[a]Ryan Bignell

# Charm baryons at finite temperature on anisotropic lattices

Gert Aarts  Chris Allton  M. Naeem Anwar  Timothy J. Burns  Benjamin Jäger

###### Abstract

Singly, doubly and triply charmed baryons are investigated at multiple
temperatures using the anisotropic FASTSUM Generation 2L ensembles. We discuss
the temperature dependence of these baryons’ spectra in both parity channels
with a focus on the confining phase. To further qualify the behaviour of these
states around the pseudocritical temperature, we investigate the effect of
chiral symmetry restoration for light quarks. We find that an estimate of the
pseudocritical temperature can still be found from positive and negative-
parity charmed baryon correlators, even when parity doubling itself is not
very evident (as expected).

##  1 Introduction

As the temperature increases, the confining hadronic phase of Quantum
Chromodynamics (QCD) smoothly [1] deforms to a deconfining quark-gluon plasma
(QGP). In this phase the lighter quarks are deconfined while the heavier charm
and bottom quarks may remain bound until higher temperature. This transition
is experimentally investigated in particle collider experiments such as the
Large Hadron Collider [2, 3, 4] and the Relativistic Heavy Ion Collider [5].

Here we examine the charmed baryon spectrum through lattice QCD simulations.
We vary the temperature by use of a series of anisotropic ensembles with
different temporal extents. Anisotropic ensembles allow fine control of
temperature and allow for a greater number of temporal data points at each
temperature which is key for spectral studies. Charm baryons are of interest
as the heavier charm quark is produced in relatively high proportions in
heavy-ion collisions and the QGP exists long enough for the charm quarks to
diffuse [6, 7] and contribute to the hadronisation process.

In particular we examine the masses of the states as a function of
temperature. The extraction of baryon masses at high temperature becomes
difficult due to both the reduced number of temporal points and the more
fundamental question of whether the hadron is a bound state at these
temperatures. The extracted baryon masses may be of interest to phenomenology
[6].

As extracting baryon masses becomes difficult at the highest temperatures, we
instead compare positive and negative parity channels [8, 9, 10] using the
correlation functions directly, to investigate the effect of the restoration
of chiral symmetry in the QGP. While parity doubling is not expected due to
the large charm quark mass, we still note a clear change of behaviour,
occurring around the thermal crossover temperature for singly and doubly
charmed baryons.

##  2 Lattice setup

Table 1: The FASTSUM Generation 2L ensembles used in this work which have lattice size $32^{3}\times N_{\tau}$ and temperature $T=1/\left(a_{\tau}N_{\tau}\right)$. The spatial lattice spacing is $a_{s}=0.1121\left(3\right)$ fm, renormalised anisotropy $\xi=a_{s}/a_{\tau}=3.453(6)$ and the pion mass $m_{\pi}=239(1)$ MeV [11]. We use $\sim 1000$ configurations and eight (random) sources for a total of $\sim 8000$ sources at each temperature. Full details of these ensembles may be found in Ref. [10, 12]. $N_{\tau}$ | 128 | 64 | 56 | 48 | 40 | 36 | 32 | 28 | 24 | 20 | 16
---|---|---|---|---|---|---|---|---|---|---|---
$T\,\,\left(\text{MeV}\right)$ | 47 | 95 | 109 | 127 | 152 | 169 | 190 | 217 | 253 | 304 | 380

We calculate two-point correlation functions using standard baryon
interpolating operators [13, 14] on the FASTSUM “Generation 2L” thermal
ensembles [10, 12]. These span a wide range of temperatures using a fixed-
scale anisotropic approach with a Wilson-clover fermion action and Symanzik-
improved gauge action [15, 16, 17, 18]. The pion mass is $239(1)$ MeV. Table 1
shows the details of these ensembles.

Excited state effects are reduced by the use of Gaussian smearing [19] at the
source and the sink. The root-mean-square radius of the smearing profile is
$\sim 6.1$ lattice sites, an amount chosen such that the ground-state
isolation for the zero temperature positive parity nucleon is good.

