The Linear Stability and Basic Reproduction Numbers for Autonomous FDEs

TL;DR

This paper establishes the stability equivalence between certain autonomous functional differential equations and their delay-free counterparts, and develops a theory of basic reproduction numbers for autonomous FDEs, with an application to tick population dynamics.

Contribution

It introduces a new stability equivalence result for autonomous FDEs and formulates a general theory of basic reproduction numbers for these equations.

Findings

Proved stability equivalence between delayed and delay-free systems.

Developed a theory of basic reproduction number $\\mathcal{R}_0$ for autonomous FDEs.

Applied the theory to a population model of black-legged ticks.

Abstract

In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $\mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $\mathcal{R}_0$.

Equations

$$\hat{L}x=L(\hat{x}),\forall x\in {\mathbb{R}}^m,$$

$$s(A)=\max\big{\{}\Re(\lambda):\lambda\ \text{is an eigenvalue of }A\big{\}}.$$

$$u_t(\theta )=u(t+\theta ),\forall \theta \in [-\tau ,0],$$

$$\frac{du(t)}{dt}=L(u_t)$$

$$T(t)\varphi =u_t(\varphi ),\forall \varphi \in C,t\ge 0.$$

$$s(L)=\max\big{\{}\Re(\lambda):\lambda\ \text{is an eigenvalue of system \eqref{autoE}}\big{\}}.$$

$$r(T(t))=e^{s(L)t},\forall t>0,$$

$$L(\varphi )=D\varphi (0)+\bar{L}(\varphi ),\forall \varphi \in C.$$

$$L=(D-aI)\varphi (0)+(a\varphi (0)+\bar{L}(\varphi )),\forall \varphi \in C.$$

$$sign(s(L))=sign(r(T(t_0))-1)=sign(r(\hat{\mathcal{F}}{\mathcal{V}}^{-1})-1).$$

$$sign(r(\hat{\mathcal{F}}{\mathcal{V}}^{-1})-1)=sign(s(\hat{\mathcal{F}}-\mathcal{V}))=sign(s(\hat{L})).$$

$$\frac{du(t)}{dt}=F(u_t)-V(u_t),$$

$$\frac{du(t)}{dt}=-V(u_t).$$

$$F(\varphi )={\int }_{-\tau }^{0}d[\eta (\theta )]\varphi (\theta ),\forall \varphi \in C,$$

$$\frac{du(t)}{dt}=-V(u_t)+g(t),t\ge 0$$

$$y(t)=[\Phi (t)\varphi ](0)+{\int }_0^{t}X(t-s)g(s)ds,\forall t\ge 0.$$

$$X^{ϵ}(t)x=[\Phi (t)h^{ϵ}(\cdot )x](0),\forall t\ge 0,x\in {\mathbb{R}}^m.$$

$$X(t)x=\underset{ϵ\to 0^{+}}{\lim }{X}^{ϵ}(t)x,\forall t\ge 0,x\in {\mathbb{R}}^m.$$

$$\frac{du(t)}{dt}=-V(u_t)+x$$

$$\underset{t\to \infty }{\lim }z(t)=\left({\int }_0^{\infty }X(s)ds\right)x=u^{\ast }={\hat{V}}^{-1}x.$$

$$\tilde{u}(t)=\{\begin{array}{ll}X(t)x,& \forall t\ge 0,\\ 0,& \forall -\tau \le t<0.\end{array}$$

$$\begin{array}{rl}{\int }_0^{\infty }F({\tilde{u}}_t)dt& ={\int }_0^{\infty }\left({\int }_{-\tau }^{0}d\eta (\theta )\tilde{u}(t+\theta )\right)dt\\ & ={\int }_{-\tau }^{0}d\eta (\theta ){\int }_0^{\infty }\tilde{u}(t+\theta )dt\\ & ={\int }_{-\tau }^{0}d\eta (\theta ){\int }_{\theta }^{\infty }\tilde{u}(s)ds\\ & ={\int }_{-\tau }^{0}d\eta (\theta )\left({\int }_{\theta }^0\tilde{u}(s)ds+{\int }_0^{\infty }\tilde{u}(s)ds\right)\\ & ={\int }_{-\tau }^{0}d\eta (\theta ){\int }_0^{\infty }X(s)xds\\ & =\hat{F}({\hat{V}}^{-1}x).\end{array}$$

