queuing theory and epidemic models
TRANSCRIPT
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8/12/2019 Queuing Theory And Epidemic Models
1/14
MATHEMATICAL BIOSCIENCES http://www.mbejournal.org/AND ENGINEERINGVolume X, Number 0X, XX 20XX pp. XXX
QUEUING THEORY AND EPIDEMIC MODELS
Carlos M. Hernandez-Suarez
Facultad de Ciencias, Universidad de Colima
Apdo. Postal 25, Colima, Colima, Mexico
Carlos Castillo-Chavez
Mathematical, Computational and Modeling Sciences CenterArizona State University, Tempe, AZ 85287-1904
Osval Montesinos Lopez
Facultad de Ciencias, Universidad de ColimaApdo. Postal 25, Colima, Colima, Mexico
Karla Hernandez-Cuevas
Facultad de Ciencias, Universidad de Colima
Apdo. Postal 25, Colima, Colima, Mexico
(Communicated by the associate editor name)
Abstract. In this work we consider every individual of a population to bea server whose state can be either busy (infected) or idle (susceptible). This
server approach allows to consider a general distribution for the duration of theinfectious state, instead of being restricted to exponential distributions. We
show that the single most important parameter in epidemiology, (R0) corre-sponds to the single most important parameter in queuing theory (the serverutilization, ). First we derive new approximations to quasistationary distribu-
tion (QSD) of SIS (Susceptible- Infected- Susceptible) and SEIS (Susceptible-Latent- Infected- Susceptible) stochastic epidemic models.
1. Introduction. There is relatively little work relating queuing theory with epi-demiology. Kendall[1] and [2]discuss the relationship betweenM/G/1 queues andbirth- death processes. Kitaev [3] works on the relation between birth and deathprocesses and the M/G/1 queues with processor sharing, similarly to [4] and [5].Ball[6] usesM/G/1 theory to find the total cost of the epidemic and most recentlyTrapman et al[7]use M/G/1 queues with processor sharing to model an SIR epi-demic with detections. We are not aware of work in which the epidemic process isconsidered an M/G/Nqueuing process implying that every individual is a serverthat can be busy (infected) or idle (susceptible). In this work we use this approachfor the SIR model, an we extend the results to an SEIR model.
In closed population stochastic epidemic models, the process reaches an absorbingstate once the population is free of infected individuals. Absorption into this statewill occur eventually, with the time to reach this state depending strongly on theinfection potential = /, where is the contact rate between individuals and
2000 Mathematics Subject Classification. Primary: 92B05; Secondary: 62J27.Key words and phrases. SIS; SEIS; Queuing theory; R0; basic reproductive number; stochastic
epidemic models.
1
MATHEMATICAL BIOSCIENCES
http://www.mbejournal.org/
AND ENGINEERING
Volume
X,
Number
OX,
XX ZOXX
pp.
X-XX
QUEUING
THEORY AND EPIDEMIC MODELS
CARLOS
M.
HERNANDEZ-SUAREZ
Facultad de
Ciencias,
Universidad
de
Colima
Apdo.
Postal
25,
Colima, Colima,
Mxico
CARLOS CASTILLO- CHAVEZ
Mathematical,
Computational
and
Modeling
Sciences Center
Arizona
State
University,
Tempe,
AZ
85287-1904
OSVAL
MoNTEs1Nos
LoPEZ
Facultad de
Ciencias,
Universidad
de
Colima
Apdo.
Postal
25,
Colima, Colima,
Mxico
KARLA HERNANDEZ-
CUEVAS
Facultad de
Ciencias,
Universidad
de
Colima
Apdo.
Postal
25, Colima, Colima,
Mxico
(Communicated
by
the associate editor
name)
ABSTRACT. In this work we consider
every
individual of a
population
to be
a server
whose
state can
be either
busy
(infected)
or
idle
(susceptible).
This
server
approach
allows to consider a
general
distribution
for
the duration
of
the
infectious
state,
instead of
being
restricted to
exponential
distributions.
We
show that the
single
most
important parameter
in
epidemiology,
(RQ)
corre-
sponds
to the
single
most
important
parameter
in
queuing theory
(the
server
utilization,
p).
First we derive new
approximations
to
quasistationary
distribu-
tion
(QSD)
of
SIS
(Susceptible-
Infected-
Susceptible)
and
SEIS
(Susceptible-
Latent- Infected-
Susceptible)
stochastic
epidemic
models.
1.
Introduction. There
is
relatively
little work
relating queuing theory
with
epi-
demiology.
Kendall
[1]
and
[2]
discuss the
relationship
between
M/
G
/
1
queues
and
birth- death
processes.
Kitaev
[3]
works
on
the relation between birth and death
processes
and the
M/ G/
1
queues
with
processor
sharing, similarly
to
[4]
and
Ball
[6]
uses
M/ G/
1
theory
to find the total cost of the
epidemic
and most
recently
Trapman
et al
[7]
use
M/ G/
1
queues
with
processor
sharing
to model an
SIR
epi-
demic with detections.
We
are
not
aware of
work
in
which the
epidemic process
is
considered an
M/
G
/N
queuing process implying
that
every
individual is a server
that
can
be
busy
(infected)
or
idle
(susceptible).
In this work
we use
this
approach
for the
SIR
model,
an we extend the results to an
SEIR
model.
In closed
population
stochastic
epidemic models,
the
process
reaches
an
absorbing
state once the
population
is
free of infected individuals.
Absorption
into
this state
will
occur
eventually,
with the
time
to reach this state
depending strongly
on
the
infection
potential
p
=
A/,a,
where A
is
the contact rate between individuals and
2000
Mathematics
Subject Classification.
Primary:
92B05;
Secondary:
62J27.
Key
words and
phrases. SIS; SEIS;
Queuing theory;
Rg;
basic
reproductive number;
stochastic
epidemic
models.
1
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8/12/2019 Queuing Theory And Epidemic Models
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2C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
the recovery rate, therefore, the process has a degenerate limiting distribution(t ) with all its weight at state 0.
The expected time to extinction generally increases with ; thus, an interestingproperty of these process is its behavior before going to absorption. Quasi-stationarydistributions provide information on the limiting distribution of the process condi-tioning on non-absorption (see [8]and[9]) that is, they allow to analyze the behaviorof the disease while it is in the endemic state. These distributions are difficult tofind, and instead, some approximations were suggested like the reflecting state 0or the one permanently infected (see [10]).
Although the current approximations to the quasi-stationary distribution of thenumber of infectives seem not to have a close known distribution, we show thata good approximation to the quasi-stationary distribution for the susceptible isPoisson distributed for a general distribution of the duration of the infectious state.Since the next natural step is to add a latent period that allows for the latency ofthe infection, we have applied this approach to approximate the quasi-stationary
distribution (bivariate) of an SEIS epidemic model. Stochastic epidemic modelsassume in general that the distribution of the infectious time is exponential. Inthis paper all approximations assume a general distribution of the duration of theinfectious period.
Here we derive the distribution of the approximation one permanently infectedfor SIS and SEIS models [10]. We use the term conditional endemic distributionfor this type of distributions in stochastic epidemic models. The paper is organizedas follows: sections 2 and 3 deal with SIS and SEIS models respectively. Section2.1 introduces the SIS model; Section 2.2 and 2.3 introduce the quasi-stationarydistribution for the SIS and its approximations. In Section 2.4 the case of a generaldistribution of the duration of the infectious state is analyzed using standard resultsfrom queuing theory. Numerical comparisons of these results via simulations arepresented in Section 2.5 and a miscellaneous result is given in Section 2.6. Section3.1 introduces the SEIS model. In Section 3.2 it is shown that the quasi-stationarydistribution of the infected individuals (latent + infective) can be approximatedwith that of an SIS model with appropriate parameters. In this section we alsoderive an approximation to the joint quasi-stationary distribution of latent andinfective, and numerical comparisons for the SEIS are presented in Section 3.3.
