key: cord-0865717-02yfqg21
authors: Kang, Hao; Ruan, Shigui
title: Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion
date: 2021-06-26
journal: J Math Biol
DOI: 10.1007/s00285-021-01634-x
sha: 3817f0c74190e3397909165e8079935d9c4dbff6
doc_id: 865717
cord_uid: 02yfqg21

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.

tions have been proposed and studied, see Chekroun and Kuniya (2020) , Di Blasio (2010) , Fitzgibbon et al. (1994 Fitzgibbon et al. ( , 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-infectioussusceptible (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 ≥ 0 at time t ≥ 0 in position x ∈ ⊂ R N , respectively, where is a bounded subset of R N and a < a + , the maximum age which is finite. Let λ(t, a, x) be the force of infection of infectious individuals to susceptible individuals of age a at time t in position x, and γ (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 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (∂ t + ∂ a )S(t, a, x) = d [LS] (t, a, x) − λ(t, a, x)S(t, a, x)− μ(a, x)S(t, a, x) + γ (a, x)I (t, a, x), t > 0, a ∈ (0, a + ], x ∈ , (∂ t + ∂ a )I (t, a, x) = d [LI ] (t, a, x) + λ(t, a, x)S(t, a, x)− (μ(a, x) + γ (a, x))I (t, a, x), t > 0, a ∈ (0, a + ], x ∈ , S(t, 0, x) = P(t, 0, x), I (t, 0, x) = 0, t > 0, x ∈ , S(0, a, x) = S 0 (a, x), I (0, a, x) = I 0 (a, x), a ∈ [0, a + ], x ∈ , in which β(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 J (x − y)u(t, a, y)dy is the rate at which individuals are arriving at position x from other places while 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.,

In addition, we assume that the force of infection λ (t, a, x) (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 Inaba ( , 2014 Inaba ( , 2017 , Kang et al. (2020a) , Kuniya and Inaba (2013) , Kuniya (2014) , Kuniya and Iannelli (2014) , Kuniya et al. (2016 Kuniya et al. ( , 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:

(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 Yang et al. ( , 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 Yang et al. ( , 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 ⊂ 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:

(1.6) where λ(t, a, x) is given by (1.3). There could be appropriate boundary conditions on ∂ 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.

Define the function spaces as follows:

with positive cones

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 ∈ L ∞ + ((0, a + )× ), then there exists a unique mild solution u(a, x), 0 ≤ τ ≤ a ≤ a + , to the evolution equation on X for any initial data φ ∈ Z

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

Then {F(τ, a)} 0≤τ ≤a≤a + is a family of linear uniformly bounded positive operators on Z and is strongly continuous in τ and a. Furthermore, We make the following assumptions on each parameter.

(i) γ and μ are belonging to L ∞ + ((0, a + ) × ) with γ + := γ L ∞ + ((0,a + )× ) and μ + := μ L ∞ + ((0,a + )× ) ; (ii) k(a, σ, x, y) = 0 for x, y ∈ R N \ or for a, σ < 0 and a, σ > a + ;

uniformly for σ ∈ [0, a + ] and y ∈ , where κ ∈ R, h ∈ R N and · denotes the usual Euclidean norm in R N +1 ;

(iv) There exists two positive functions k 1 ∈ L 1 + ((0, a + ) × ) and k 2 ∈ L ∞ + ((0, a + ) × ) respectively such that

for all (a, x), (σ, y) ∈ R + × , where ∈ (0, 1); (v) k 1 (a, x) has positive upper and lower bounds k + and k − such that k

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 

there exists a positive integer p = p(ψ, f ) such that f , L n ψ > 0 for all n ≥ p. r (L) denotes the spectral radius of L ∈ B(Y ), σ (L) denotes the spectrum of L, and σ 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 be a positive operator from E + to itself and L := [0] be its Fréchet derivative at 0. If

is compact and bounded; (iii) L has a positive eigenvector v 0 ∈ Y + \ {0} associated with an eigenvalue λ 0 > 1; (iv) L has no eigenvector in Y + associated with the eigenvalue 1, then has at least one nontrivial fixed point in Y + .

