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Bull Math Biol (2015) 77:1237–1255DOI
10.1007/s11538-015-0084-6
ORIGINAL ARTICLE
An Epidemic Patchy Model with Entry–Exit Screening
Xinxin Wang1 · Shengqiang Liu1 · Lin Wang2 ·Weiwei Zhang1
Received: 25 April 2014 / Accepted: 30 April 2015 / Published
online: 15 May 2015© Society for Mathematical Biology 2015
Abstract Amulti-patch SEIQR epidemicmodel is formulated to
investigate the long-term impact of entry–exit screening measures
on the spread and control of infectiousdiseases. A threshold
dynamics determined by the basic reproduction number �0
isestablished: The disease can be eradicated if �0 < 1, while
the disease persists if�0 > 1. As an application, six different
screening strategies are explored to examinethe impacts of
screening on the control of the 2009 influenza A (H1N1) pandemic.We
find that it is crucial to screen travelers from and to high-risk
patches, and it is notnecessary to implement screening in all
connected patches, and both the dispersal ratesand the successful
detection rate of screening play an important role on determiningan
effective and practical screening strategy.
Keywords Epidemicmodel · Patchy environment ·Threshold dynamics
·Dispersal ·Entry–exit screening
1 Introduction
With the rapidly growing travel among cities and countries,
newly emerging infectiousdiseases and many re-emerging
once-controlled infectious diseases have the trend tospread
regionally and globallymuch faster than ever before (Jones et al.
2008). Interna-tional travel has been amajor factor causing the
SARS outbreak in 2003 (Lipsitch et al.
B Shengqiang [email protected]
1 Academy of Fundamental and Interdisciplinary Sciences, Harbin
Institute of Technology, 3041#,2 Yi-Kuang Street, Nan-Gang
District, Harbin 150080, China
2 Department of Mathematics and Statistics, University of New
Brunswick, Fredericton,NB E3B 5A3, Canada
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1238 X. Wang et al.
2003; Ruan et al. 2006), the A (H1N1) influenza pandemic in 2009
(Tang et al. 2010,2012; Yu et al. 2012), and the outbreak of a
novel avian-origin influenza A(H7N9)(Gao et al. 2013). To better
understand how travel among patches (a patch could be assmall as a
community village, or as large as a country or even a continent)
influencesthe spread of infectious diseases, many deterministic
models involving the interactionand dispersal of meta-populations
in two or multiple patches have been proposed andinvestigated. See
for example, Allen et al. (2007), Alonso and McKane (2002), Arinoet
al. (2005), Arino and van denDriessche (2003a, b, 2006), Bolker
(1999), Brauer andvan den Driessche (2001), Brauer et al. (2008),
Brown and Bolker (2004), Eisenberget al. (2013), Gao and Ruan
(2013), Hethcote (1976), Hsieh et al. (2007), Sattenspieland
Herring (2003) and references therein.
Being aware that travel can quickly bring infectious diseases
from one patch toanother, it is natural for the authorities to
implement traveler health screeningmeasuresat borders for exit and
entry when an outbreak occurs. For example, the World
HealthOrganization (WHO) declared the influenza A (H1N1) outbreak a
pandemic in Juneof 2009. As a response, many countries implemented
the entry–exit health screeningmeasures for travelers (Ainseba and
Iannelli 2012; Cowling et al. 2010). In China, anational
surveillance was established which includes the border entry
screening: Anyone entering China was required to undergo screening
at the border. Moreover, allpatients with suspected A (H1N1) pdm09
virus infection were admitted to designatedhospitals for
containment (Yu et al. 2010). Exit screening was also conducted by
thescreening of travelers at Mexican airports before they boarded
flights out of Mexico(Khan et al. 2013). There are several broad
approaches to border screening, includingscan of travelers by
thermal scanners for elevated body temperature, observation
oftravelers by alert staff for influenza symptoms (e.g., cough) as
well as collection ofhealth declaration forms (Cowling et al. 2010;
Li et al. 2013). During the 2009 A(H1N1) pandemic, it was shown
that about one-third of confirmed imported H1N1cases were
identified through entry screening to Hong Kong and Singapore
(Cowlinget al. 2010), while for China, the detection rate of entry
screening was about 45.56%(1027 detected cases over the total 2254
imported cases) (Li et al. 2013).
Practically, the screening process is very complicated, and many
questions shouldbe considered. For example, should ‘exit
screening’, or ‘targeted entry screening’ or‘indiscriminate entry
screening’ (Khan et al. 2013) be implemented? When to initiateand
when to discontinue the measures? As screening measures may have
tremendousimpacts on travel and trade and hence result in huge
consequences in public healthand economy, it is of great importance
to assess the effectiveness and impact of theentry–exit screening
measures on the spread and control of infectious diseases. In
thisregard, a recent study analyzed the effectiveness of border
entry screening in Chinaduring this pandemic (Yu et al. 2012), and
another paper performed a retrospectiveevaluation for the entry and
exit screening of travelers flying out of Mexico during theA (H1N1)
2009 pandemic (Khan et al. 2013).
Mathematical models have been developed by many researchers over
the pastdecades to evaluate the effectiveness of screening. For
example, Gumel et al. (2004)formulated a model to investigate the
long-term control strategies of SARS, where itwas suggested that
the eradication of SARS would require the implementation of
areliable and rapid screening test at the entry points in
conjunction with optimal isola-
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An Epidemic Patchy Model with Entry–Exit Screening 1239
tion. Hyman et al. (2003) formulated models with random
screening to estimate theeffectiveness of control measures on the
spread of HIV and other sexually transmit-ted diseases. In Nyabadza
and Mukandavire (2011), screening strategy through HIVcounseling
was incorporated into the model to qualify its impact on prevention
ofHIV/AIDS epidemic in South Africa. For other epidemic models with
screening, werefer to Ainseba and Iannelli (2012), Hove-Musekwa and
Nyabadza (2009), Liu et al.(2011), Liu and Takeuchi (2006), Liu and
Stechlinski (2013). Most of these studiesfocus on evaluations of
screening in a single patch, but little attention has been paidto
the effectiveness of screening on the travelers among patches (Liu
et al. 2011; Liuand Takeuchi 2006).