##  3 Masses

The zero-momentum projected two-point correlation function has contributions
from multiple positive and negative parity energy eigenstates. Accordingly, we
fit the correlator using functions of the form

$\displaystyle
G\left(\tau\right)=\sum_{\alpha=1}^{N}\,A_{\alpha}\text{e}^{-a_{\tau}E^{+}_{\alpha}\tau/a_{\tau}}+B_{\alpha}\text{e}^{-a_{\tau}E_{\alpha}^{-}\left(N_{\tau}-\tau/a_{\tau}\right)},$
(1)

where the number of exponentials is allowed to vary $N=1,2,3$;
$E_{\alpha}^{+}$ are the positive parity energies and $E_{\alpha}^{-}$ are the
negative parity energies. $N$ is set by examination of the Gaussian Bayes
factor [20, 21] which tells us that additional exponential terms would not
increase the quality of the fit.

Figure 1: Highest weighted fit results for the $N_{\tau}=64$
${}^{\frac{3}{2}}\Omega_{ccc}$ positive parity energy as a function of fit
window. The weights of Ref. [22] are also shown, as is the resulting model
averaged fit value. Both sides of the correlator are fit symmetrically e.g.
“10-26:38-54” fits points $[10,26]$ and $[38,54]$ simultaneously. Figure 2:
Effective mass for the $N_{\tau}=64$ ${}^{\frac{3}{2}}\Omega_{ccc}$ state
showing the results of the two model averaging methods described in Refs. [22,
23].

In order to ensure a robust determination of the masses, we make use of the
model averaging methods presented in Refs. [22, 23]. Here all possible
temporal subsets of the correlator (“fit-windows”) are considered and the
resulting fits are averaged with some weight. Outlying fits are excluded via a
consideration of the 30% of fits with the highest weight from the Akaike
information criterion weights [23]. The final mass is produced by picking the
averaged result which most clearly represents the underlying fits. An example
of this is shown in Figures 1 and 2 for the positive parity $N_{\tau}=64$
${}^{\frac{3}{2}}\Omega_{ccc}$ state. In Figure 1 we show the highest weighted
fits using the “p-value” method [22] where the fit-window is shown above and
below. As we fit both sides of the correlator, the label reflects this. The
averaged result encompasses the clear majority of fit weights and is not an
outlier or unstable result. A systematic uncertainty from the choice of
averaging method is also considered.[24, 25]

Figure 3: The ground-state charm baryon spectrum from the $N_{t}=128$ zero-
temperature ensemble. Positive parity states are shown by “diamonds”
$\left(\blacklozenge\right)$ and the negative parity by “stars”
$\left(\star\right)$. Inner error bars represent the statistical uncertainty
and the outer incorporates a systematic from the choice of averaging method.
The grey bands represent experimental results from the Particle Data Group
[26] with uncertainties from different charge states added in quadrature. For
the $\Omega_{c}$ we use the spin-parity assignments suggested in Refs. [27,
28]. We select $\Omega_{c}^{3/2-}=\Omega_{c}(3065)$ and include a systematic
uncertainty encompassing the $\Omega_{c}(3050)$ mass due to the former’s
better known width.

A description of the charm baryon states investigated in this study is
provided in Figure 3 which displays the ground-state charm baryon spectrum on
the “zero temperature” $N_{\tau}=128$ ensemble. The zero-temperature results
are in good agreement with the experimental measurements for the states
containing only $s$ or $c$ quarks as these quarks have been tuned to their
physical values [24]. States containing $u$ or $d$ quarks are heavier than
nature which is expected on these ensembles. The emphasis in this work is not
on extreme precision zero-temperature spectroscopy [24, 25] and so Figure 3 is
only a guide to the charmed baryons considered in this work.

Figure 4: Left: Mass spectrum of $J^{P}={1/2}{}^{+}$ baryons as a function of
temperature. Dashed lines are zero-temperature experimental results [26] to
guide the eye. The inner shaded region represents the statistical uncertainty,
and the outer incorporates the systematic from the choice of averaging method.
The same vertical scale is used in each subplot, making comparison of the
uncertainties possible. Some states have increasing uncertainties around and
past $T_{pc}$. This may indicate these states becoming unbound at these
temperatures. Right: Mass spectrum of $J^{P}={1/2}{}^{-}$ baryons.

We extend the analysis of Figure 3 to non-zero temperature for the $J=1/2$
charmed baryons in Figure 4. Here the positive and negative parity sectors are
shown separately, enabling an examination of the temperature dependence. It is
clear that within each parity sector, different channels display qualitatively
similar behaviour. Figure 4 suggests that the effect of temperature decreases
with charmness $C$.