In the case where L is a majorant of (that is, L is a linear operator such that (φ) ≤ Lφ for any φ ∈ 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 be a positive operator from Y + to itself and L be its compact and nonsupporting majorant. Then, has no trivial fixed point in Y + provided r (L) ≤ 1.

Let P(t, a, x) be the total population density at time t ≥ 0 with age a ≥ 0 in position x ∈ . Then the total population P is governed by the McKendrick equation with nonlocal diffusion (see Kuniya and Oizumi (2015) ): (t, a, x) , t>0, a∈(0, a + ], x∈ ,

1) where μ(a, x) represents the mortality of an individual of age a in position x, β(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.

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

where F is the solution operator of (2.1) with f being replaced by μ. Substituting it into the second equation of (3.2), one obtains the following integral equation:

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

and investigate the existence of a nontrivial fixed point ψ * 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.

Here, r(·) 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 β, 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 ψ * ∈ L 1

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

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

Once the existence of a positive fixed point ψ * (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 * ∈ X + \ {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 * ∈ X + \ {0}. Further, we assume that such a steady state has been attained from the initial time and bounded; that is,

Now we consider model (1.1). Since it follows from P(t, a, x) = P * (a, x) for all t ≥ 0 that S(t, a, x) = P * (a, x) − I (t, a, x), system (1.1) can be reduced to a single

Note that λ(·, ·|ψ) ∈ L ∞ + ((0, a + ) × ) for a fixed ψ ∈ E by Assumption 2.2-(iv), (v). Then system (3.5) can be rewritten as a semilinear Cauchy problem

where {F(σ, a)} 0≤σ ≤a≤a + is defined in Lemma 2.1 via replacing f by μ. From (3.10) we immediately have e t A E ⊂ E and if ψ ≤ φ, then e t A ψ ≤ e t A φ since the comparison principle of operators F(σ, a) for any 0 ≤ σ ≤ a ≤ a + holds, where ≤ denotes the usual ordering in L 1 ((0, a + ), Z ).

We also note that if α > 0 the resolvent (I − α A) −1 is given by the formula:

Then again by the comparison theorem for {F(σ, a)} 0≤σ ≤a≤a + , we have that

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

Using the fact that γ and λ 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.

Note that in the proof of Proposition 3.3, we will obtain α < 1 γ + +λ + , where λ + is the upper bound of λ; that is, λ[a, x|ψ] < λ + for any ψ ∈ E in [0, a + ] × . Note that λ + can be obtained by

Using α appeared in Proposition 3.3, we can write problem (3.9) as d dt

(3.14)

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

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

Since we can see that E is invariant with respect to the iteration process, that is I n+1 ∈ E if I n ∈ 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 ∈ E. Then Eq. (3.9) has a unique mild solution

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 n ξ be the iterates (3.16) with I 0 replaced by ξ I 0 . Since 0 ≤ ξ ≤ 1, we have

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

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

(4.1) It is obvious that the trivial disease-free steady state solution I * (a, x) ≡ 0 always exists. In what follows, we investigate the existence of a nontrivial endemic steady state solution

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

k(a, σ, x, y)I * (σ, y)dydσ, a∈(0, a + ], x∈ .