In Liu et al. (2011) and Liu and Takeuchi (2006), the entry–exit
screening processis incorporated into an SIS model between two
cities with transport-related infection.It is shown that the
entry–exit screening measures have the potential to eradicatethe
disease induced by the transport-related infection when the disease
is otherwiseendemicwhen both cities are isolated. Note that
themodels in Liu et al. (2011), Liu andTakeuchi (2006) do not
include a latent compartment, while many infectious diseasesdo
undergo latent stages before they show obvious symptoms and become
activelyinfectious. In this paper, we will incorporate the
entry–exit screening measures intoa compartmental deterministic
model with multiple patches and study the impacts ofthe entry–exit
screening measures on the spread and control of the 2009 influenza
A(H1N1).
The rest of this paper is organized as follows. In Sect. 2,
based on the models ofGerberry and Milner (2009) (also Feng 2007;
Hethcote et al. 2002; Hsu and Hsieh2005; Safi and Gumel 2010; Tang
et al. 2010), we formulate a multi-patch modelto examine how
entry–exit screening measures impact the spread and control of
pan-demic infectious diseases among patches. In Sect. 3, we
identify the basic reproductionnumber and establish a global
threshold dynamics for our model. In Sect. 4, we applythe results
in Sect. 3 to the case study of 2009 influenza A (H1N1) pandemic:
Weshow the sensitivities of the basic reproduction number upon the
variation of otherparameters before we discuss the impacts of
various screening strategies on the con-trol of influenza A (H1N1).
We conclude this paper in Sect. 5 with a summary anddiscussion.
2 The n-Patch SEIQR Model with Entry–Exit Screenings
In this section, we model the transmission dynamics of a disease
in a population withn patches by taking into consideration
entry–exit screening among patches. Within asingle patch, ourmodel
is based on that ofGerberry andMilner (2009) (also Feng
2007;Hethcote et al. 2002; Hsu and Hsieh 2005; Safi and Gumel 2010)
with a susceptible–exposed–infectious–isolation–recovered
structure. Hereafter, the subscript i refers topatch i . Our patchy
model is motivated by that of Tang et al. (2010).
For patch i (i = 1, 2, . . . , n), the population is divided
into five disjoint com-partments, namely, compartments of
susceptible, exposed (infected but not infectiousand have not yet
developed clinical symptoms), infective, isolated, and recovered.We
use Si , Ei , Ii , Qi and Ri to denote the corresponding population
sizes, respec-
123
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1240 X. Wang et al.
Fig. 1 Flow chart of the disease transmission process between
patch i and patch j
tively. Let μSi > 0, μEi > 0, μ
Ii > 0, μ
Qi > 0, μ
Ri > 0 denote the death rates of
the individuals in the five classes, respectively. Throughout
the paper, we alwaysassume that μSi ≤ min{μEi , μQi , μIi , μRi }.
The total population of patch i is givenby Ni = Si + Ei + Ii + Qi +
Ri . It is assumed that all newly born individuals aresusceptible,
and the birth rate Bi (Ni ) satisfies the following assumptions
(see Tangand Chen 2002)
(A1) Bi (Ni ) > 0 for Ni > 0;(A2) Bi (Ni ) is continuously
differentiable and B ′i (Ni ) < 0;(A3) μSi > Bi (∞).
Let γ Ii > 0 and γQ
i > 0 denote the recovery rates of infectious individuals
incompartments Ii and Qi , respectively.Weused Ai j ≥ 0, i �= j to
denote the dispersal (ortravel) rate from patch j to patch i (thus
d Aii = 0) for A = S, E, I, Q, R. We define thedispersal matrix by
D A = (˜d Ai j )n×n , with ˜d Ai j = d Ai j , i �= j and ˜d Aii =
−
∑nj=1, j �=i d Aji .
It is assumed that DE and DI are irreducible. We denote by βi
> 0 the diseasetransmission rate and assume the standard
incidence rate of infection. Let θei j ∈ [0, 1)denote the
probability that an infectious individual in patch j traveling to
patch i issuccessfully detected by the entry screening in patch i .
This individual is then isolatedin patch i . Similarly, we use θoi
j ∈ [0, 1) to denote the probability that an infectiousindividual
in patch j leaving for patch i is detected by the exit screening in
patch jand is then put into isolation in patch j . We assume that a
susceptible individual entersthe exposed class after being infected
by an infectious individual and after 1
αi> 0
time units of latent period, the individual becomes infectious
and thus can infect othersusceptible individuals. A flow chart of
the transmission process between patches iand j is sketched in Fig.
1, where δoi j = 1 − θoi j and δei j = 1 − θei j .
As shown in Fig. 1, we neglect the transport-related infection
and assume thatindividuals do not change their disease states when
they travel from one patch to
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An Epidemic Patchy Model with Entry–Exit Screening 1241
another; moreover, for the sake of focusing our study on the
entry–exit screening, weignore the contact tracing. Based on the
above assumptions, our model can then bedescribed by the following
system
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Ṡi = Bi (Ni )Ni − βi Si IiNi − μSi Si −n∑
j=1d Sji Si +
n∑
j=1d Si j S j ,
Ėi = βi Si IiNi − αi Ei − μEi Ei −n∑
j=1d Eji Ei +
n∑
j=1d Ei j E j ,
İi = αi Ei − γ Ii Ii − μIi Ii −n∑
j=1d Iji Ii +
n∑
j=1δei jδ
oi j d
Ii j I j ,
Q̇i = −γ Qi Qi − μQi Qi +n∑
j=1θoji d
Iji Ii +
n∑
j=1θei jδ
oi j d
Ii j I j ,
Ṙi = γ Ii Ii + γ Qi Qi − μRi Ri −n∑
j=1d Rji Ri +
n∑
j=1d Ri j R j ,
i = 1, 2, . . . , n.
