Some models in epidemic control

MATHEMATICAL

383

BIOSCIENCES

Some Models in Epidemic HOWARD

hf. T4YLOR**

Department

of Operations

Ithaca.

New

Control*

Research,

C~YHPII T’nzwrsil~

York

Communicated

bv Samuel

Karlin

rZBSTR.4CT

A population

having members that are either susceptible

or immune to a given

disease is studied. It is assumed that the number of susceptibles increases stochastically as a pure birth process until either a decision is made to vaccinate or the disease strikes, problem

which

confers

immunity

is to balance the cost of population

of epidemic

outbreak

on all surviving

vaccination

and to arrive at a vaccination

long-run time-average

sum of these two cost factors.

this general structure

are studied,

1.

INTRODUCTION

The

control

AND

models

developed

model

Kahrs,

and Baker

of bovine

point in time, some cattle Immunity BVD

arises either

attack.

to increase

The

The

versus the costs and risks

schedule

that minimizes

the

Several specific models within

one relevant to bovine

virus diarrhea.

SUMMARY

epidemiological Robson,

including

all susceptibles members.

[l].

in this virus

are suggested (BVD)

by

developed

an by

In any given herd of dairy cows at any

will be immune

from vaccination

number

article

diarrhea

of susceptible

over time, to some extent

to BVD

while others

or from survival cattle

because

will not.

of a previous

in a given herd tends

individual

cattle

may lose

* This work was conducted under the auspices of the Center for Environmental Quality Management, Cornell University, and was supported in part by PHS grant ES-00098 and 1 TO1 ES 00130-01. ** On leave to the Department of Statistics, from

September

to August

1, 1968,

University

Mathemafical Copyright

0

of California, Berkeley,

31, 1969. Biosciences

1968 by -4merican Elsevier

3, 383-

Publishing

398 (1968)

Company,

Inc.

1-I. N.

384 their immunity,

but more importantly

immune cattle die or are slaughtered It is assumed until either

that the increase the susceptible

If BVD strikes infected,

and thus

of an epidemic

schedule

minimize

will

some

cattle

continues

or the disease

strikes.

cattle very quickly

become

of susceptible

is to balance

the costs and risks that

of susceptible

are vaccinated

all susceptible

the number

The $voblem

because

and may be replaced by susceptibles.

in number

cattle

a herd, virtually

drops to zero.

in the case of BVD,

TAYLOR

cattle

in the herd again

the cost of herd vaccircation

outbreak

and to arrive

the long-run

time-average

wersus

at a vaccination sum

of these

two

cost factors. Thus, in general, fraction

we have a picture

of its members

susceptible

members

corrective

action

of the epidemic specify

susceptible increases

is taken, occurs,

disease.

stochastically

which entails

which entails

when to take corrective

of these

of a finite population

to a given

over

a certain

another

action

time, cost,

cost.

and Sabin vaccines

smallpox,

polio, and rabies.

in the population

effectively

has dimmed,

some function

became

However,

zero.

Shortly

But

The number of susceptibles that

an educational

see the relevance without

situations, described

this.

action

the number

will fit

this

timing rule considered.

corrective

since the last total Other

Our variations

action

immunization, models

number of susceptibles

assume

only on the time

by vaccination

continuous

surveillance,

is assumed known at all times.

on this information.

law

and in the type

at any time is not

can be based

whether

research

in the

Some of our models assume

in the population timing

we

and similar

further differ

increase in number of susceptibles

of susceptibles

recently

Clearly

within the general framework

with the desire of at least stimulating

the stochastic

is based

in the population

for example,

While it is too much to hope

our models

for epidemics.

the memory

is not felt to be

aimed at correcting

modifications

models

so that

City,

of our model to this problem.

some

above

of corrective

outbreak.

campaign

we do present several variations

control

governing

New York

of susceptibles

now that

the need for vaccination

urgent by many individuals. launched

at least

after the Salk

the number

has risen to such an extent

timing

is to

involve similar considerations,

for polio were developed,

of polio epidemics

known

or an outbreak

so as to minimize

many disease control situations

among them perhaps

that

of

either

The problem

The only specific disease we study in detail is BVD.

into

until

two cost factors.

superficially,

that

with some

The number

or epidemic where

the

In this case, action

SOME

MODELS

Section examines

IS

2 presents in detail

RVD.

Subsequent

common

situation

model “.

EPIDEMIC

385

CONTROI,

the general

formulation

the set of assumptions sections

consider

where outbreaks

most

closely

variations

and

Section

GENERAL

approximating

extensions.

