\beta _1](x)=\int _{0}^{a^+}\int _{\Omega }\beta _2(a, y)\left( {\mathcal {F}}(0, a)\beta _1\right) (y)dyda\beta _1(x)=r(K)\beta _1(x), \end{aligned}$$

and hence, $r(K)=1$ implies that the following Lotka’s characteristic equation holds:

$$\begin{aligned} \int _{0}^{a^+}\int _{\Omega }\beta _2(a, y)\left( {\mathcal {F}}(0, a)\beta _1\right) (y)dyda=1. \end{aligned}$$

Once the existence of a positive fixed point $\psi ^*(x)$ of operator K is guaranteed from Proposition 3.2, it is not other than the nontrivial solution $P^*(0, x)$ of integral equation (3.3) and thus, the existence of a nontrivial steady state solution $P^*\in X_+\setminus \{0\}$ of system (3.1) is shown. Therefore, in the rest of this paper, we assume that $r(K)=1$ and hence system (3.1) has a nontrivial steady state solution $P^*\in X_+\setminus \{0\}$. Further, we assume that such a steady state has been attained from the initial time and bounded; that is,

$$\begin{aligned} P(t, a, x)\equiv P^*(a, x), t\ge 0, \text { and } \; \sup _{(a, x)\in (0, a^+)\times \Omega }P^*(a, x)<\infty . \end{aligned}$$

3.2 SIS epidemic model

Now we consider model (1.1). Since it follows from $P(t, a, x)=P^*(a, x)$ for all $t\ge 0$ that $S(t, a, x)=P^*(a, x)-I(t, a, x)$, system (1.1) can be reduced to a single equation for I(t, a, x) as

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)I(t, a, x)=d[{\mathcal {L}}I](t, a, x)+\lambda (t, a, x)(P^*(a, x)-I(t, a, x))-&{}(\gamma (a, x)+\mu (a, x))I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in {\overline{\Omega }},\\ I(t, 0, x)=0,&{}t>0, x\in {\overline{\Omega }}\\ I(0, a, x)=I_0(a, x)\in X_+,&{}a\in [0, a^+], x\in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(3.5)

where $\lambda (t, a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I(t, \sigma , y)dyd\sigma$.

Now let us define a linear operator $A: D(A)\subset X\rightarrow X$ as

$$\begin{aligned}&A\psi :=-\partial _a\psi +d{\mathcal {L}}\psi -\mu \psi ,\nonumber \\&D(A)=\{\psi \in X:\psi \in W^{1, 1}((0, a^+), L^1(\Omega )), \psi (0, x)=0\}. \end{aligned}$$

(3.6)

Note that $\psi (0, x)$ is well-defined for any $\psi \in W^{1,1}((0, a^+), L^1(\Omega ))$ since the Sobolev embedding $W^{1,1}((0, a^+), L^1(\Omega ))\subset C([0, a^+], L^1(\Omega ))$ holds. Let E be a closed convex set defined by

$$\begin{aligned} E:=\{\psi \in X_+: 0\le \psi (a, x)\le P^*(a, x), \text {a.e.}\}. \end{aligned}$$

(3.7)

Define $F: E\subset X\rightarrow X$ of nonlinear bounded operators by

$$\begin{aligned} F(\psi )(a, x):=\lambda (a, x | \psi )(P^*(a, x)-\psi (a, x))-\gamma (a, x)\psi (a, x),\quad \psi \in E, \end{aligned}$$

(3.8)

where

$$\begin{aligned} \lambda (a, x | \psi ):=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\psi (\sigma , y)dyd\sigma . \end{aligned}$$

Note that $\lambda (\cdot , \cdot | \psi )\in L^\infty _+((0, a^+)\times \Omega )$ for a fixed $\psi \in E$ by Assumption 2.2-(iv), (v). Then system (3.5) can be rewritten as a semilinear Cauchy problem

$$\begin{aligned} \frac{d}{dt}I(t)=AI(t)+F(I(t)), \quad I(0)=I_0. \end{aligned}$$

(3.9)

We can see that operator A is the infinitesimal generator of a $C_0$-semigroup $\{e^{tA}\}_{t\ge 0}$ defined by

$$\begin{aligned} (e^{tA}\psi )(a, x)={\left\{ \begin{array}{ll} \left( {\mathcal {F}}(a-t, a)\psi (a-t)\right) (x), &{}\quad t<a,\\ 0, &{}\quad t\ge a, \end{array}\right. } \end{aligned}$$

(3.10)

where $\{{\mathcal {F}}(\sigma , a)\}_{0\le \sigma \le a\le a^+}$ is defined in Lemma 2.1 via replacing f by $\mu$. From (3.10) we immediately have $e^{tA}E\subset E$ and if $\psi \le \phi$, then $e^{tA}\psi \le e^{tA}\phi$ since the comparison principle of operators ${\mathcal {F}}(\sigma , a)$ for any $0\le \sigma \le a\le a^+$ holds, where $\le$ denotes the usual ordering in $L^1((0, a^+), Z)$.

We also note that if $\alpha >0$ the resolvent $(I-\alpha A)^{-1}$ is given by the formula:

$$\begin{aligned} {[}(I-\alpha A)^{-1}f](a, x)=\frac{1}{\alpha }\int _{0}^{a}e^{-\frac{1}{\alpha }(a-s)}\left( {\mathcal {F}}(s, a)f(s)\right) (x)ds. \end{aligned}$$

(3.11)

Then again by the comparison theorem for $\{{\mathcal {F}}(\sigma , a)\}_{0\le \sigma \le a\le a^+}$, we have that

$$\begin{aligned}&(I-\alpha A)^{-1}E\subset E,\nonumber \\&\psi \le \phi \Rightarrow (I-\alpha A)^{-1}\psi \le (I-\alpha A)^{-1}\phi , \quad \forall \psi , \phi \in E. \end{aligned}$$

(3.12)

In fact, we can check that $u\equiv P^*(a, x)$ and $u\equiv 0$ are a pair of super-solution and sub-solution of the following nonlocal problem with Dirichlet boundary condition

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u(a, x)}{\partial a}=d(J*u-u)(a, x)-\mu (a, x)u(a, x), &{}\quad a>0, x\in {\overline{\Omega }},\\ u(0, x)=u_0(x)\in E, &{}\quad x\in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(3.13)

Thus ${\mathcal {F}}(s, a)f(s, x)\in E$ for any $f\in E$. Using the fact that $\gamma$ and $\lambda$ are a.e. bounded and the same kind of argument as in Busenberg et al. (1991, Proposition 3.1), it is easy to show the following proposition.

Proposition 3.3

Let Assumption 2.2 be satisfied. Then the nonlinear bounded operator $F: E\rightarrow X$ defined in (3.8) is Lipschitz continuous and there exists a constant $\alpha \in (0, 1)$ such that if $\psi \in E$ then $\psi +\alpha F(\psi )\in E$, and if $\psi \le \phi$ then $\psi +\alpha F(\psi )\le \phi +\alpha F(\phi )$.

Note that in the proof of Proposition 3.3, we will obtain $\alpha <\frac{1}{\gamma ^++\lambda ^+}$, where $\lambda ^+$ is the upper bound of $\lambda$; that is, $\lambda [a, x | \psi ]<\lambda ^+$ for any $\psi \in E$ in $[0, a^+]\times {\overline{\Omega }}$. Note that $\lambda ^+$ can be obtained by

$$\begin{aligned} \lambda ^+:=k^+\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)P^*(\sigma , y)dyd\sigma . \end{aligned}$$

Using $\alpha$ appeared in Proposition 3.3, we can write problem (3.9) as

$$\begin{aligned} \frac{d}{dt}I(t)=\left( A-\frac{1}{\alpha }\right) I(t)+\frac{1}{\alpha }(I(t)+\alpha F(I(t)), \quad I(0)=I_0. \end{aligned}$$

(3.14)

Now we are in a position to look for a mild solution $I\in E$ of (3.14), which is given by a solution of the integral equation

$$\begin{aligned} I(t)=e^{-\frac{1}{\alpha }t}e^{tA}I_0+\frac{1}{\alpha }\int _{0}^{t}e^{-\frac{1}{\alpha }(t-s)}e^{(t-s)A}[I(s)+\alpha F(I(s))]ds. \end{aligned}$$

(3.15)

Equation (3.15) provides the following scheme of the standard iterative procedure for obtaining a mild solution $I\in E$:

$$\begin{aligned}&I^0(t)=I_0,\nonumber \\&I^{n+1}(t)=e^{-\frac{1}{\alpha }t}e^{tA}I_0+\frac{1}{\alpha }\int _{0}^{t}e^{-\frac{1}{\alpha }(t-s)}e^{(t-s)A}[I^n(s)+\alpha F(I^n(s))]ds,\nonumber \\&\quad n=0, 1, 2,\dots . \end{aligned}$$

(3.16)

Since we can see that E is invariant with respect to the iteration process, that is $I^{n+1}\in E$ if $I^n\in E$, according to the argument in Busenberg et al. (1991, Theorem 3.2), we can prove the following theorem.