(4.2)

Viewing f (a, x) = λ * (a, x) + γ (a, x) + μ(a, x) in Lemma 2.1 and solving the first equation of (4.1), we obtain

where F λ * is the solution operator of (4.1) as in Lemma 2.1 with f being replaced by λ * + γ + μ (note that λ * + γ + μ ∈ L ∞ + ((0, a + ) × )). Substituting (4.3) into the third equation of (4.1), we obtain the following integral equation of λ * (a, x):

Solving the first equation of (4.2), we obtain

where F 0 is the solution operator of (4.1) as in Lemma 2.1 with f being replaced by γ + μ. Substituting (4.5) into the third equation of (4.2), we obtain the following integral equation of λ * (a, x):

Observe that we have F λ * (σ, a) ≤ F 0 (σ, a) for all 0 ≤ σ ≤ a ≤ a + in the operator sense by Lemma 2.1. Now we define a nonlinear operator on X by

(4.7) and investigate the existence of a nontrivial positive fixed point φ * ∈ X + \ {0} such that (φ * ) = φ * , which is no other than the desired nontrivial positive solution λ * ∈ X + \ {0} of (4.4).

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

(4.8) One can see that M is the Fréchert derivative of at φ = 0 and a positive linear majorant of . 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 ; that is, if r (M) > 1, then has a positive nontrivial fixed point. We prove the following lemma about M.

Proof First we can see that M is positive since {F 0 (τ, σ )} 0≤τ ≤a is positive. Next we prove the compactness. Note by Assumption 2.2-(i) that for φ ∈ B, where B is a uniformly bounded subset in X ,

where we used Minkowski inequality and Assumption 2.2-(iii). Thus M : X → 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 ∈ X * on X ,

and hence we see from the positivity of P * , k − and {F 0 (τ, σ )} 0≤τ ≤σ ≤a + that the functional f 0 is strictly positive (that is, f 0 , φ > 0 for all φ ∈ X + \ {0}). Now we have

Note that e ∈ X + , then we have 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, 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 is a positive operator from the positive cone X + to itself and M is the positive linear majorant of , we can apply Theorem 2.4 to conclude that has no nontrivial fixed point in X + provided r (M) ≤ 1. This completes the proof.

The existence of a nontrivial fixed point of implies the existence of a nontrivial steady state solution I * ∈ X + \ {0}; that is, the existence of a nontrivial steady state solution (S * , I * ) ∈ X 2 + \ {0, 0} of the original system (1.1). Further, I * must be in E \ {0}, otherwise I * ≡ 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.

(i) If r (M) ≤ 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 + .

In this section we show that the threshold value r (M) obtained in the previous section is equal to the basic reproduction number 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 , x) , t > 0, a ∈ (0, a + ], x ∈ ,

where v(t, a, x) = P * (a, x) a + 0 k(a, σ, x, y)I (t, σ, y)dydσ denotes the density of newly infected population at the linear invasion phase. Integrating along the characteristic line t − a =constant, we obtain

Here note that it is suffices to consider t > a + from Assumption 2.2-(ii). Now plugging it into the equation

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

Then following the arguments in Diekmann et al. (1990) , we obtain the next generation operator in X as Proof Considering the calculation of (5.2) conversely, we see that the right hand side of (5.2) is

(5.5) Let us define a linear operator Q on X by

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

(i) If R 0 ≤ 1, then system (1.1) has only the trivial steady state (S * , I * ) = (P * , 0) in X 2 + ; (ii) If R 0 > 1, then system (1.1) has at least one nontrivial steady state (S * , I * ) in X 2 + .

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

(6.1) From (3.12) and Proposition 3.3, we can see that T (E) ⊂ E and T ψ ≤ T φ for any ψ ≤ φ and ψ, φ ∈ 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 a + 0 k 2 (σ, y)I * (σ, y)dydσ > 0.

(6.2) Proof If (6.2) is not satisfied, then

and I * satisfies the following equations

x ∈ . (6.3) Thus I * is identically zero by the uniqueness of solution for the above linear equation, which is a contradiction.

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

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

where c 1 , c 2 are positive. Hence I * satisfies the inequality

Since I * (0, x) = 0, we can find positive 1 , 2 such that (6.4) is satisfied.