The

are rare but costly is studied in every

XODEL

AND

NOTATION

Let N be the size of the population t > 0 let X(t)

be the number

under consideration

of susceptible

members.

and for any

We suppose

= 0 and that X(t) for t > 0 evolves as a pure birth stochastic

([2], page 177) at least until corrective of the

disease

occurs.

Under

of (X(t);

t 3 0) is governed

negative

parameters,

P[X(t ijX(t)

= i] = 1 -

remains

there,

terms

exposure interval

for

that

exposures

members

distributed

to the first

where 0 is known. outbreaks

1 where

Once

o(dt)

X(t) = N it

or an outbreak.

We will

(and outbreaks,

in the population)

of the susceptible

process.

program

those

We let T be

value of T, which may be

duration

in K(i)

between

outbreaks,

action

is taken, costs,

will exhibit

we assume

case,

the

susceptible.

K(i)

If

Often

a cost K(i)

is

the corrective the cost of the

of scale in i, the number

= k, + k,. i, where

K, is a

and k, is a direct cost per suscep-

susceptible

of testing

for example,

economies

action program

When cost

is l/O.

this case in the limit.

are the costs of administering

as in the affine

the

7

time

the expected

fixed cost per corrective include

(In Section

Note that

as well as the direct

innoculated.

inthe

P [T > t] = e- ” for t > 0

Often K(i)

of susceptibles,

with

and that

in a small

of susceptibles.)

are rare, or 0 is small, and we will examine Included

occur

times,

of an outbreak

number

provided

and assume

as the average

vaccine itself.

ascertain

action

that

+ At) =

exposure

If X(t) = i and corrective incurred.

At.

interoccurrence

to allow the probability

to depend on the current

interpreted

P[X(t

i = 0, 1,. . ., N -

to the disease

process is independent

the time

and

less than

of course, until corrective

exponentially

we generalize

tible

and with the interpretation

of order

are some susceptible

dependent

behavior 1) of non-

t > 0) the susceptible process.

We assume

might

the stochastic

(A,; i = 0, 1, . . . , A’ -

known,

I.,(At) + o(At)

that

process

action is taken or until an outbreak assumption

by a set

assumed

remainder

call (X(t);

action

this

+ At) = i + 11X(t) = ;] = A,(dt) + o(dt)

represents

there

3

as a limit.

THE

X(0)

of the model.

process

all of the

X(t) = i and Mathematical

is not observable,

k,

population

members

to

an outbreak

occurs,

we

Biosciences

3, 383 - 398 (1968)

11. RI.

386 assume

costs

nonfatal

are

human

among

other

to obtain

things.

balancing made

We will consider

time

computing

control

it

cost

random)

time

or observable

is,

of course,

Decisions

of corrective corrective

action

is taken

action

is taken

cost of operation

divided

Let

T(b) bc

if a policy

value

at the random whenever

That is, T(6) = B.

= 0 and evol\,es until

of X(t)

is again

action if T(b) < T or by an outbreak identical

generating

is given by the expected cycle

time.

VI(b)= ~l~(X(T(b))&~,,

reduced

to

if T(b) 3

7‘.

to the first but statisticalll-

The process repcats,

by the expected

thercl of rules

case, where rule b calls

of such cycles and thus, by the law of large numbers, cycle

rules, where

The second family

begins with X(0)

In either case a new cycle, statistically

average

b.

Rules arc specified by an integer

or zmobservublc

w h en the current

of it, is begun.

the

of rules we call the continuous

in the population.

In either case, the population

independent

with

would then btl

action timing

by a parameter

‘Th a t 15, ‘. action

we call the fixed-time

T(b)}

subjecti\

outbreaks

for action b time units after the last total immunization. S = min{T,

decisions

Here we assume the number of susceptibles

T(b) = inf{t: X(t) = b}.

zero either by corrective

impossible

under a variety

these policies,

b < N and rule Ir calls for corrective

considered

In

effect.

is known at all times.

are b or more susccptibles

c(i).

of work,

Yet

policies

program.