Figure 5: Left: Mass spectrum of $J^{P}={3/2}{}^{+}$ baryons as a function of
temperature. Details as in Figure 4. Right: Mass spectrum of
$J^{P}={3/2}{}^{-}$ baryons. Here positive and negative parity plots have
different vertical scales.

In Figure 5 a similar picture emerges for the $J=3/2$ baryons. In this figure,
the positive and negative parity plots do not share the same vertical scale
however each subplot is the same. Here however the positive parity, $C=3$
$\Omega_{ccc}$ mass is unaffected by temperature within uncertainties. This
agrees with the pattern observed for the $J=1/2$ sector where the $C=1$ states
displayed greater temperature dependence than the $C=2$.

Determining the mass at non-zero temperature is a difficult task. This can be
observed by the increasing uncertainty present in the preceding figures. The
extracted masses suggest that above the transition temperature of $\sim 167$
MeV [29] the positive and negative parity sectors show greater temperature
dependence. The suitability of the exponential ansatz of Eq.
$\left(\ref{eqn:expFit}\right)$ at high temperatures is an interesting
problem. Already at $N_{\tau}=32$ ($T=190$ MeV) the exponential fits do not
seem to be describing all the states well. In particular the $\Xi_{cc}$ is
well described at this temperature, but not at the next hottest. The lighter
states are ill-behaved at lower temperatures, which may be an indication they
are no longer bound states at those temperatures. Nevertheless, we show the
obtained masses, including the large uncertainty produced by our analysis - a
sign of the breakdown of the ansatz. This problem could be better investigated
with e.g. a spectral function function analysis or a correlator with more
ground-state isolation before the interference of the positive and negative
parity terms in the centre of the lattice. Such isolation could be achieved
via a better determination of the correlator on these ensembles, or an
ensemble with more temporal points at the same temperature. Both these methods
are under investigation.

In summary, our results suggest that the temperature dependence is greater for
negative parity states and in the $J=3/2$ sectors.

##  4 Parity Doubling

Chiral symmetry is expected to-be restored in the QGP. For light and strange
baryons, this results in a parity-doubling signal at the level of the
correlators [8, 9]. For charmed baryons for the temperatures studied in this
work, such a signal is not expected, since the charm quark mass breaks chiral
symmetry explicitly. Nevertheless, it is interesting to study whether
restoration of chiral symmetry for lighter quarks leads to visible effects in
the charmed baryon sector.

We therefore turn to the parity doubling $R$-quantity considered in Refs. [30,
31]. This method has the advantage that no fits are required, enabling high
statistical precision. We form the quantities

$\displaystyle\mathcal{R}\left(\tau\right)$
$\displaystyle=\frac{G^{+}\left(\tau\right)-G^{-}\left(\tau\right)}{G^{+}\left(\tau\right)+G^{-}\left(\tau\right)},$
(2) $\displaystyle R\left(n_{0}\right)$
$\displaystyle=\frac{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,\mathcal{R}\left(\tau_{n}\right)/\sigma_{\mathcal{R}}^{2}\left(\tau_{n}\right)}{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,1/\sigma_{\mathcal{R}}^{2}\left(\tau_{n}\right)},$
(3)

where $G\left(N_{\tau}a_{t}-\tau\right)=G^{-}\left(\tau\right)$ is the
negative parity correlator, $G\left(\tau\right)=G^{+}\left(\tau\right)$ the
positive parity correlator, and $\sigma_{\mathcal{R}}\left(\tau_{n}\right)$
denotes the statistical error for $\mathcal{R}\left(\tau_{n}\right)$. The sum
over time slices $\tau_{n}\in\left[n_{0}=4,N_{\tau}/2-1\right]$ is chosen such
that excited state contributions and lattice artefacts at small $\tau_{n}$ are
suppressed. Small shifts in $n_{0}$ do not have a qualitative effect on the
results.

Figure 6: Left: $R$-quantity of Eq. $\left(\ref{eqn:Rratio}\right)$ for
$J=\frac{1}{2}$ baryons. The lines are cubic splines used to find the
transition temperature which are shown in the vertical bands. Right:
$R$-quantity of Eq. $\left(\ref{eqn:Rratio}\right)$ for $J=\frac{3}{2}$
baryons.