2. The SIS Model.
2.1. Introduction. The susceptible-infected-susceptible (SIS) stochastic epidemicmodel attempts to reproduce the behavior of epidemics running through a popula-tion with no vital dynamics; that is, no births and deaths occur. In this model it isassumed that no individuals are removed from circulation either by immunization
or isolation. The deterministic version of the SIS model was introduced in [11]and has been fully analyzed since then. Its stochastic counterpart, also called thestochastic logistic epidemic model (see[12] and [13]) was introduced early in [14],and has been applied similarly to study the transmission of rumors (see [15]). How-ever, most of the relevant results concerning this model have been in the epidemicscontext.
In the SIS model, susceptible individuals may become infected by contact withinfective individuals, and hence, it is assumed that there is no incubation periodfor the disease; that is, infected individuals become infectious immediately. After
2C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
,LL
the
recovery rate, therefore,
the
process
has a
degenerate limiting
distribution
(t
->
oo)
with all its
weight
at state
0.
The
expected
time
to
extinction
generally
increases
with
p;
thus,
an
interesting
property
of these
process
is its behavior before
going
to
absorption. Quasi-stationary
distributions
provide
information on
the
limiting
distribution
of
the
process
condi-
tioning
on
non-absorption
(see[8]
and
that
is,
they
allow to
analyze
the behavior
of
the disease while
it is in
the endemic state. These distributions
are diilicult
to
find,
and
instead,
some
approximations
were
suggested
like the
e ec ng
state
0
or
the one
permanently
n ec ed
(see[10]).
Although
the current
approximations
to the
quasi-stationary
distribution of the
number of infectives seem
not to
have a close known
distribution,
we show that
a
good approximation
to the
quasi-stationary
distribution for the
susceptible
is
Poisson distr ibuted
for a
general
distribution
of
the durat ion
of
the
infectious
state.
Since
the next natural
step
is
to add a latent
period
that allows for the
latency
of
the
infection,
we have
applied
this
approach
to
approximate
the
quasi-stationary
distribution
(bivariate)
of
an
SEIS
epidemic
model.
Stochastic
epidemic
models
assume in
general
that the distribution
of
the
infectious time is
exponential.
In
this
paper
all
approximations
assume a
general
distribution of the duration of the
infectious
period.
Here
we
derive the distribution
of
the
approximation
one
permanently
n ec ed
for
SIS
and
SEIS
models
[10].
We
use the term cond ona endemic d s bu on
for this
type
of distributions
in
stochastic
epidemic
models. The
paper
is
organized
as follows: sections 2 and
3
deal with
SIS
and
SEIS
models
respectively.
Section
2.1
introduces the
SIS
model;
Section
2.2
and
2.3
introduce the
quasi-stationary
distribution for the
SIS
and its
approximations.
In
Section
2.4 the case of a
general
distribution
of
the durat ion
of
the
infectious
state
is
analyzed
using
standard results
from
queuing theory.
Numerical
comparisons
of these results
via
simulations are
presented
in
Section
2.5
and
a
miscellaneous result
is
given
in
Section
2.6.
Section
3.1
introduces the
SEIS
model. In
Section 3.2
it is shown that the
quasi-stationary
distribution
of
the
infected
individuals
(latent
+
infective)
can
be
approximated
with that of an
SIS
model with
appropriate parameters.
In this
section
we also
derive an
approximation
to
the
joint quasi-stationary
distribution of latent and
infective,
and numerical
comparisons
for the
SEIS
are
presented
in
Section 3.3.
2. The SIS Model.
2.1.
Introduction. The
susceptible-infected-susceptible
(SIS)
stochastic
epidemic
model
attempts
to
reproduce
the behavior of
epidemics running through
a
popula-
tion with no vital
dynamics;
that
is,
no births and deaths occur. In this model it is
assumed that
no
individuals
are
removed from circulation either
by
immunization
or
isolation. The deterministic
version
of
the
SIS
model
was
introduced
in
[11]
and has been
fully analyzed
since
then. Its stochastic
counterpart,
also called the
stochastic
logistic epidemic
model
(see[12]
and
[13])
was introduced
early
in
[14],
and has been
applied similarly
to
study
the
transmission of rumors
(see[15]).
I-Iow-
ever,
most of the relevant results
concerning
this model have been in the
epidemics
context.
In the
SIS
model, susceptible
individuals
may
become infected
by
contact with
infective
individuals,
and
hence,
it is
assumed that there
is no
incubation
period
for the
disease;
that
is,
infected individuals become infectious
immediately.
After
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QUEUING THEORY AND EPIDEMIC MODELS 3
some time, they become healthy and susceptible again. The process of contagion isassumed to be driven by the homogeneous mixing of individuals in the population.
We let I(t) and S(t) be the number of infective and susceptible respectively attimet, and note that since the population is closed, the state of the process at timetcan be fully described by eitherI(t) orS(t). It is customary to follow I(t).
I(t) takes values on = {0, 1, 2, . . . , N }, N being the population size. TheSIS stochastic epidemic model is a discrete space, continuous time Markov Chain.In particular, it is a unidimensional continuous time birth- death process. Upondefining
Pj,k(, t + ) = P(I(t + ) = k | I(t) = j), j, k ,
the transition probabilities are
Pk,k+1(t, t + ) = k(N k)/N+ o()
Pk,k1(t, t + ) = k+ o(). (1)
From the expressions above, one can see that the mass action law plays an importantrole and results from the assumption of homogeneous mixing.
One way to dissect the process given by (1) is by specifying the following tworules:
i) Every individual comes into contact with another at random intervals whichare independent and identically distributed random variables. The distribution ofthese intervals is exponential with parameter . If a contact involves a susceptibleindividual and an infectious one, the probability of infection is .
ii) The duration of the infectious state is an exponential random variable withparameter.
Fromi), allk infective individuals come into close contact with others accordingto a Poisson process with parameter k. Since every one of these contacts could be
with a susceptible individual with probability (N k)/N, by thinning the Poissonprocess, we conclude that for fixed k, contacts between infective and susceptibleindividuals that will end up in a infection of the susceptible occur according to aPoisson process with parameter k(N k)/N. Letting = yields the firstequation of (1). The second equation follows directly from ii).
2.2. The Quasi-stationary distribution. We are interested in the future of theSIS epidemic, which depends strongly on the ratio =/, called the transmissionfactor, basic reproduction ratio, basic reproduction number or infection potential.The deterministic model has a threshold at = 1, and results in an endemic infectionof size N(1 1/) if this value is greater than 1. In the stochastic model, since{0}, the disease free state is an absorbing state that can be reached with positiveprobability, the process will end up in this state for any value of given a sufficient
amount of time, for any initial number of infected as long as N is finite. Thecertainty of extinction imposes a problem if our interest is in the long-time behaviorof the epidemic, conditioning in the process not being in the epidemic state.
Let
j = limt
Xj(t)
t ,
Xj(t) being the time spent in state j up to time t, j . Thus, j denotes theproportion of time spent in state j as t . is called the stationary distributionof the process. Ifpj(t) denotes the probability that the process is in state j at time
QUEUING
THEORY AND EPIDEMIC MODELS
3
some
time, they
become
healthy
and
susceptible again.
The
process
of
contagion
is
assumed to be driven
by
the
homogeneous
mixing
of individuals in the
population.
We
let I
(t)
and
S
(t)
be the number
of infective
and
susceptible respectively
at
time
t,
and note that since the
population
is
closed,
the state of the
process
at time
t
can
be
fully
described
by
either I
(t)
or
S
It
is
customary
to
follow
I
I(t)
takes values on Q
=
{0,1,2,
_ _ _
,N},
N
being
the
population
size. The
SIS
stochastic
epidemic
model
is a
discrete
space,
continuous time
Markov
Chain.
In
particular,
it is a unidimensional continuous time birth- death
process. Upon
defining
1%,k(,f+5)
=
P(I(f+5)
=
if
I IO?)=j),
Lk
6
Q,
the
transition
probabilities
are
Pk,k+1(t,t
+
5)
=
)6k(N
_
15)N
+
5(5)
Pk,kL1(t,t
+
5)
=
M515
5(5) (1)
From the
expressions
above,
one can see that the mass action law
plays
an
important
role and results from the
assumption
of
homogeneous mixing.