$$\mathcal{R}_{0}=r\big{(}\hat{F}\hat{V}^{-1}\big{)}.$$

$$sign({\mathcal{R}}_0-1)=sign(s(\hat{F}-\hat{V})).$$

$$sign(s(F-V))=sign(s(\widehat{F-V}))=sign(s(\hat{F}-\hat{V})).$$

$$\frac{du}{dt}=\frac{1}{\mu }F(u_t)-V(u_t)$$

$${\mathcal{R}}_0(\mu )=\frac{1}{\mu }{\mathcal{R}}_0,\forall \mu >0.$$

$$sign({\mathcal{R}}_0(\mu )-1)=sign({\lambda }^{\ast }(\mu ))=sign(r(Q_{\mu }(t_0))-1).$$

$$\frac{du}{dt}=\hat{F}u-\hat{V}u=\big{(}\hat{F}-\hat{V}\big{)}u.$$

$$\underset{n\to \infty }{\lim }\Vert (\frac{1}{\delta }M{)}^n{\Vert }^{\frac{1}{n}}=r(\frac{1}{\delta }M)=\frac{1}{\delta }r(M)<1,$$

Full text

Dedicated to Professor Yihong Du on the occasion of his 60th birthday

Key words and phrases: Autonomous FDEs, linear stability, exponential growth bound, basic reproduction number, threshold dynamics.

MSC: 34K06, 34K30, 37C65 , 37L15, 92D25.

**Short title: ** Stability and $\mathcal{R}_{0}$ for autonomous FDEs

1 Introduction

The basic reproduction number (ratio) $\mathcal{R}_{0}$ is one of the most important concepts in population biology. In epidemiology, $\mathcal{R}_{0}$ is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual, and $\mathcal{R}_{0}$ is also a commonly used measure of the effort needed to control an infectious disease. The theory of $\mathcal{R}_{0}$ has been greatly developed for various evolution systems, see survey and review papers [6, 7, 2, 17, 22] and references therein. Among these research works, Diekmann, Heesterbeek and Metz [3] introduced the next generation operators approach; van den Driessche and Watmough [18] gave a formula of $\mathcal{R}_{0}$ and proved the local stability in terms of $\mathcal{R}_{0}$ for compartmental ordinary differential equation (ODE) models; Bacaër and and Guernaoui [1] proposed a general definition of $\mathcal{R}_{0}$ for population models in a periodic environment; Wang and Zhao [19] characterized $\mathcal{R}_{0}$ and proved the local stability in terms of $\mathcal{R}_{0}$ for periodic compartmental ODE models; Thieme [15] presented the theory of spectral bounds and reproduction numbers for infinite-dimensional population structure; Wang and Zhao [20] gave a biologically meaningful definition of $\mathcal{R}_{0}$ for reaction-diffusion models and found its relation to the principal eigenvalue of an associated elliptic eigenvalue problem; and Inaba [9] employed a generation evolution operator to give a new definition of $\mathcal{R}_{0}$ for structured populations in heterogeneous environments.

Regarding population models with time delay, Zhao [21] established the theory of $\mathcal{R}_{0}$ for periodic time-delayed compartmental systems where the internal transition does not involve time delay, and give a formula of $\mathcal{R}_{0}$ in the autonomous case of such systems (see [21, Corollary 2.1]). Liang, Zhang and Zhao [10] further generalized this theory to periodic abstract FDEs so that it can be applied to spatial population models whose solution maps are not compact. More recently, Huang, Wu and Zhao [8] extended the theory developed in [10] to more general periodic abstract FDEs where the internal transition term also has time delay. Note that any autonomous evolution system can be regarded as an $t_{0}$-periodic one for any given positive number $t_{0}$. As a straightforward consequence of this theory, a general formula of $\mathcal{R}_{0}$ was provided for autonomous FDEs (see [8, Corollary A.8]).

