An epidemic model with density dependent parameters and vaccination

AN EPIDEMIC MODEL

WITH DENSITY DEPENDENT

PARAMETERS AND VACCINATION

Q.J.A.KHANandB.S.BHATT

Department

ofMathematics andComputing

Collegeof Science Sultan QaboosUniversity P O Box36, Postal Code 123 AI-Khod

Muscat,SULTANATEOF OMAN

(Received July29,1993andin revisedformDecember28, 1994)

ABSTRACT. Amodel originally suggested by Greenhalgh 12] and later modifiedbythat same author [13,14] is considered under the assumption that the transmission coefficient is inversely proportional to the total population size Thepurpose ofthisstudy istosee theeffect ofthis density

dependenttransmission coefficient on thestabilitycriteriafortheequilibrium ofthemodel equations. It

isfoundthatGreenhalgh’s resultsarestillvalid

KEY WORDS AND PHRASES. Epidemic model, density-dependent death rate and transmission

coefficient,vaccination,stability

1992AMSSUBJECTCLASSIFICATION CODES. 92B05

1. INTRODUCTION.

Thispaperuses arelatively simpledeterministic mathematical modeltodescribe infectious diseases like

measles, rubella, chickenpoxandmumps Thebookby Bailey [4]describes much of thebackground in the area ofepidemicmodelsupto 1975. Weareinterested inlookingat amodelwherethe parameters

which describe the transmission of the disease and the death rate ofindividuals are dependent onthe

number ofindividuals inthe population. We also take into account the fact that infected individuals suffer a higher death rate than other individuals becausethey have the disease Some models for a

populationwith adensitydependentdeathratehave beenstudiedbyNisbetandGumey [20] Anderson

[1] has analyzed anepidemic model with birth and death in which infected individuals suffer ahigher

death ratethanother individuals. Heconsiders the deathrate of individualsto bedensity independent

Dietz and Schenzle [10] and Mollison [19] consideredthe transmission coefficient to bedependenton

population density Anderson etal [2]have discussed models for rabies withadensitydependentdeath rate McLeanand Anderson 17,18]also incorporatedthisfeatureindiscussingamodel ofmeasles Gao

and Hethcote 11 studiedSIRS/SISmodelswith restrictedpopulation growth by logistic equation dueto

density dependence inboth birth anddeathrates They also considereddensitydependent transmission coefficient (inversely proportional to the total population size) They obtained four equilibria and discussed theirstability results

Busenberg and Van den Driesche [8], Diekmann and Kretzschmav [9], Greenhalgh [12-14], May and Anderson[3],Pugliese[21,22]andTuljapurkarandMerd,th-John[23]

Greenhalgh [12] described a mathematical model for a disease where the death rate is a monotonicallyincreasingfunctionof the number of individuals in the population and infected individuals sufferahigher deathratethanother individuals Thesameauthor in[13,14]studied the modified model witha class ofndividualswho areincubatingthe disease and vaccination ofsusceptibleindividuals In

our modelwehave considereddensity dependenceinthe transmission coefficient(inversely proportional

to the total population) together with the vaccination ofsusceptible individuals Considering density dependenceof the transm,ssionparameter, wecan relax theassumption thatthenumber ofcontacts per

unit timeper susceptibleindividual increaseslinearlywiththe populationsize 2. MATHEMATICALMODEL

Weexaminea modelmwhich anindividual startsasa susceptible, catchesthe diseaseand afterashort infectiousperiodbecomespermanentlyimmunetoit Weassumetheindividualswho aresusceptibleare vaccinated at aconstant per capitalratec The spread of thedisease ismodeled bya setof differential equations which describe the transfer of individuals between these classes The system of ordinary

differentialequationswhich describethespread ofthe disease is asfollows

dx b

rN xy cx

f(N)x

(2 la)

dt N

dy b

xy-f(N)y-

(u

+

c)y (2 lb)

dt N

dz

uy

+

cz

f(N)z

(2

lc)

dt dN

rN-

f(N)N-

cy (2 ld)

dt

with suitable initialconditions,where oneof the equationsisredundantsince

x(t)

+

y(t)

+

z(t)