One
way
to dissect the
process given
by
(1)
is
by
specifying
the
following
two
rules:
Every
individual comes into contact with another at random intervals which
are
independent
and
identically
distributed random variables. The distribution
of
these intervals is
exponential
with
parameter
,3_
If a contact involves a
susceptible
individual and
an infectious
one,
the
probability
of infection is 0.
The duration of the infectious state is an
exponential
random variable with
parameter
,u_
From
i),
all kr infective individuals come
into
close contact with others
according
to
a
Poisson
process
with
parameter B
k.
Since
every
one of
these contacts could be
with a
susceptible
individual with
probability
(N
-
lc)/N
,
by thinning
the Poisson
process,
we
conclude that
for fixed
k,
contacts between
infective
and
susceptible
individuals that will end
up
in
a infection of the
susceptible
occur
according
to a
Poisson
process
with
parameter
,30k(N
-
lc)/N
_
Letting
A
=
,BH
yields
the first
equation
of
The second
equation
follows
directly
from
2.2.
The
Quasi-stationary
distribution.
We
are
interested
in
the
future of
the
SIS
epidemic,
which
depends strongly
on the
ratio
p
=
A//5,
called the
transmission
factor,
basic
reproduction
ratio,
basic
reproduction
number or infection
potential.
The deterministic model has
a
threshold at
p
=
1,
and results
in an
endemic
infection
of size N
(1
-
1/p)
if this value is
greater
than 1. In the stochastic
model,
since
{0},
the disease
free
state
is an
absorbing
state that
can
be reached with
positive
probability,
the
process
will end
up
in this state for
any
value of
p
given
a sufiicient
amount
of
time,
for
any
initial number
of infected
as
long
as
N
is
finite.
The
certainty
of extinction
imposes
a
problem
if our interest is in the
long-time
behavior
of
the
epidemic, conditioning
in
the
process
not
being
in
the
epidemic
state.
Let
X- t
Tfj
=
L,
>oo
If
Xj
(t)
being
the time
spent
in state
j up
to time
t,
j
6
Q.
Thus,
Trj
denotes the
proportion
of time
spent
in
state
j
as
t
-> oo.
II
is
called the
stationary
distribution
of the
process.
If
pj
(t)
denotes the
probability
that the
process
is in state
j
at time
-
8/12/2019 Queuing Theory And Epidemic Models
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4C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
tthen it is possible to give an alternative representation for j
j = limt
pj(t).
The stationary distribution for the SIS epidemic model is degenerate with all itsmass in state {0}, which is true for any finite value of. However, when 1it is reasonable to assume that the disease will be in an endemic state for a while,and any information on the behavior of the process previous to extinction would beuseful in the understanding of the epidemic. This interest led to the developmentof the concept of quasi-stationary distributions.
Quasi- stationary distributions (QSD) are limiting distributions conditioning onthe process not being in an absorbing state. If we let
Q= {q1, q2,...,qN}
denote the quasi-stationary distribution of the SIS epidemics, then
qj = limtP(I(t) = j | I(t) = 0}
= limt
pj(t)
1 p0(t).
Observe that when 1,Q is a conditional endemic state distribution. No simpleexpression exists to calculate Q, although Nasell (see [16]) proposed a numericalalgorithm for its calculation. Most of the relevant work regarding the calculationof analytical expressions for the QSD of the SIS model is based on approximationmethods. Two of these approximations were suggested by Kryscio and Lefevre (see[10]) and analyzed in detail in [16], see also [13]. The common characteristic ofthese approximations is that the process is modified in such a way that it lacks theabsorbing state {0} and thus the possibility of degenerate distributions is avoided.In one approximation the number of infected in the population is at least one, and
it is called the SIS model with one permanently infected individual. In this processevery recovery rate j = j is replaced by (j 1), while the infection rates areunchanged. In the second approximation, the only rate that is changed is 1, whichis replaced by zero. This latter approximation is referred to as the SIS model withthe origin removed or reflecting state approximation model (see [17]).
Andersson and Britton [18] studied the SIR and SEIR epidemic models withdemography and obtained expressions for the time to extinction starting from theQSD. In their work, the duration of the infectious state was not restricted to bean exponential distribution but instead a gamma distribution. Ovaskainen [16]improved previous approximations for the QSD and the time to extinction for twocases: when N and when the basic reproduction ratio R0 . Nasell[19]gave approximations for the QSD and the time to extinction for the stochasticversion of the Verhulst epidemic model, depending onR0being greater, smaller or in
the transition region. For the first two regions, the approximations yielded normaland geometric distributions respectively. Nasell [20]derived approximations to theQSD for SI, SIS SIR and SEIR epidemic models with demography forN large. Forthe regionRo > 1,Nasell concluded that the normal distribution yielded reasonableapproximations to the QSD.
A different approach to obtain the reflecting state approximation model wassuggested in [21] to obtain the time to extinction starting from a general state k.Stefanov [22] and Ball [23] derived higher moments for the time to extinction forthe SIS starting from a general state k, as well as other useful functionals.
4C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
t then
it is
possible
to
give
an alternative
representation
for
Trj
Wd
=
gf1goPj(1f)-
The
stationary
distribution for the
SIS
epidemic
model is
degenerate
with all its
mass in
state
[0],
which
is
true
for
any
finite
value
of
p.
However,
when
p
>
1
it is reasonable to assume that the disease will be in an endemic state for a
while,
and
any
information on
the behavior
of
the
process
previous
to
extinction
would be
useful in the
understanding
of the
epidemic.
This interest led to the
development
of
the
concept
of
quasi-stationary
distributions.
Quasi- stationary
distributions
(QSD)
are
limiting
distributions
conditioning
on
the
process
not
being
in an
absorbing
state.
If we
let
Q
:
lqla
q2> a
denote the
quasi-stationary
distribution of the
SIS
epidemics,
then
qi
=
,gH;,P
1, Q
is
a conditional endemic state distribution. No
simple
expression
exists
to calculate
Q,
although
Nasell
(see[16])
proposed
a
numerical
algorithm
for
its
calculation. Most of the relevant work
regarding
the calculation
of
analytical
expressions
for the
QSD
of the
SIS
model is based on
approximation
methods. Two
of
these
approximations
were
suggested by Kryscio
and
Lefevre
(see
[10])
and
analyzed
in detail in
[16],
see also
[13].
The common characteristic of
these
approximations
is
that the
process
is
modified
in
such a
way
that
it
lacks the
absorbing
state
[0]
and thus the
possibility
of
degenerate
distributions is avoided.
In
one
approximation
the number
of infected in
the
population
is
at least
one,
and
it is
called the
SIS
model with
one
permanently
infected
individual. In this
process
every recovery
rate
,aj
=
/I
j
is
replaced by
(j
-
1)/1,
while the
infection
rates
are
unchanged.
In the second
approximation,
the
only
rate that is
changed
is
pl,
which
is
replaced by
zero.
This latter
approximation
is referred
to
as
the
SIS
model with
the
origin
removed or
reflecting
state
approximation
model
(see[17]).
Andersson
and Britton
[18]
studied the
SIR
and
SEIR
epidemic
models with
demography
and obtained
expressions
for the
time
to
extinction
starting
from the
QSD.
In their
work,
the durat ion of the infectious
state
was
not
restricted
to
be
an
exponential
distribution but instead a
gamma
distribution.
Ovaskainen
[16]
improved
previous approximations
for
the
QSD
and the
time
to
extinction for
two
cases: when N -> oo and when the basic
reproduction
ratio
R0
-> oo. Nasell
[19]
gave approximations
for the
QSD
and the time
to
extinction for the stochastic
version
of the
Verhulst
epidemic model, depending
on
R0
being greater,
smaller
or in
the
transition
region.
For the
first
two
regions,
the
approximations
yielded
normal
and
geometric
distributions
respectively.
Nasell
[20]
derived
approximations
to the
QSD
for
SI,
SIS SIR
and
SEIR
epidemic
models with
demography
for N
large.
For
the
region
R0
>
1,
Nasell
concluded that the normal distribution
yielded
reasonable
approximations
to the
QSD.