In this paper, we will first show that the stability of a linear autonomous and cooperative FDE can be determined by that of an associated autonomous and cooperative system without time delay from a perspective of the stability equivalence related to $\mathcal{R}_{0}$. This result further develops the earlier works in [12, 14, 16] and also of its own interest. Then we will introduce next generation operators directly for general autonomous FDEs to define $\mathcal{R}_{0}$ rather than regarding them as time-periodic systems, and prove the stability equivalence in terms of $\mathcal{R}_{0}$, which makes the formulas given in [21, Corollary 2.1] and [8, Corollary A.8] have more intuitive biological meaning.

The rest of this paper is organized as follows. In the next two sections, we consider autonomous FDEs on $\mathbb{R}^{m}$ and abstract autonomous FDEs on an ordered Banach space $X$, respectively. In section 4, as an illustrative example, we briefly study a time-delayed population model of black-legged ticks to obtain a threshold type result on its global dynamics in terms of $\mathcal{R}_{0}$.

2 Autonomous FDEs on $\mathbb{R}^{m}$

Let $\tau\in\mathbb{R}_{+}$ be given, and $C=C([-\tau,0],\mathbb{R}^{m})$ equipped with the maximum norm $\|\cdot\|_{C}$ and the positive cone $C_{+}=C([-\tau,0],\mathbb{R}^{m}_{+})$. Then $(C,C_{+})$ is an ordered Banach space. Let $\mathcal{L}(C,\mathbb{R}^{m})$ be the space of all bounded and linear operators from $C$ to $\mathbb{R}^{m}$. For any $L\in\mathcal{L}(C,\mathbb{R}^{m})$, we define $\hat{L}\in\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{m})$ by

[TABLE]

where $\hat{x}(\theta)=x,\forall\theta\in[-\tau,0]$. Clearly, $\hat{L}$ can be regarded as an $m\times m$ matrix. Recall that the stability modulus of an $m\times m$ matrix $A$ is defined as

[TABLE]

For a continuous function $u:[-\tau,\sigma)\to\mathbb{R}^{m}$ with $\sigma>0$, we define $u_{t}\in C$ by

[TABLE]

for any $t\in[0,\sigma)$.

Let $L\in\mathcal{L}(C,\mathbb{R}^{m})$ be given. By the general theory of linear FDEs in [5, Section 8.1], it follows that for any $\phi\in C$, the linear system

[TABLE]

admits a unique solution $u(t,\phi)$ on $[0,\infty)$ with $u_{0}=\phi$. Let $T(t)$ be the solution semigroup of linear system (2.1), that is,

[TABLE]

We define the stability modulus of linear system (2.1) as

[TABLE]

Assume that $L=(L_{1},\cdots,L_{m})$ satisfies the following quasimonotone condition:

1. (K)

$L_{i}(\phi)\geq 0$ whenever $\phi\geq 0$ and $\phi_{i}(0)=0$.

Then [14, Theorem 5.1.1] implies that $T(t)$ is monotone semiflow on $C$. Further, if $L$ satisfies (K) and the irreducibility conditions (R) and (I), then $s(L)$ is the principal eigenvalue of system (2.1) (see [14, Theorem 5.5.1]).

The following result improves [14, Corollary 5.5.2], where the irreducibility conditions (R) and (I) were assumed.

Theorem 2.1**.**

Assume that $L=(L_{1},\cdots,L_{m})$ satisfies condition (K). Then s(L) and $s(\hat{L})$ have the same sign.

Proof.

Given $t_{0}>0$, we may regard system (2.1) as a linear $t_{0}$-periodic system. By the spectral mapping theorem for linear autonomous FDEs (see, e.g., [5, Lemma 7.6.1]), it follows that

[TABLE]

and hence, $sign(s(L))=sign(r(T(t_{0}))-1)$.