Theorem 3.4

Let Assumption 2.2 be satisfied and $I_0\in E$. Then Eq. (3.9) has a unique mild solution $I(t)={\mathcal {S}}(t)I_0\in C([0, \infty ), E)$, where $\{{\mathcal {S}}(t)\}_{t\ge 0}$ defines a $C_0$-semigroup in X with the following properties:

• (i) ${\mathcal {S}}(t)E\subset E$;

• (ii) If $I_0\le I_1$, then ${\mathcal {S}}(t)I_0\le {\mathcal {S}}(t)I_1$;

• (iii) If $\xi \in (0, 1)$, then $\xi {\mathcal {S}}(t)I_0\le {\mathcal {S}}(t)(\xi I_0)$.

Proof

For the completeness, we provide the proof based on Busenberg et al. (1991, Theorem 3.2). In fact, (i) and (ii) can be obtained easily by the arguments before the theorem. Thus we are only concerned on (iii). Let $I_\xi ^n$ be the iterates (3.16) with $I_0$ replaced by $\xi I_0$. Since $0\le \xi \le 1$, we have

$$\begin{aligned} \xi I+\alpha F(\xi I)\ge \xi (I+\alpha F(I)), \end{aligned}$$

so that

$$\begin{aligned} I_\xi ^n\ge \xi I^n, \end{aligned}$$

and going to the limit, we obtain (iii). $\square$

4 Existence of nontrivial steady states

Now in this section we investigate the existence of steady states of (3.5). Let $I^*(a, x)$ be a (time-independent) steady states solution of system (3.5). Then it satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _aI^*(a, x)=d[{\mathcal {L}}I^*](a, x)+\lambda ^*(a, x)(P^*(a, x)-I^*(a, x))-(\gamma +\mu )I^*(a, x),&{}\quad a{\in }(0, a^+], x{\in }{\overline{\Omega }},\\ I^*(0, x)=0, &{}\quad x\in {\overline{\Omega }},\\ \lambda ^*(a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I^*(\sigma , y)dyd\sigma ,&{}\quad a{\in }(0, a^+], x{\in }{\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(4.1)

It is obvious that the trivial disease-free steady state solution $I^*(a, x)\equiv 0$ always exists. In what follows, we investigate the existence of a nontrivial endemic steady state solution $I^*\in E\setminus \{0\}$, where E is defined in (3.7).

Considering the linearized equation of (4.1) at $I^*=0$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _aI^*(a, x)=d[{\mathcal {L}}I^*](a, x)+\lambda ^*(a, x)P^*(a, x)-(\gamma +\mu )I^*(a, x),&{}\quad a{\in }(0, a^+], x{\in }{\overline{\Omega }},\\ I^*(0, x)=0, &{}\quad x\in {\overline{\Omega }},\\ \lambda ^*(a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I^*(\sigma , y)dyd\sigma ,&{}\quad a{\in }(0, a^+], x{\in }{\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(4.2)

Viewing $f(a, x)=\lambda ^*(a, x)+\gamma (a, x)+\mu (a, x)$ in Lemma 2.1 and solving the first equation of (4.1), we obtain

$$\begin{aligned} I^*(a, x)=\int _{0}^{a}\left( {\mathcal {F}}_{\lambda ^*}(\sigma , a)(\lambda ^*P^*)(\sigma )\right) (x)d\sigma , \end{aligned}$$

(4.3)

where ${\mathcal {F}}_{\lambda ^*}$ is the solution operator of (4.1) as in Lemma 2.1 with f being replaced by $\lambda ^*+\gamma +\mu$ (note that $\lambda ^*+\gamma +\mu \in L^\infty _+((0, a^+)\times \Omega )$). Substituting (4.3) into the third equation of (4.1), we obtain the following integral equation of $\lambda ^*(a, x)$:

$$\begin{aligned} \lambda ^*(a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_{\lambda ^*}(\tau , \sigma )(\lambda ^*P^*)(\tau )\right) (y)d\tau dyd\sigma . \end{aligned}$$

(4.4)

Solving the first equation of (4.2), we obtain

$$\begin{aligned} I^*(a, x)=\int _{0}^{a}\left( {\mathcal {F}}_0(\sigma , a)(\lambda ^*P^*)(\sigma )\right) (x)d\sigma , \end{aligned}$$

(4.5)

where ${\mathcal {F}}_0$ is the solution operator of (4.1) as in Lemma 2.1 with f being replaced by $\gamma +\mu$. Substituting (4.5) into the third equation of (4.2), we obtain the following integral equation of $\lambda ^*(a, x)$:

$$\begin{aligned} \lambda ^*(a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\tau , \sigma )(\lambda ^*P^*)(\tau )\right) (y)d\tau dyd\sigma . \end{aligned}$$

(4.6)

Observe that we have ${\mathcal {F}}_{\lambda ^*}(\sigma , a)\le {\mathcal {F}}_0(\sigma , a)$ for all $0\le \sigma \le a\le a^+$ in the operator sense by Lemma 2.1. Now we define a nonlinear operator $\Phi$ on X by

$$\begin{aligned} {[}\Phi \phi ](a, x):=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_{\phi }(\tau , \sigma )(\phi P^*)(\tau )\right) (y)d\tau dyd\sigma ,\quad \phi \in X, \end{aligned}$$

(4.7)

and investigate the existence of a nontrivial positive fixed point $\phi ^*\in X_+\setminus \{0\}$ such that $\Phi (\phi ^*)=\phi ^*$, which is no other than the desired nontrivial positive solution $\lambda ^*\in X_+\setminus \{0\}$ of (4.4).

We can check that $\Phi$ is bounded and positive by Assumption 2.2 and the boundedness of $P^*$. Next, consider the following linear integral operator M on X:

$$\begin{aligned} {[}M\phi ](a, x):=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\tau , \sigma )(\phi P^*)(\tau )\right) (y)d\tau dyd\sigma ,\quad \phi \in X. \end{aligned}$$

(4.8)

One can see that M is the Fréchert derivative of $\Phi$ at $\phi =0$ and a positive linear majorant of $\Phi$. We will show that the spectral radius r(M) of operator M plays the role of a threshold value for the existence of a nontrivial fixed point of operator $\Phi$; that is, if $r(M)>1$, then $\Phi$ has a positive nontrivial fixed point. We prove the following lemma about M.

Lemma 4.1

Let Assumption 2.2 be satisfied and M be defined by (4.8). Then M is positive, bounded, compact and nonsupporting.

Proof

First we can see that M is positive since $\{{\mathcal {F}}_0(\tau , \sigma )\}_{0\le \tau \le a}$ is positive. Next we prove the compactness. Note by Assumption 2.2-(i) that for $\phi \in {\mathcal {B}}$, where ${\mathcal {B}}$ is a uniformly bounded subset in X,

$$\begin{aligned}&\Vert [M\phi ](\cdot +\kappa , \cdot +h)-[M\phi ](\cdot , \cdot )\Vert _X\nonumber \\&\quad \le \int _{0}^{a^+}\!\int _{\Omega }\left[ \int _{0}^{a^+}\!\int _{\Omega }|k(a+\kappa , \sigma , x+h, y)-k(a, \sigma , x, y)|\nonumber \right. \\&\left. \qquad \times \int _{0}^{\sigma }\!\left( {\mathcal {F}}_0(\sigma , \tau )(\phi P^*)(\tau )\right) (y)d\tau dyd\sigma \right] dxda\nonumber \\&\quad \le \int _{0}^{a^+}\!\int _{\Omega }\left[ \int _{0}^{a^+}\!\int _{\Omega }|k(a+\kappa , \sigma , x+h, y)-k(a, \sigma , x, y)|dxda\right] \nonumber \\&\qquad \times \int _{0}^{\sigma } \left( {\mathcal {F}}_0(\sigma , \tau )(\phi P^*)(\tau )\right) (y)d\tau dyd\sigma \nonumber \\&\quad \rightarrow 0\quad \text {as}\quad \Vert (\kappa , h)\Vert \rightarrow 0, \end{aligned}$$

(4.9)

where we used Minkowski inequality and Assumption 2.2-(iii). Thus $M: X\rightarrow X$ is compact by Kolmogorov compactness theorem, see Brezis (2010, Theorem 4.26). Finally, we prove that M is nonsupporting. By Assumption 2.2-(iv), we can define the following linear functional $f_0\in X^*$ on X,

$$\begin{aligned} \langle f_0, \phi \rangle :=\int _{0}^{a^+}\int _{\Omega }\epsilon k^-k_2(\sigma , y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\tau , \sigma )(\phi P^*)(\tau )\right) (y)d\tau dyd\sigma , \end{aligned}$$

and hence we see from the positivity of $P^*, k^-$ and $\{{\mathcal {F}}_0(\tau , \sigma )\}_{0\le \tau \le \sigma \le a^+}$ that the functional $f_0$ is strictly positive (that is, $\langle f_0, \phi \rangle >0$ for all $\phi \in X_+\setminus \{0\}$). Now we have

$$\begin{aligned} M\phi \ge \langle f_0, \phi \rangle e, \quad e(a, x)\equiv 1, \quad \forall (a, x)\in [0, a^+]\times \Omega . \end{aligned}$$

Note that $e\in X_+$, then we have

$$\begin{aligned} M^{n+1}\phi\ge & {} \langle f_0, M^n\phi \rangle e\nonumber \\\ge & {} \langle f_0, \langle f_0, M^{n-1}\phi \rangle e\rangle e\nonumber \\= & {} \langle f_0, e\rangle \langle f_0, M^{n-1}\phi \rangle e\nonumber \\\ge & {} \langle f_0, e\rangle ^n\langle f_0, \phi \rangle e, \quad n=1,2,\dots . \end{aligned}$$

(4.10)

Thus, for any $M^*\in X^*_+\setminus \{0\}$ and $\phi \in X_+\setminus \{0\}$, it follows that

$$\begin{aligned} \langle M^*, M^n\phi \rangle \ge \langle f_0, e\rangle ^{n-1}\langle f_0, \phi \rangle \langle M^*, e\rangle >0, \quad n=1, 2,\dots , \end{aligned}$$

which implies that M is nonsupporting. The proof is complete. $\square$

Similar to Lemma 4.1, one can show that $\Phi$ defined by (4.7) is also compact. Under Lemma 4.1, we can apply the fixed point theorem (Theorem 2.3) to prove the following proposition.