A different
approach
to obtain the
reflecting
state
approximation
model was
suggested
in
[21]
to obtain the time to extinction
starting
from a
general
state k.
Stefanov
[22]
and Ball
[23]
derived
higher
moments
for
the
time
to
extinction for
the
SIS
starting
from a
general
state
k,
as well as other useful functionals.
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QUEUING THEORY AND EPIDEMIC MODELS 5
In the notation we use hereq(1)j denotes the approximation toqj when using one
permanently infected individual andq(0)j denotes the reflecting state approximation
model.For the SIS, it has been shown in [10] that when 1 andN ,
Nj =1 qj =N
j=1 q(0)j . Also, in[16]it is proved that for >1, the distribution of the number
of infectives under both the reflecting state 0 approximation and the one with onepermanently infected individual are approximately normal with mean N(1 1/)and variance N/ when N and is fixed.
2.3. The approximation to the quasi-stationary distribution of the num-ber of infectives. In the following calculations we derive an approximation for theQSD of the susceptible individuals. We establish a new result: that the distribu-tion of the QSD of the number of susceptible is closely approximated by a Poissontruncated at zero when N is large and N/ tends to a constant.
Let Q = {q1, q2,...,qN} be the stationary distribution of the approximation tothe QSD when there is one permanently infected individual. Here j = (j 1)andj =j(N j)/N, j = 1, 2, . . . , N . We use the recurrence relation
q(0)n =123...n1
234...nq
(0)1 , n 2
that we obtain when considering one permanently infected individual for an SISmodel with j = (j 1) and j =j(N j)/N, j = 1, 2, . . . , N . Hence
q(0)k =q
(0)1
(k1)
(k 1)!
k1j=1
j (2)
with = / yields,
q(0)k =q
(0)1
(/N)k1
(N k)!
(N 1)!.
Definepk = q(0)Nk as the QSD approximation to the number of susceptible. Thus
pk = q(0)1
(/N)Nk1
k! (N 1)!
since
q(0)1 =
(N 1)!(/N)N1
Nj=1
(/N)j/j!
1
we have
pk = (N/)k
k!N
j=1(N/)j/j!
,
whose limit when N and N/ tends to a constant, is
pk = k
k!(eN/ 1), (3)
a truncated Poisson random variable with parameter N 1. Nasell[16] has alreadyestablished that the approximations to the QSD for the infective individuals yieldeda normal distribution with mean N(1 1) and varianceN/ for N and constant. From (3) we can see that when N/ moderately large, usingNk = pk,the approximation to the QSD for the susceptible is a Poisson random variable,which in turn can be approximated with a normal distribution with mean and
QUEUING
THEORY AND EPIDEMIC MODELS
5
_
1
_ _ _
In the
notat1on
we use here
q;
)
denotes the
approx1mat1on
to
qj
when
us1ng
one
permanently
infected individual and
q0)
enotes the
reflecting
state
approximation
model.
For the
SIS,
it has been shown in
[10]
that when
p
>
1 and N ->
oo,
E]V:1
j
=
Ejil
q0).
lso,
in
[16]
it is
proved
that for
p
>
1,
the distribution of the number
of infectives under both the
reflecting
state
0
approximation
and the one with one
permanently
infected
individual
are
approximately
normal with
mean
N
(1
-
1
/ p)
and variance
N/
p
when N -> oo and
p
is fixed.
2.3.
The
approximation
to the
quasi-stationary
distribution of the num-
ber of infectives. In the
following
calculations we derive an
approximation
for the
QSD
of
the
susceptible
individuals.
We
establish
a new
result: that the distribu-
tion of the
QSD
of the number of
susceptible
is
closely approximated by
a Poisson
truncated at
zero
when N
is
large
and
N/
p
tends to
a
constant.
Let
Q
=
{q1,
qg,
...,qN}
be the
stationary
distribution of the
approximation
to
the
QSD
when there
is one
permanently
infected
individual. Here
,aj
=
M(
-
1)
and
Aj
=
)j(N
-
j)/N,
j
=
1,2,
_ _
.,N.
We
use the recurrence relation
qgbo)
A1A2A3...An_1
q())
n
2
2
M2M3M4---Mn
that we obtain when
considering
one
permanently
infected individual for an
SIS
model with
,aj
=
,u(j
-
1)
and
Aj
=
)j(N-j)/N,
j
=
1,2,...,N.
Hence
) U
k
_q1(k_1)!_1
J:
with
p
=
A//I
yields,
(0) _ (0)
(P/N)kI1
_
qk
_
ql
(N_ ky
(N 1)!.
Define
pk
=
qglk
as the
QSD approximation
to the number of
susceptible.
Thus
N
N-lc-1
Pk
=
ql) (p/
,
(N
_
U!
filo)
(N-
1)!(/0/N)N11:(/0/NW/j!
we have
k
_
(N/P)
pk
_
N
.
_
1
kl
Ej=1(N/PV]/J!
whose limit when N -> oo and
N/
p
tends to a
constant,
is
Tk
pk:
a
truncated Poisson random variable with
parameter
N
p 1
asell
[16]
has
already
established that the
approximations
to the
QSD
for the infective individuals
yielded
a normal distribution with mean N
(1
-
p11)
and
variance
N/
p
for N
->
oo and
p
constant. From
(3)
we can see that when N
/
p
moderately large,
using
7rN,k
=
pk,
the
approximation
to the
QSD
for
the
susceptible
is a
Poisson random
variable,
which in turn can be
approximated
with a normal distribution with mean and
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6C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
Figure 1. The gamma distributions used for the duration of theinfectious state. and are the shape and scale parameters re-spectively. All distributions have the same expected value: 2.
variance N/; hence, that of the infected is normal with mean N(1 1) andvariance N/.
2.4. The approximation to the quasi-stationary distribution of the num-ber of susceptible for non- exponential duration of the illness state: an
application of queuing theory. An SIS epidemic process can also be seen asa queuing process with state dependent arrival rate. We can think ofN servers,where a busy server corresponds to an infected individual. Individuals are servedat a rate, and the arrival rate isj (Nj)/Nwhen the number of busy servers is
j. We will use this analogy to derive an approximation to the QSD of the numberof susceptible individuals .The constant hazard rate characteristic of the exponential distribution makes it dif-
ficult to adapt for most diseases. The idea of assuming that the probability that aperson will be cured in the nexts units of time given that she/he has been infectedduring a timet is independent oft is not very realistic. In this section we derive anapproximation to the QSD of SIS models when the illness state is non-exponential.From queuing theory, (see [24]) the limiting distribution of the number of busyservers in a system M/G/N, with state dependent arrival rate is given by
k = 1(E[S])
k1
k!
k1j=0
j , (4)
where E[S] is the expected value of the service time. Thus, (2) is is a particularcase of (4) with the proviso that state 0 is never visited. Hence, the approximation
to the QSD for the number of busy servers (infectives) depends on the durationof the illness state only through its first moment. The general expression for theapproximation to the QSD of the number of susceptible becomes
pk = N/(E[S])k
k!
eN/(E[S]) 1 . (5)
Observe that, forN(E[S])1 moderately large,pk can be approximated by a Pois-son distribution with parameter N(E[S])1. We refer to the next section fornumerical evaluations.
6C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
/
5
/
H
/
f
.
1
=
_ 1= Q
4
`
I
D
T
FIGURE
1.
The
gamma
distributions used
for
the durat ion
of
the
infectious state. cz and
B
are the
shape
and scale
parameters
re-
spectively.
All distributions have the same
expected
value: 2.
variance
N/
p;
hence,
that
of
the
infected is
normal with
mean
N
(1
-
pil)
and
variance
N/
p.
2.4. The
approximation
to
the
quasi-stationary
distribution
of
the
num-
ber of
susceptible
for non-
exponential
duration of the illness state: an
application
of
queuing theory.
An
SIS
epidemic
process
can also be seen as
a
queuing
process
with state
dependent
arrival rate.
We
can
think of N
servers,
where
a
busy
server
corresponds
to
an infected
individual. Individuals
are
served
at a rate
M,
and the arrival rate
is
Aj
(N
-
j) /N
when the number of
busy
servers
is
j.