Since $L$ satisfies (K), [14, Lemma 5.1.2] implies that there exist a diagonal matrix $D=diag(a_{1},\cdots,a_{m})$ and $\bar{L}\in\mathcal{L}(C,\mathbb{R}^{m})$ with $\bar{L}(C_{+})\subset\mathbb{R}^{m}_{+}$ such that

[TABLE]

Fix a real number $a>\max\{0,a_{1},\cdots,a_{m}\}$. Thus, we can write $L$ as

[TABLE]

By [21, Theorem 2.1 and Corollary 2.1] with $\mathcal{F}(\phi)=a\phi(0)+\bar{L}(\phi)$ and $\mathcal{V}=aI-D$, it follows that

[TABLE]

On the other hand, the well-known result on basic reproduction numbers for ODE systems [18, Theorem 2] (see also [15, Theorem 2.3]) gives rise to

[TABLE]

Now (2.3) and (2.4) show that $sign(s(L))=sign(s(\hat{L}))$. ∎

Next, we consider the following autonomous linear FDE:

[TABLE]

where $F\in\mathcal{L}(C,\mathbb{R}^{m})$ and $V\in\mathcal{L}(C,\mathbb{R}^{m})$. System (2.5) may come from the equations of infectious variables in the linearization of a given time-delayed compartmental epidemic model at a disease-free equilibrium. As such, $m$ is the total number of the infectious compartments, and the newly infected individuals at time $t$ depend linearly on the infectious individuals over the time interval $[t-\tau,t]$, which is described by $F(u_{t})$. Further, the internal transition of individuals in the infectious compartments (e.g., natural and disease-induced deaths, and movements among compartments) is governed by the linear time-delayed system:

[TABLE]

Without loss of generality, we assume that $F:C\to\mathbb{R}^{m}$ is given by

[TABLE]

where $\eta(\theta)$ is an $m\times m$ matrix function which is measurable in $\theta\in\mathbb{R}$ and normalized so that $\eta(\theta)=0$ for all $\theta\geq 0$ and $\eta(\theta)=\eta(-\tau)$ for all $\theta\leq-\tau$. Moreover, $\eta(\theta)$ is continuous from the left in $\theta$ on $(-\tau,0)$, and the variation of $\eta(\cdot)$ on $[-\tau,0]$ is bounded. Clearly, $F$ is also well defined on $L^{1}([-\tau,0],\mathbb{R}^{m})$.

Let $Q(t)$ and $\Phi(t)$ be the solution semigroups of linear systems (2.5) and (2.6), respectively. According to [13, Theorem 1], there is a family of continuous matrix-valued functions $X(t)$ on $[0,\infty)$ such that for any $\phi\in C$ and $g\in C([0,\infty),\mathbb{R}^{m})$, the unique solution $y(t)$ with initial data $y_{0}=\phi$ of the following inhomogeneous system

[TABLE]

satisfies

[TABLE]

Choose a family of functions $h^{\epsilon}\in C^{\infty}([-\tau,0],[0,1])$ with $\epsilon\in(0,\tau]$ such that $h^{\epsilon}$ is nondecreasing on $[-\tau,0]$; $h^{\epsilon}(\theta)=0,\,\forall\theta\in[-\tau,-\epsilon]$ and $h^{\epsilon}(0)=1$; and $h^{\tilde{\epsilon}}(\theta)\leq h^{\epsilon}(\theta),\ \forall 0<\tilde{\epsilon}\leq\epsilon\leq\tau$ and $\theta\in[-\tau,0]$. Define a family of linear operators on $\mathbb{R}^{m}$ by

[TABLE]

It then follows from [13, Theorem 1] that

[TABLE]

Thus, $z(t):=X(t)x$ may be regarded as a solution of system (2.6) on $[0,\infty)$ with the initial data satisfying $z_{0}(\theta)=0$ for all $\theta\in[-\tau,0)$ and $z_{0}(0)=x$.

Let $s(-V)$ be the stability modulus of linear system (2.6). Then we have the following observation.

Lemma 2.1**.**

If $s(-V)<0$, then $\int_{0}^{\infty}X(s)ds=\hat{V}^{-1}$.

Proof.