Let ξ ∈ (0, 1) and I ∈ E such that ξ I * + I ∈ E and a + 0 k 2 (σ, y)I (σ, y)dydσ > 0. Then there exists a constant 3 > 0 such that

(6.7)

Proof Observe that (α < 1 γ + +λ + ) ξ I * + I + α F(ξ I * + I ) = ξ I * + I + α(P * − ξ I * − I )λ[·, ·|ξ I * + I ] − αγ (ξ I * + I )

Then from (3.11), (3.12) and (6.5), we obtain (6.7). ∈ (0, 1) . Then there exists a constant 4 > 0 such that

Then we have the desired (6.8) from (3.11), (3.12) and definition of m 1 .

Proof Let u * = v * be two nontrivial fixed points. Let ξ = sup{k ∈ R + : v * ≥ ku * }. From (6.4), we can see that ξ > 0. If ξ < 1, we have from Lemma 6.4 that

If 4 is small enough, we can assume that ξ(1 − ξ) 4 m 1 , ξu * + ξ(1 − ξ) 4 m 1 ∈ E. Then we can apply Lemma 6.3 (taking

which implies by (6.4) that v * ≥ ξ u * + (1 − ξ) But this contradicts the definition of ξ . Therefore, we have ξ ≥ 1 and v * ≥ u * . By using the same approach, we can show that u * ≥ v * . Then u * = v * .

In this subsection we analyze local stability of the nontrivial steady state I * ∈ E. Note that I * ∈ L ∞ + ((0, a + ) × ) since I * ∈ E implies that I * (a, x) ≤ P * (a, x) . Consider the linearization of system (4.1) at the nontrivial steady state I * ∈ E in the following

where λ * (a, x) = where G : E ⊂ X → E is defined as

[Gψ](a, x) := λ(a, x|ψ)(P * (a, x) − I * (a, x)).

Then by using the same argument in Lemma 4.1, one can show that G is compact. Now we can check that A generates a nilpotent semigroup and its perturbed semigroup by the compact operator G is eventually compact. Hence,

where ω 1 (A) represents the essential growth bound of operator A. It follows that the stability of I * is determined by the eigenvalues of A+ G. By solving the characteristic equation ( × F λ * (τ, σ )(λ(ψ)(P * − I * ))(τ ) (y)dτ dydσ.

(7.5)

Define an operator : X → X by

(7.6)

Here we should emphasize that F λ * (τ, σ ) is a linear operator since λ * is fixed now and that F λ * (τ, σ ) is applied to the whole ϕ(P * − I * )(τ, y) rather than only ϕ(τ, y). Note that is well defined due to the boundedness of P * and I * . Then it is easy to check that is also compact and nonsupporting by Lemma 4.1, thus has a simple eigenvalue r ( ) corresponding to a simple and positive eigenvector. Moreover the mapping → r ( ) : R → (0, ∞) is continuous and strictly decreasing from +∞ to 0 (see Lemma 3.4 of Kang et al. (2020b) ). This implies that there exists a unique 0 associated with a positive eigenvector ϕ 0 ∈ X + such that r ( 0 ) = 1. Thus 0 ϕ 0 = ϕ 0 and hence 0 is an eigenvalue of A + G. In what follows, we will show that such 0 is also a principal eigenvalue, i.e. 0 > | | for any other

Once it is true, we have the following theorem.

Theorem 7.1 Let Assumption 2.2 be satisfied. Then it is locally stable when 0 < 0, which is equivalent to r ( 0 ) < 1; Otherwise, I * is unstable when 0 > 0, which is equivalent to r ( 0 ) > 1.

Since is compact, it follows that σ ( ) \ {0} = σ P ( ) \ {0}. Note that { ∈ C : 1 ∈ σ P ( )} = { ∈ C : ∈ σ P (A + G)}. Hence, it is sufficient to prove that for any ∈ σ P ( ) \ { 0 }, 0 > | |.