Proposition 4.2

Let Assumption 2.2 be satisfied and $\Phi , M$ be defined by (4.7) and (4.8), respectively.

• (i) If $r(M)\le 1$, then $\Phi$ only has the trivial fixed point $\phi \equiv 0$ in $X_+$;

• (ii) If $r(M)>1$, then $\Phi$ has at least one nontrivial fixed point $\phi ^*\in X_+\setminus \{0\}$.

Proof

First note that conditions (i) and (ii) of Theorem 2.3 are obviously satisfied. From Lemma 4.1, we can apply the theory of nonsupporting operators (see Inaba (1990) or Marek (1970)) to prove that $r(M)>1$ is an eigenvalue of operator M and there exists a corresponding positive nonzero eigenvector and M does not have any eigenvector associated with eigenvalue 1. Hence, conditions (iii) and (iv) of Theorem 2.3 follow and, consequently, $\Phi$ has at least one nontrivial fixed point in $X_+$.

Next, we prove (i). From Lemma 4.1, we see that M is compact and nonsupporting. Since $\Phi$ is a positive operator from the positive cone $X_+$ to itself and M is the positive linear majorant of $\Phi$, we can apply Theorem 2.4 to conclude that $\Phi$ has no nontrivial fixed point in $X_+$ provided $r(M)\le 1$. This completes the proof. $\square$

The existence of a nontrivial fixed point of $\Phi$ implies the existence of a nontrivial steady state solution $I^*\in X_+\setminus \{0\}$; that is, the existence of a nontrivial steady state solution $(S^*, I^*)\in X_+^2\setminus \{0, 0\}$ of the original system (1.1). Further, $I^*$ must be in $E\setminus \{0\}$, otherwise $I^*\equiv 0$ from (4.1). In conclusion, from Proposition 4.2, the following theorem can be obtained as one of the main results of this paper.

Theorem 4.3

Let Assumption 2.2 be satisfied and M be defined in (4.8).

• (i) If $r(M)\le 1$, then system (1.1) only has the trivial steady state $(S^*, I^*)=(P^*, 0)$ in $X_+^2$;

• (ii) If $r(M)>1$, then system (1.1) has at least one nontrivial steady state $(S^*, I^*)$ in $X_+^2$.

5 Basic reproduction number

In this section we show that the threshold value r(M) obtained in the previous section is equal to the basic reproduction number ${\mathcal {R}}_0$ for system (1.1). Following the idea from Diekmann et al. (1990) and Kuniya and Oizumi (2015), linearizing system (3.5) at the disease-free steady state, we obtain the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t+\partial _a)I(t, a, x)=d[{\mathcal {L}}I](t, a, x)+v(t, a, x)-(\gamma +\mu )I(t, a, x),&{}t>0, a\in (0, a^+], x\in {\overline{\Omega }},\\ I(t, 0, x)=0,&{}t>0, x\in {\overline{\Omega }}\\ I(0, a, x)=I_0(a, x),&{}a\in [0, a^+], x\in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(5.1)

where $v(t, a, x)=P^*(a, x)\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I(t, \sigma , y)dyd\sigma$ denotes the density of newly infected population at the linear invasion phase. Integrating along the characteristic line $t-a=$constant, we obtain

$$\begin{aligned} I(t, a, x)= \int _{0}^{a}\left( {\mathcal {F}}_0(\sigma , a)v(t-a+\sigma , \sigma )\right) (x)d\sigma . \end{aligned}$$

Here note that it is suffices to consider $t>a^+$ from Assumption 2.2-(ii). Now plugging it into the equation of v, we have

$$\begin{aligned} v(t, a, x)= & {} P^*(a, x)\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\tau , \sigma )v(t-\sigma +\tau , \tau )\right) (y)d\tau dyd\sigma \nonumber \\= & {} P^*(a, x)\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\sigma -\rho , \sigma )v(t-\rho , \sigma -\rho )\right) (y)d\rho dyd\sigma \nonumber \\= & {} P^*(a, x)\int _{\Omega }\int _{0}^{a^+}\int _{\rho }^{a^+}k(a, \sigma , x, y)\left( {\mathcal {F}}_0(\sigma -\rho , \sigma )v(t-\rho , \sigma -\rho )\right) (y)d\sigma d\rho dy.\nonumber \\ \end{aligned}$$

(5.2)

Defining a linear operator $B(\rho , x, y): X\rightarrow L^1(0, a^+)$ by

$$\begin{aligned} {[}B(\rho , x, y)\varphi (\cdot , y)](a):=\int _{\rho }^{a^+}k(a, \sigma , x, y)\left( {\mathcal {F}}_0(\sigma -\rho , \sigma )\varphi (\sigma -\rho )\right) (y)d\sigma , \quad \varphi \in X, \end{aligned}$$

we can rewrite (5.2) as the abstract integral equation in X:

$$\begin{aligned} v(t, x)=P^*(\cdot , x)\int _{\Omega }\int _{0}^{a^+}B(\rho , x, y)v(t-\rho , y)d\rho dy. \end{aligned}$$

Then following the arguments in Diekmann et al. (1990), we obtain the next generation operator in X as

$$\begin{aligned} {[}{\mathcal {G}}\varphi ](a, x)&:=P^*(a, x)\int _{\Omega }\int _{0}^{a^+}[B(\rho , x, y)\varphi ](a)d\rho dy\nonumber \\&=P^*(a, x)\int _{\Omega }\int _{0}^{a^+}\int _{\rho }^{a^+}k(a, \sigma , x, y)\nonumber \\&\quad \times \left( {\mathcal {F}}_0(\sigma -\rho , \sigma )\varphi (\sigma -\rho )\right) (y)d\sigma d\rho dy, \end{aligned}$$

(5.3)

and the basic reproduction number ${\mathcal {R}}_0$ is defined by its spectral radius

$$\begin{aligned} {\mathcal {R}}_0:=r({\mathcal {G}}). \end{aligned}$$

(5.4)

Then we have the following proposition inspired from Kuniya and Oizumi (2015, Proposition 4.1).

Proposition 5.1

Let $M, {\mathcal {G}}$ and ${\mathcal {R}}_0$ be defined in (4.8), (5.3) and (5.4) respectively. It follows that

$$\begin{aligned} r(M)=r({\mathcal {G}})={\mathcal {R}}_0. \end{aligned}$$

Proof

Considering the calculation of (5.2) conversely, we see that the right hand side of (5.2) is

$$\begin{aligned} {[}{\mathcal {G}}\varphi ](a, x)=P^*(a, x)\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\left( {\mathcal {F}}_0(\tau , \sigma )\varphi (\tau )\right) (y)d\tau dyd\sigma . \end{aligned}$$

(5.5)

Let us define a linear operator Q on X by

$$\begin{aligned}{}[Q\psi ](a, x)=\psi (a, x)P^*(a, x), \quad \psi \in X. \end{aligned}$$

Then, comparing (4.8) with (5.5), we see that

$$\begin{aligned} M=Q^{-1}{\mathcal {G}}Q. \end{aligned}$$

Since $r(Q^{-1}{\mathcal {G}}Q)=r({\mathcal {G}})={\mathcal {R}}_0$, we obtain $r(M)=r({\mathcal {G}})={\mathcal {R}}_0$. $\square$

Theorem 5.2

Let Assumption 2.2 be satisfied and ${\mathcal {R}}_0$ be the basic reproduction number defined in (5.4).

• (i) If ${\mathcal {R}}_0\le 1$, then system (1.1) has only the trivial steady state $(S^*, I^*)=(P^*, 0)$ in $X_+^2$;

• (ii) If ${\mathcal {R}}_0>1$, then system (1.1) has at least one nontrivial steady state $(S^*, I^*)$ in $X_+^2$.