For any given $x\in\mathbb{R}^{m}$, it is easy to see that the autonomous system

[TABLE]

has a unique equilibrium $u^{*}=\hat{V}^{-1}x$. Since $s(-V)<0$, we conclude that $u^{*}$ is globally attractive for system (2.9) in $C$. Note that $z(t):=\int_{0}^{t}X(t-s)xds=\left(\int_{0}^{t}X(s)ds\right)x$ is the unique solution on $[0,\infty)$ of (2.9) with $z_{0}=0$. It then follows that

[TABLE]

Since $x\in\mathbb{R}^{m}$ is arbitrary, we obtain $\int_{0}^{\infty}X(s)ds=\hat{V}^{-1}$. ∎

To introduce the basic reproduction number for system (2.5), throughout this section we assume that

1. (A1)

$F$ is positive in the sense that $F(C_{+})\subset\mathbb{R}^{m}_{+}$. 2. (A2)

$-V$ satisfies condition (K) and $s(-V)<0$.

Let $x\in\mathbb{R}^{m}_{+}$ be the initial infected individuals distributed among $m$ compartments. The distribution of these individuals under the internal evolution at time $t\geq 0$ is $X(t)x$. Define a function

[TABLE]

Then the new infection of the infected individuals at time $t$ is $F(\tilde{u}_{t})$. Thus, the distribution of the total new infection is

[TABLE]

It follows that the next generation matrix for system (2.5) is $\hat{F}\hat{V}^{-1}$. Accordingly, we define the basic reproduction number for system (2.5) as the spectral radius of $\hat{F}\hat{V}^{-1}$, that is,

[TABLE]

Let $\lambda^{*}=s(F-V)$ be the stability modulus of linear system (2.5). Then we have the following stability result in terms of $\mathcal{R}_{0}$.

Theorem 2.2**.**

$\mathcal{R}_{0}-1$* and $\lambda^{*}$ have the same sign.*

Proof.

Since \mathcal{R}_{0}=r\big{(}\hat{F}\hat{V}^{-1}\big{)}, it follows from [18, Theorem 2] (see also [15, Theorem 2.3]) that

[TABLE]

By Theorem 2.1 with $L=F-V$, we see that

[TABLE]

Thus, we have $sign(\mathcal{R}_{0}-1)=sign(s(F-V))=\lambda^{*}$. ∎

Let $Q_{\mu}(t)$ be the solution semigroup of the following linear system with parameter $\mu>0$:

[TABLE]

and $\lambda^{*}(\mu)=s(\frac{1}{\mu}F-V)$ be its stability modulus. Clearly, $Q_{1}(t)=Q(t)$ and $\lambda^{*}(1)=\lambda^{*}$. As a consequence of Theorem 2.2, we have the following observation.

Theorem 2.3**.**

If $\mathcal{R}_{0}>0$, then $\mu=\mathcal{R}_{0}$ is the unique positive solution of $r(Q_{\mu}(t_{0}))=1$ for any given $t_{0}>0$, and also the unique positive solution of $\lambda^{*}(\mu)=0$.

Proof.

Let $\mathcal{R}_{0}(\mu)$ be the basic reproduction number of system (2.11). Clearly, there holds

[TABLE]

Since $\mathcal{R}_{0}>0$, $\mathcal{R}_{0}$ is the unique solution of $\mathcal{R}_{0}(\mu)=1$. Let $t_{0}>0$ be given. By Theorem 2.2 and the spectral mapping theorem for linear autonomous FDEs (see [5, Lemma 7.6.1]), it then follows that

[TABLE]

This implies two desired statements. ∎

Remark 2.1**.**

In view of (2.10) and the theory of $\mathcal{R}_{0}$ for ODE systems in [18], it follows that $\mathcal{R}_{0}$ defined for FDE system (2.5) is also the basic reproduction number of the following ODE system:

[TABLE]

Remark 2.2**.**

In the case where $\tau=0$, $\mathcal{R}_{0}$ in (2.10) reduces to the formula of $\mathcal{R}_{0}$ for ODE systems in [18], and in the case where $V(\phi)=V\phi(0)$ for a square matrix $V$, $\mathcal{R}_{0}$ in (2.10) is also consistent with the formula of $\mathcal{R}_{0}$ given in [21, Corollary 2.1].

3 Abstract autonomous FDEs

In this section, we extend the results in section 2 to a large class of abstract autonomous FDEs. We start with a simple observation on the spectral radius of positive linear operators.

