queuing theory and epidemic models

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


2/14

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


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


3/14

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


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


4/14


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.


K.

HERNANDEZ

t then

it is

possible

to

give

an alternative

representation

for

Trj

Wd

=

gf1goPj(1f)-

The

stationary


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.


5/14


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


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


6/14


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.


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.


7/14


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


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)


8/14


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.


9/14


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


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:


10/14


=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


11/14


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


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


12/14


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

/


13/14


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


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

processes,

Advances in

Applied

Probability

10

(3) (1978)

570-586.


14/14


[10] R. J. Kryscio, C. Lefevre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl.

Probab. 26 (4) (1989) 685694.[11] W. Kermack, A. McKendrick, Contributions to the mathematical theory of epidemicsiii.

further studies of the problem of endemicity, Bulletin of mathematical biology 53 (1) (1991)

89118.[12] R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population

model, Adv. in Appl. Probab. 14 (4) (1982) 687708. doi:10.2307/1427019.[13] I. Nasell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci.

156 (1-2) (1999) 2140, epidemiology, cellular automata, and evolution (Sofia, 1997). doi:10.1016/S0025-5564(98)10059-7 .

[14] G. Weiss, M. Dishon, On the asymptotic behavior of the stochastic and deterministic models

of an epidemic, Math. Biosci 11 (1971) 261265.[15] D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J.

Mathematical Sociology 4 (2) (1976) 187199.[16] O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl.

Probab. 38 (4) (2001) 898907.

[17] I. Nasell, On the quasi-stationary distribution of the Ross malaria model., Mathematicalbiosciences 107 (2) (1991) 187.

[18] H. Andersson, T. Britton, Stochastic epidemics in dynamic p opulations: quasi-stationarityand extinction, J. Math. Biol. 41 (6) (2000) 559580. doi:10.1007/s002850000060.

[19] I. Nasell,

queuing theory and epidemic models

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