6 Uniqueness of the nontrivial steady state

In this section we discuss the uniqueness of the nontrivial steady state $I^*\in E$. Note that $I^*$ is a steady state of (3.5) if and only if it satisfies

$$\begin{aligned} TI^*:=(I-\alpha A)^{-1}(I+\alpha F)I^*=I^*. \end{aligned}$$

(6.1)

From (3.12) and Proposition 3.3, we can see that $T(E)\subset E$ and $T\psi \le T\phi$ for any $\psi \le \phi$ and $\psi , \phi \in E$.

We need several lemmas to prove the uniqueness. First we state a necessary condition which should be satisfied by any nontrivial solution of (6.1).

Lemma 6.1

Let $I^*$ be a nontrivial solution of (6.1). Then

$$\begin{aligned} \int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I^*(\sigma , y)dyd\sigma >0. \end{aligned}$$

(6.2)

Proof

If (6.2) is not satisfied, then

$$\begin{aligned} \int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I^*(\sigma , y)dyd\sigma =0,\quad \forall (a, x)\in (0, a^+)\times {\overline{\Omega }}, \end{aligned}$$

and $I^*$ satisfies the following equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _aI^*(a, x)=d{\mathcal {L}}I^*(a, x)-(\gamma (a, x)+\mu (a, x))I^*(a, x),&{}\quad a\in (0, a^+], x\in {\overline{\Omega }},\\ I^*(0, x)=0,&{}\quad x\in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(6.3)

Thus $I^*$ is identically zero by the uniqueness of solution for the above linear equation, which is a contradiction. $\square$

We next give an estimate on the nontrivial fixed point of T.

Lemma 6.2

Let Assumption 2.2 be satisfied and $I^*$ be a nontrivial solution to (6.1). Then

$$\begin{aligned} \epsilon _1m(a, x)\le I^*(a, x)\le \epsilon _2m(a, x), \end{aligned}$$

(6.4)

where

$$\begin{aligned} m(a, x):=\int _{0}^{a}\left( {\mathcal {F}}(\sigma , a)(k_1P^*)(\sigma )\right) (x)d\sigma , \end{aligned}$$

(6.5)

$\epsilon _1$ and $\epsilon _2$ are positive constants depending on $I^*$ and $\{{\mathcal {F}}(\sigma , a)\}_{0\le \sigma \le a\le a^+}$ is from Lemma 2.1 via replacing f by $\mu$.

Proof

As $I^*$ is a nontrivial solution, (6.2) is satisfied so that

$$\begin{aligned} c_1k_1(a, x)\le \lambda [a, x | I^*]\le c_2k_1(a, x), \end{aligned}$$

where $c_1, c_2$ are positive. Hence $I^*$ satisfies the inequality

$$\begin{aligned}&c_1k_1(a, x)P^*(a, x)-(\lambda ^++\gamma ^+)I^*(a, x)\nonumber \\&\quad \le \partial _aI^*(a, x)-d{\mathcal {L}}I^*(a, x)+\mu (a, x)I^*(a, x)\nonumber \\&\quad \le c_2k_1(a, x)P^*(a, x). \end{aligned}$$

(6.6)

Since $I^*(0, x)=0$, we can find positive $\epsilon _1, \epsilon _2$ such that (6.4) is satisfied. $\square$

Lemma 6.3

Let Assumption 2.2 be satisfied and $I^*\in E$ be a nontrivial fixed point of T. Let $\xi \in (0, 1)$ and $I\in E$ such that $\xi I^*+I\in E$ and $\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(\sigma , y)dyd\sigma >0$. Then there exists a constant $\epsilon _3>0$ such that

$$\begin{aligned} T(\xi I^*+I)\ge \xi I^*+\epsilon _3(1-\xi )m. \end{aligned}$$

(6.7)

Proof

Observe that ($\alpha <\frac{1}{\gamma ^++\lambda ^+}$)

$$\begin{aligned}&\xi I^*+I+\alpha F(\xi I^*+I)\\&\quad =\xi I^*+I+\alpha (P^*-\xi I^*-I)\lambda [\cdot , \cdot | \xi I^*+I]-\alpha \gamma (\xi I^*+I) \\&\quad = \xi I^*+\alpha F(\xi I^*)+I(1-\alpha \lambda [\cdot , \cdot | \xi I^*+I]-\alpha \gamma )+\alpha (P^*-\xi I^*)\lambda [\cdot ,\cdot | I] \\&\quad \ge \xi I^*+\alpha \xi F(I^*)+\alpha (P^*-\xi I^*)\lambda [\cdot , \cdot | I] \\&\quad \ge \xi (I^*+\alpha F(I^*))+\alpha \epsilon (P^*-\xi I^*)k_1\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(\sigma , y)dyd\sigma . \end{aligned}$$

Then from (3.11), (3.12) and (6.5), we obtain (6.7). $\square$

Lemma 6.4

Let Assumption 2.2 be satisfied and $I^*\in E$ be a nontrivial fixed point of T and let $\xi \in (0, 1)$. Then there exists a constant $\epsilon _4>0$ such that

$$\begin{aligned} T(\xi I^*)\ge \xi I^*+\xi (1-\xi )\epsilon _4m_1, \end{aligned}$$

(6.8)

where $m_1(a, x):=\int _{0}^{a}\left( {\mathcal {F}}(\sigma , a)I^*(\sigma )\right) (x)d\sigma$.

Proof

Observe that

$$\begin{aligned}&\xi I^*+\alpha F(\xi I^*)\\&\quad =\xi (I^*+\alpha F(I^*))+\alpha \xi (1-\xi )I^*\lambda [\cdot , \cdot | I^*] \\&\quad \ge \xi (I^*+\alpha F(I^*))+\alpha \xi (1-\xi )\epsilon I^*k_1\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I^*(\sigma , y)dyd\sigma \\&\quad \ge \xi (I^*+\alpha F(I^*))+\alpha \xi (1-\xi )\epsilon I^*k^-\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I^*(\sigma , y)dyd\sigma . \end{aligned}$$

Then we have the desired (6.8) from (3.11), (3.12) and definition of $m_1$. $\square$

Theorem 6.5

Let Assumption 2.2 be satisfied, then T has at most one nontrivial fixed point.

Proof

Let $u^*\ne v^*$ be two nontrivial fixed points. Let $\xi =\sup \{k\in {\mathbb {R}}_+: v^*\ge ku^*\}$. From (6.4), we can see that $\xi >0$. If $\xi <1$, we have from Lemma 6.4 that

$$\begin{aligned} v^*=T^2(v^*)\ge T^2(\xi u^*)\ge T(\xi u^*+\xi (1-\xi )\epsilon _4m_1). \end{aligned}$$

If $\epsilon _4$ is small enough, we can assume that $\xi (1-\xi )\epsilon _4m_1, \xi u^*+\xi (1-\xi )\epsilon _4m_1\in E$. Then we can apply Lemma 6.3 (taking $I=\xi (1-\xi )\epsilon _4m_1$) to obtain

$$\begin{aligned} v^*\ge T(\xi u^*+\xi (1-\xi )\epsilon _4m_1)\ge \xi u^*+(1-\xi )\epsilon _3m, \end{aligned}$$

which implies by (6.4) that

$$\begin{aligned} v^*\ge \xi u^*+(1-\xi )\frac{\epsilon _3}{\epsilon _2}u^*. \end{aligned}$$

But this contradicts the definition of $\xi$. Therefore, we have $\xi \ge 1$ and $v^*\ge u^*$. By using the same approach, we can show that $u^*\ge v^*$. Then $u^*=v^*$. $\square$

7 Local stability of the nontrivial steady state

7.1 Local stability

In this subsection we analyze local stability of the nontrivial steady state $I^*\in E$. Note that $I^*\in L^\infty _+((0, a^+)\times \Omega )$ since $I^*\in E$ implies that $I^*(a, x) \le P^*(a, x)$. Consider the linearization of system (4.1) at the nontrivial steady state $I^*\in E$ in the following

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t\!+\!\partial _a)I(t, a, x)\!=\!d[{\mathcal {L}}I](t, a, x)\!+\!\lambda (t, a, x | I)(P^*\!-\!I^*)(a, x)-&{}\lambda ^*(a, x)I(t, a, x)\!{-}\!(\gamma {+}\mu )I(t, a, x),\\ &{}t>0, a\in (0, a^+], x\in {\overline{\Omega }},\\ I(t, 0, x)=0,&{}t>0, x\in {\overline{\Omega }}\\ I(0, a, x)=I_0(a, x),&{}a\in [0, a^+], x\in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(7.1)

where $\lambda (t, a, x | I)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I(t, \sigma , y)dyd\sigma$. Define a linear operator ${\mathcal {A}}: D({\mathcal {A}})\subset X\rightarrow X$ as

$$\begin{aligned}&{\mathcal {A}}\psi :=-\partial _a\psi +d{\mathcal {L}}\psi -\lambda ^*(a, x)\psi -(\gamma (a, x)+\mu (a, x))\psi ,\nonumber \\&D({\mathcal {A}})=\{\psi \in X:\psi \in W^{1, 1}((0, a^+), L^1(\Omega )), \psi (0, x)=0\}, \end{aligned}$$