Lemma 3.1**.**

Let $(W,P)$ be an ordered Banach space with the positive cone $P$ being solid (i.e., $Int(P)\neq\emptyset$), and $M$ be a bounded and linear operator on $W$. If $M$ is positive (i.e., $M(P)\subset P$) and $Me\gg 0$ for some $e\gg 0$, then $r(M)>0$.

Proof.

Since $\lim_{\delta\to 0}\delta e=0\ll Me$, we can fix a sufficiently small number $\delta>0$ such that $Me\gg\delta e$. Now we show that $r(M)\geq\delta$. Assume, by contradiction, that $r(M)<\delta$. Since

[TABLE]

there exists $n_{0}>0$ such that $\|(\frac{1}{\delta}M)^{n}\|<1$ for all $n\geq n_{0}$. Let $\bar{M}=(\frac{1}{\delta}M)^{n_{0}}$. Clearly, $\|\bar{M}\|<1$ and $h:=\bar{M}e-e\gg 0$. Since $(I-\bar{M})e=-h$, it follows that

[TABLE]

which contradicts $e\gg 0$. ∎

Let $(X,X_{+})$ be an ordered Banach space with the positive cone $X_{+}$ being normal and solid. Let $\tau\in\mathbb{R}_{+}$ be given, and $E=C([-\tau,0],X)$ equipped with the maximum norm $\|\cdot\|_{E}$ and the positive cone $E_{+}=C([-\tau,0],X_{+})$. Then $(E,E_{+})$ is an ordered Banach space. Let $\mathcal{L}(E,X)$ be the space of all bounded and linear operators from $E$ to $X$. For any $L\in\mathcal{L}(E,X)$, we define $\hat{L}\in\mathcal{L}(X,X)$ by

[TABLE]

where $\hat{x}(\theta)=x,\forall\theta\in[-\tau,0]$.

We consider the following abstract autonomous FDE:

[TABLE]

where $A$ is a closed linear operator in $X$ with a dense domain $D(A)$ and $B\in\mathcal{L}(E,X)$. Assume that

1. (H)

$A:D(A)\to X$ generates a strongly continuous positive semigroup $T_{A}(t)$ on $X$, and $B$ is positive in the sense that $B(E_{+})\subset X_{+}$.

By the standard semigroup theory (see, e.g., [11]), it follows that for any $\phi\in E$, system (3.1) has a unique mild solution $u(t,\phi)$ on $[0,\infty)$ with $u_{0}=\phi$, and its solution maps generate a positive semigroup $\mathcal{T}(t)$ on $E$.

Recall that the exponential growth bound of the semigroup $\mathcal{T}(t)$ is defined as

[TABLE]

and the spectral bound of the closed linear operator $A+\hat{B}$ in $X$ is defined as

[TABLE]

where $\sigma(A+\hat{B})$ is the spectrum of $A+\hat{B}$.

The following result is a generalization of Theorem 2.1 to abstract FDE (3.1) on $X$.

Theorem 3.1**.**

Assume that $A$ and $B$ satisfy (H), and $T_{A}(t)$ is compact on $X$ for each $t>0$. Then $\omega(\mathcal{T})$ and $s(A+\hat{B})$ have the same sign.

Proof.

We first choose a large real number $b>0$ such that the exponential growth bound of the semigroup $e^{-bt}T_{A}(t)$ is negative. Let $-V:=A-bI$ and $F(\phi):=b\phi(0)+B(\phi),\forall\phi\in E$. Then we can write equation (3.1) as

[TABLE]

For any given $t_{0}>0$, we regard (3.1) as an $t_{0}$-periodic equation. In view of [15, Proposition A.2], we have

[TABLE]

We fix an element $e\in Int(X_{+})$. Since $\lim_{t\to 0^{+}}T_{A}(t)e=e\gg 0$, there exists $\delta>0$ such that $T_{A}(t)e\gg 0$ for all $t\in[0,\delta]$. From [15, Theorem 3.12], we further see that

[TABLE]

This, together with Lemma 3.1, implies that $r((bI+\hat{B})\circ(bI-A)^{-1})>0$. Let $B_{t_{0}}$ be the ordered Banach space of all continuous and $t_{0}$-periodic functions from $\mathbb{R}$ to $X$. Following [10], we define a linear operator $\tilde{\mathfrak{L}}$ on $B_{t_{0}}$ by