We
will use this
analogy
to derive an
approximation
to the
QSD
of the number
of
susceptible
individuals .
The constant hazard rate characteristic of the
exponential
distribution makes it dif-
ficult
to
adapt
for
most
diseases. The idea
of
assuming
that the
probability
that
a
person
will be cured in the next s units of time
given
that
she/
he has been infected
during
a time
t
is
independent
of
t
is
not
very
realistic. In this
section we
derive
an
approximation
to the
QSD
of
SIS
models when the illness state is
non-exponential
From
queuing theory,
(see [24])
the
limiting
distribution of the number of
busy
servers in a
system
M/
G
/N
,
with state
dependent
arrival rate
is
given
by
Trk
=7r1 liii}j,
(4)
where E
[S]
is
the
expected
value
of
the
service time.
Thus,
(2)
is is a
particular
case of
(4)
with the
proviso
that state
0
is never visited.
Hence,
the
approximation
to
the
QSD
for
the number
of
busy
servers
(infectives)
depends
on
the duration
of the illness state
only through
its
first moment. The
general expression
for the
approximation
to
the
QSD
of the number of
susceptible
becomes
N/ E S
pk
I
ky
(@N/oElS1>1)'
(5)
Observe
that,
for N
()E[S])T1
moderately large,
pk
can be
approximated by
a Pois-
son
distribution with
parameter
N
(AE
[S])t1.
We
refer
to the next
section for
numerical evaluat ions.
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QUEUING THEORY AND EPIDEMIC MODELS 7
Figure 2. Simulations of an SIS stochastic model (histogram) andits Poisson approximation (solid line) with mean N/. Here N =50, = 4, = 10, = 20, that is, = 2. Time = 2000 units. The
simulations for = 1, = 2 and = 2, = 4, yielded the sameplots as expected since they all had the same .
2.5. Simulations. In this section, we simulate the approximation one perma-nently infected to the SIS epidemic model with N= 50 and >1. We assume agamma distribution for the duration of the infectious state with three sets of pa-rameters. Only the distribution of susceptibles is plotted against the approximated(5). Figure 1 shows the gamma distributions used for the duration of the illnessstate for a shape parameter and scale parameter . The goal was to analyze theeffect of different distributions for the duration of the infectious state keeping theaverage time of the infectious state (and ) constant. As expected from (5), thethree gamma distributions yielded the same distribution and only one is plotted.
Figure 2 shows the time spent in every state in the SIS epidemic model with a per-manent infected individual (histogram) and the Poisson approximation (solid line).
The simulations were performed in MATLAB. These were implemented as fol-lows: for each parameter set, a stochastic SIS epidemic was simulated during 2000units of time, and the time spent in each state during this interval was recorded. Ifthe epidemic went to extinction before reaching 2000 units of time, it was discarded,although due to the used this would not happen very often. The distribution ofthe time spent in each state is plotted against the Poisson approximation given by(5). It can be seen in Figure 2 that the approximation is good in every case exceptin the right tails, which is due to the time at which the simulations stopped, andwill tend to be more precise when the simulation time is increased. As expectedfrom (5), the approximation to the QSD is insensitive to higher moments of the
duration of the infectious state.
2.6. The expected time elapsed between two infections of a particularindividual. Queuing theory is useful to perform calculations of some quantitiesin epidemic processes. It is known that for epidemic process in equilibrium theproportion of the populationi on a given state i is proportional to the mean timei that a typical individual spends in that state [25]. This means that
i = i/L, (6)
QUEUING
THEORY AND EPIDEMIC MODELS
7
f
/
llll
lllll
FIGURE
2.
Simulations
of
an
SIS
stochastic model
(histogram)
and
its Poisson
approximation
(solid line)
with mean
N/
p.
Here N
=
50,
A
=
4,cu
=
10,5
=
20,
that
is,
p
=
2.
Time
=
2000
units.
The
simulations
for
cz
=
1,
B
=
2
and
cz
=
2,
B
=
4, yielded
the
same
plots
as
expected
since
they
all had the
same
p.
2.5.
Simulations. In this
section,
we simulate the
approximation
one
perma-
nently
n ec ed
to the
SIS
epidemic
model with N
=
50
and
p
>
1.
We
assume a
gamma
distribution for the durat ion of the infectious state with three sets of
pa-
rameters.
Only
the distribution of
susceptibles
is
plotted
against
the
approximated
(5).
Figure
1
shows the
gamma
distributions used for the duration of the illness
state
for a
shape parameter
cz
and scale
parameter
B.
The
goal
was
to
analyze
the
effect of different distributions for the duration of the infectious state
keeping
the
average
time of the infectious
state
(and p)
constant.
As
expected
from
(5),
the
three
gamma
distributions
yielded
the
same
distribution and
only
one is
plotted.
Figure
2
shows the
time
spent
in
every
state
in
the
SIS
epidemic
model with
a
per-
manent
infected
individual
(histogram)
and the Poisson
approximation
(solid line)
The simulat ions
were
performed
in MATLAB.
These
were
implemented
as fol-
lows: for each
parameter set,
a stochastic
SIS
epidemic
was simulated
during
2000
units of
time,
and the
time
spent
in
each state
during
this interval
was
recorded.
If
the
epidemic
went to
extinction
before
reaching
2000
units
of
time,
it
was
discarded,
although
due to the
p
used this would not
happen very
often.
The distribution
of
the time
spent
in each state is
plotted against
the Poisson
approximation given by
(5).
It
can
be
seen in
Figure
2
that the
approximation
is
good
in
every
case
except
in
the
right tails,
which
is
due to the
time
at which the simulations
stopped,
and
will tend to be
more
precise
when the simulation
time is
increased.
As
expected
from
(5),
the
approximation
to the
QSD
is insensitive to
higher
moments of the
duration
of
the
infectious
state.
2.6.
The
expected
time
elapsed
between two infections of a
particular
individual.
Queuing
theory
is useful
to
perform
calculations
of some
quantities
in
epidemic processes.
It
is
known that for
epidemic process
in
equilibrium
the
proportion
of the
population
Tfi
on a
given
state
i is
proportional
to
the mean time
'ri
that a
typical
individual
spends
in
that state
[25].
This means that
Tfi
=
'ri/L, (6)
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8C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
where Lis the mean life time of an individual. Now, suppose that we have a situationin which there are Nservers (one for each member of the population), an arrival
ratef() and a service rate, and that we are interested in the long-run proportionof busy servers. If service means being infected, theniin (6) is
1. Define a cycleas the passage through the susceptible and infected states, and let L be the averagetime of a cycle, thus, individuals complete a cycle at a rate N L1 which impliesthat they become infected at a rate N L1 which we define as f(). Therefore,L1 =f()/Nand (6) can be rewritten as:
i= 1f()/N
or
N i= 1f().
In other words
Average number of busy servers = mean service time arrival rate
which is the Littles law, perhaps the simplest and best known of equalities inQueuing Theory. This equality comes in turn from the fact that when in equilibriainfections (arrivals) and recoveries (services) must occur at the same rate. Thus, atequilibrium the probability that a particular individual (server) is infected (busy)at a particular time is alsoi. In the endemic equilibria, the proportion of infectedis precisely the probability that an individual is found infected, thus i = 11 . Since in equilibrium the rate at which a particular servers becomes busyis i(1 i), that is (1 /), then the inter arrival time between infectionsof a particular individual is given by (1 1). Observe that is the serverutilization in queuing systems. The server utilization is the long run proportionof time a server is busy. Here is R0,the basic reproductive number, that is theexpected number of secondary cases of infection originated by an infected individualin a population of susceptible. This lead us to an interesting finding: the singlemost important parameter in epidemiology, (R0) corresponds to the single mostimportant parameter in queuing theory (the server utilization, ).
3. The SEIS Model.
3.1. Introduction. A natural generalization of the SIS model is to consider ad-ditional epidemiological states. Here we consider the possibility that an infectiveindividual undergoes a latent state before becoming infectious. Thus, individualswho become infected are not able to transmit the disease for a period of time. Sinceinfected individuals in the latent state can not transmit the disease or acquire ad-ditional infection, they play no role in the transmission of the epidemic but ratherserve as buffers or reservoirs of infection. This model is referred as the SEIS modelor SLIS model .