The first family case.

of

component.

optimal

are indexed that

value

lost man-days

of epidemic

of their

two families

the members

in the population b with 0<

diseases

for this

and severity

by b is used.

surveillance

expected

on the costs c(i) and, among

with some knowledge

specified

an

may reflect

fatal

costs of the optimal

(possibly

costs

value

the frequency

in each family the

For

We suggest

of assumptions

having

such

an objective

must be made.

known

incurred cases

I-.\YI,OR

a sequence

the long-run

time)-

cost during any such

We let

7

(1)

;,

and w(b) = E [min{ T, T(6)) / where I,

is the indicator

and zero otherwise.

function

Our average

(3)

of the event A and is one if A occurs cost as a function

of b is then

SOME

MODELS

IN

EPIDEMIC

387

CONTROL

(4) In what follows evaluate

we consider

allowing

j(b),

particular

us to choose

cases of this general

a policy

b = b* that

model this

minimizes

expression. a. Continuous Cnder

surveillance

continuous

surveillance

is known

at all times.

population memoryless

nature

the parametric

the

Because

of the exponential

family

susceptibles

the level b.

A pure birth characterized

process

process

0, 1,. . ., n and for which

the states exponentially

distributed

times being statistically in this section

random

that

independent

with parameters

nentially

distributed

the sojourn

Thus,

b}.

through

time in state

([it], page 179).

Most of our results distributed

then min{U,

,u + Y and P[U

k is an

AA,all sojourn

and the easily

exponentially

of

A,,, . . . , An is also

successively

with parameter

p and Y, respectively,

with parameter

If we let W(k) = inf(t: > lW41

we consider

when first the number

evolves

variable

fact that if U and V are independent

verified random

V> is expo-

< V] = ,u/(,u + Y).

X(t) = k} for k = 0, 1, . . . , IV, then

= P[T

> W(l)]

X’.‘X =..Ig

x P[T

> W(2)\T > W(l)]

P[T>FV(k)(T>W(K-1)] for

[&J

For k = 0 we adopt the convention T(b) = W(b)

action

follow from this characterization

variables

p;T

involved,

(X(1); t >, 0) with parameters

as a stochastic

of the

of this and because

distributions

to take corrective

in the

of susceptibles

of rules given by T(b) = inf{t: X(t) 3

policy or rule b says reaches

number

that

k = 1,. . ., N.

n;&_,[AJ(e

(5)

+ &)I = 1.

Since

3, 383-398

(1968)

we have

Mathematical

Biosciences

H. M. T.AYLOlI

388 Also V,(b)

=

E

iC(x(

h -

Of considerable

T‘))IT

< T(b)]

1

importance

is the linear

case where

c(i) = c1 * I, which

yields

(8) Finally,

let S = min(T,

t] = te

ot + SOsee- OSds = O-~l[I -

T(b)}

so that

w(b) = E[Sl. Thus

b. Fixed Under is not

&lE!

E[S/T(b)

=

I}

zc(b) = E{E[SlT(b)

_

Easily

e-- “1.

1 _

rp

fJ?‘@))

Time

fixed-time

observable,

rules, the number and thus

T(b) = b for b > 0.

We

of susceptibles

we consider

the family

in the population of rules

given

by

have

v,(b) = E [K(X(l’@)))Z,(,,< T: = e- ObE[K(X(b))]. In the important

linear

case with K(i)

= k, + k, . i we have

(10)

SOME

MODELS

IN

EPIDEMIC

CONTROL

fPb[kl + &m(b)]

v,(b) = where

m(t) = E[X(t)]

389

for t > 0.

Similarly,

u,(b) = E [C(X(T))I,,

And again

in the linear

T(b)]

E [c(X(t))]W

=

(11)

Otdt.

(12)

case when c(i) = cr * i we have

v,(b) =

cl

5

m(t)f3eeet dt.

(13)

0 Finally

for S = min{T,

we have

T(b)}

w(b) = E[S]

= 0-l[l As a brief aside, let us compute is deterministic, process

there

unobservable form

say,

cases,

COb].

(14)

the average cost when the X(t) process

X(t) = m(t).

is no basic

T(b) = b.

-

With

difference

a deterministic

between

and we will consider

the

susceptible

observable

the unobservable

and

the

rules of the

We have v,(b) = X(m(b))(l

v2(b) =

- eCBb),

c(m(t))W

” dt,

0

and zo(b) = I!?r[l When

costs

are exactly unobservable formulas).

are linear, equivalent situation

K(i)

eMBb]

= k, + k,i

to Eqs. (ll), (which

The interesting

-

and

c(i) = c,i,

these

(13), and (14), describing differ

conclusion

cost case, the fact that the susceptible

from

the

formulas

the stochastic

stochastic

observable

is that in the unobservable process is stochastic

Mathematical

Biosciences

linear

is unimportant. 3, 383 - 398 (1968)

FJ.