As is clear from Eq. $\left(\ref{eqn:mcRratio}\right)$, $R\rightarrow 0$ when
positive and negative correlators become degenerate, which coincides with
chiral symmetry restoration. Conversely, when the states are non-degenerate
with $m_{+}\ll m_{-}$, $R$ will be close to one. This is indeed the behaviour
we see in Figure 6.

Due to the heavier mass of the charm quarks, chiral symmetry restoration is
not expected, even at our highest temperatures. Again this is the effect seen
in Figure 6 with it being particularly notable that the $R$-quantity increases
monotonically with the number of charm quarks.

We also observe an intriguing similarity in the behaviour of the $R$-quantity
for states (of a given $J$) which have the same charm content, and which
belong to the same $SU(3)$ flavour multiplets, namely:

$\displaystyle C=+2,\,\quad SU(3)\,\,\mathbf{3}:\,\quad$
$\displaystyle\Omega_{cc}\left(ccs\right),\,\Xi_{cc}\left(ccu\right)$
$\displaystyle C=+1,\,\quad SU(3)\,\,\mathbf{\overline{}}{3}:\,\quad$
$\displaystyle\Lambda_{c}\left(udc\right),\,\Xi_{c}\left(usc\right)$
$\displaystyle C=+1,\,\quad SU(3)\,\,\mathbf{6}:\,\quad$
$\displaystyle\Sigma_{c}\left(udc\right),\,\Xi_{c}^{\prime}\left(usc\right),\,\Omega_{c}\left(ssc\right)$

A cubic spline interpolation of the data points for each hadron is performed;
this enables the inflection point of the curves to be found. For singly-
charmed baryons with $J=1/2$ in Figure 6 left we find the inflection point to
be near $T_{pc}$; interestingly, the doubly charmed $J=1/2$ baryons do not
have an inflection point. The ability of these inflection points to describe
the transition temperature is present even when parity doubling is not nearly
manifest, as for the $\Lambda_{c}(udc)$, as seen in Figure 6 left.

We repeat this exercise for the $J=3/2$ baryons in Figure 6 right. Here we
note that doubly charmed $J=3/2$ baryons display an inflection point, again
around $T_{pc}$. This is aligned with the results observed for masses, where
the $J=3/2$ states were more affected by temperature than the $J=1/2$ states.

Figure 7: Inflection points of baryon $R$-quantity of Figure 6. The blue band
is the pseudocritical temperature from the renormalised chiral condensate [29]
adjusted for the change in lattice spacing [11, 12].

The inflection points found are presented in Figure 7 alongside the
computation of the pseudocritical temperature as obtained via the inflection
point of the renormalised chiral condensate (blue band) [29]. Excellent
precision and good agreement with the previous measurement is seen.

##  5 Conclusions & Future Work

A variety of singly, doubly and triply charmed baryons have been investigated
using lattice QCD. We calculated two-point correlation functions at a range of
temperatures using the FASTSUM “Generation 2L” anisotropic ensembles. Baryon
masses were extracted by performing multi-exponential fits to all possible fit
windows and considering two different methods with which to weigh the results.
In the hadronic phase and just above the crossover, it is possible to extract
ground state masses using the method described. At higher temperatures a more
sophisticated spectral function analysis would be required.

To investigate the effect of chiral symmetry restoration, the correlator
$R$-quantity was considered, to examine the difference between positive and
negative parity correlation functions as a function of temperature. Despite
the absence of parity doubling (due to the large charm quark mass), a
crossover effect can nevertheless be observed, with inflection points close to
the pseudocritical temperature.

In future work in preparation, we will examine more sophisticated methods of
exploiting the correlators directly, as recently performed for the $D$-mesons
in Ref. [12]. We anticipate providing the data and analysis tools at that
time. In order to improve these results, one could construct a correlator
which had more overlap with the ground-state or repeat the analysis on
correlators with more temporal points at the same temperature.