For any ∈ σ P ( ), there is an eigenfunction ϕ such that ϕ = ϕ . Then we have |ϕ | = | ϕ | ≤ Re |ϕ |. Let f Re be a positive eigenfunctional corresponding to the eigenvalue r ( Re ) of Re . We obtain that

Hence, r ( Re ) ≥ 1 and Re ≤ 0 since r ( ) is strictly decreasing for ∈ R and r ( ) = 1. If Re = 0 , then 0 |ϕ | = |ϕ |. In fact, if 0 |ϕ | > |ϕ |, taking duality pairing with the eigenfunctional f 0 corresponding to the eigenvalue r ( 0 ) = 1 on both sides yields f 0 , 0 |ϕ | = f 0 , |ϕ | > f 0 , |ϕ | , which is a contradiction. Then we can write that |ϕ | = cϕ 0 , where ϕ 0 is the eigenfunction corresponding to the eigenvalue r ( 0 ) = 1. Without loss of generality, we assume that c = 1 and write ϕ (a, x) = ϕ 0 (a, x)e iα (a,x) for some real function α(a, x). Substituting this relation into we have

From Heijmans (1986, Lemma 6.12), we obtain that −Im (σ − τ ) + α(τ, y) = θ for some constant θ . By ϕ = ϕ , we have e iθ 0 ϕ 0 = ϕ 0 e iα , so θ = α(a, x), which implies that Im = 0. Then, there is no element ∈ σ P ( ) such that Re = 0 and = 0 . Therefore, the result is proved.

Remark 7.2 Note that 0 ≤ M in the sense of positive operators since F λ * ≤ F 0 and that when I * ≡ 0, 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 ( 0 ) < 1 < r (M), there exists a unique nontrivial steady state (S * , I * ) for (1.1) which is locally stable. Otherwise, if r ( 0 ) > 1, there exists a unique nontrivial steady state (S * , I * ) for (1.1) which is locally unstable.

Now let us study the effects of small diffusion rate d on the principal eigenvalue 0 . We write I * d , d 0 , d and F d λ * (τ, σ ) for I * , 0 , and F λ * (τ, σ ), respectively, to highlight the dependence on d. Proof Consider the following eigenvalue problem

(7.7) Solving the first equation of (7.7), we obtain ψ(a, x) = a 0 e − (a−σ ) e − a σ (λ * (η,x)+γ (η,x)+μ(η,x))dη λ(σ, x|ψ)(P * − I * )(σ, x)dσ.

Plugging it into the equation of λ, we have (λ(η,x)+γ (η,x)+μ(η,x) )dη . Define an operator H : X → X by 

We would like to mention that it is not easy in general to obtain the monotonicity of d 0 with respect to d. However, in some specific cases, for example if the transmission rate k and birth rate β depend only on age, and the recovery rate γ and mortality rate μ are separated (i.e. γ (a, x) = γ 1 (a) + γ 2 (x) and μ(a, x) = μ 1 (a) + μ 2 (x)) and I * ≡ 0, then d → d 0 is strictly decreasing. Moreover, one can prove that d 0 → −∞ as d → ∞ 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 ( ) > 1, then I * d →Ĩ as d → 0, whereĨ ≡ 0 satisfies the following equation

k(a, σ, x, y)I (σ, y)dydσ, a ∈ (0, a + ], x ∈ (7.9) and : X → X is a linear and bounded operator defined in (7.12).

Proof Solving the first equation of (7.9), we obtain (λ(ρ,x)+γ (ρ,x)+μ(ρ,x) )dρ dσ.

Plugging it into the equation of λ, we have (λ(ρ,y) +γ (ρ,y)+μ(ρ,y))dρ dτ dydσ.