390

\I. T.\YJ,OJ<

\\‘\:e have carried the general analysis about as far as is possible without making

further

based

assumptions.

on a careful

In this section

study

describing

general

model under

First, within

this

disease. these

upon exposure

the herd immunity virus stimulates and presence represents

\\:e then

particular

infected

of sufficient

within

numbers

we must specify

infection

behavior

Only detailed Even extend once

family

knowledge

though

the duration

well beyond infected,

reinfection

of active

BVD

will gradually

occurs.

revert

The rate

These,

In order to continue

parameters

(A,; i = 0,

of the susceptible observable

,

process or not so

rules may be considered.

can answer

the life cspectation

most of the time

(most of the time).

timing

of the disease

so that

in the serum of cattle

is continuously

of action

time,

of cattle with KVD

antibody

of our model.

the appropriate

the stochastic

the appropriate

of the>

all individuals

short

Infection

of antibodies

and we must decide if this process that

be

most

analysis

virus,

a relatively

of measurable

to subsequent

ii’ - 1) that govern

thr

herd to RVD

of course, are the general characteristics our analysis

continue

level is forced to loo”/;.

the production

immunity

must

under consideration.

assumptions.

of a dairy

the herd become

such assumptions disease

rexien- RVD and derive the assumptions

we briefly

closely

To be valid,

of the particular

these

immunity

of a dairy

to a state

questions.

is estimated

cow, a dairy

of susceptibility

at which the herd becomes

to

herd, unless

susceptible

is

related 0111~7 to the rate of rcmo\A of immune cattle and their replacement by susceptible

cattle.

These

several

sources,

because

of loss of colostrally

including

were susceptible received immunity never mean

that

occurs

lose their cattle rate

(susceptible)

dams were

the fact

immunity

during

eventually

order (immune)

which is in general and Baker.

where the average

cattle

susceptible

and thus decline

immunized

the observation

2 and that Robson,

The

actively

period.

randomly”

the!, adult

in herd

individuals We assume

in time at some

are replaced

agreement

from

calves that

and some purchased

susceptible.

that

rate A should vary linearly

become immunity,

were not immune

in the herd “purely

calves,

may be recruited

maternal

in their colostrum,

chance

despite

that

transferred

cattle

are replaced

model of Kahrs, replacement

by

calves their

no KVD antibody

replacements

that

because

susceptible

by younger

with the deterministic

It should be noted that the mean with the herd size N, say, il = yS

life of a cow in a herd is y-1.

Thus

we arrive

at a

SOME

MODELS

susceptible

IS

EPIDEMIC

process 1.

0, 1,. . ., X -

with

antibody

the control

procedure

Thus

observable difficulty,

no clinical

cattle

of susceptible

of unvaccinated this

signs

observable

acquired

observer.

To be consistent

loo:/;

after

recent

assumption a disease

cattle,

would

(after steady

cattle

a slight

of epidemic the herd is to occur

and thus

be unknown

an outbreak

is

the

to the

to occur only

Thus, to apply our model in the case of BVD

we assume all replacement

calves are susceptible

clinical signs of an epidemic of the

level

we must consider

all

replacements.

creates

outbreak

in the infected

immunity

if clinical signs are present.

rules

that

i =

that

to reach

It is possible, however, for a BVD outbreak

naturally

reaches

the number

of BVD

We have supposed

again 100~~~immune.

timing

that

long enough

which will call for a minor change in the definition

outbreak. with

assume case

for

(after all maternally

and are the only susceptible

the number

in the

A, = A = yN

are susceptible

has been operating

and equals

rate

it is clear we may also assume

is lost)

we may

Unfortunately,

birth

calf replacements

transferred state).

a constant

Furthermore,

recent unvaccinated

391

COSTROL

form:

are observed “When

the

until vaccinated

in the herd. number

or until

We consider action

of replacement

calves

b, \Taccinate them.”

Substituting

A, = 2 in Eqs.

(6),

(7), and (9), we get

(15) (16) and

(17) Using

Eq.

(4) for the average

cost per unit

(18) For

any given

minimized

choice

numerically.

of K(b)

and c(j)

the

expression

If c(j) = ci * j, we use Eq. Jlathemnticcll

above

may

be

(8), obtaining

Biosciences

3, 383-398

(1968)

H.

