Simultaneous control of measles and rubella by multidose vaccination schedules

ELSEVIER

Simultaneous Control of Measles and Rubella by Multidose Vaccination Schedules NIELS G. BECKER AND VLADIMIR ROUDERFER School of Statistics, La Trobe University, Bundoora, Australia Received 28 October 1994; revised 28 January 1995

ABSTRACT There is currently a preference for using measles-mumps-rubella vaccine to simultaneously control these three diseases. Here an age-specific transmission model is used to investigate the consequences, on cases of measles and congenital rubella syndrome, of switching from a one-dose vaccination with this vaccine to a two-dose vaccination schedule. The model allows for a period of maternally acquired immunity and assumes that infection leads to permanent immunity, while vaccine-induced immunity is allowed to wane. The vaccination coverage at the second dose is expressed in terms of availability for vaccination, which depends on whether the individual received the first dose and the age of the individual. It is found that the optimal age for the first vaccination is not very sensitive to variations in the force of infection and is close to age 1 year for both measles and rubella. However, the optimal age for a second vaccination, offered indiscriminately, depends significantly on the age-specific forces of infection. This emphasizes that decisions about immunization schedules require reliable information about age-specific forces of infection in the community. In some circumstances control may be significantly more effective when the ages for the second dose differ for measles and rubella. It is found that the addition of a catch-up vaccination, offered to previously unvaccinated children at school entry, makes it more feasible to find a common age for the second dose that controls both measles and rubella effectively.

1.

INTRODUCTION

Single-dose vaccination p r o g r a m s with high coverage dramatically r e d u c e d the incidence of measles a n d rubella in developed n a t i o n s for some time after their i n t r o d u c t i o n , b u t e l i m i n a t i o n of these diseases from c o m m u n i t i e s has n o t occurred, a n d recently there has b e e n a r e s u r g e n c e in their incidence [1-4]. R u b e l l a incidence nearly d o u b l e d in the U n i t e d States d u r i n g 1989-1990, twenty years after the advent of mass i m m u n i z a t i o n [4]. N a t i o n a l serosurveys in the U n i t e d States show a significant increase in seronegativity to measles a n d rubella despite the r e q u i r e m e n t of evidence of i m m u n i z a t i o n at school entry [2, 3, 5]. I n MATHEMATICAL BIOSCIENCES 131:81-102 (1996) 0025-5564/96/$15.00 © Elsevier Science Inc., 1996 SSDI 0025-5564(95)00034-B 655 Avenue of the Americas, New York, NY 10010

82

NIELS G. BECKER AND VLADIMIR ROUDERFER

1993-1994, large outbreaks of measles occurred in some parts of Australia [6]. In response to these recent trends, some countries have introduced, or are considering, a two-dose vaccination strategy. In 1989 the United States recommended a schedule of two doses of measles-mumps-rubella (MMR) vaccine administered at ages 1 and 5 years [7]. Canada decided on a two-dose strategy in 1993 [8], with the first dose at age 1 year and the second vaccination age in the range 1.5-5 years. Australia recently adopted a policy with two doses of MMR vaccine, with the first dose to be administered at age 1 year and the second dose in the age range 10-16 years, with encouragement for a catch-up vaccination at school entry for children with no previous vaccination history [9]. Considerations of vaccination schedules are complicated by the current preference for the use of MMR vaccine for the simultaneous control of these three diseases [7]. The use of trivalent vaccines clearly has administrative and cost advantages, but it is important to ensure that there exists a practical one- or two-dose vaccination schedule for MMR vaccine that enables effective control of measles, mumps, and rubella. It is not clear that such simultaneous control with MMR vaccine is feasible, since for measles the aim is to minimize the overall incidence of measles cases whereas for rubella the aim is to minimize the number of children born with congenital rubella syndrome (CRS), which requires minimization of the number of cases of rubella among women of childbearing age. Here we consider the feasibility of simultaneously controlling measles and rubella, as judged by the number of cases of measles and the number of births of children with CRS, with a common one- or two-dose vaccination schedule. We consider this problem with the aid of an age-specific transmission model. Age-specific transmission models have been used extensively to assess vaccination strategies against measles and rubella since the pioneering work of Knox [10]. Anderson and May [11-13], McLean and Anderson [14, 15], Hethcote [16, 17], and Schenzle [18, 19] formulated different models for this purpose, with the common assumption that both disease-induced immunity and vaccineinduced immunity are lifelong. The effect of waning immunity on vaccination schedules was studied by Katzmann and Dietz [20] and by Rouderfer et al. [21]. It is the waning of vaccine-induced immunity that casts doubt on the feasibility of simultaneous control of measles and rubella with a childhood vaccination schedule, since a significant number of women may have lost vaccine-induced immunity by the time they reach childbearing age.