## Acknowledgments

This work is supported by STFC grant ST/T000813/1. This work used the DiRAC
Extreme Scaling service at the University of Edinburgh, operated by the
Edinburgh Parallel Computing Centre and the DiRAC Data Intensive service
operated by the University of Leicester IT Services on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS
capital funding via STFC capital grants ST/R00238X/1, ST/K000373/1 and
ST/R002363/1 and STFC DiRAC Operations grants ST/R001006/1 and ST/R001014/1.
DiRAC is part of the UK National e-Infrastructure. We acknowledge the support
of the Swansea Academy for Advanced Computing, the Supercomputing Wales
project, which is part-funded by the European Regional Development Fund (ERDF)
via Welsh Government, and the University of Southern Denmark and ICHEC,
Ireland for use of computing facilities. This work was performed using PRACE
resources at Cineca (Italy), CEA (France) and Stuttgart (Germany) via grants
2015133079, 2018194714, 2019214714 and 2020214714. M. N. A. acknowledges
support from The Royal Society Newton International Fellowship.

## References

  * [1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo __,_Nature_ 443 (2006) 675–678, [hep-lat/0611014].
  * [2] N. Armesto and E. Scomparin __,_Eur. Phys. J. Plus_ 131 (2016) 52, [arXiv:1511.02151].
  * [3] W. Busza, K. Rajagopal and W. van der Schee __,_Ann. Rev. Nucl. Part. Sci._ 68 (2018) 339–376, [arXiv:1802.04801].
  * [4] J. Berges, M. P. Heller, A. Mazeliauskas and R. Venugopalan __,_Rev. Mod. Phys._ 93 (2021) 035003, [arXiv:2005.12299].
  * [5] P. Braun-Munzinger, V. Koch, T. Schäfer and J. Stachel __,_Phys. Rept._ 621 (2016) 76–126, [arXiv:1510.00442].
  * [6] X. Yao and B. Müller __,_Phys. Rev. D_ 97 (2018) 074003, [arXiv:1801.02652].
  * [7] M. Begel et al.__,arXiv:2209.14872.
  * [8] G. Aarts, C. Allton, D. De Boni, S. Hands, B. Jäger, C. Praki et al.__,_JHEP_ 06 (2017) 034, [arXiv:1703.09246].
  * [9] G. Aarts, C. Allton, D. De Boni and B. Jäger __,_Phys. Rev. D_ 99 (2019) 074503, [arXiv:1812.07393].
  * [10] G. Aarts et al.__,_Phys. Rev. D_ 105 (2022) 034504, [arXiv:2007.04188].
  * [11] D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards and C. E. Thomas __,_Phys. Rev. Lett._ 123 (2019) 042002, [arXiv:1904.03188].
  * [12] G. Aarts, C. Allton, R. Bignell, T. J. Burns, S. C. García-Mascaraque, S. Hands et al.__,arXiv:2209.14681.
  * [13] D. B. Leinweber, W. Melnitchouk, D. G. Richards, A. G. Williams and J. M. Zanotti __,_Lect. Notes Phys._ 663 (2005) 71–112, [nucl-th/0406032].
  * [14] SciDAC, LHPC, UKQCD collaboration, R. G. Edwards and B. Joo __,_Nucl. Phys. B Proc. Suppl._ 140 (2005) 832, [hep-lat/0409003].
  * [15] R. G. Edwards, B. Joo and H.-W. Lin __,_Phys. Rev. D_ 78 (2008) 054501, [arXiv:0803.3960].
  * [16] B. Sheikholeslami and R. Wohlert __,_Nucl. Phys._ B259 (1985) 572.
  * [17] K. Symanzik __,_Nucl. Phys._ B226 (1983) 187–204.
  * [18] K. Symanzik __,_Nucl. Phys._ B226 (1983) 205–227.
  * [19] S. Gusken __,_Nucl. Phys. Proc. Suppl._ 17 (1990) 361–364.
  * [20] G. P. Lepage, B. Clark, C. T. H. Davies, K. Hornbostel, P. B. Mackenzie, C. Morningstar et al.__,_Nucl. Phys. B Proc. Suppl._ 106 (2002) 12–20, [hep-lat/0110175].
  * [21] P. Lepage and C. Gohlke __, Dec., 2021. 10.5281/zenodo.5777652.
  * [22] E. Rinaldi, S. Syritsyn, M. L. Wagman, M. I. Buchoff, C. Schroeder and J. Wasem __,_Phys. Rev. D_ 99 (2019) 074510, [arXiv:1901.07519].
  * [23] W. I. Jay and E. T. Neil __,_Phys. Rev. D_ 103 (2021) 114502, [arXiv:2008.01069].
  * [24] Hadron Spectrum collaboration, L. Liu, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, P. Vilaseca et al.__,_JHEP_ 07 (2012) 126, [arXiv:1204.5425].
  * [25] M. Padmanath __,_PoS_ CHARM2020 (2021) 014, [arXiv:2109.04748].
  * [26] Particle Data Group collaboration, P. Zyla et al.__,_PTEP_ 2020 (2020) 083C01.
  * [27] H.-Y. Cheng __,_Chin. J. Phys._ 78 (2022) 324–362, [arXiv:2109.01216].
  * [28] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu and S.-L. Zhu __,arXiv:2204.02649.
  * [29] G. Aarts et al.__,_PoS_ LATTICE2019 (2019) 075, [arXiv:1912.09827].
  * [30] S. Datta, S. Gupta, M. Padmanath, J. Maiti and N. Mathur __,_JHEP_ 02 (2013) 145, [arXiv:1212.2927].
  * [31] G. Aarts, C. Allton, S. Hands, B. Jäger, C. Praki and J.-I. Skullerud __,_Phys. Rev. D_ 92 (2015) 014503, [arXiv:1502.03603].