(7.2)

where $\lambda ^*(a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I^*(\sigma , y)dyd\sigma$. Rewrite equation (7.1) as

$$\begin{aligned} \frac{d}{dt}I(t)={\mathcal {A}}I(t)+G(I(t)),\quad I(0)=I_0, \end{aligned}$$

(7.3)

where $G: E\subset X\rightarrow E$ is defined as

$$\begin{aligned}{}[G\psi ](a, x):=\lambda (a, x | \psi )(P^*(a, x)-I^*(a, x)). \end{aligned}$$

Then by using the same argument in Lemma 4.1, one can show that G is compact. Now we can check that ${\mathcal {A}}$ generates a nilpotent semigroup and its perturbed semigroup by the compact operator G is eventually compact. Hence, $\omega _1({\mathcal {A}}+G)=\omega _1({\mathcal {A}})=-\infty$, where $\omega _1(A)$ represents the essential growth bound of operator A. It follows that the stability of $I^*$ is determined by the eigenvalues of ${\mathcal {A}}+G$. By solving the characteristic equation $({\mathcal {A}}+G)\psi =\Lambda \psi$ and viewing $f(a, x)=\lambda ^*(a, x)+\gamma (a, x)+\mu (a, x)$ in Lemma 2.1, we obtain

$$\begin{aligned} \psi (a, x)=\int _{0}^{a}e^{-\Lambda (a-\sigma )}\left( {\mathcal {F}}_{\lambda ^*}(\sigma , a)(\lambda (\psi )(P^*-I^*))(\sigma )\right) (x)d\sigma . \end{aligned}$$

(7.4)

Plugging (7.4) into the equation of $\lambda$, we have

$$\begin{aligned} \lambda (a, x | \psi )&=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }e^{-\Lambda (\sigma -\tau )}\nonumber \\&\quad \times \left( {\mathcal {F}}_{\lambda ^*}(\tau , \sigma )(\lambda (\psi )(P^*-I^*))(\tau )\right) (y)d\tau dyd\sigma . \end{aligned}$$

(7.5)

Define an operator $\Gamma _\Lambda : X\rightarrow X$ by

$$\begin{aligned} \Gamma _\Lambda \varphi :=\int _{0}^{a^+}\int _{\Omega }k(\cdot , \sigma , \cdot , y)\int _{0}^{\sigma }e^{-\Lambda (\sigma -\tau )}\left( {\mathcal {F}}_{\lambda ^*}(\tau , \sigma )(\varphi (P^*-I^*))(\tau )\right) (y)d\tau dyd\sigma .\nonumber \\ \end{aligned}$$

(7.6)

Here we should emphasize that ${\mathcal {F}}_{\lambda ^*}(\tau , \sigma )$ is a linear operator since $\lambda ^*$ is fixed now and that ${\mathcal {F}}_{\lambda ^*}(\tau , \sigma )$ is applied to the whole $\varphi (P^*-I^*)(\tau , y)$ rather than only $\varphi (\tau , y)$. Note that $\Gamma _\Lambda$ is well defined due to the boundedness of $P^*$ and $I^*$. Then it is easy to check that $\Gamma _\Lambda$ is also compact and nonsupporting by Lemma 4.1, thus has a simple eigenvalue $r(\Gamma _\Lambda )$ corresponding to a simple and positive eigenvector. Moreover the mapping $\Lambda \rightarrow r(\Gamma _\Lambda ): {\mathbb {R}}\rightarrow (0, \infty )$ is continuous and strictly decreasing from $+\infty$ to 0 (see Lemma 3.4 of Kang et al. (2020b)). This implies that there exists a unique $\Lambda _0$ associated with a positive eigenvector $\varphi _0\in X_+$ such that $r(\Gamma _{\Lambda _0})=1$. Thus $\Gamma _{\Lambda _0}\varphi _0=\varphi _0$ and hence $\Lambda _0$ is an eigenvalue of ${\mathcal {A}}+G$. In what follows, we will show that such $\Lambda _0$ is also a principal eigenvalue, i.e. $\Lambda _0>|\Lambda |$ for any other $\Lambda \in \sigma _P({\mathcal {A}}+G)\setminus \{\Lambda _0\}$, where $\sigma _P(A)$ represents the point spectrum of A. Once it is true, we have the following theorem.

Theorem 7.1

Let Assumption 2.2 be satisfied. Then it is locally stable when $\Lambda _0<0$, which is equivalent to $r(\Gamma _0)<1$; Otherwise, $I^*$ is unstable when $\Lambda _0>0$, which is equivalent to $r(\Gamma _0)>1$.

Proof

Since $\Gamma _\Lambda$ is compact, it follows that $\sigma (\Gamma _\Lambda )\setminus \{0\}=\sigma _P(\Gamma _\Lambda )\setminus \{0\}$. Note that $\{\Lambda \in {\mathbb {C}}: 1\in \sigma _P(\Gamma _\Lambda )\}=\{\Lambda \in {\mathbb {C}}: \Lambda \in \sigma _P({\mathcal {A}}+G)\}$. Hence, it is sufficient to prove that for any $\Lambda \in \sigma _P(\Gamma _\Lambda )\setminus \{\Lambda _0\}$, $\Lambda _0>|\Lambda |$.

For any $\Lambda \in \sigma _P(\Gamma _\Lambda )$, there is an eigenfunction $\varphi _\Lambda$ such that $\Gamma _\Lambda \varphi _\Lambda =\varphi _\Lambda$. Then we have $|\varphi _\Lambda |=|\Gamma _\Lambda \varphi _\Lambda |\le \Gamma _{{\mathrm{Re}}\Lambda }|\varphi _\Lambda |$. Let $f_{{\mathrm{Re}}\Lambda }$ be a positive eigenfunctional corresponding to the eigenvalue $r(\Gamma _{{\mathrm{Re}}\Lambda })$ of $\Gamma _{{\mathrm{Re}}\Lambda }$. We obtain that

$$\begin{aligned} \langle f_{{\mathrm{Re}}\Lambda }, \Gamma _{{\mathrm{Re}}\Lambda }|\varphi _\Lambda |\rangle =r(\Gamma _{{\mathrm{Re}}\Lambda })\langle f_{{\mathrm{Re}}\Lambda }, |\varphi _\Lambda |\rangle \ge \langle f_{{\mathrm{Re}}\Lambda }, |\varphi _\Lambda |\rangle . \end{aligned}$$

Hence, $r(\Gamma _{{\mathrm{Re}}\Lambda })\ge 1$ and ${\mathrm{Re}}\Lambda \le \Lambda _0$ since $r(\Gamma _\Lambda )$ is strictly decreasing for $\Lambda \in {\mathbb {R}}$ and $r(\Gamma _\Lambda )=1$. If ${\mathrm{Re}}\Lambda =\Lambda _0$, then $\Gamma _{\Lambda _0}|\varphi _\Lambda |=|\varphi _\Lambda |$. In fact, if $\Gamma _{\Lambda _0}|\varphi _\Lambda |>|\varphi _\Lambda |$, taking duality pairing with the eigenfunctional $f_{\Lambda _0}$ corresponding to the eigenvalue $r(\Gamma _{\Lambda _0})=1$ on both sides yields $\langle f_{\Lambda _0}, \Gamma _{\Lambda _0}|\varphi _\Lambda |\rangle =\langle f_{\Lambda _0}, |\varphi _\Lambda |\rangle >\langle f_{\Lambda _0}, |\varphi _\Lambda |\rangle ,$ which is a contradiction. Then we can write that $|\varphi _\Lambda |=c\varphi _0$, where $\varphi _0$ is the eigenfunction corresponding to the eigenvalue $r(\Gamma _{\Lambda _0})=1$. Without loss of generality, we assume that $c=1$ and write $\varphi _\Lambda (a, x)=\varphi _0(a, x)e^{i\alpha (a, x)}$ for some real function $\alpha (a, x)$. Substituting this relation into

$$\begin{aligned} \Gamma _{\Lambda _0}\varphi _0=\varphi _0=|\varphi _\Lambda |=|\Gamma _\Lambda \varphi _\Lambda |, \end{aligned}$$

we have

$$\begin{aligned}&\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }e^{-\Lambda _0(\sigma -\tau )}\left( {\mathcal {F}}_{\lambda ^*}(\tau , \sigma )(\varphi _0(P^*-I^*))(\tau )\right) (y)d\tau dyd\sigma \\&\quad =\left| \int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\right. \int _{0}^{\sigma }e^{-\Lambda _0(\sigma -\tau )}e^{-{i\mathrm{Im}}\Lambda (\sigma -\tau )} \\&\qquad \times \left. \left( {\mathcal {F}}_{\lambda ^*}(\tau , \sigma )(\varphi _0(P^*\!-\!I^*))(\tau )\right) (y)e^{i\alpha (\tau , y)}d\tau dyd\sigma \right| . \end{aligned}$$

From Heijmans (1986, Lemma 6.12), we obtain that $-{{\mathrm{Im}}}\Lambda (\sigma -\tau )+\alpha (\tau , y)=\theta$ for some constant $\theta$. By $\Gamma _\Lambda \varphi _\Lambda =\varphi _\Lambda$, we have $e^{i\theta }\Gamma _{\Lambda _0}\varphi _0=\varphi _0 e^{i\alpha }$, so $\theta =\alpha (a, x)$, which implies that ${{\mathrm{Im}}}\Lambda =0$. Then, there is no element $\Lambda \in \sigma _P(\Gamma _\Lambda )$ such that ${\mathrm{Re}}\Lambda =\Lambda _0$ and $\Lambda \ne \Lambda _0$. Therefore, the result is proved. $\square$

Remark 7.2

Note that $\Gamma _0\le M$ in the sense of positive operators since ${\mathcal {F}}_{\lambda ^*}\le {\mathcal {F}}_0$ and that when $I^*\equiv 0$, $\Gamma _0=M$. It follows that if $r(M)<1$, there exists only the trivial steady state $(P^*, 0)$ for (1.1) which is locally stable. Further, if $r(\Gamma _0)<1<r(M)$, there exists a unique nontrivial steady state $(S^*, I^*)$ for (1.1) which is locally stable. Otherwise, if $r(\Gamma _0)>1$, there exists a unique nontrivial steady state $(S^*, I^*)$ for (1.1) which is locally unstable.