Quasi-stationary distributions for SEIS models are two-dimensional arrays, m,n,where m,n denotes the limiting proportion of time in which there were m latentand n infective individuals conditioning in the process not being in the absorbingstate. The first effort to analyze the joint QSD of an epidemic model is reported in[17]with a stochastic version of the Ross malaria model.
It is possible to define a QSD for the total number of infective individuals kwhere k = m+n. In this section it is shown how the QSD for the total infectedin a population in an SEIS model can be derived from that of an SIS model. Fromthis last result the two-dimensional QSD follows directly.
SC. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
where L
is
the mean life
time
of an individual.
Now, suppose
that we have a
situation
in which there are N servers
(one
for each member of the
population),
an arrival
rate
and
a
service
rate
,li,
and that
we are
interested
in
the
long-run
proportion
of
busy
servers. If service means
being infected,
then
'ri
in
(6)
is
,lf1.
Define a
cycle
as
the
passage through
the
susceptible
and
infected
states,
and let L be the
average
time of a
cycle, thus,
individuals
complete
a
cycle
at a rate N L11 which
implies
that
they
become
infected
at
a
rate NL which
we define as
Therefore,
L 1
=
and
(6)
can be rewritten as:
Wi
=
lflfw)/N
or
NTU
=
Milf(/3)
In other words
Average
number of
busy
servers
=
mean service time >< arrival rate
which
is
the L e s
aw perhaps
the
simplest
and best known
of
equalities
in
Queuing
Theory.
This
equality
comes in turn from the fact that when in
equilibria
infections
(arrivals)
and
recoveries
(services)
must
occur
at the
same
rate.
Thus,
at
equilibrium
the
probability
that a
particular
individual
(server)
is
infected
(busy)
at
a
particular
time is
also
Tfi.
In the endemic
equilibria,
the
proportion
of infected
is
precisely
the
probability
that an individual is found
infected,
thus
rr,
=
1
-
pil
_
Since
in
equilibrium
the rate at which
a
particular
servers
becomes
busy
is
)rri(1
-
Tri),
that
is
/i(1
-
,li/A),
then the
inter
arrival
time
between infections
of a
particular
individual
is
given
by
,li(1
-
p 1
Observe
that
p
is
the
server
utilization in
queuing
systems.
The server utilization is the
long
run
proportion
of time a server is
busy.
Here
p
is
R0,the
bas c
reproductive numbe
that
is
the
expected
number of
secondary
cases of infection
originated by
an infected individual
in a
population
of
susceptible.
This lead us
to
an
interesting finding:
the
single
most
important parameter
in
epidemiology,
(RO)
corresponds
to the
single
most
important
parameter
in
queuing
theory
(the
server
utilization,
p).
3.
The SEIS Model.
3.1.
Introduction.
A
natural
generalization
of
the
SIS
model
is
to consider ad-
ditional
epidemiological
states. Here we consider the
possibility
that an infective
individual
undergoes
a latent
state
before
becoming
infectious.
Thus,
individuals
who become infected are not able to transmit the disease for a
period
of time.
Since
infected
individuals
in
the latent state
can
not
transmit
the disease
or
acquire
ad-
ditional
infection, they play
no role
in
the
transmission
of the
epidemic
but rather
serve as buffers or reservoirs of infection. This model is referred as the
SEIS
model
or
SLIS
model _
Quasi-stationary
distributions
for
SEIS
models
are
two-dimensional
arrays,
rrmm,
where
frm,"
denotes the
limiting proportion
of
time in
which there were m latent
and n infective individuals
conditioning
in the
process
not
being
in the
absorbing
state. The
first effort
to
analyze
the
joint
QSD
of an
epidemic
model
is
reported
in
[17]
with a stochastic version of the
Ross
malaria model.
It
is
possible
to define a
QSD
for the total number of infective individuals
rrk
where k
=
m
+
n. In this section it is shown how the
QSD
for the total infected
in a
population
in an
SEIS
model
can
be derived
from
that
of an
SIS
model. From
this last result the two-dimensional
QSD
follows
directly.
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QUEUING THEORY AND EPIDEMIC MODELS 9
The state space of the process can be stated in terms of the number of latentand infective individuals, (E,I) as:
= {(e, i); e + i N; e, i Z+}
The following diagram shows the transitions between the different epidemiologicalstates:
SSI/N E
1E I
2I S
3.2. The quasi-stationary distribution of the SEIS model. Define:
Pi,j;k,m(t, t + ) = P(E(t + ) = k, I(t + ) = m | E(t) = i, I(t) = j)
where E(t) and I(t) are random variables that denote the number of latent andinfective individuals at time t, respectively. Clearly {k, m}, {i, j} . The instan-taneous transition probabilities are
Pk,m;k+1,m(t, t + ) = m(N k m)/N+ o(),
Pk,m;k1,m+1(t, t + ) = 1k + o(),Pk,m;k1,m1(t, t + ) = 2m + o(),
while all other events are assumed to occur with probability o(). In section 3.2.1an approximation to the QSD of the total number of infected people (latent +infective) is derived. In section 3.2.2 the joint QSD distribution of the latent andinfective is analyzed.
3.2.1. The quasi-stationary distribution of the total number of infected in the SEISmodel. We introduce the random variable I(t) = E(t) + I(t) which denotes thetotal number of infected individuals at timet. Define also:
Pj,k(t, t + ) = limt
P(I(t + ) = k | I(t) = j), j, k , (7)
and call pk
thek-th element of the QSD of this process. To derive the approxima-tion to the quasi-stationary distribution, it suffices to start from the instantaneoustransition probabilities whent in (7), and considering thatj in this expressionis k, k 1 or k+ 1.
Now choose an arbitrary but fixed time t. From renewal theory (see Ref. [26],pp. 80-86) as t , the probability that an individual is in the infective stategiven that he is infected (latent or infective) is
I=12 (
11 +
12 )
1 (8)
and conditioning in exactly k infected, the number of infective individuals followsa Binomial distribution with parameters k andI, that is
limt
P(I(t) = j | I(t) = k) =
k
j
jI(1 I)
kj . (9)
Using these results,the instantaneous transition probabilities are
limt
P(I(t + ) = k+ 1 | I(t) = k) =
limt
kj=0
P(infection in(t, t + )| E(t) = k j, I(t) = j)P(E(t) = k j, I(t) = j)
=k
j=0
(j(N k)/N)
k
j
jI(1 I)
kj + o()
QUEUING
THEORY AND EPIDEMIC MODELS
9
The state
space
of the
process
can be stated
in
terms of the number of latent
and infective
individuals,
(E,I)
as:
Q={(e,i);
e-'f-iN;
e,i6Z+}
The
following diagram
shows the transitions between the different
epidemiological
states:
ASI N
E I
Sl
E Vi
S
3.2.
The
quasi-stationary
distribution
of
the SEIS model.
Define:
Pi,j;k,m(t,t
+
6)
=
P(E(t
+
6)
=
k,I(t
+
6)
=
m
| E(t)
=
i,I(t)
=
j)
where
E(t)
and I
(t)
are random variables that denote the number of latent and
infective individuals at
time
t, respectively. Clearly
{k, m}, {i,
j
}
6
Q. The
instan-
taneous
transition
probabilities
are
Pk,m;k+1,m(t,
+
6)
=
)5m(N
-
kr
-
m)/N
+
0(5),
Pk m k_1 m+1
f
-lr
6)
=
/1,1619
lr
0(6),
Pk,m;k1,m-1(ta
`l`
:
//$26777167771
l`
0(6)>
while all other events
are
assumed to
occur
with
probability
0(5).
In
section
3.2.1
an
approximation
to the
QSD
of the total number of infected
people
(latent
-l-
infective)
is
derived. In
section
3.2.2
the
joint QSD
distribution
of
the latent and
infective
is
analyzed.
3.2.1.
The
quasi-stationary
distribution
of
the total number
of infected
in
the
SEIS
model.
