WORDS OMAN

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 due^to 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

Mathematiciansworking in theoretical ecology have considereddensity dependencein deathrate and transmission coefficient recently. One may refer to Brauer [5,6], Busenberg and Hadeler [7],

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 deathratesufferedbyinfected}individuals, 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 theoremforthepossible equilibriumsolutions

THEOREM Let

, ,

^{and denote the}equilibrium 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

-

^c

^{+ f(N)}

^{=0, "2=}

^c+f(N)

⁽³²⁾

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(/’)- _r . ^{+ c} _f(’)/" (u + ^f() a) _a

0⁰_0. (3 6b)^{(3 6c)(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

=

c

+ f ^{(N) =o,} =

^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}

and fromequation(36d)

(r f (N)

)N

Equatingtheseexpressionsfor

,

^{we get an}equation 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 the^theorem 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 positive^rootof

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 monotonically decreasingin(,itfollowsthat

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])

^{indirectmethodwedetermine}the 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

=

_c

_{+ f (N)}

^=0,

=

_c

_{+ f (N)}

Consider a smallperturbation aboutthisequilibriumlevel

X’--+Xl,

Y=+Yl, Z+’+Z

^{and N N}

+n.

Substituting theseintothedifferentialequationswhich describethespread of the disease,andusing the approximation

f(N +

^nl)

f(N) + nlf’(N) + O(nl)

we get thestabilitymatrix

A

as

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

N ⁰

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;}

(ii) ala2

>

a3. (4

9)

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, Wland_w2areall 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)

+ -(r-

b

f(N))(u +

^a

+ f(N)).

Now_vl

>

0if

b(r f(N)) (f(l’) + c) (u +

^a

+ f(.l’)) + -(u

b

+

^a

+ f(l’)) >

^O.

Withthehelpof equation

(3

4),theaboveinequality reduces^tothefollowing^form

---(r-f(.))-b__{o f(’)(f(l)(u f(/r))+u-l-a)-r(f()’) + +

(u +

^a

+ f(N)) + -(u

b

+

^a

+ f(l’)) >

^O.

(u +

^a

+ f())2 (b a)

bra

or

a(f() +

^u

+ a) >

O. (4

12)

Hence

vl

>

0 if

(u +

^a

+ f())2(b a) >

bra. (4 13) Fromequation

(3 4)

(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,thereforethe}above inequalityistrueandhence_a.2

>

0

Using

equauon

(35)thetermsinsidethebracket can be written as

Thus

w

⁰

Now_w2

>

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}

UlV2

+

^u,2v

we >

^{0 Wehave}U2Vl w2 hencethe inequality

5. SUMMARYANDCONCLUSIONS

Inthispaper^wehave 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 Local}stability 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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