7.2 Effects of small diffusion

Now let us study the effects of small diffusion rate d on the principal eigenvalue $\Lambda _0$. We write $I_d^*, \; \Lambda _0^d, \; \Gamma _\Lambda ^d$ and ${\mathcal {F}}_{\lambda ^*}^d(\tau , \sigma )$ for $I^*, \; \Lambda _0, \; \Gamma _\Lambda$ and ${\mathcal {F}}_{\lambda ^*}(\tau , \sigma )$, respectively, to highlight the dependence on d.

Theorem 7.3

Let Assumption 2.2 be satisfied. Then we have

$$\begin{aligned} \Lambda _0^d\rightarrow \Lambda _0^0 \;\; \text { as } \;\; d\rightarrow 0, \end{aligned}$$

where $\Lambda _0^0$ satisfies $r(H_{\Lambda _0^0})=1$ and $H_\Lambda$ is defined in (7.8).

Proof

Consider the following eigenvalue problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _a\psi (a, x)=-\Lambda \psi (a, x)-(\lambda ^*(a, x)+\gamma (a, x)+\mu (a, x))\psi (a, x)+[G\psi ](a, x),&{}a{\in }(0, a^+], x{\in }{\overline{\Omega }},\\ \psi (0, x)=0,&{}x\in {\overline{\Omega }},\\ \lambda (a, x | \psi )=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\psi (\sigma , y)dyd\sigma ,&{}a{\in }[0, a^+], x{\in }{\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(7.7)

Solving the first equation of (7.7), we obtain

$$\begin{aligned} \psi (a, x)=\int _{0}^{a}e^{-\Lambda (a-\sigma )}e^{-\int _{\sigma }^{a}(\lambda ^*(\eta , x)+\gamma (\eta , x)+\mu (\eta , x))d\eta }\lambda (\sigma , x | \psi )(P^*-I^*)(\sigma , x)d\sigma . \end{aligned}$$

Plugging it into the equation of $\lambda$, we have

$$\begin{aligned} \lambda (a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }e^{-\Lambda (\sigma -\tau )}\Pi _{\lambda ^*}(\tau , \sigma , y)\lambda (\tau , y)(P^*-I^*)(\tau , y)d\tau dyd\sigma , \end{aligned}$$

where $\Pi _\lambda (\tau , \sigma , x):=e^{-\int _{\tau }^{\sigma }(\lambda (\eta , x)+\gamma (\eta , x)+\mu (\eta , x))d\eta }$. Define an operator $H_\Lambda : X\rightarrow X$ by

$$\begin{aligned} H_\Lambda \varphi:= & {} \int _{0}^{a^+}\int _{\Omega }k(\cdot , \sigma , \cdot , y)\int _{0}^{\sigma }e^{-\Lambda (\sigma -\tau )}\Pi _{\lambda ^*}(\tau , \sigma , y)\varphi (\tau , y)\nonumber \\&\times (P^*-I^*)(\tau , y)d\tau dyd\sigma ,\quad \varphi \in X. \end{aligned}$$

(7.8)

By using a similar argument as in Lemma 4.1, we have that $H_\Lambda$ is compact and nonsupporting for every $\Lambda \in {\mathbb {R}}$. Hence, $\Lambda \rightarrow r(H_\Lambda ): {\mathbb {R}}\rightarrow (0, \infty )$ is strictly decreasing with $\lim \limits _{\Lambda \rightarrow -\infty }r(H_\Lambda )=\infty$ and $\lim \limits _{\Lambda \rightarrow \infty }r(H_\Lambda )=0$. Thus there exists a unique $\Lambda _0^0$ such that $r(H_{\Lambda _0^0})=1$. Moreover, $\Lambda _0^0$ is the principal eigenvalue of (7.7) by the argument in Theorem 7.1.

Now since ${\mathcal {F}}_{\lambda ^*}^d(\tau , \sigma )$ is continuous with respect to d in the sense of operators and ${\mathcal {F}}_{\lambda ^*}^d(\tau , \sigma )$ converges to $\Pi _{\lambda ^*}(\tau , \sigma , x)$ as $d\rightarrow 0$, $\Gamma _\Lambda ^d$ converges to $H_\Lambda$ as $d\rightarrow 0$. Thus, by Krein-Rutman theorem, we have $r(\Gamma _\Lambda ^d)\rightarrow r(H_\Lambda )$ as $d\rightarrow 0$, see Degla (2008, Proposition 2.1). Hence the result is concluded by the continuity of $d\rightarrow r(\Gamma _\Lambda ^d)$. $\square$

Corollary 7.4

Let Assumption 2.2 be satisfied. Then the nontrivial steady state $I^*_d$ is locally stable for each $0<d\ll 1$ if $\Lambda _0^0<0$.

Proof

The proof follows from Theorems 7.1 and 7.3. $\square$

Remark 7.5

We would like to mention that it is not easy in general to obtain the monotonicity of $\Lambda _0^d$ with respect to d. However, in some specific cases, for example if the transmission rate k and birth rate $\beta$ depend only on age, and the recovery rate $\gamma$ and mortality rate $\mu$ are separated (i.e. $\gamma (a, x)=\gamma _1(a)+\gamma _2(x)$ and $\mu (a, x)=\mu _1(a)+\mu _2(x)$) and $I^*\equiv 0$, then $d\rightarrow \Lambda _0^d$ is strictly decreasing. Moreover, one can prove that $\Lambda _0^d\rightarrow -\infty$ as $d\rightarrow \infty$ under the above assumptions. We will study these in details in our forthcoming papers.

At the end of this section, we investigate the asymptotic behavior of the steady state $I^*_d$ in terms of the diffusion rate. Once we know $I^*_d$, then $S^*$ can be easily obtained via $S^*=P^*-I^*_d$.

Theorem 7.6

Let Assumption 2.2 be satisfied. If $r(\Xi )>1$, then $I^*_d\rightarrow {\tilde{I}}$ as $d\rightarrow 0$, where ${\tilde{I}}\not \equiv 0$ satisfies the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _aI(a, x)=\lambda (a, x)(P^*(a, x)-I(a, x))-(\gamma (a, x)+\mu (a, x))I(a, x),&{}\quad a\in (0, a^+], x\in {\overline{\Omega }},\\ I(0, x)=0, &{}\quad x\in {\overline{\Omega }},\\ \lambda (a, x)=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)I(\sigma , y)dyd\sigma ,&{}\quad a\in (0, a^+], x\in {\overline{\Omega }} \end{array}\right. } \end{aligned}$$

(7.9)

and $\Xi : X\rightarrow X$ is a linear and bounded operator defined in (7.12).

Proof

Solving the first equation of (7.9), we obtain

$$\begin{aligned} I(a, x)=\int _{0}^{a}\lambda (\sigma , x)P^*(\sigma , x)e^{-\int _{\sigma }^{a}(\lambda (\rho , x)+\gamma (\rho , x)+\mu (\rho , x))d\rho }d\sigma . \end{aligned}$$

Plugging it into the equation of $\lambda$, we have

$$\begin{aligned} \lambda (a, x)&=\int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y) \int _{0}^{\sigma }\lambda (\tau , y)\nonumber \\&\quad \times P^*(\tau , y)e^{-\int _{\tau }^{\sigma }(\lambda (\rho , y)+\gamma (\rho , y)+\mu (\rho , y))d\rho }d\tau dyd\sigma . \end{aligned}$$

(7.10)

Define a nonlinear operator $\Psi$ on X by

$$\begin{aligned} {[}\Psi \phi ](a, x):= & {} \int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\phi (\tau , y)\nonumber \\&\times P^*(\tau , y)e^{-\int _{\tau }^{\sigma }(\phi (\rho , y)+\gamma (\rho , y)+\mu (\rho , y))d\rho }d\tau dyd\sigma ,\quad \phi \in X,\nonumber \\ \end{aligned}$$

(7.11)

and investigate the existence of a nontrivial positive fixed point $\phi ^*\in X_+\setminus \{0\}$ such that $\Psi (\phi ^*)=\phi ^*$, which is no other than the desired nontrivial positive solution $\lambda \in X_+\setminus \{0\}$ of (7.10).