We
introduce the random variable I
*
(t)
=
E(t)
+
I
(t)
which denotes the
total number
of infected
individuals at
time
t.
Define
also:
P,t,.
oo,
the
probability
that an individual is in the infective state
given
that he
is infected
(latent
or
infective)
is
91
=
uf(/H1
+
#FYI
(8)
and
conditioning
in
exactly
kr
infected,
the number
of infective
individuals
follows
a Binomial distribution with
parameters
kr and
01,
that
is
gn;oP
=
du
-
eff.
Using
these
results,the
instantaneous transition
probabilities
are
tlim
P(I*(t+ 5)
=
k
-+-
1|I*(t)= k)
=
4)OO
lc
tlim
ZP(infection
n(t,
+
5)| Eu)
=
k
-
j,
I(t)
=
j)P(E()
=
k
-
j,
I(t)
=
j)
4)OO
1-0
=
i:
(A6j(N
_
k)/N)
9(1
@,)'H
+
0(5)
j:
-
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10C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
=kI(N k)/N+ o(). (10)
Similarly,
limt
P(I(t + ) = k 1 | I(t) = k) =
limt
kj=0
P(recovery in(t, t + )| E(t) = k j, I(t) = j)P(E(t) = k j, I(t) = j)
=k
j=0
j
k
j
jI(1 I)
kj + o()
=kI. (11)
Equations (10) and (11) are formally equivalent to those given in (1) with I
and2 I replacing and respectively. Therefore, an approximation to the QSDof the total number of susceptible individuals in an SEIS model can be obtaineddirectly from that of an SIS model (Sec. 2.3), which is given by a truncated Poissondistribution with parameter N 2/:
pk = (N 2/)
k
k!
eN2/ 1 (12)
In Section 2.4 we found that the QSD for the number of infective individuals(busy servers) in an SIS epidemic model depends on the duration of the infectiousperiod (service time) only through the first moment. Since an SEIS epidemic modelcan be seen as a queuing system with two phases on each server, it is natural toexpect the same behavior in this case. Note that the infection rate is I and E(S),the mean duration in the system is 11 +
12 .
According to (5), the parameter of the Poisson distribution for the number ofsusceptible becomes
N/(IE[S]) =N/(I[11 +
12 ]).
Using (8), the mean number of infected becomes N 2/, as expected. Thus, thejoint QSD has its mean at
{E, I} = {N 2(1 I)/,N2I/}.
It is important to observe the resulting lack of dependence of the above result onhigher moments of the duration of the latent and infectious period.
3.2.2. The marginal joint quasi-stationary distribution of the number of latent and
infective individuals in the SEIS model. The QSD joint distribution for the num-ber of latent and infective individuals can be derived from expression ( 12). WhenN 2/ is large the number of susceptible is given approximately by a Poisson dis-tribution with parameter N 2/. Under the assumption thatN 2/ is large wederive an approximation to pm,n , the joint QSD for the number of latent and in-fective individuals respectively.
The joint distribution is defined as
pm,n = limt
P{E(t) = m, I(t) = n},
101. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
=
A6k0I(N
-
k)/N
-'r
0(5). (10)
Similarly,
lim
P(I*(t-'f-5)
=
lc-
1
| I*(t)
=
k)
=
k
t&mo;:P(recovery
n(t,t-'r 5)| E(t)
=
lc
-j,
I(t)
=
j)P(E(t)
=
k
-j,
I(t)
=
j)
=
fjl
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QUEUING THEORY AND EPIDEMIC MODELS 11
which can be rewritten as
pm,n = limt
P{E(t) = m | E(t) + I(t) = m + n}P{E(t) + I(t) = m + n}
= limt
P{E(t) = m | I(t) = m + n}P{I(t) = m + n}
= limt
P{E(t) = m | I(t) = m + n}pNmn
with pNmn given in (12). using (9) we conclude that
pm,n= p
Nmn
m + n
m
(1 I)
mnI. (13)
The computation of the marginal QSD of the number of latent and infective isstraightforward. Definethe marginal QSDs for the number of latent and infectiveby
m = limt
P(E(t) = m)
and n= P(E(t) = n),
thus
m =
Nk=m
P(E(t) = m | I(t) = k)P(I(t) = k)
= pNk
Nk=m
k
m
(1 I)
mkmI
and using a similar argument
n= p
Nk
Nk=m
k
n
(1 I)
knnI
These results are evaluated through simulations in the next section.
3.3. Simulations. In this section, we simulate several SEIS epidemic models withdifferent values of , 1 and 2, with N = 100, allowing for one permanentlyinfected individual. We use an exponential distribution for the duration of thelatent and infectious states. Both observed and theoretical distributions are plottedfor a) the number of susceptible and b) the joint (infected, latent) distribution. Acontour diagram proves to be very useful for comparing the theoretical and observed
joint distributions. For every set of parameters, a simulation of an SEIS epidemicmodel was run for 5000 units of time, and the proportion of time spent in each statewas recorded. Again, if the epidemic went to extinction before reaching 5000 unitsof time, it was discarded. The approximation to the QSD for the total numberof susceptible is shown in a histogram with its Poisson approximation. An insert
shows the joint distribution of latent and infected using a contour diagram togetherwith the approximation (13).
The approximation to the QSD for the total number of susceptible and for thejoint distribution in Figures 3-5 seems very good. In Figure 3 the population sizeis N= 100,with = 4, 1 = 1/2 and 2 = 1. Figure 4 was constructed using thesame population size but with = 6, 1 = 1 and2 = 1/2, that is, the relationshipbetween the duration of the latent and infective states is reversed compared to theprevious case. Since the duration of the infected state has been increased as well asthe infection rate, the joint distribution has moved towards a higher proportion of
QUEUING
THEORY AND EPIDEMIC MODELS
11
which can be
rewritten
as
pm,"
=
tgr1oP{E(t)
m
| E(t)
+
I(t)
=
m
+
n}P{E(t)
-'r
I(t)
=
m
+
n}
=
t&r1oP{E(t)=m|I*(t)=m-'f-n}P{I*(t)=m+n}
-
lim
P{E(t)
-
m
| I*(t)
-
m
+
n}p*
f
_ _
N
with
pfvhmnn
given
in
(12).
using
(9)
we conclude that
pm,"
=pN.,...()
The
computation
of the
marginal QSD
of the number of latent and infective is
straightforward.
Define
the
marginal QSD s
or
the number
of
latent and
infective
by
am
=
t&r1oP(E(t)
m)
and
Bn
=
P(E(f)
=
H),
thus
am
=
Z
P(E(t)
=
m
| I*(t)
=
k)P(I*(t)
=
k)
k:m
_
=|=
if:
)m9k-m
PN-lc
m
I
I
krm
and
using
a similar
argument
=|=
N
k
k-n n
5
pN-k
Z
n
(1
_
91)
91
krm
These resul ts are evaluated
through
simulations in the next section.
3.3.
Simulations. In this
section,
we simulate several
SEIS
epidemic
models with
different
values
of
A,
,ul
and
pg,
with N
=
100,
allowing
for one
permanently
infected individual.
We
use an
exponential
distribution for the duration of the
latent and
infectious
states. Both observed and theoretical distributions
are
plotted
for
a)
the number of
susceptible
and
b)
the
joint
(infected, latent)
distribution. A
contour
diagram
proves
to
be
very
useful for
comparing
the theoretical and observed
joint
distributions. For
every
set of
parameters,
a simulation of an
SEIS
epidemic
model
was run for
5000
units of
time,
and the
proportion
of time
spent
in
each state
was recorded.
Again,
if the
epidemic
went to
extinction
before
reaching
5000
units
of
time,
it was discarded. The
approximation
to
the
QSD
for the total number
of
susceptible
is
shown
in a
histogram
with
its
Poisson
approximation.
An
insert
shows the
joint
distribution
of
latent and
infected
using
a
contour
diagram together
with the
approximation
(13).
The
approximation
to the
QSD
for the total number of
susceptible
and for the
joint
distribution
in
Figures
3-5
seems
very good.