(7.10)

Define a nonlinear operator on X by (7.11) and investigate the existence of a nontrivial positive fixed point φ * ∈ X + \ {0} such that (φ * ) = φ * , which is no other than the desired nontrivial positive solution λ ∈ X + \ {0} of (7.10). One can check that is bounded and positive. Next, consider the following linear integral operator on X : (γ (ρ,y) +μ(ρ,y))dρ dτ dydσ, φ ∈ X . (7.12)

We can see that is the Fréchert derivative of at φ = 0 and a positive linear majorant of . We show that the spectral radius r ( ) of operator is a threshold value for the existence of a nontrivial fixed point of operator . Then by using similar arguments as in Lemma 4.1 and Proposition 4.2, we obtain that if r ( ) > 1, then has a positive nontrivial fixed point; that is, (7.9) has a nontrivial solutionĨ in E. Moreover, by using similar arguments as in Sect. 6 or in Busenberg et al. (1991, Theorem 4.5) , one can show thatĨ is unique. Next we prove the limit result. Since F d λ (τ, σ ) is continuous with respect to d in the sense of operators and F d λ (τ, σ ) converges to λ (τ, σ, x) as d → 0, it follows that converges to as d → 0, where is defined in (4.7) and F d λ is defined in Lemma 2.1 with f being replaced by λ + γ + μ. Thus by the uniqueness of a nontrivial fixed point, we have a unique nontrivial fixed point of converging to the unique one of as d → 0; that is I * d →Ĩ .

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

If this condition is satisfied, we say that I 0 , with support in [0, a + ] × , is a nontrivial initial data.

Theorem 8.1 Assume that no nontrivial steady state exists, then

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 ξ > 0 such that

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

Now since α < 1 (λ + +γ + ) , we have

so that

Plugging it into (8.2), we obtain

We make the following claim.

There exists s ∞ such that h(s) > 0 for s > s ∞ .

To prove this claim, we first prove another claim. 

×h(s)ds > 0 for all x ∈ , because by (8.5) the integrand is positive on some interval contained in I . Finally, fixing t ∈ [s 1 − a + , s 2 + a + ], noting that k 2 is positive on [0, a + ] × , and using (8.4), we know that I (t)(a, x) is positive for a > 0 and a ∈ R := [t − s 2 , t + a + − s 1 ]. But our choice of t yields t − s 2 < a + , t + a + − s 1 > a + − a + = 0 so that R ∩ [0, a + ] = ∅ 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 δ, 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 ∞ such that I (t)(a, x) > 0 for a > 0, x ∈ and t > t ∞ .

In fact, if t > s ∞ + 2a + , then t − a > s ∞ . By (8.3), we have

Now let t 0 > s ∞ + a + . Then t 0 − a > s ∞ and

where h − := inf s∈[s ∞ ,t 0 ] h(s). Finally, by (6.4) we have

where ξ = 2 e − 1 α t 0 h − and the proof is complete.

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 S(t)I 0 → I * as t → ∞. Consequently, S(t)ξ I * and 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 a + t k 2 (σ, y) (F(σ − t, σ )I 0 (σ − t)) (y)dydσ = 0 for all t ≥ 0.

Let us show that the iterates I n (t) defined in (3.16) satisfy a + s k 2 (a, y)I n (t)(a − s, y)dyda = 0 for all t ≥ 0, s ≥ 0. 

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, R 0 := r (G), where 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 G. Theorem 5.2 stated that 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 R 0 ≤ 1 and has at least one nontrivial steady state (S * , I * ) if R 0 > 1. For the operator defined by (7.6), which also depends on age and nonlocal diffusion implicitly, there exists a unique (principal) eigenvalue 0 such that r ( 0 ) = 1. Thus, Theorem 7.1 indicates that r ( 0 ) plays as another threshold for the stability of the nontrivial steady state; namely, it is locally stable when r ( 0 ) < 1 and unstable when r ( 0 ) > 1. Rewrite the principal eigenvalue as d 0 to represent the fact that it depends on the diffusion rate d. If the transmission rate k, birth rate β depend only on age and the recovery rate γ , mortality rate μ are separated and I * ≡ 0, then d → d 0 is strictly decreasing. Moreover, d 0 → −∞ as d → ∞. 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, Lu := 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 = 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.

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We are very grateful to the anonymous reviewers for their helpful comments and constructive suggestions which helped us to improve the presentation of the paper.