(1)
Throughout this paper, we use the following notation. For a
vector x ∈ Rn, weuse diag(x) to denote the n × n diagonal matrix,
whose diagonal elements are thecomponents of x. We use the ordering
in Rn generated by the cone Rn+, and that is,x ≤ y if y − x ∈ Rn+,
x < y if x ≤ y and x �= y and finally x y means xi < yi
forany index i . We use (S(t), E(t), I (t), Q(t), R(t)) to denote
the vector
(S1(t), . . . , Sn(t), E1(t), . . . , En(t), I1(t), . . . ,
In(t),
Q1(t), . . . , Qn(t), R1(t), . . . , Rn(t))T ∈ R5n+ ,
and we denote A0 =(S(0), E(0), I (0), Q(0), R(0)), then for any
nonnegative ini-tial condition A0, it follows from the standard
existence and uniqueness theorem forordinary differential equations
(Perko 2001) that system (1) admits a unique solution(S(t), E(t), I
(t), Q(t), R(t)).
To show the existence of the disease-free equilibrium (DFE), we
let Ei = Ii =Qi = Ri = 0, i = 1, 2, . . . , n in (1) to get
Ṡi = Bi (Si )Si −n
∑
j=1d Sji Si +
n∑
j=1d Si j S j − μSi Si , Si (0) > 0, i = 1, 2, . . . , n.
(2)
In order for system (2) to have a positive equilibrium, we
assume that
(A4) s(diag(Bi (0) − μSi ) + DS) > 0, where s(·) represents
the stability modulus ofan n ×n matrix and is defined by s(M) :=
max{Reλ, λ is an eigenvalue of M}.
Biologically,Assumption (A4) guarantees that the number of
susceptible populationis positive when there is no infective
individual.
Following similar arguments as those in Wang and Zhao (2004),
Zhao and Jing(1996), we have
123
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1242 X. Wang et al.
Lemma 1 Under the assumptions (A1)–(A4), system (2) admits a
unique positiveequilibrium (S0)T = (S01 , S02 , . . . , S0n ),
which is globally asymptotically stable forR
n+\{0}.Our preliminary results are the following two lemmas,
whose proofs are similar to
those given in Wang and Zhao (2004) and thus are omitted.
Lemma 2 Under the assumptions (A1)–(A4), system (1) admits a
unique DFE P0 =(S0, 0, . . . , 0) with (S0)T = (S01 , S01 , . . . ,
S0n ) ∈ Rn+ and 0T = (0, . . . , 0) ∈ Rn+.
Lemma 3 If (A1)–(A4) hold, then system (1) is dissipative, that
is, there exists anN∗ > 0 such that every forward orbit (S(t),
E(t), I (t), Q(t), R(t)) eventually entersthe set G := {(S(t),
E(t), I (t), Q(t), R(t)) ∈ R5n+ :
∑ni=1(Si (t) + Ei (t) + Ii (t) +
Qi (t) + Ri (t)) ≤ N∗} and G is positively invariant with
respect to system (1).
3 Model Analysis
3.1 The Basic Reproduction Number
In this section, we identify the basic reproduction number, �0
for system (1).Following the procedure introduced in van den
Driessche and Watmough (2002),�0 = ρ(FV −1), where ρ represents the
spectral radius of a matrix. Here F = F3n×3nand V = V3n×3n
represent new infections and transition terms, respectively. They
aregiven by
F3n×3n =⎛
⎝
0 F12 00 0 00 0 0
⎞
⎠ , V3n×3n =⎛
⎝
V11 0 0V21 V22 00 V32 V33
⎞
⎠
with
F12 =
⎛
⎜
⎜
⎜
⎝
β1 0 . . . 00 β2 . . . 0...
.... . .
...
0 0 . . . βn
⎞
⎟
⎟
⎟
⎠
V11 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
α1 + μE1 +n∑
j=1d Ej1 − d E12 . . . − d E1n
− d E21 α2 + μE2 +n∑
j=1d Ej2 . . . − d E2n
......
. . ....
− d En1 − d En2 . . . αn + μEn +n∑
j=1d Ejn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
123
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An Epidemic Patchy Model with Entry–Exit Screening 1243
V21 =
⎛
⎜
⎜
⎜
⎝
−α1 0 . . . 00 −α2 . . . 0...
.... . .
...
0 0 . . . −αn
⎞
⎟
⎟
⎟
⎠
V22 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
γ I1 + μI1 +n∑
j=1d Ij1 − δe12δo12d I12 . . . −δe1nδo1nd I1n
−δe21δo21d I21 γ I2 + μI2 +n∑
j=1d Ij2 . . . −δe2nδo2nd I2n
......
. . ....
−δen1δon1d In1 − δen2δon2d In2 . . . γ In + μIn +n∑
j=1d Ijn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
V32 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−n∑
j=1θoj1d
Ij1 − θe12δo12d I12 . . . −θe1nδo1nd I1n
−θe21δo21d I21 −n∑
j=1θoj2d
Ij2 . . . −θe2nδo2nd I2n
......
. . ....
−θen1δon1d In1 − θen2δon2d In2 . . . −n∑
j=1θojnd
Ijn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
and
V33 =
⎛
⎜
⎜
⎜
⎜
⎝
γQ1 + μQ1 0 . . . 00 γ Q2 + μQ2 . . . 0...
.... . .
...