N(t)

x(t)

represents thepopulation(ordensity)ofthe susceptible classattimet,

y(t) represents thepopulation of theinfectedclassattimet,

and

z(t)

represents thepopulation of theimmuneclassattimet, e.those individualswho have had

thedisease,have recoveredandarepermanentlyimmune

In ourmodel, risthe birth rate, isthe transmissioncoefficient,

f(N)

isthedensitydependent

death rate taken as a continuous, strictly monotonic increasing function of N (Greenhalgh [12-14],

considered

-

constant), cistheadditional deathratesufferedbyinfectedindividuals, andv is therate atwhich infected individualsbecome immune, sothat

v-1

istheaverageinfectiousperiodinthe

absenceof adeathrate The probabilitythatasusceptibleindividualmeetsand becomes infectedbyan

(At)

+

o(At)

The per capitarateof vaccination ofsusceptible

infectious individual in

[t,

+

At]

is

individuals isc, sothatin asmalltime interval

[t,

+

At]

the number of susceptible individualswho are vaccinated iscxAt

+

o(xt)

Thisterm cxmustbe subtracted from the equation(2. la)correspondingto the fractionof susceptible individuals whoarevaccinatedand addedtoequation(2 c)correspondingto new immune individuals. We are interested inperforming anequilibrium and stability analysis of this model Thestability analysishelpsustodeterminethelong-termbehaviorof thesystem, e whetherthe diseasepersists

3. EQUILIBRIUM ANALYSIS.

The first stepis to examinethe possible equilibrium solutions of these equations Firstofall, we shallsuppose that

f(o)

-imf(N)

>

r, so that if the populationsize islarge enoughthe deathrate exceeds the birthrate Settingallofthe time derivativetozeroinsystem(2 1),wededuce thefollowing

THEOREM Let

,

,

and denote theequilibrium numbersofsusceptible,infectedandimmune individualsrespectively Let Ndenote thetotal numberofindividualsatequilibrium

There are threepossible equilibria

(i) When there is no disease present because thepopulation hasdiedout

===N-0.

(31)

(ii) If

f(c)

> r

> f(0),

thepopulationhasreachedanequilibriumlevelbut the dsease hasdied out

f(N)N

cN

-

=0, "2= (32)

c

+

f(N)

c+f(N)

and

r=

f(N).

(33)

(iii) The disease is present and the equilibrium values of susceptible, infected and immune

individuals,are

(u

+

+

f

(N) )N

(r

f

(N)

)N

b Y= a

.{ub(r

f(1"))

+

ca(.

+

a

+

f())}

"

ab

f

()

(34)

Population valueNsatisfiesthe equation

b

(f(N)

+

c)(u

+

a

+

f(N))

....

(3 5)

a

f(N)

Thisequilibriumexists ifand onlyif b

_>

PROOF. Thistheorem is proved alongsimilar lines tothe corresponding results for therelated modelsinGreenhalgh

[12-14]

Setting thetimederivatives to zero insystem(21),wededuce that

r

b’

c

rf(/)

0 _(36a)

b’-

f(/’)-

.

(u

+

a)

0 (3 6b)

+ c

f()

0 (3 6c)

r

f(’)/"

a

0. (3 6d)

From equation (3 6b) either

-

0 or fVl+f"+)r NOW 0 implies from equation (3 6d)

that N-0 or

r--f(N)

Hence

ifr:/=f(N)

then =y=z=N=0

Ifr=f(N)

then it is

straightforwardtoshow thatequations

(36a)

and

(36c)

yield

f(N)N

cN

=

=o,

=

c

+ f

(N)

c

+

f

(N)

anequilibriumsolutionfor any value of If f()+("+)thenfrom equation(3 6a)wehave

r

c

f(r)

r./"

(c

+

f(/))

Y _b’

f(N)

+

u

+

a b

(r

f

(N)

)N

Equatingtheseexpressionsfor

,

we get anequation forNwhich canbe reducedto b

(f(N)

+

c)(u

+

a

+

f(N))

c

f(l’-

+

(.

+

a,

r)f()

Thustheequilibriumvalues5,

,

2 andNmustsausfythevaluesgivenin(i), (ii)and(iii) of thetheorem The first equilibrium is always possible The second equilibrium is well defined if and only if

f(0) <

r

< f(c)

The thirdequilibriumexists ifandonlyif

(r

+

c)(u

+

c

+

r)

b.