392

JI. TA1YLC)II

(19) and then

eIK(b) - @I r c 2

_

(1 + e/n)b and when K(b)

I

l ’

= k, + k,b, we have

(21) We

attempt

to zero,

to minimize

which implies

in Eq.

that

(21) by differentiating

the optimal

l-(1+;) “‘_ib* Since

clinically

observed

0 may be assumed

small,

assumed large relative

outbreaks

-

and

equating

b = b* satisfies iT!X)ln(l

of BVD

+ T). are relatively

and at least for large herds,

(22) infrequent,

A = yN may be

to 0. If 0/A is small and we use the approximations

and

then

in the linear

case,

Eq.

(22) reduces

to (23)

Let Baker

us consider

a simple

numerical

example.

Robson,

Kahrs,

[3] report a decline of about 20% per year in herd immunity,

if we consider a herd of 80 cows, yields a replacement 16 cows per year.

They

also estimate

that BVD

and

which,

rate il = 0.2 x 80 = epidemics

occur on the

average once every 3-5 years within a dairy herd, which implies & < 8 < Mathematical

Biosciences

3, 383-398

(196X)

i

SO_ME MODELS

IS

EPIDEMIC

393

CONTROL

and we will take 0 = $. Pritchard 141 reports a mortality rate of from We will use a mortality rate of 0% to 20% of the susceptible cattle. lo:/, and value a cow at $400. We thus have an expected cost of infection that is linear in the number of susceptibles i with coefficient ci = 0.1 x $400 = $40. Finally, the veterinarian charges $10 per visit and $5 per vaccination and thus k, = 10 and k, = 5. Then k,/(c, - k,) = lo/35 and (21/e) - 1 = 127, which substituted into Eq. (23) give b* g (36.28)““; that is, b* is a little over 6. If we substitute into the exact equation (21) in the vicinity of b = 6, we get f(4) = 130.60, f(5) = 126.95, f(6) = 126.13, f(7) = 126.45, and f(8) = 127.86. The exact minimum occurs at b = 6 but, as can be seen, the total cost curve is very flat in the vicinity of the minimum. 4.

MODEL

2:

OBSERVABLE

DECREASING

BIRTH

RATE

In this model we consider a situation under continuous surveillance but having a decreasing linear birth rate given by I, = 3,. (A’ - i) for i = 0, 1,. . .) N. Physically such a situation would arise if there were no herd turnover or recruitment of susceptibles but instead the duration of immunity of an individual was exponentially distributed with mean duration 1-l and the immunity durations of individuals in the population were independent random variables. Substituting AI = A(N - i) in Eqs. (6), (7), and (9), we get

O-l %(b)= K(b) ,IJ,

@-_)A 0

+

(N

_

ip

[

h-‘C. [

(24)

’

],[_Wk~~

0

u2(b) = F”(I) (N -

1

i)A

k=,)

8

+

(N -

1,

(25)

k)3,

and w(b) = B-1 I 1 -

In the linear

case where

(26)

’

1 c(i) = c,i we have

Mathenzatical

Biosciences

3, 383-

398

(1968)

Let us examine

the linear case where epidemic

but expensive.

outbreaks

We let 0 --f 0 and cr + CC keeping

c,t)

are infrequent constant,

c,0 = CC. Then

say,

-4lso Ill(b) --f K(b) and from Eq.

(29)

(7) iI -

v,(b)'ajl-l

2

1

h -- 1

_A

j7,

(S -

j)-l -- b

(SO)

j_“N-j=

Placing average

the limits (28), (29), and (30) into Eq. (5), we get the approximate cost

(31) Since ln[N/(N obtain

-

b)] 3

the further

To minimize

l/(N -

c;_,t

approximation,

j) 3 In [(S + l)/(i\: -- b + l)], we

valid for N large,

in Eq. (32) we again differentiate

and equate

to zero, which

yields j_J&‘(b*)

_

aj

=

!%!!)___~_ - ab* X -

When

K(b) = k, + k,b,

lVIathemzfical

Biosciem~s

this simplifies

3, 383-

398

(1968)

to

b*

(33)

SOME

MODELS

IN

EPIDEhIIC

If we have reason to suppose that we might use the approximation which

gives the approximate

395

COSTROL

the optimal b* is small relative to K, ln[N/(w - b*)] 2 b*/N + &(b*/N)2,

solution (34)

5.