[TABLE]

It is easy to see that

[TABLE]

In view of [10, Lemma 2.4], we obtain $r(\tilde{\mathfrak{L}})=r((bI+\hat{B})\circ(bI-A)^{-1})$. By [10, Theorem 3.7 and Proposition 3.9], it then follows that

[TABLE]

Let $\mathcal{A}=A+\hat{B}$, $\mathcal{B}=A-bI$ and $\mathcal{C}=bI+\hat{B}$. Clearly, $\mathcal{A}=\mathcal{C}+\mathcal{B}$ and $s(\mathcal{B})<0$. Since $\mathcal{A}$ and $\mathcal{B}$ generates positive $C_{0}$-semigroups on $X$, [15, Theorem 3.12] implies that they are resolvent-positive. Thus, [15, Theorem 3.5] gives rise to

[TABLE]

Since $-\mathcal{C}\mathcal{B}^{-1}=(bI+\hat{B})\circ(bI-A)^{-1}$, it follows from (3.3) and (3.4) that $\omega(\mathcal{T})$ and $s(A+\hat{B})$ have the same sign. ∎

Let $T_{N}(t)$ be the strongly continuous semigroup generated by a closed linear operator $N$ in $X$ with a dense domain $D(N)$, and $L,\mathcal{F}\in\mathcal{L}(E,X)$. Next, we consider the following autonomous linear FDE:

[TABLE]

where $-\mathcal{V}\phi:=N\phi(0)+L\phi$. It then follows that (3.5) generates a semigroup $U(t)$ on $E$, and the autonomous linear FDE

[TABLE]

generates a semigroup $\Psi(t)$ on $E$, respectively.

To introduce the basic reproduction number for system (3.5), throughout this section we assume that

1. (H1)

$\mathcal{F}$ is positive in the sense that $\mathcal{F}(E_{+})\subset X_{+}$. 2. (H2)

$T_{N}(t)$ is a positive semigroup, $L$ is positive, and $\omega(\Psi)<0$.

With the help of [13, Theorem 1], as applied to (3.6), we may use the essentially same arguments as those for (2.5) to derive the next generation operator as $\hat{\mathcal{F}}\circ\hat{\mathcal{V}}^{-1}$, where $-\hat{\mathcal{V}}:=N+\hat{L}$. Thus, we define the basic reproduction number for system (3.5) to be $\mathcal{R}_{0}=r(\hat{\mathcal{F}}\circ\hat{\mathcal{V}}^{-1})$.

The following result shows that the stability of the zero solution for system (3.5) can be determined by the sign of $\mathcal{R}_{0}-1$.

Theorem 3.2**.**

Assume that $T_{N}(t)$ is compact on $X$ for each $t>0$. Then $\mathcal{R}_{0}-1$ and $\omega(U)$ have the same sign.

Proof.

By [15, Theorem 3.5] with $\mathcal{B}=-\hat{\mathcal{V}}$ and $\mathcal{C}=\hat{\mathcal{F}}$, it follows from that

[TABLE]

On the other hand, Theorem 3.1 with $A=N$ and $B=L+\mathcal{F}$ implies that

[TABLE]

Thus, we have $sign(\mathcal{R}_{0}-1)=sign(\omega(U))$. ∎

Remark 3.1**.**

Since $\omega(U)=\frac{\ln r(U(t_{0}))}{t_{0}}$ for any given $t_{0}>0$, it is easy to see that Theorem 3.2 is a straightforward consequence of [8, Theorem A.5, Corollary A.8 and Proposition A.9].

Let $U_{\mu}(t)$ be the solution semigroup of the following linear system with parameter $\mu>0$:

[TABLE]

Then we have the following result which characterizes $\mathcal{R}_{0}$.

Theorem 3.3**.**

Assume that $T_{N}(t)$ is compact on $X$ for each $t>0$. If $\mathcal{R}_{0}>0$, then $\mu=\mathcal{R}_{0}$ is the unique positive solution of $r(U_{\mu}(t_{0}))=1$ for any given $t_{0}>0$, and also the unique positive solution of $\omega(U_{\mu})=0$.