In
Figure
3
the
population
size
is N
=
100,with
A
=
4,
/11
=
1/2
and
/12
=
1.
Figure
4 was constructed
using
the
same
population
size
but with A
=
6,
,ul
=
1
and
,ug
=
1/2,
that
is,
the
relationship
between the duration of the latent and infective states is reversed
compared
to the
previous
case.
Since
the durat ion
of
the
infected
state has been increased
as
well
as
the infection rate
A,
the
joint
distribution has moved towards a
higher proportion
of
-
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12C. HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND K. HERNANDEZ
Figure 3. Simulations of a stochastic SEIS epidemic model (his-togram) and its Poisson approximation with mean N/ (solid line).For this N = 100, = 4, 1 = 1/2, 2 = 1, = 4.Time = 5000
units. Insert shows the contour diagram of the approximation tothe joint QSD (solid line) and its approximation (dotted line).
Figure 4. Simulations of a stochastic SEIS epidemic model (his-togram) and its Poisson approximation with mean N/ (solid line).For this N= 100, = 6, 1 = 1, 2 = 1/2, = 12.Time = 5000units. Insert shows the contour diagram of the approximation tothe joint QSD (solid line) and its approximation (dotted line).
infected individuals than in the previous case. Figure 5 usesN= 100, = 6, 1 = 3and 2 = 1/2, that is, the duration of the latent state has been reduced comparedto the previous case, which increases the number of infective.
4. Conclusions. Here we derived the exact distribution of the approximation one
permanently infected individual to the QSD of the number of susceptible in an SISmodel for N keeping N/ constant. This is a Poisson random variable withparameter N/. It was shown that this holds for a general distribution for theduration of the latent state with finite mean. Since Poisson distributions with alarge mean can be approximated with a normal distribution, we can approximatethe QSD of the number of susceptible with a normal distribution, and from herethe approximation for the QSD of the number of infective is also normal.
Regarding the SEIS epidemic model, suitable good approximations were derivedfor both the total number of infected and the joint distribution of the latent and
121 HERNANDEZ AND C. CASTILLO-CHAVEZ AND O. MONTESINOS AND
K.
HERNANDEZ
FIGURE
3. Simulations
of a stochastic
SEIS
epidemic
model his-
(
togram)
and
its
Poisson
approximation
with
mean
N/
p
(solid line).
For this N
=
100,
A
=
4,/11
=
1/2,/12
=
1,p
=
4.'I`ime
=
5000
units.
Insert shows the
contour
diagram
of
the
approximation
to
the
joint QSD
(solid line)
and
its
approximation
(dotted line).
I
I
I
/
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QUEUING THEORY AND EPIDEMIC MODELS 13
Figure 5. Simulations of a stochastic SEIS epidemic model (his-togram) and its Poisson approximation with mean N/ (solid line).For this N= 100, = 6, 1 = 3, 2 = 1/2, = 12.Time = 5000
units. Insert shows the contour diagram of the approximation tothe joint QSD (solid line) and its approximation (dotted line).
infected. It was shown that the approximation to the QSD of the SEIS epidemicmodel with infection rate and mean time 11 and
12 respectively for the latent
and infected state is the same than that of an SIS epidemic model with infectionrate and mean recovery rate 12 . The simulations show that the approximationis accurate in both SIS and SEIS epidemic models, in agreement with (12)for thelatent plus infected and with (13) for the joint distribution. Since these are limitdistributions, it is important to run the simulations for a long time in order to obtainan appropriate sample from the targeted distributions. This can be easily seen inFigures 4 and 5, that differ only in the parameter 1, but are almost identical. It is
a matter of further study to analyze if epidemic models of the form S E1 E2 E3 ... Ek I Sbehave similarly, that is, if they only depend on the infectionand recovery rate.
REFERENCES
[1] D. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society.Series B (Methodological) (1951) 151185.
[2] D. Kendall, Stochastic processes occurring in the theory of queues and their analysis by themethod of the imbedded markov chain, The Annals of mathematical statistics 24 (3) (1953)338354.
[3] M. Kitaev, The m/g/1 processor-sharing model: transient behavior, Queueing Systems 14 (3)(1993) 239273.
[4] H. Andersson, T. Britton, Stochastic epidemic models and their statistical analysis, SpringerVerlag, 2000.
[5] T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, J. Appl. Probab.20 (2) (1983) 390394.
[6] F. Ball, P. Donnelly, Strong approximations for epidemic models, Stochastic processes and
their applications 55 (1) (1995) 121.[7] P. Trapman, M. Bootsma, A useful relationship between epidemiology and queueing theory:
The distribution of the number of infectives at the moment of the first detection, MathematicalBiosciences 219 (1) (2009) 1522.
[8] J. N. Darroch, E. Seneta, On quasi-stationary distributions in absorbing continuous-time
finite Markov chains, J. Appl. Probability 4 (1967) 192196.[9] J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied
Probability 10 (3) (1978) 570586.
QUEUING
THEORY AND EPIDEMIC MODELS
13
f
,
.
{,Qf}Q2;f;>,
~
l
I I
l
f
]
FIGURE
5. Simulations
of a stochastic
SEIS
epidemic
model
(his-
togram)
and
its
Poisson
approximation
with
mean
N/
p
(solid line).
For this N
=
100,
A
=
5,/L1
=
3,
/L2
=
1/2,p
=
12.Time
=
5000
units.
Insert shows the
contour
diagram
of
the
approximation
to
the
joint QSD
(solid line)
and
its
approximation
(dotted line).
infected.
It
was
shown that the
approximation
to the
QSD
of
the
SEIS
epidemic
model with infection rate A and mean
time
,ufl
and
/L51
espectively
for the latent
and infected
state
is the same than that of an
SIS
epidemic
model with infection
rate A and mean
recovery
rate
/L51.
The simulations show that the
approximation
is accurate in both
SIS
and
SEIS
epidemic models,
in
agreement
with
(12)
for the
latent
plus
infected and with
(13)
for the
joint
distribution.
Since
these are limit
distributions,
it is
important
to run the simulations for a
long
time in order to obtain
an
appropriate
sample
from
the
targeted
distributions. This
can
be
easily
seen in
Figures
4 and
5,
that differ
only
in the
parameter
pl,
but are almost identical. It is
a
matter
of further
study
to
analyze
if
epidemic
models
of
the
form
S
-
E1
-
E2
-
E3
-
-
Ek
-
I
-
S
behave
similarly,
that
is,
if
they only depend
on the infection
and
recovery
rate.
REFERENCES
[1]
D.
Kendall,
Some
problems
in the
theory
of
queues,
Journal
of the
Royal
Statistical
Society.
Series
B
(Methodological) (1951)
151~185.
[2]
D.
Kendall,
Stochastic
processes
occurring
in the
theory
of
queues
and their
analysis by
the
method of the imbedded markov
chain,
The Annals of mathematical statistics 24
(3) (1953)
338-354.
[3]
M.
Kitaev,
The
m/ g/
1
processor-sharing
model: t ransient
behavior,
Queueing Systems
14
(3)
(1993)
239-273.
[4]
H.
Andersson,
T.
Britton,
Stochastic
epidemic
models and their statistical
analysis,
Springer
Verlag,
2000.
[5]
T.
Sellke,
On
the
asymptotic
distribution
of
the size
of
a
stochastic
epidemic,
J.
Appl.
Probab.
20
(2) (1983)
390-394.
[6]
F.
Ball,
P.
Donnelly,
Strong
approximations
for
epidemic models,
Stochastic
processes
and
their
applications
55
(1) (1995)
1-21.
[7]
P.
Trapman,
M.
Bootsma,
A useful
relationship
between
epidemiology
and
queueing theory:
The distr ibution
of
the number
of infectives at
the
moment of
the
first
detection,
Mathematical
Biosciences
219
(1) (2009)
15-22.
[8]
J.
N.
Darroch,
E.
Seneta,
On
quasi-stationary
distributions in
absorbing
continuous-time
finite
Markov
chains,
J.
Appl.
Probability
4
(1967)
192-196.
[9]
J.
Cavender,
Quasi-stationary
distributions of birth-and-death
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