MODEL

3:

FIXED-TIME

COi%STANT

BIRTH

RATE

In this section we consider an infinite population model with Ai = A fori = 0, I,. . and in which the susceptible process is considered unobservable so that we examine action timing rules of the form I‘(b) = b for b > 0. We emphasize that now b is in units of time (as opposed to the continuous surveillance situation, where b is in units of number of susceptibles). We will only consider the linear costs K(b) = k, + k,b and c(i) = cri. Of course, (X(t); t > 0) is now the well-known Poisson process for which m(t) = At. Substituting into Eqs. (ll), (13), and (14), we get v,(b) = E - O6[k, + kJb 1,

(35)

[l -- (1 T b@c”“;, v,(b) = cl ‘; ( 1

(36)

w(b) = @‘[I

(37)

and -

e-Oh].

W’e may again form an average cost, according to Eq. (4), and attempt We will immediately go, however, to minimize through differentiation. to the limiting case G’+ 0 and ci + CCwith cc = Bc,. In the limit we get o,(b) = k, + k,lb, vg(b) = xAb2/2, and w(b) = b, which yields ,f(b) = k,b-l

Differentiating 6.

MODEL

4:

and equating FIXED-TIME

+ k,i + 5;”

to zero yields

DECREASlh-G

BIRTH

b* = (dk,/d)‘@ RATE

As our last model we will consider the case AI = &N - i) under rules of the form T(b) = b for b > 0. It is well known that in this case P[X(t) = (e-““)“-3(1

~_ c-y1

and from this we easily MatAematzcal Biosciences

see that 3, 383-398

nz(t) = (1968)

N(1 -

e-i’).

Considering

and substituting

into

linear

Eqs.

tll(b) = exp(-

(II),

costs (I3),

K(i)

= k, $ k,i

and

c(i) -= c,i

and (14) yields

Oh)(k, + k,N 11 -~- exp( -

1Jj) j},

(3X)

and W(b) = O-111 -1s earlier,

exp(--

Ob) ;.

let us allow 0 +O

(40)

and c1

11,(b) = k, f

+ 00 with c( = Oc,.

k&(1

-

e

In the limit

jiJ),

(41)

and u’(b) = b, implying

e

Differentiating

and equating

)-‘I)/

(43)

to zero yields

(44) If we have reason to expect side of Eq.

that b*l is small, which from the right-hand

(44) would occur if N were large,

(XI*)~/~ and dropping

terms

of higher

order,

using em-“’ 2 1 -

Rb* +

we would ha\-e

(45)

7.

DEPENDENT

OUTBREAK

PROBABILITIES

In many situations ability current model

it may be more realistic

of an outbreak

occurring

level of the susceptible under continuous

Mathematical

Bio.sciences

3,

process.

surveillance 383-

to assume

in a small time interval

398

(1968)

In this section

and the assumption

that the probdepends on the we analyze

our

SOME

MODELS

IN

EPIDElMIC

JTt < T < t + AtIT 2 t, X(t)

397

COSTROL

= i] = H@)

i = 0, . . . ) iv.

+ cl(&),

(4’3) In most cases, the parameters susceptibles,

{e,} will be nondecreasing

the more likely an outbreak.

Equation

in i;

the more

(5) easily generalizes

to (47) from which (48) and

(49) To obtain w(b) =E [S] where S = min{ T, T(b)) we let f(i) = E [SIX(O) = i]. By

conditioning

on the random

variable

min{T,

T(i + l)},

we obtain

the recursion

with

the

boundary

which,

when

to Eq.

(9).

condition

f(b) = 0.

Oj = 0 for j = 0, . . , N may

Solving

the

be shown

recursion

yields

to be equivalent

8. REMARKS

We have

considered

several

an epidemic control model. ical epidemiology epidemiology

variations

Although

is well advanced,

is just

in a general

the descriptive

formulation

of

theory of mathemat-

the studv of control

in mathematical

beginning. Mathematical Uzosciences 3,

383-

398

(1968)

In our model we see several linear decreasing immunity

for individuals.

would be of interest.

tiblcs would become infective to many diseases in human be always effective. is temporarily new strains

compare

The

The situation

populations.

to the particular

the results

under

clinically

such a model

Our

distributed an arbitrq-

in which not all suscep occurred

would be closes

vaccination

after an attack

strain

it would be of interest and without

of sucll

Similarly,

appear to which the entire population with

study.

to exponentially study

when an outbrtak

L4pparently in influenza,

immune

in the case of RVD, of outbreaks,

areas for further

birth rate model corresponds

lifetimes

distribution

interesting

that

may not

an individual

attacked

him.

is susceptible.

to csplicitl!~ observed with ours.

HIIt

Finall!-,

allow two types

symptoms,

and to