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

83

Here we focus on simultaneously vaccinating against measles and rubella with an MMR vaccine, allowing for waning of vaccine-induced immunity. In contrast to most previous work, we avoid the unrealistic assumption that subjects are selected independently for their second vaccination. We also investigate the consequences of a catch-up vaccination at school entry. Section 2 describes the main features of the age-specific transmission model used for this analysis. Details of the model are given in [22]. The assignment of values to parameters of the model is described in Section 3. Optimal two-dose vaccination schedules are computed for measles and rubella in Section 4. It is found that the optimal age for the second dose may differ significantly for measles and rubella, depending on the forces of infection, so that effective control of the two diseases may not be feasible with a common second dose. In Section 5 we show that when a catch-up vaccination for previously unvaccinated children is added at school entry, a common schedule for two doses offered indiscriminately becomes more feasible. Conclusions and recommendations are given in Section 6. 2.

MODEL FORMULATION

The characteristics of measles and rubella are very similar, and so their dynamics can be described by transmission models having the same form, but different parameter values need to be assigned for the two diseases (see Section 3). The form of the transmission model used here is taken from [22]. 2.1. THE DYNAMICS OF DISEASE TRANSMISSION The model allows for a period of maternally acquired immunity beginning at birth. It is assumed that infection leads to permanent immunity, whereas vaccine-induced immunity may wane over time. Vaccination coverage at the second dose is a function of the subjects' availability for vaccination, which depends on whether the individuals received the first dose, the time elapsed since the first dose, and school attendance. At any point in time the whole population is partitioned into seven classes: two classes of individuals who are protected by maternal antibodies (~'0, ~1), two classes of susceptibles (~'0, S~1), whose who are infectious ( J ) , those who have recovered from the disease and are permanently immune (~'), and those who have vaccine-induced immunity ( ~ ) . The subscripts 0 and 1 on 9 and S ~ distinguish individuals who were "never vaccinated" and those who were "previously vacci-

84

NIELS G. BECKER AND VLADIMIR ROUDERFER

nated," respectively. This distinction is required to accommodate different levels of second-dose vaccine uptake for those never vaccinated and those previously vaccinated. The seven classes and transitions between them are depicted in Figure 1. The transmission between these classes is described by the system of partial differential equations

D P o ( a , t ) = - ( a + t z a ) P o, DPl(a,t)=-(a+txa)el, DS0(a , t) = aP o - [ h(a, t) + tza] So, DSl(a,t ) = aP 1 + O V - [ )t(a,t)+ tza]S1, Dl(a,t) = h(a,t)( S o + $1) -(3~ + tza)I, D R ( a , t ) = T I - t x , R, D V ( a , t ) = - ( O + txa)V,

(1)

where D denotes the differential operator 3 / 3 a + c)/3t. The parameters a, /z~, 7, and 0 are rates of loss of maternally acquired immunity, mortality, recovery from disease, and loss of vaccine-induced immunity, respectively. The remaining quantity in (1) is the force of infection, given by h ( a , / ) = f f ~ ( a , a ' ) I ( a ' , t ) da', where ~(a,a') is the per capita number of susceptibles of age a infected by an infective of age a' per unit time.

a~

1 OV

~D1 .... .=.fu,] 1/I=P1 v&ccana~tlon ]

lu,.Po

[

$1

v~cdnstionTl.ov ........

~u, . , ~ i . . , i . .

I

]~,,So FIG. 1.

A(a,t)Sl

q l" t==I

~=R

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

85

2.2. JUMP CONDITIONS ARISING FROM VACCINATION The system of equations (1) is accompanied by a set of jump conditions corresponding to each age at which vaccination occurs. Let av denote an age at which a fraction c o of the previously unvaccinated individuals and a fraction c 1 of the previously vaccinated individuals are vaccinated. The jump conditions at av then take the form

Po( av+ , t ) = (1 - co)Po( a v- ,t), Pl(av + , t ) = P l ( a v - , t ) + coPo(a ~- ,t) (2)

S0(av+ , t ) = (1 - co)So(a v- ,t), Sl(a~+, t) = [1 - c1(1 - f ) ] $1( av- , t) + cofSo( a v - , t) V(a~+ ,t) = V ( a ~ - , t ) + (1 - f ) [ c o S o ( a v - , t )