0 0 . . . γ Qn + μQn
⎞
⎟
⎟
⎟
⎟
⎠
Both V −111 and V−122 are nonsingular M-matrices, which shows
their inverses are
nonnegative. This allows us to simplify the basic reproduction
number as follows:
�0 = ρ(F3n×3n · V −13n×3n) = ρ(−F12V −122 V21V −111 ). (3)
Under some special circumstances, we have the following
result.
Proposition 1 If αi = α, γ Ii = γ I , μEi = μE , μIi = μI , d Ai
j = d for A = E, I ,θoi j = θei j = θ ∈ [0, 1], for ∀ i, j = 1, 2,
. . . , n with i �= j , then
limd→0+
�0 = maxi
αβi
(α + μE )(γ I + μI ) . (4)
123
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1244 X. Wang et al.
Proof Note that in this special case,
V11 =
⎛
⎜
⎜
⎜
⎝
α + μE + (n − 1)d − d . . . − d− d α + μE + (n − 1)d . . . −
d...
.... . .
...
− d − d . . . α + μE + (n − 1)d
⎞
⎟
⎟
⎟
⎠
and
V22 =
⎛
⎜
⎜
⎜
⎝
γ I + μI + (n − 1)d − (1 − θ)2d . . . − (1 − θ)2d− (1 − θ)2d γ I
+ μI + (n − 1) . . . − (1 − θ)2d...
.... . .
...
− (1 − θ)2d − (1 − θ)2d . . . γ I + μI + (n − 1)d
⎞
⎟
⎟
⎟
⎠
Thus, limd→0 V −111 = 1α+μE Id and limd→0 V −122 = 1γ I +μI Id,
where Id denotes then×n identitymatrix.Therefore, using
(3),weobtain limd→0+ �0=max
i
αβi(α+μE )(γ+μI ) .
�
3.2 Global Stability of the DFE
It follows from van den Driessche and Watmough (2002, Theorem 2)
that the DFE islocally asymptotical stable if �0 < 1 and is
unstable if �0 > 1.Theorem 1 Assume that (A1)–(A4) are
satisfied. If �0 < 1, then the solution(S(t), E(t), I (t), Q(t),
R(t)) of (1) satisfies
limt→∞(S(t), E(t), I (t), Q(t), R(t)) = P
0.
Proof By Lemma 3, there exists a T ′ > 0 and when t > T
′:⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Ėi ≤ βi Ii − αi Ei − μEi Ei −n∑
j=1d Eji Ei +
n∑
j=1d Ei j E j ,
İi = αi Ei − γ Ii Ii − μIi Ii −n∑
j=1d Iji Ii +
n∑
j=1δei jδ
oi j d
Ii j I j ,
Q̇i = −γ Qi Qi − μQi Qi +n∑
j=1θoji d
Iji Ii +
n∑
j=1θei jδ
oi j d
Ii j I j .
(5)
Set M1 = F − V and �0 < 1, it follows that s(M1) < 0. Let
vT = (v1, . . . , vn) bea positive eigenvector associated with
s(M1). Choose k > 0 small enough such thatkv ξ . Since kv is a
positive eigenvector associated with s(M1), the solution of
thecomparison system (5) is given by kves(M1)(t−T ′). According to
the comparison prin-ciple (Smith 1995, 2008), (E(t), I (t), Q(t)) ≤
kves(M1)(t−T ′), for t > T ′. Therefore,
123
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An Epidemic Patchy Model with Entry–Exit Screening 1245
we get (E(t), I (t), Q(t)) → (0, 0, 0) as t → ∞. Further using
the equations for Riof model (1), we can prove that limt→∞ R(t) =
0.
Next we prove that limt→∞ S(t) = S
0. Let Φ(t) : Rn+ → Rn+ be the solution semi-flow of (1), that
is, Φ(t)(S0, E0, I0, Q0, R0) = (S(t), E(t), I (t), Q(t), R(t))
withthe nonnegative initial value (S0, E0, I0, Q0, R0).
Given (S0, E0, I0, Q0, R0) ∈ G with Si0 �= 0, it easily follows
that S(t) ∈ Int(Rn+),∀t > 0. Let ω = ω(S0, E0, I0, Q0, R0) be
the omega limit set of Φ(t). Since(E(t), I (t), Q(t), R(t)) → (0,
0, 0, 0) as t → ∞, there holds ω = ω̃×{(0, 0, 0, 0)}.We claim that
there must be ω̃ �= {0}. Otherwise, if this is not true, i.e., ω̃ =
{0},then we must have lim
t→∞(S(t), E(t), I (t), Q(t), R(t)) = (0, 0, 0, 0, 0). By
Assump-tion (A4), we can choose a small η > 0, such that
s(diag(Bi (0) − μSi ) + DS −ηdiag(1)) > 0, where 1 = (1, . . . ,
1). It follows that there exists a t1 > 0such that Bi (Ni (t)) −
βi Ii (t)Ni (t) ≥ Bi (0+) − η for ∀t ≥ t1, i = 1, . . . , n.
ThenS(t)T = (S1(t), S2(t), . . . , Sn(t)) satisfies
Ṡi (t) > (Bi (0+) − ηi )Si (t) −n∑
j=1d Sji Si (t) +
n∑
j=1d Si j S j (t) − μSi Si (t). (6)
Let ωT = (ω1, . . . , ωn) be a positive eigenvector of the
matrix diag(Bi (0) − μSi ) +DS −ηdiag(1) associated with the
eigenvalue s(diag(Bi (0)−μSi )+ DS −ηdiag(1)).Choose a small number
α > 0 such that S(t1) > αω. Then the comparison
principle(Smith 1995, 2008) implies that
S(t) ≥ αωes(diag(Bi (0) − μSi ) + DS − ηdiag(1))(t − t1), ∀t ≥
t1
and then Si (t) → ∞, i = 1, 2 . . . n, a contradiction. It is
easy to see thatΦ1(t)|ω(S(t),0, 0, 0, 0) = (Φ1(t)S(t), 0, 0, 0, 0),
where Φ1(t) is the solution semi-flow of system(2). By Hirsch et
al. (2001, Lemma 2.10), ω is an internal chain transitive set
forΦ(t), and hence, ω1 is an internal chain transitive set for
Φ1(t). Since ω1 �= 0 andS0 is globally asymptotically stable for
(1) in Rn+\{0}, we have ω1 ∩ W s(S0) �= ∅.By (Hirsch et al. 2001,
Theorem 3.1 and Remark 4.6), we then get ω1 = S0, provingω = {P0}.