,swell definedusingthefollowinglemma

LEMMA. Theequation b_ _{f(9)+iu+c,-r)fl)-"r} has a unique positive root

N,

ifand onlyif

f() #,

where

+

istheunique positiverootof

t

0Here

That valueof Nsatisfiesr

_>

f(N,)ensuring ")spositiveifand onlyif

(r

+

c)(

+

a

+

r)

b>

PROOF. Consider theequation

b

(:(N)

+

)(,

+

,

+

:(N))

c

f()2

+

(,

4-

r)f(//)

r Withthetransformation

f(N)

Then

b

(

+

)(.

+

+

)

+(.+-)-(+c)(++)

Consider g() _{+(.+_)_.,..} Here g(’) has roots c, c rootsof

Q()

O,where

Q()

2

+

(.

+

r)

The asymptotes arethe

Q(()

hastworealroots(_ and

(+

givenby

1

(a

+

t,-

r)

-4- 1

_,

+,

[(

+.

)

+

4.]

The function g() is negative for 0

<

<

+

and monotonically decreasing for

>

+.

Hence the equation

b

has aunique positiveroot(if andonly if

f(o) > (+

Otherwise, if

f(c)

<

(.

this equationhas no

positiverootfor( Since

f(oo) >

(+

denote the uniquerootby(andletNbethe correspondingvalue of N For the solution to be feasible we require

f()

_<

r or

<

r as __b is constant and g(() is

ismonotonically increasingin(andzeroat([seethegraph ofg(()] Hencer

_>

(ifandonlyif

g()

_>

0.

This condition isequivalentto

b>

(r

+

c)(u

+

c

+

r) Thiscompletestheproof ofthe lemma

b

4. STABILITY

Stability Analysis of Equilibrium (i).

By

Liapunov’s(Jordanand Smith

15])

indirectmethodwedeterminethe stabilitybehaviorof the system of differential equations

(2 1)

which describe the spread of the disease at the equilibrium

2 N 0 Considerthe Liapunov function

L

z

+

y

+

zwhichleadsto

L’=(r-f(N))N-ay<(r-f(O))N-ay<O

when N>0 and

r<f(0).

(41)

We conclude that the zero solution of

(2 1)

is globally asymptotic stable (GAS) for r<

f(0)

since

L

<

0and unstable forr

>

f(0).

Thereforeitfollows that thezerosolutionoftheoriginalsystem(2 1)

isGASforr

< f(0)

andunstable forr

> f(0)

Analysis of Equilibrium(ii).

Theequilibrium valuesare

f(N)N

cN

=

=0,

=

c

+

f

(N)

c

+

f

(N)

Consider a smallperturbation aboutthisequilibriumlevel

X’--+Xl,

Y=+Yl,

Z+’+Z

and N N

+n.

Substitutingthese into the differential equationswhichdescribethespread ofthedisease, and usingthe

approximation

f(

+

n

f()

+

nf’()

+

o(n)

we get thestabilitymatrixAas

whichgivesthe characteristicequationas

(f(N)

+

A)(c

+

f(N)

+

A)

(f()

+

u

+

a) A

(I’().

+

A)

O. (4 3)

From equations (32), (3 3) and (4.3) we get the equilibrium (ii) to be locally stable for small perturbationsif

Ro

<

1and locally unstableif

Ro

>

Iwhere

Ro

(4 4)

(,:

+

,,.)(,,-

+

,,,

+

)

Local Stability ofEquilibrium (iii).

Usingthe sameprocedureasfor equilibrium(ii),weget thestabilitymatrix6’asfollows

---f(9)

-4

0

N

0 0

C N

u

f(gr)

0 -a 0

,.

+

f’

(9)’

-j,

)y-f’

r

f(.r)

f’(/)2

(45)

The determinantofC-

AI

is

IC-

All

(f(Fr)

+

A)IDI,

where

----f(N)-a D=

0

-A -1

)Y-o r-

f()

f(l)ll

,k

(4.6)

The correspondingcharacteristicequation for

D

is

(4.7)

where

(4.8)

TheRouth-Hurwitz(May

16])

stabilitycriteria forthethirdordersystemis

(i) a >0,

a2>0

and as>0;

Nowwe willshowthe positivity of allthe constantsappearingin(4 8)

Let

(4 10)

Rewriting equation(4 8)as

al 7dl -I--a2 731 _[_

ft

(.//")732

a3 Wl -[-

f’()w2

(4 ll)

Since

f(N)

is monotonicincreasingin

N, f’(N) >

0 Hencetoshow(i)of equation(4 9)we willshow

thatul,u2, Vl, v2, Wlandw2areall positive

Using equation

(3

5)we get

U

--(r

+f(N)

+c,

wherer

> f(N)

and from equation(3 4)b

>

a,thereforeu

>

0

Thensinceu2 N

>

0,it mustbe thatal

>

0

Now731 canberewritten as

{

b

(r

f())

+

f()

+

c}

(r

f(l’))(u

+

a

+

f())

731

(7"-

f

(N)

b

+

-(r-

f(N))(u

+

a

+

f(N)).