Proof.

Let $\mathcal{R}_{0}(\mu)$ be the basic reproduction number of system (3.7). It then follows that

[TABLE]

Since $\mathcal{R}_{0}>0$, $\mathcal{R}_{0}$ is the unique solution of $\mathcal{R}_{0}(\mu)=1$. For any given $t_{0}>0$, there holds $\omega(U_{\mu})=\frac{\ln r(U_{\mu}(t_{0}))}{t_{0}}$. In view of Theorem 3.2, we have

[TABLE]

This gives rise to the desired two statements. ∎

Remark 3.2**.**

For any given $\mu>0$, we can use [10, Lemma 2.5] to compute $r(U_{\mu}(t_{0}))$ numerically. Thus, the method of bisection can be employed to solve $r(U_{\mu}(t_{0}))=1$, which gives the value of $\mathcal{R}_{0}$ due to Theorem 3.3.

Remark 3.3**.**

In Theorems 3.1, 3.2 and 3.3, the compactness condition for the semigroups $T_{A}(t)$ and $T_{N}(t)$ can be weakened via the verification of assumptions (H3), (H4) and (H5) in [10] (see also (C3), (C4) and (C5) in [8]).

4 An application

In this section, we apply the theory of $\mathcal{R}_{0}$ in section 2 to a time-delayed population model and obtain a threshold type result on its global dynamics in terms of $\mathcal{R}_{0}$. Clearly, one can also apply the theory of $\mathcal{R}_{0}$ in section 3 to some reaction-diffusion models with time delay.

Gourley et al. [4] proposed a nonlocal spatial model of black-legged ticks to study the role of white-tailed deer in their geographic spread. The spatially homogeneous version of this model is governed by the following time-delayed differential system:

[TABLE]

where $L(t)$, $N(t)$, $A_{q}(t)$ and $A_{f}(t)$ are the population densities of larvae, nymphs, questing adults and female fed adults at time $t$, respectively. All parameters $b$, $r_{i}$, $d_{i}$ and $\tau_{i}$ are positive numbers, and we refer to Table 1 in [4] for their biological meanings. The nonlinear function $g(L)=\frac{N_{cap}L}{h+L}$ with two positive constants $N_{cap}$ and $h$.

Linearizing system (4.1) at $(0,0,0,0)$, we obtain the following linear system:

[TABLE]

Let $\tau=\max\{\tau_{1},\tau_{2}\}$ and $C=C([-\tau,0],\mathbb{R}^{4})$. We define $F,V\in\mathcal{L}(C,\mathbb{R}^{4})$ as follows

[TABLE]

and

[TABLE]

Then system (4.2) is of the form (2.5). It is easy to see that

[TABLE]

and

[TABLE]

By (2.10) and a straightforward computation, we obtain

[TABLE]

Note that the function $g(x)$ is strictly sublinear (i.e, subhomogeneous) on $[0,\infty)$ in the sense that $g(sx)>sg(x)$ for all $x>0$ and $s\in(0,1)$. It follows that the solution maps of system (4.1) are sublinear (subhomogeneous) on $C([-\tau,0],\mathbb{R}^{4}_{+})$ (see [23]). Further, one can easily verify that system (4.1) has a unique positive equilibrium $u^{*}$ in the case where $\mathcal{R}_{0}>1$.

As a consequence of Theorems 2.1 and 2.2 and [23, Theorem 3.2], we have the following threshold dynamics for system (4.1).

Theorem 4.1**.**

Let $\mathcal{R}_{0}$ be given in (4.3). Then the following statements are valid.

1. (i)

If $\mathcal{R}_{0}\leq 1$, then the zero solution is globally asymptotically stable in $C([-\tau,0],\mathbb{R}^{4}_{+})$; 2. (ii)

If $\mathcal{R}_{0}>1$, then system (4.1) admits a unique positive equilibrium $u^{*}$ and $u^{*}$ is globally asymptotically stable in $C([-\tau,0],\mathbb{R}^{4}_{+})\setminus\{0\}$.