This proves Theorem 1. �
3.3 Disease Persistence and Existence of the Endemic
Equilibrium
It follows from van den Driessche and Watmough (2002, Theorem 2)
that the DFEP0 is unstable if �0 > 1. The following result shows
that �0 > 1 actually impliesthat model (1) admits at least one
endemic equilibrium (EE) and the disease persistsuniformly.
Define X = {(S, E, I, Q, R) : Si ≥ 0, Ei ≥ 0, Ii ≥ 0, Qi ≥ 0, Ri
≥ 0, i =1, 2, . . . , n}, X0 = {(S, E, I, Q, R) : Ei > 0, Ii
> 0, i = 1, 2, . . . , n}, ∂ X0 =X\X0.
123
-
1246 X. Wang et al.
Theorem 2 Suppose the assumptions (A1)–(A4) hold and the basic
reproductionnumber �0 > 1. Then system (1) is uniformly
persistent, that is, there exists a positiveconstant � > 0 such
that every solution A(t) of (1) with
A0 ∈ Rn+ × (Rn+\{0}) × (Rn+\{0}) × Rn+ × Rn+satisfies
limt→∞ inf(Ei (t), Ii (t)) ≥ (�, �), i = 1, . . . , n.
Moreover, there exists at least one EE.
Proof Let Φ(t) : X → X be the solution flow associated with
system (1), that is,Φ(t)(A0) = A(t).
It follows from Lemma 3 that X is positively invariant and
system (1) is pointdissipative. By the comparison principle for
cooperative systems (Smith 1995, 2008),it follows that (E, I ) �
(0, 0). This implies that X0 is positively invariant, thus ∂ X0is
relatively closed in X .
Set M∂ = {A0 ∈ X : A(t) ∈ ∂ X0, ∀t ≥ 0}. We claim that
M∂ = {(E(t), I (t)) = (0, 0), ∀t ≥ 0}.
Suppose on the contrary, i.e., there exists a t0 ≥ 0 such that
E(t0) > 0, or I (t0) > 0.Without loss of generality, we
assume that E(t0) > 0. We partition {1, 2, . . . , n} intotwo
sets, Z1 and Z2, such that
Ei (t0) = 0,∀i ∈ Z1, and Ei (t0) > 0,∀i ∈ Z2.
Next we claim that Ei (t) > 0 for t0 < t < t0 + �0, ∀i
= 1, · · · , n for �0 > 0small enough, which could be easily
verified provided that Z1 = ∅. Assume Z1 �= ∅,since Z2 �= ∅ as
E(t0) > 0. Then for ∀i ∈ Z1,
Ėi (t0) = βi Si (t0)Ii (t0)Ni (t0) − αi Ei (t0) −n∑
j=1d Eji Ei (t0) +
n∑
j=1d Ei j E j (t0) − μEi Ei (t0)
= βi Si (t0)Ii (t0)Ni (t0) +∑
j∈Z2d Ei j E j (t0).
According to the irreducibility of DE , there is always a chain
j1, j2, . . . , jn withj1 = j, jn = i , which ensures that ∑ j∈Z2 d
Ei j E j (t0) > 0 ⇒ Ėi (t0) > 0, and hence,there exists an
�0 > 0 such that Ei (t) > 0 for t0 < t < t0+�0.
Clearly, we can restrict�0 > 0 small enough such that Ei (t)
> 0, t0 < t < t0 + �0 for ∀i ∈ Z1∪Z2. Thisproves Ei (t)
> 0, i = 1, . . . , n, t0 < t < t0 + �0.
Using the equation of system (1) for Ii , we obtain that either
Ii (t0) > 0, or Ii (t0) = 0and İi (t0) > 0 holds true, for
both cases, we have Ii (t) > 0, t0 < t < t0 + �1, ∀i =1, ·
· · , n for �1 > 0 sufficiently small. This results in A(t) /∈ ∂
X0 for t0 < t < t0 +min{�0, �1}, which contradicts the
assumption of M∂ , this proves E(t) = 0, ∀t ≥ 0.
123
-
An Epidemic Patchy Model with Entry–Exit Screening 1247
Repeating the same way on I (t), we can show that I (t) = 0∀t ≥
0, and thus we verifythe above claim on M∂ .
Next we show that the solution A(t) through A0 ∈ X0
satisfies
lim supt→∞
max1≤i≤n{Ei (t), Ii (t)} > �2.
for some positive constant �2.Suppose this is not true, there
must exist a T1 > 0 such that
0 < max1≤i≤n{Ei (t), Ii (t)} ≤ �2, for all t ≥ T1,
then it follows from the equations of Qi , Ri , i = 1, . . . , n
of system (1) that ∃k0 > 0,such that (Qi (t), Ri (t)) <
(k0�2, k0�2). As the boundedness of S, we can restrictβi Si Ii
Ni≤ ξ(�2). Then we have
Ṡi = Bi (Ni )Ni − βi Si IiNi
−n
∑
j=1d Sji Si +
n∑
j=1d Si j S j − μSi Si
≥ Bi (Si + (2k0 + 2)�2)Si − (μSi + ξ(�2))Si −n
∑
j=1d ji Si +
n∑
j=1di j S j
Consider the following auxiliary equation
Ṡi = Bi (Si + (2k0 + 2)�2)Si − (μSi + ξ(�2))Si −n
∑
j=1d Sji Si +
n∑
j=1d Si j S j ,
(S1(T1), . . . , Sn(T1)) = (S1(T1), . . . , Sn(T1)), i = 1, . .