+ clS~(a ~ - , t ) ] ,

where f is the proportion of primary failures among vaccinations of susceptibles. The jump conditions simplify for the first vaccination because at that age there are no previously vaccinated individuals, so that Pl(av - , t) = Sl(av - , t) = 0. The jump conditions also simplify for a catch-up vaccination because c I = 0 when only previously unvaccinated individuals are vaccinated. 2.3. AVAILABILITY FOR THE SECOND VACCINATION For a second vaccination, offered indiscriminately to all who are available, the coverage levels c o and c~ need careful attention. It is usually assumed that a second-dose vaccination is taken up independently of whether the first dose was taken up. This tends to give optimal vaccination ages a I and a 2 that are close together, but it is unrealistic to expect availability for vaccination to be independent when the second dose is offered very soon after the first dose. We must also acknowledge that availability for vaccination is high during the compulsory school years but then declines to a lower level for adults. Data are potentially available to formulate the availability for vaccination, but we are not aware of any published data and so rely on a model formulation. A major aim is to demonstrate that data of this kind are necessary for making decisions on vaccination strategies. We are guided by the form for second-dose availability formulated in [22], but we have simplified the curves to be piecewise linear. Among previously vaccinated individuals, the availability for the second dose is 100% at the age of the first dose and is assumed to decline linearly until the age of school entry, assumed to be 5 years. The availability of individuals not vaccinated at the first vaccination age is assumed to increase linearly from zero at the first vaccination age. There is a

86

NIELS G. BECKER AND VLADIMIR ROUDERFER

sudden increase in the availability of both groups at school entry. For the compulsory school years we assume that the coverage levels for each of the groups never vaccinated (0) and previously vaccinated (1) for constants. At the end of the compulsory school age, taken to be 15 years, we assume a linear decline in availability for both groups until age 20 years, after which availability again becomes a constant for each group. This last feature, which seems realistic, may be important here because of our desire to protect women of childbearing ages from infection with rubella. The algebraic expressions for these second-dose coverage levels at age a, given that the first dose was given at age a~, are

(j + [ ( 1 - j ) b o - jbl]( a - al)

for a l < a < 5 ,

JCjl

for 5 ~< a ~< 15,

I

cj( a,al) = ]0.2(cj2 - Cjl)a + 4cjl -3cj2

for 15 < a < 20, for a >i 20,

~,cj2

for j = 0 and 1. These expressions assume that the optimal age for the first dose is before school entry, as is always the case in practice. The form for the coverage levels prior to school entry ensures that at ages soon after the first dose there is only a small change in the availability for vaccination. The graphs of these two coverage levels for the second dose are given in Figure 2 for the case a 1 = 1,

b 0 = 0.05,

b I = 0.0125,

C01 = 0 . 8 0 ,

Cll = 0.98,

C02 = 0.15, c12 = 0.25.

These values seem plausible, but they should ideally be determined from data. Such data do not seem to be available, but they could be collected. Note that the coverage level c o, for unvaccinated individuals, is always less than c 1 in Figure 2. This is thought appropriate because the group not vaccinated with the first dose contains individuals who are never available for vaccination, for health, religious, or other reasons. The overall achievable coverage level for the second dose at age a is given by [1 - ~ ( a 1)]c0(a, a 1) + 7 r ( a l ) c l ( a , a 1), where "rr(a I ) is the achieved coverage level for the first dose at age a 1 and co(a,a 1) and cl(a,a 1) are the two coverage levels given in Figure 2. For example, if we achieve 0.80 coverage with the first dose at age a~ = 1, then we can achieve an overall coverage of 0.944 during the compulsory school years and a coverage of 0.23 at any age after age 20 years. The graph of the overall

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

87

\

~o Q 2

3

4

5

1

L

6

7

i

i

i

i

i

i

i

i

i

i

i

i

8

9

10

11

12

13

14

15

16

17

18

19

Age a t s e c o n d

i

i

2 0 21

i

22

i

i

23 2"t 25

dose

FIG. 2. Percentage of individuals assumed available for vaccination with the second dose, as a function of age a 2. (×) q(a,1), the availability for those vaccinated at age 1 year; (0) 1), the availability for those not vaccinated at age 1 year; (z~) overall availability, that is, possible overall coverage.

co(a,

coverage for the second dose, given 0.80 coverage with the first dose, is shown in Figure 2. The coverage levels produced by this choice of p a r a m e t e r values is thought to be plausible for a developed country, and we use these functions of second-dose availability for the analyses in this paper. 2.4.