One can check that $\Psi$ is bounded and positive. Next, consider the following linear integral operator $\Xi$ on X:

$$\begin{aligned} {[}\Xi \phi ](a, x):= & {} \int _{0}^{a^+}\int _{\Omega }k(a, \sigma , x, y)\int _{0}^{\sigma }\phi (\tau , y)\nonumber \\&\times P^*(\tau , y)e^{-\int _{\tau }^{\sigma }(\gamma (\rho , y)+\mu (\rho , y))d\rho }d\tau dyd\sigma ,\quad \phi \in X. \end{aligned}$$

(7.12)

We can see that $\Xi$ is the Fréchert derivative of $\Psi$ at $\phi =0$ and a positive linear majorant of $\Psi$. We show that the spectral radius $r(\Xi )$ of operator $\Xi$ is a threshold value for the existence of a nontrivial fixed point of operator $\Psi$. Then by using similar arguments as in Lemma 4.1 and Proposition 4.2, we obtain that if $r(\Xi )>1$, then $\Psi$ has a positive nontrivial fixed point; that is, (7.9) has a nontrivial solution ${\tilde{I}}$ in E. Moreover, by using similar arguments as in Sect. 6 or in Busenberg et al. (1991, Theorem 4.5), one can show that ${\tilde{I}}$ is unique.

Next we prove the limit result. Since ${\mathcal {F}}_{\lambda }^d(\tau , \sigma )$ is continuous with respect to d in the sense of operators and ${\mathcal {F}}_{\lambda }^d(\tau , \sigma )$ converges to $\Pi _\lambda (\tau , \sigma , x)$ as $d\rightarrow 0$, it follows that $\Phi$ converges to $\Psi$ as $d\rightarrow 0$, where $\Phi$ is defined in (4.7) and ${\mathcal {F}}_{\lambda }^d$ is defined in Lemma 2.1 with f being replaced by $\lambda +\gamma +\mu$. Thus by the uniqueness of a nontrivial fixed point, we have a unique nontrivial fixed point of $\Phi$ converging to the unique one of $\Psi$ as $d\rightarrow 0$; that is $I^*_d\rightarrow {\tilde{I}}$. $\square$

8 Global stability

In this section we discuss the global stability of system (3.5). First we introduce a condition as follows

$$\begin{aligned} \int _{t}^{a^+}\int _{\Omega }k_2(a, y)\left( {\mathcal {F}}(a-t, a)I_0(a-t)\right) (y)dyda>0 \quad \text {for some}\quad t\ge 0. \end{aligned}$$

(8.1)

If this condition is satisfied, we say that $I_0$, with support in $[0, a^+]\times {\overline{\Omega }}$, is a nontrivial initial data.

Theorem 8.1

Assume that no nontrivial steady state exists, then

$$\begin{aligned} \forall I_0\in E, \quad {\mathcal {S}}(t)I_0\rightarrow 0 \quad \text {as}\quad t\rightarrow \infty . \end{aligned}$$

The proof is identical to that of Theorem 5.1 in Busenberg et al. (1991). So we omit it.

Proposition 8.2

Let Assumption 2.2 be satisfied and $I_0$ be a nontrivial initial data. Then there exist $t_0>0$ and $\xi >0$ such that

$$\begin{aligned} {\mathcal {S}}(t_0)I_0\ge \xi I^*. \end{aligned}$$

Proof

The proof is inspired by Proposition 5.2 in Busenberg et al. (1991). Noting (3.15), we have

$$\begin{aligned} I(t)\ge \frac{1}{\alpha }\int _{0}^{t}e^{-\frac{1}{\alpha }(t-s)}e^{(t-s)A}[I(s)+\alpha F(I(s))]ds. \end{aligned}$$

(8.2)

Now since $\alpha <\frac{1}{(\lambda ^++\gamma ^+)}$, we have

$$\begin{aligned} I(s)+\alpha F(I(s))\ge w(s), \end{aligned}$$

where

$$\begin{aligned} w(s)(a, x)=\alpha \epsilon k_1(a, x)P^*(a, x)\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(s)(\sigma , y)dyd\sigma , \end{aligned}$$

so that

$$\begin{aligned}&e^{(t-s)A}w(s)(a, x) \\&\quad \ge \!{\left\{ \begin{array}{ll} \alpha \epsilon \left( {\mathcal {F}}(a+s-t, a)(k_1P^*)(a+s-t)\right) (x)\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(s)(\sigma , y)dyd\sigma &{}\text {for } a>t-s,\\ 0 &{}\text {for } a<t-s. \end{array}\right. } \end{aligned}$$

Plugging it into (8.2), we obtain

$$\begin{aligned} I(t)(a, x)\ge \epsilon e^{-\frac{1}{\alpha }t}\int _{(t-a)\vee 0}^{t}\left( {\mathcal {F}}(a+s-t, a)(k_1P^*)(a+s-t)\right) (x)h(s)ds \end{aligned}$$

(8.3)

with

$$\begin{aligned} h(s)=\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(s)(\sigma , y)dyd\sigma . \end{aligned}$$

We make the following claim. $\square$

Claim 8.3

There exists $s_\infty$ such that $h(s)>0$ for $s>s_\infty$.

To prove this claim, we first prove another claim.

Claim 8.4

If $h(s)>0$ for $s\in [s_1, s_2]$, then $h(s)>0$ for $s\in [s_1-a^+, s_2+a^+]$.

Note first that $I(t)(a, x)>0$ for all $x\in {\overline{\Omega }}$ if

$$\begin{aligned} a>0, \quad t>s_1, \quad s_1-a^+<t-a<s_2. \end{aligned}$$

(8.4)

In fact, if these are satisfied, then $(t-a)\vee s_1<t\wedge s_2$. Setting $I=[(t-a)\vee s_1, t\wedge s_2]$, we have

$$\begin{aligned} I\subset [s_1, s_2],\quad \{a-t+I\}\cap [0, a^+]\ne \emptyset , \end{aligned}$$

(8.5)

where $\{a-t+I\}:=\{a-t+i: i\in I\}$. So it follows from (8.3) that

$$\begin{aligned} I(t)(a, x)\ge & {} \epsilon e^{-\frac{1}{\alpha }t}\int _{(t-a)\vee s_1}^{t\wedge s_2} \left( {\mathcal {F}}(a+s-t, a)(k_1P^*)(a+s-t)\right) (x)\\&\times h(s)ds>0 \quad \text {for all}\, x\in {\overline{\Omega }}, \end{aligned}$$

because by (8.5) the integrand is positive on some interval contained in I.

Finally, fixing $t\in [s_1-a^+, s_2+a^+]$, noting that $k_2$ is positive on $[0, a^+]\times {\overline{\Omega }},$ and using (8.4), we know that I(t)(a, x) is positive for $a>0$ and $a\in R:=[t-s_2, t+a^+-s_1].$ But our choice of t yields

$$\begin{aligned} t-s_2<a^+, \quad t+a^+-s_1>a^+-a^+=0 \end{aligned}$$

so that $R\cap [0, a^+]\ne \emptyset$ and $h(t)>0$.

Now Claim 8.4 is proven. Note that if $I_0$ is nontrivial, then from (3.15) we see that for some positive constant $\delta$,

$$\begin{aligned} h(s)= & {} \int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I(s)(\sigma , y)dyd\sigma \ge \delta \int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)(e^{sA}I_0)(\sigma , y)dyd\sigma \\= & {} \delta \int _{s}^{a^+}\int _{\Omega }k_2(\sigma , y)\left( {\mathcal {F}}(\sigma -s, \sigma )I_0(\sigma -s)\right) (y)dyd\sigma >0 \end{aligned}$$

for s in some interval $[s_1, s_2]$. Thus, iterating Claim 8.4 we have for any integer n that

$$\begin{aligned} h(s)>0 \quad \text {in}\quad [s_1-na^+, s_2+na^+]. \end{aligned}$$

So if n is sufficiently large, successive intervals are overlapping, and consequently Claim 8.3 is desired.

We now use Claim 8.3 to the prove following result.

Claim 8.5

There exists $t_\infty$ such that $I(t)(a, x)>0$ for $a>0, x\in {\overline{\Omega }}$ and $t>t_\infty$.