. , n . (7)
By the comparison principle Smith (1995), Smith (2008), we have
S(t) ≥S(t),∀t ≥ T1, here ST (t) = (S1(t), . . . , Sn(t)). By (7),
there exists a constant vec-tor S0(�2) > 0 such that (S0)T (�2)
is globally asymptotically stable for (7) and thatS0(0) = S0. Then
there exists a sufficiently small ηwith ηT = (η1, . . . , ηn) ∈ Rn+
anda sufficiently large T > 0, such that S0(�2) > S0−η, and
that S(t) ≥ S0(�2) ≥ S0−ηfor all t ≥ T + T1. As a consequence, SiNi
= 1 −
Ei +Ii +Qi +RiNi
≥ 1 − (2k0+2)�2S0i −ηi
,∀t ≥T + T1, hence we have
Ėi = βi Si IiNi
− αi Ei −n
∑
j=1d Eji Ei +
n∑
j=1d Ei j E j − μEi Ei
≥ βi Ii (1 − (2k0 + 2)�2S0i − ηi
) − (αi + μEi )Ei −n
∑
j=1d Eji Ei +
n∑
j=1d Ei j E j .
123
-
1248 X. Wang et al.
Consider the following auxiliary system:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙̃Ei = βi˜Ii (1 − (2k0+2)�2S0i −ηi ) − αi˜Ei − μEi ˜Ei −
n∑
j=1d Eji ˜Ei +
n∑
j=1d Ei j ˜E j ,
˙̃Ii = αi ˜Ei − γi˜Ii − μIi ˜Ii −n∑
j=1d Iji˜Ii +
n∑
j=1δei jδ
oi j d
Ii j
˜I j ,
˙̃Qi = −γi ˜Qi − μIi ˜Qi +n∑
j=1θoji d
Qji˜Ii +
n∑
j=1θei jδ
oi j d
Qi j
˜I j .
(8)
Since �0 > 1, we know that the matrix (1 − (2k0+2)�2S0i −ηi
)M1 has a positive eigenvalues((1− (2k0+2)�2
S0i −ηi)M1) with a positive eigenvector. Let ṽT = (v1, . . . ,
vn) be a positive
eigenvector associated with s((1− (2k0+2)�2S0i −ηi
)M1), then the solution of Eq. (8) is given
by k1ṽes((1− (2k0+2)�2
S0i −ηi)M1)
. Again, by the comparison principle (Smith 1995, 2008),
wehave
(E(t), I (t), Q(t)) ≥ k1ṽes((1− (2k0+2)�2
S0i −ηi)M1)t
, ∀t ≥ T .
Then (Ei (t), Ii (t), Qi (t)) → (∞,∞,∞), i = 1, 2, . . . , n,
for t → ∞, which leadsto a contradiction.
Note that (S0)T is globally asymptotically stable in Rn+\{0} for
system (2). By theafore-mentioned claim, it then follows that (0,
0, 0, 0, 0) and P0 are isolated invariantsets in X , W s((0, 0, 0,
0, 0)) ∩ X0 = ∅, and W s(P0) ∩ X0 = ∅. Clearly, every orbitin M∂
converges to either (0, 0, 0, 0, 0) or P0, and (0, 0, 0, 0, 0) and
P0 are acyclic inM∂ . By Hirsch et al. (2001, Theorem 4.3 and
Remark 4.3), we conclude that system(1) is uniformly persistent
with respect to (X0, ∂ X0). By Zhao (1995, Theorem 2.4),system (1)
has at least one equilibrium (S∗, E∗, I ∗, Q∗, R∗) ∈ X0, with E∗ �
0 andI ∗ � 0. We further show that S∗ ∈ Rn+\{0}. Suppose S∗ = 0, by
the sum of equationsof Ei in (1), we get 0 =
n∑
i=1(αi + μi )E∗i , and hence E∗ = 0, a contradiction. Thus
the proof is complete. 2. �
4 Case Study for the 2009 Influenza A (H1N1)
In this section, we use numerical simulations to explore the
impacts of various screen-ing strategies on the control of 2009
influenza A (H1N1) pandemic.
4.1 Sensitivity Analysis of �0 on Parameters for a Two-Patch
Model
We consider a special case of system (1) with n = 2 as
follows:
123
-
An Epidemic Patchy Model with Entry–Exit Screening 1249
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Ṡ1 = B1(N1)N1 − β1S1 I1N1
− μS1 S1 − d S21S1 + d S12S2,Ė1 = β1S1 I1
N1− α1E1 − μE1 E1 − d E21E1 + d E12E2,
İ1 = α1E1 − γ I1 I1 − μI1 I1 − d I21 I1 + δe12δo12d I12 I2,Q̇1
= −γ Q1 Q1 − μQ1 Q1 + θo21d I21 I1 + θe12δo12d I12 I2,Ṙ1 = γ I1 I1
+ γ Q1 Q1 − μR1 R1 − d R21R1 + d R12R2,Ṡ2 = B2(N2)N2 − β2S2 I2
N2− μS2 S2 − d S12S2 + d S21S1,
Ė2 = β2S2 I2N2
− α2E2 − μE2 E2 − d E12E2 + d E21E1,İ2 = α2E2 − γ I2 I2 − μI2
I2 − d I22 I2 + δe21δo21d I21 I1,Q̇2 = −γ Q2 Q2 − μI2Q2 + θo12d I12
I2 + θe21δo21d I21 I1,Ṙ2 = γ I2 I2 + γ Q2 Q2 − μR2 R2 − d R12R2 +
d R21R1.