# The Linear Stability and Basic Reproduction Numbers for Autonomous FDEs

Xiao-Qiang Zhao
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL A1C 5S7, Canada
E-mail<EMAIL_ADDRESS>Research supported in part by the NSERC of Canada
(RGPIN-2019-05648).

###### 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
$\mathcal{R}_{0}$.

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

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

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

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

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

$u_{t}(\theta)=u(t+\theta),\quad\forall\,\theta\in[-\tau,0],$

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

$\dfrac{du(t)}{dt}=L(u_{t})$ (2.1)

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,

$T(t)\phi=u_{t}(\phi),\quad\forall\phi\in C,\,t\geq 0.$ (2.2)

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

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

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

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

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

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

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

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

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

$sign(s(L))=sign(r(T(t_{0}))-1)=sign(r(\hat{\mathcal{F}}\mathcal{V}^{-1})-1).$
(2.3)

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

$sign(r(\hat{\mathcal{F}}\mathcal{V}^{-1})-1)=sign(s(\hat{\mathcal{F}}-\mathcal{V}))=sign(s(\hat{L})).$
(2.4)

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

Next, we consider the following autonomous linear FDE:

$\dfrac{du(t)}{dt}=F(u_{t})-V(u_{t}),$ (2.5)

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:

$\dfrac{du(t)}{dt}=-V(u_{t}).$ (2.6)

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

$F(\phi)=\int_{-\tau}^{0}d[\eta(\theta)]\phi(\theta),\quad\forall\phi\in C,$

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

$\dfrac{du(t)}{dt}=-V(u_{t})+g(t),\quad t\geq 0$ (2.7)

satisfies

$y(t)=[\Phi(t)\phi](0)+\int_{0}^{t}X(t-s)g(s)ds,\quad\forall t\geq 0.$

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

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

It then follows from [13, Theorem 1] that

$X(t)x=\lim_{\epsilon\rightarrow 0^{+}}X^{\epsilon}(t)x,\,\,\forall t\geq
0,x\in\mathbb{R}^{m}.$ (2.8)

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

$\dfrac{du(t)}{dt}=-V(u_{t})+x$ (2.9)

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

$\lim_{t\to\infty}z(t)=\left(\int_{0}^{\infty}X(s)ds\right)x=u^{*}=\hat{V}^{-1}x.$

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

$\tilde{u}(t)=\begin{cases}X(t)x,&\forall\,t\geq 0,\\\ 0,&\forall\,-\tau\leq
t<0.\end{cases}$

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

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

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,

$\mathcal{R}_{0}=r\big{(}\hat{F}\hat{V}^{-1}\big{)}.$ (2.10)

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

$sign(\mathcal{R}_{0}-1)=sign(s(\hat{F}-\hat{V})).$

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

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

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 $\mu>0$:

$\frac{du}{dt}=\frac{1}{\mu}F(u_{t})-V(u_{t})$ (2.11)

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

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

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

$sign(\mathcal{R}_{0}(\mu)-1)=sign(\lambda^{*}(\mu))=sign(r(Q_{\mu}(t_{0}))-1).$

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:

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

######  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

$\lim_{n\to\infty}\|(\frac{1}{\delta}M)^{n}\|^{\frac{1}{n}}=r(\frac{1}{\delta}M)=\frac{1}{\delta}r(M)<1,$

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

$e=(I-\bar{M})^{-1}(-h)=-\sum_{i=0}^{\infty}\bar{M}^{i}(h)\leq 0,$

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

$\hat{L}x=L(\hat{x}),\quad\forall x\in X,$

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

We consider the following abstract autonomous FDE:

$\dfrac{du(t)}{dt}=Au(t)+B(u_{t})$ (3.1)

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

$\omega(\mathcal{T})=\inf\\{\tilde{\omega}:\exists M\geq 1\ \text{such that
$\|\mathcal{T}(t)\|\leq Me^{\tilde{\omega}t}$, $\forall t\geq 0$}\\},$

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

$s(A+\hat{B})=\sup\\{Re\,\lambda:\,\,\lambda\in\sigma(A+\hat{B})\\},$

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

$\dfrac{du(t)}{dt}=F(u_{t})-Vu(t).$ (3.2)

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

$\omega(\mathcal{T})=\frac{\ln r(\mathcal{T}(t_{0}))}{t_{0}}.$

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

$(bI-A)^{-1}e=-(A-bI)^{-1}e=\int_{0}^{\infty}e^{-bt}T_{A}(t)edt\gg 0.$

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

$[\tilde{\mathfrak{L}}(v)](t)=F\int_{0}^{\infty}e^{-bs}T_{A}(s)v(t-s+\cdot)ds,\quad\forall
t\in\mathbb{R},\,v\in B_{t_{0}}.$

It is easy to see that

$\tilde{\mathfrak{L}}(x)=F\int_{0}^{\infty}e^{-bs}T_{A}(s)xds=(bI+\hat{B})\circ(bI-A)^{-1}x,\,\,\forall
x\in X.$

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

$sign(\omega(\mathcal{T}))=sign(r(\mathcal{T}(t_{0}))-1)=sign(r((bI+\hat{B})\circ(bI-A)^{-1})-1).$
(3.3)

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

$sign(s(A+\hat{B}))=sign(s(\mathcal{A}))=sign(r(-\mathcal{C}\mathcal{B}^{-1})-1).$
(3.4)

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:

$\dfrac{du}{dt}=Nu(t)+L(u_{t})+\mathcal{F}(u_{t})=\mathcal{F}(u_{t})-\mathcal{V}(u_{t})$
(3.5)

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

$\dfrac{du}{dt}=-\mathcal{V}(u_{t})$ (3.6)

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

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

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

$sign(\omega(U))=sign(s(N+\hat{L}+\hat{\mathcal{F}}))=sign(s(\hat{\mathcal{F}}-\hat{\mathcal{V}})).$

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 $\mu>0$:

$\frac{du}{dt}=\frac{1}{\mu}\mathcal{F}(u_{t})-\mathcal{V}(u_{t}).$ (3.7)

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

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

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

$sign(\mathcal{R}_{0}(\mu)-1)=sign(\omega(U_{\mu}))=sign(r(U_{\mu}(t_{0}))-1).$

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:

$\displaystyle
L^{\prime}(t)=br_{4}e^{-d_{4}\tau_{1}}A_{f}(t-\tau_{1})-(d_{1}+r_{1})L(t),$
$\displaystyle N^{\prime}(t)=r_{1}g(L(t))-(d_{2}+r_{2})N(t),$ $\displaystyle
A_{q}^{\prime}(t)=r_{2}N(t)-(d_{3}+r_{3})A_{q}(t),$ (4.1) $\displaystyle
A_{f}^{\prime}(t)=\frac{r_{3}}{2}e^{-d_{3}\tau_{2}}A_{q}(t-\tau_{2})-(d_{4}+r_{4})A_{f}(t),$

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:

$\displaystyle
L^{\prime}(t)=br_{4}e^{-d_{4}\tau_{1}}A_{f}(t-\tau_{1})-(d_{1}+r_{1})L(t),$
$\displaystyle N^{\prime}(t)=r_{1}g^{\prime}(0)L(t)-(d_{2}+r_{2})N(t),$
$\displaystyle A_{q}^{\prime}(t)=r_{2}N(t)-(d_{3}+r_{3})A_{q}(t),$ (4.2)
$\displaystyle
A_{f}^{\prime}(t)=\frac{r_{3}}{2}e^{-d_{3}\tau_{2}}A_{q}(t-\tau_{2})-(d_{4}+r_{4})A_{f}(t),$