Nowvl

>

0if

b

b(r

f(N))

(f(l’)

+

c)

(u

+

a

+

f(.l’))

+

-(u

+

a

+

f(l’))

>

O.

Withthehelpof equation

(3

4),theaboveinequality reducestothefollowingform

---(r-f(.))-b__{o

f(’)(f(l)(u

f(/r))+u-l-a)-r(f()’)

+

+

b

(u

+

a

+

f(N))

+

-(u

+

a

+

f(l’))

>

O.

(u

+

a

+

f())2

(b

a)

bra or

a(f()

+

u

+

a)

>

O. (4

12)

Hence

(u

+

a

+

f(N))(r- f(N))

Since <

/,

from equaUon(6)r

f()

<

a,so that

r-

f(N)

_<

a+u+ f(N).

(4 4)

(4 5)

Inequalities (4 14)and(4 15)combinedtoshow that theinequality(4 13)istruei.e

v

>

0

Now

_,

>

0 f

b

+

9f(-/-

5

+

9

>

0,

or

(b

a)

+

N

f(N)

+

cN

>

0

Fromequation(34),b

>

a,thereforetheabove inequalityistrueandhencea.2

>

0

Using

equauon

(35)thetermsinsidethebracket can be written as

Thus

w

0

Noww2

>

0ifthe sumof alltermsinsidethebracketispositive. Using equation

(3

5),thetermsinside

thebracket become

a

--(r

f(lr))

(f(/r)

+

c)

+

--(u

+

a

+

f())

(.

+

a

+

f(/))

(4.16)

Wehavealready shown above that thesumofalltermsof(4

16)

ispositive

Hence,a3

>

0

Nowwehavetoshowthata _a2 a3

>

0.

alag. a3

(UlVl

Wl)

+

f/()(UlV2

+

U2Vl

w2)

+

f"2(l)(u2v2)

where

w

0 and ul, u2,v,v2,w2all are positive i.e. to show aa2

as >

0 we will show that

5. SUMMARYANDCONCLUSIONS

Inthispaperwehave studieda simplemathematical epidemicmodel with vaccination There are three possible equilibrium situations which arise Equilibrium wthpopulation extinct will beglobally asymptotically stable ifr

_< f(0)

and will be unstable ifr

> f(0)

Equilibriumwithsteady population

b,

<

andlocallyunstable

andno diseasepresentwillbelocally stabletosmallperturbationsif

if_..+(

>

1 Localstability oftheequilibriumwith disease present is examinedanalyticallyandis

found stable In this paper we have extended the work ofGreenhalgh [12-14] It is unrealistic to consider the transmissionparameteraconstantbecausethis assumes contactperunit timeper susceptible

individual increaseslinearlywiththepopulationsize Thisassumptioncanberelaxed bytaking adensity

dependent transmission parameter Here we consider the transmission coefficient as inversely

proportional to the total number of individuals in the population Gao and Hethcote

[11]

have also

considered density dependence in transmission coefficient similar to ours but they have taken density

dependent restrictedgrowthduetoadecreasingbirthrateandanincreasing deathrateasthepopulation

size increasestowardsits carrying capacity They have obtained four equilibria The first equilibrium

when disease fades out and population size tendsto zeroand showed that it is always a saddle The

second equilibriumwhere thepopulationsize isupthecarrying capacityisfoundtobeGASunder certain

threshold conditions The third equilibrium occurs when thebirthrate is density independent and the deathrate isdensitydependent and isLASwhen certain conditions are met The fourthequilibriumis

achieved when thebirthrateisdensitydependentand the deathrate isdensity independent andit isalso LAS under differentthreshold conditions TheSIRSmodels reduceto SIS models when theimmunity

lossrateis zero All the stabilityresults ofSIRS models holdgoodfor SIS models providedthere is some inflow into thesusceptibleclass Wecanmake our model more realisticbythe introduction of a class ofindividualswhoare incubating thediseaseand taking intoaccount the fact thatimmunity may

onlyprovide temporary protection

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