(9)
The basic reproduction number of system (9) is given by
�0∣
∣
(9) = ρ(
−F̃12Ṽ −122 Ṽ21Ṽ −111)
(10)
where
F̃12 =⎛
⎝
β1 0
0 β2
⎞
⎠ ; Ṽ21 =⎛
⎝
γ I1 + μI1 + d I21 − δe12δo12d I12−δe21δo21d I21 γ I2 + μI2 + d
I12
⎞
⎠ ;
Ṽ21 =(−α1 0
0 −α2
)
; Ṽ11 =(
α1 + μE1 + d E21 − d E12− d E21 α2 + μE2 + d E12
)
.
Note that the basic reproduction number �0 defined in (10)
involves a group ofparameters. To identify to which parameters �0
is sensitive, we carry out a sensitivityanalysis by evaluating the
partial rank correlation coefficients (PRCCs) for all
inputparameters against the output variable �0 (Blower and
Dowlatabadi 1994; Wu et al.2013). We take parameter values as in
Table 1. Moreover, we denote β = β1 =β2, α = α1 = α2, μK = μK1 =
μK2 ; d K = d K12 = d K21, (K = S, I, E, R), and we setγ = 1/6.56 =
γ I1 = γ I2 , d I = 0.5× d S, d E = 0.9× d S , where d S = 0.135;
we alsoassume θe = 0.37 = θe21 = θe12, θo = θo21 = θo12 with θo =
0.3 × θe.
We show that the infection rateβ is themost sensitive parameter
of�0 of system (9).For other coefficients, Fig. 2 shows that the
six parameters with most impacts on �0are the rate of progression
to latent (α), the recovery rate (γ ), the entry screening
rate(θe), the exit screening rate (θo), the susceptible dispersal
rate (d S), and the infecteddispersal rate (d I ).
123
-
1250 X. Wang et al.
Table 1 The parameter estimates for the 2009 influenza A (H1N1)
pandemic
Parametersi, j = 1, 2 i �= j
Definitions Values References
βi The infection rate (days−1) in
patch i0.3936
Tang et al. (2012)
αi Rate of progression to latent(days−1) in patch i
0.55Tang et al. (2012)
γ Ii Recovery rate (days−1) in
patch i1/7.48–1/6
Tang et al. (2012),Xiao et al. (2015)
μSi The natural death rate(days−1) in patch i
3.805 × 10−5Wang and Wang(2012)
μIi , (μEi ) Infectious (Exposed) death
rate (days−1) in patch i5.307 × 10−5
Xu et al. (2011)
θei j Entry screening rate oftravelers from patch j to i
0.33–0.4556Cowling et al. (2010),Li et al. (2013)
θoi j Exit screening rate oftravelers from patch j to i
0.165–0.2228 Estimated
d Si j , (dRi j ) Dispersal rate (days
−1) of thesusceptible (recovered)from patch j to i
0.0135–0.135Tang et al. (2010)
d Ii j , (dEi j ) Dispersal rate (days
−1) of theinfectious (exposed) frompatch j to i
0.0122–0.122 Estimated
Fig. 2 Partial rank correlation coefficients illustrating the
dependence of �0 on each parameter
4.2 Impacts of Various Screening Strategies
For the model (1), if patch i is not connected to any other
patch, then the patch-specificbasic reproduction number is given
by
�0i = βiαi(αi + μEi )(γ Ii + μIi )
.
123
-
An Epidemic Patchy Model with Entry–Exit Screening 1251
If �0i < 1, then we call patch i a low-risk patch, while if
�0i > 1, then we call it ahigh-risk patch (Allen et al. 2007).
Therefore, the disease persists in isolated high-riskpatches and
dies out in isolated low-risk patches. During the 2009 H1N1
pandemic,Mexico could be regarded as a high-risk patch, and it is
believed that the pandemic wascaused by the global connection with
Mexico (Khan et al. 2013; Smith et al. 2009).
To examine the impacts of various screening strategies on the
control of diseases,we assume there are one high-risk patch,
labeled as patch 1, and 4 low-risk patches,labeled as patches 2, 3,
4, and 5, and the overall basic reproduction number�0 is largerthan
1.
For simulation purpose, we take parameter values as in Table 1:
μSi = μRi =3.805 × 10−5, μEi = μIi = 5.307 × 10−5, αi = 0.55 for i
= 1, . . . , 5, and β1 =0.3936 � βi = 0.15, γ I1 = 1/7.48 < γ Ii
= 1/6 for i = 2, . . . , 5. For this set ofparameter values, �01 =
2.772 � 1, �02 = �03 = �04 = �05 = 0.900 < 1.That is, patch 1 is
a high-risk patch, and the remaining 4 patches are low-risk
patches.Considering reduced dispersal rates for exposed and
infectious individuals, we taked Ei j = 0.9 × d Si j , d Ii j = 0.5
× d Si j , i �= j. Further we assume that patch 5 has
lesscommunication with patch 1. For simplicity, we assume d S15 = d
S51 = 0.1 × d S ,d S1 j = d Sj1 = d S for j = 2, 3, 4 and d Si j =
d Sji = d S for i, j = 2, 3, 4, 5 and i �= j .
The question we want to address is: what types of screening
strategies are capableof preventing the endemic disease? To answer
this question, we explore the outcomesof several possible screening
strategies.
We first consider Strategy I, “indiscriminate entry–exit
screening” strategy: regard-less of the risk level and dispersal
rates, same strength of entry and exit screeningsare implemented to
each patch. That is, θoi j = θei j = θ for i, j = 1, . . . , 5.
Forthe parameter values taken, we find numerically that this
strategy is the most effec-tive of the six strategies considered.
Moreover, the detection rate needs to be higheras the dispersal
rates decrease, and when the dispersal rate d S < dc1 =
0.090,the screening strategy cannot control the disease even when
the detection is perfect(see Fig. 3a).