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

$F(\phi)=\left(br_{4}e^{-d_{4}\tau_{1}}\phi_{4}(-\tau_{1}),0,0,0\right)^{T},\quad\forall\phi=(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\in
C,$

and

$V(\phi)=\left(\begin{array}[]{c}(d_{1}+r_{1})\phi_{1}(0)\\\
-r_{1}g^{\prime}(0)\phi_{1}(0)+(d_{2}+r_{2})\phi_{2}(0)\\\
-r_{2}\phi_{2}(0)+(d_{3}+r_{3})\phi_{3}(0)\\\
-\frac{r_{3}}{2}e^{-d_{3}\tau_{2}}\phi_{3}(-\tau_{2})+(d_{4}+r_{4})\phi_{4}(0)\end{array}\right),\quad\forall\phi=(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\in
C.$

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

$\hat{F}=\left(\begin{array}[]{cccc}0&0&0&br_{4}e^{-d_{4}\tau_{1}}\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right)$

and

$\hat{V}=\left(\begin{array}[]{cccc}d_{1}+r_{1}&0&0&0\\\
-r_{1}g^{\prime}(0)&d_{2}+r_{2}&0&0\\\ 0&-r_{2}&d_{3}+r_{3}&0\\\
0&0&-\frac{r_{3}}{2}e^{-d_{3}\tau_{2}}&d_{4}+r_{4}\end{array}\right).$

By (2.10) and a straightforward computation, we obtain

$\mathcal{R}_{0}=r\big{(}\hat{F}\hat{V}^{-1}\big{)}=\frac{1}{2}bg^{\prime}(0)e^{-(d_{4}\tau_{1}+d_{3}\tau_{2})}\prod_{i=1}^{4}\frac{r_{i}}{d_{i}+r_{i}}.$
(4.3)

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
$C([-\tau,0],\mathbb{R}^{4}_{+})\setminus\\{0\\}$.

## References

  * [1] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53(2006), 421-436.
  * [2] J. M. Cushing and O. Diekmann, The many guises of $R_{0}$ (a didactic note), J. Theoret. Biol., 404 (2016), 295-302.
  * [3] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28(1990), 365-382.
  * [4] S. A. Gourley, X. Lai, J. Shi, W. Wang, Y. Xiao and X. Zou, Role of white-tailed deer in geographic spread of the blacklegged tick Ixodes scapularis: analysis of a spatially nonlocal model, Math. Biosci. Eng., 15 (2018), 1033-1054.
  * [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
  * [6] J. A. P. Heesterbeek, A brief history of $R_{0}$ and a recipe for its calculation, Acta Biotheoretica, 50(2002), 189-204.
  * [7] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2(2005), 281-293.
  * [8] M. Huang, S.-L. Wu and X.-Q. Zhao, The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay, J. Differential Equations, in review (a revised version).
  * [9] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65(2012), 309-348.
  * [10] X. Liang, L. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), Journal of Dynamics and Differential Equations, 31(2019), 1247-1278.
  * [11] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321(1990), 1-44.
  * [12] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413(1991), 1-35.
  * [13] K. Schumacher, On the resolvent of linear nonautonomous partial functional differential equations, J. Differential Equations, 59(1985), 355-387.
  * [14] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, Amer. Math. Soc., Providence, RI, 1995.
  * [15] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal of Applied Mathematics, 70(2009), 188-211.
  * [16] H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis, RWA, 2(2001), 145-160.
  * [17] van den Driessche, Infectious Disease Modelling, 2017, Reproduction numbers of infectious disease models, Infectious Disease Modelling, 2 (2017), 288-303.
  * [18] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180(2002), 29-48.
  * [19] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynamics and Differential Equations, 20(2008), 699-717.
  * [20] W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11(2012), 1652-1673.
  * [21] X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, Journal of Dynamics and Differential Equations, 29(2017), 67-82.
  * [22] X.-Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer, New York, 2017.
  * [23] X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior of some cooperative systems of functional differential equations, Canadian Appl. Math. Quart., 4(1996), 421-444.