In fact, if $t>s_\infty +2a^+$, then $t-a>s_\infty$. By (8.3), we have

$$\begin{aligned} I(t)(a, x)\ge & {} \epsilon e^{-\frac{1}{\alpha }t}\int _{t-a}^{t}\left( {\mathcal {F}}(a+s-t, a)(k_1P^*)(a+s-t)\right) (x)h(s)ds\nonumber \\= & {} \epsilon e^{-\frac{1}{\alpha }t}\int _{0}^{a}\left( {\mathcal {F}}(\sigma , a)(k_1P^*)(\sigma )\right) (x)h(\sigma +t-a)d\sigma >0. \end{aligned}$$

(8.6)

Now let $t_0>s_\infty +a^+$. Then $t_0-a>s_\infty$ and

$$\begin{aligned} I(t_0)(a, x)\ge & {} \epsilon e^{-\frac{1}{\alpha }t_0}\int _{t_0-a}^{t_0}\left( {\mathcal {F}}(a+s-t_0, a)(k_1P^*)(a+s-t_0)\right) (x)h(s)ds\nonumber \\> & {} \epsilon e^{-\frac{1}{\alpha }t_0}h_-\int _{t_0-a}^{t_0}\left( {\mathcal {F}}(a+s-t_0, a)(k_1P^*)(a+s-t_0)\right) (x)ds\nonumber \\= & {} \epsilon e^{-\frac{1}{\alpha }t_0}h_-\int _{0}^{a}\left( {\mathcal {F}}(s, a)(k_1P^*)(s)\right) (x)ds=\epsilon e^{-\frac{1}{\alpha }t_0}h_-m(a, x), \end{aligned}$$

(8.7)

where $h_-:=\inf _{s\in [s_\infty , t_0]}h(s)$. Finally, by (6.4) we have

$$\begin{aligned} I(t_0)(a, x)\ge \xi I^*(a, x), \end{aligned}$$

where $\xi =\frac{\epsilon }{\epsilon _2}e^{-\frac{1}{\alpha }t_0}h_-$ and the proof is complete. $\square$

Theorem 8.6

Let Assumption 2.2 be satisfied. Let $I^*$ be the unique nontrivial steady state. Then for any nontrivial initial data $I_0$ we have

$$\begin{aligned} {\mathcal {S}}(t)I_0\rightarrow I^* \quad \text {as}\quad t\rightarrow \infty . \end{aligned}$$

(8.8)

If $I_0$ is not nontrivial, then

$$\begin{aligned} {\mathcal {S}}(t)I_0=0\quad \text {for}\quad t\ge a^+. \end{aligned}$$

(8.9)

Proof

Inspired by Theorem 5.3 in Busenberg et al. (1991), let $\xi$ and $t_0$ be as in previous proposition. We have

$$\begin{aligned} \xi I^*\le {\mathcal {S}}(t_0)I_0\le P^* \end{aligned}$$

and

$$\begin{aligned} {\mathcal {S}}(t)\xi I^*\le {\mathcal {S}}(t+t_0)I_0\le {\mathcal {S}}(t) P^*. \end{aligned}$$

(8.10)

Now

$$\begin{aligned} \xi I^*=\xi {\mathcal {S}}(t)I^*\le {\mathcal {S}}(t)\xi I^*,\quad {\mathcal {S}}(t)P^*\le P^*. \end{aligned}$$

Consequently, ${\mathcal {S}}(t)\xi I^*$ and ${\mathcal {S}}(t)P^*$ are respectively nondecreasing and nonincreasing with respect to t and converge to a nontrivial steady state which coincides with $I^*$ by the uniqueness Theorem 6.5. Then the first result (8.8) follows from (8.10).

Next let

$$\begin{aligned} \int _{t}^{a^+}\int _{\Omega }k_2(\sigma , y)\left( {\mathcal {F}}(\sigma -t, \sigma )I_0(\sigma -t)\right) (y)dyd\sigma =0\quad \text {for all}\quad t\ge 0. \end{aligned}$$

Let us show that the iterates $I^n(t)$ defined in (3.16) satisfy

$$\begin{aligned} \int _{s}^{a^+}\int _{\Omega }k_2(a, y)I^n(t)(a-s, y)dyda=0\quad \text {for all}\quad t\ge 0,\,s\ge 0. \end{aligned}$$

(8.11)

In fact, if (8.11) is true, then

$$\begin{aligned} \lambda [a, x | I^n(t)(\cdot , \cdot )]\le k_1(a, x)\int _{0}^{a^+}\int _{\Omega }k_2(\sigma , y)I^n(t)(\sigma , y)dyd\sigma =0 \end{aligned}$$

and

$$\begin{aligned} I^n(t)+\alpha F(I^n(t))\le I^n(t). \end{aligned}$$

This implies that

$$\begin{aligned} I^{n+1}(t)\le e^{tA}I_0+c\int _{0}^{t}e^{(t-\sigma )A}I^n(\sigma )d\sigma , \end{aligned}$$

where $c>0$ and consequently,

$$\begin{aligned}&\int _{s}^{a^+}\int _{\Omega }k_2(\sigma , y)I^{n+1}(t)(\sigma -s, y)dyd\sigma \\&\quad \le \int _{s+t}^{a^+}\int _{\Omega }k_2(\sigma , y)\left( {\mathcal {F}}(\sigma -t-s, \sigma -s)I_0(\sigma -t-s)\right) (y)dyd\sigma \\&\qquad +c\int _{0}^{t}\int _{s+t-\eta }^{a^+}\int _{\Omega }k_2(\sigma , y)\left( {\mathcal {F}}(\sigma -t\right. -s+\eta , \sigma -s)\\&\left. \qquad \times I^n(\eta )(\sigma -t-s+\eta )\right) (y)dyd\sigma d\eta =0. \end{aligned}$$

Thus, (8.11) is true by induction, where we used the positivity of $k_2$ and strong positivity of semigroup ${\mathcal {F}}(\sigma , a)$ for $a-\sigma \ge 0$,

$$\begin{aligned} \int _{s+t-\eta }^{a^+}&\int _{\Omega }k_2(\sigma , y)\left( {\mathcal {F}}(\sigma -t-s+\eta , \sigma -s)I^n(\eta )(\sigma -t-s+\eta )\right) (y)dyd\sigma =0,\\& \forall \eta \ge 0. \end{aligned}$$

Finally (8.11) implies that

$$\begin{aligned} \int _{s}^{a^+}\int _{\Omega }k_2(a, y)I(t)(a-s, y)dyda=0\quad \text {for all}\quad t\ge 0, \, s\ge 0. \end{aligned}$$

So if $t>a^+$, then

$$\begin{aligned} I(t)\le c\int _{0}^{t}e^{(t-s)A}I(s)ds \quad \text {in}\quad [0, a^+]\times {\overline{\Omega }}, \end{aligned}$$

which implies the second result (8.9), see the argument in Theorem 5.3 in Busenberg et al. (1991). $\square$

9 Discussion

In this paper, we studied an age-structured epidemic model with nonlocal (convolution) diffusion which describes the spatial spread (via long-distance travel) of an infectious disease in a host population with age structure. We first analyzed the well-posedness of the model, then investigated the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and finally studied the local and global stability of this nontrivial steady state. Moreover, we considered the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the diffusion rate.

Recall that the basic reproduction number is defined by the spectral radius of the next generation operator, ${\mathcal {R}}_0:=r({\mathcal {G}}),$ where ${\mathcal {G}}$ is given by (5.3). We would like to point out that the effects of age and nonlocal diffusion were implicitly included in the expression of the operator ${\mathcal {G}}.$ Theorem 5.2 stated that ${\mathcal {R}}_0$ is a threshold for the existence of a nontrivial steady state; namely, system (1.1) has only the trivial steady state $(P^*, 0)$ if ${\mathcal {R}}_0\le 1$ and has at least one nontrivial steady state $(S^*, I^*)$ if ${\mathcal {R}}_0>1$. For the operator $\Gamma _\Lambda$ defined by (7.6), which also depends on age and nonlocal diffusion implicitly, there exists a unique (principal) eigenvalue $\Lambda _0$ such that $r(\Gamma _{\Lambda _0})=1$. Thus, Theorem 7.1 indicates that $r(\Gamma _0)$ plays as another threshold for the stability of the nontrivial steady state; namely, it is locally stable when $r(\Gamma _0)<1$ and unstable when $r(\Gamma _0)>1$. Rewrite the principal eigenvalue as $\Lambda _0^d$ to represent the fact that it depends on the diffusion rate d. If the transmission rate k, birth rate $\beta$ depend only on age and the recovery rate $\gamma$, mortality rate $\mu$ are separated and $I^*\equiv 0$, then $d\rightarrow \Lambda _0^d$ is strictly decreasing. Moreover, $\Lambda _0^d\rightarrow -\infty$ as $d\rightarrow \infty$.

We would like to mention again that the analysis of (1.1) with nonlocal diffusion is also valid for (1.6) with Laplace diffusion; that is, ${\mathcal {L}}u:=\Delta u$, see Kuniya and Oizumi (2015) for the existence of steady states for an age-structured SIS epidemic model with Laplace diffusion. It is also interesting to investigate the principal eigenvalue of the age-structured operator coupled with more complicated nonlocal diffusion operator in SIS epidemic models (for example including cost parameter and diffusion range in our forthcoming paper), and use it as a threshold to study the dynamics behavior of SIS epidemic models, see Yang et al. (2019). Note that we assumed $d_I=d_S$ in (1.1), it is also valuable to consider different diffusion rates, for example, $d_I\ne d_S$. In addition, the method of monotone and positive operators will not be valid if we consider more complicated epidemic models, for example SIR or SEIR age-structured models with nonlocal diffusion (which might be suitable to describe the geographic spread of COVID-19), since the system will lose monotonicity. Thus, it is necessary to study such more complicated epidemic models by using a different approach. We leave these for future consideration.