Strategy II is called the “indiscriminate entry screening”
strategy: only entry screen-ing of each patch is implemented. That
is, θei j = θ for i, j = 1, ..., 5. By comparingFig. 3a with Fig.
3b, we find that Strategy II can control the disease if Strategy I
can,but Strategy II requires higher successful detection rate of
infectious travelers.
Strategy III in consideration is called the “targeted entry
screening”: only applyentry screening for travel from and to the
high-risk patch, patch 1. That is, θej1 =θe1 j = θ for j = 2, 3, 4,
5. As shown in Fig. 3c, the disease will die out if the
dispersalrate d S is higher than a critical value dc2 (in our
simulation, dc2 = 0.095).
In practice, a natural question to be asked is: do we need to
implement screeningin all patches? To answer this question, we
examine Strategy IV, “selective entryscreening”: only apply entry
screening to the high-risk patch and patches that arehighly
connected to the high-risk patch (i.e., those patches with larger
dispersal rates).In our simulation, we assume θej1 = θe1 j = θ, j =
2, 3, 4. Figure 3d indicates thatthis strategy can also eliminate
the disease if the dispersal rate d S is higher than thecritical
value dc3 = 0.097. This implies that there is no need to implement
screeningat all patches.
123
-
1252 X. Wang et al.
(a)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θoij = θeij = θ, i, j = 1, · · · , 5
R0
(b)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θeij = θ, i, j = 1, · · · , 5
R0
(c)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θe1j = θej1 = θ, j = 2, · · · , 5
R0
(d)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θe1j = θej1 = θ, j = 1, · · · , 4
R0
(e)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θej1 = θ, j = 2, · · · , 5
R0
(f)
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
θe1j = θ, j = 1, · · · , 5
R0
Fig. 3 Plots of �0 under screening Strategies I–VI. Parameters:
γ I1 = 1/7.48; γ I2 = γ I3 = γ I4 = γ4 =1/6, β1 = 0.3936;β2 = β3 =
β4 = β5 = 0.15.The black solid line: d S = 0.09; The pink solid
linewith ∗:d S = 0.095; the blue dotted line: d S = 0.097; the red
dot-and-dash line d S = 0.107; the green dashedline: d S = 0.135
(Color figure online)
It is worth mentioning Strategy V, “one-way entry screening”:
applying entryscreening at low-risk patches to individuals
traveling from the high-risk patch only.In our simulations, we take
θej1 = θ for j = 2, 3, 4, 5, and all other screening ratesare set
to be zero. Then for our chosen parameter values, the dispersal
rate d S must besufficiently large, larger than dc4 = 0.135 in our
simulations (See Fig. 3e). However,another “one-way entry
screening”, Strategy VI: applying entry screening at the high-risk
patch to travelers from all low-risk patches is capable of
eradicating the disease.
123
-
An Epidemic Patchy Model with Entry–Exit Screening 1253
In our simulation, we choose θe1 j = θ for j = 2, 3, 4, 5, and
we can lower �0 to beless than 1 provided that d S > dc5 = 0.107
(see Fig. 3f).
5 Summary and Discussion
In this paper, we have investigated a general multi-patch
SEIQRmodel with entry–exitscreenings. Our theoretical results,
Theorems 1 and 2, were established by appealingto the comparison
principle and uniform persistence principle. Our results show
thatthe disease dies out when �0 < 1 and persists when �0 >
1. Thus the basic repro-duction number is a vital index to measure
the level of the disease (Bauch and Rand2000; Clancy and Pearce
2013). Our Proposition 1 provides a theoretical confirmationfor the
simulation results obtained in Sattenspiel and Herring’s (2003):
When the dis-persal rates are very low, screening must be highly
effective to alter disease patternssignificantly.
We have also explored six different screening strategies to
examine how the screen-ing impacts the control of influenza A
(H1N1). Our numerical results show that itis crucial to screen
travelers from and to high-risk patches, and it is not necessaryto
implement screening in all connected patches, though the minimum
number ofpatches that should implement screening depends critically
on the dispersal rates andthe accuracy of screening process.
During the 2009 influenza A (H1N1) pandemic in mainland China,
besides theisolation of those detected infected individuals from
the border screening, the indi-viduals who had been exposed to
those who had been detected and isolated infectedwas traced and
received medical observations (Yu et al. 2012), it is interesting
to studythe combined effects of contact tracing and border
screening, which we leave as ourfuture work.
Acknowledgments The authors would like to extend their thanks to
Professor Mark Lewis and the threeanonymous referees for their
valuable comments and suggestions which led to a significant
improvementin this work. The authors thank Professor Sanyi Tang for
his very kind help in the sensitivity analysis andthank Professor
Philip Maini and Dr. Jinzhi Lei for their helpful advice on the
revision of this manuscript.The revision of this work was partially
completed during XW’s visit to Wolfson Centre for
MathematicalBiology, the University of Oxford, and she would like
to acknowledge the hospitality received there. S. L.is supported by
the National Natural Science Foundation of China (No. 11471089) and
the FundamentalResearch Funds for the Central Universities (Grant
No. HIT. IBRSEM. A. 201401) and LW is partiallysupported by NSERC
of Canada.
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http://dx.doi.org/10.1038/srep07838http://dx.doi.org/10.1136/bmj.c4779
An Epidemic Patchy Model with Entry--Exit ScreeningAbstract1
Introduction2 The n-Patch SEIQR Model with Entry--Exit Screenings3
Model Analysis3.1 The Basic Reproduction Number3.2 Global Stability
of the DFE3.3 Disease Persistence and Existence of the Endemic
Equilibrium
4 Case Study for the 2009 Influenza A (H1N1)4.1 Sensitivity
Analysis of 0 on Parameters for a Two-Patch Model4.2 Impacts of
Various Screening Strategies
5 Summary and DiscussionAcknowledgmentsReferences