THE QUANTITIES TO BE MINIMIZED

In the case of measles, the effect of changing to a new vaccination schedule at time t = 0 is judged in terms of T

IT =Yf0

fo I(a,t)dtda,

which is the total number of individuals infected over the time interval of duration T years immediately following the change to the new vaccination schedule, where T might be 10 or 20 years. Rubella is considered a milder disease than measles, with fewer serious complications [23]. The main concern about rubella is associated with congenital rubella syndrome (CRS), which results in serious disease in approximately 80% of infants born to women who had infection

88

NIELS G. BECKER AND VLADIMIR ROUDERFER

during the first trimester of pregnancy [24]. A measure of the cumulative number of CRS births over the period (0, T) is given by C r = 0.5 x 0 . 8 x 0 . 3 3 x y

f0 f0T$ ( a ) I ( a , t ) d t d a ,

where ~.(a) is the fertility of females in the population as a function of age a and therefore indicates the probability that a woman of age a is pregnant. In contrast, Anderson and May [11] and Anderson and Grenfell [24] used the ratio of the number of cases of CRS after the start of vaccination to the number of CRS cases before the start of immunization as their criterion. This ratio seems an appropriate criterion when long-term outcomes are the basis for comparison. As argued in [22], we consider short-term and medium-term consequences more appropriate to the assessment of vaccination schedules and therefore focus on the quantity C T for rubella. 3.

A S S I G N M E N T OF P A R A M E T E R V A L U E S

3.1. THE INITIAL CONDITIONS As most developed and many developing countries have maintained a relatively high level of vaccination of infants for some time [25], we use the stationary solution of system (1) corresponding to vaccination of 80% of infants at the age of 1 year as initial conditions for both measles and rubella. 3.2. DEMOGRAPHIC PARAMETERS For our applications we consider a community of 1 million individuals, so all quantitative results are therefore per 1 million population. We assume that the population is in equlibrium, that is, the number of births matches the number of deaths, and that O /xa =

for a < 75, for a >~ 75,

that is, all people live to the age of 75 years and then die. This implies the boundary conditions P o ( O , t ) = N O= 106/75, P l ( O , t ) = So(O,t ) = S l ( O , t ) = I ( O , t ) = R ( O , t ) = O,

for the system of equations (1).

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

89

3.3. THE INFECTIOUS PERIODS All rate parameters are expressed in terms of a time unit of 1 year. For example, the rate of recovery during the infectious period of measles is 3' = 52 year -1, so that the mean infectious period is 1 week [13, 26]. For rubella the mean infectious period is about 11.5 days, which corresponds to 3, = 31.74 year-1 [24]. 3.4.

LOSS OF M A T E R N A L L Y ACQUIRED IMMUNITY

The rate of loss of maternally acquired immunity is chosen as a = 4 for both diseases, which means that the average duration of the period with maternally acquired immunity is 3 months and only 5% of offspring retain their immunity beyond 9 months of age [13, 16, 24]. 3.5.

PRIMARY AN D SECONDARY VACCINATION FAILURES

Generally, vaccines against measles and rubella induce seroconversion in about 95% of recipients, which corresponds to a primary vaccine failure rate of about 5% [1, 27-29]. Failures occur in part due to the presence of maternal antibodies at age 1 year, when vaccination is recommended in most countries. Therefore the "net" primary failure rate for susceptible infants is about 3%. There is strong evidence that secondary vaccine failures occur, although estimates of their extent vary considerably. A U.S. study indicated that 7-33% of previously vaccinated individuals failed to demonstrate immunity against rubella [2]. Secondary vaccine failures rates in the range of 5-7% have been observed for measles [30, 31]. These data motivate the choice of primary vaccine failure parameter f = 0.03 and rate of loss of vaccine-induced immunity (secondary failure rate) 0 = 0.005 for both measles and rubella. With 0 = 0.005, about 5% of individuals immunized by vaccination lose their immunity over a 10-year period. 3.6.

TRANSMISSION PARAMETERS

To specify transmission rates 8(a,a'), we partition the population into the five age groups 0-5 years, 5-9 years, 10-14 years, 15-19 years, and 20-75 years. In each case we then set /3(a, a') = 8ij, a constant, for any a and a' in age groups i and j, respectively. In our application we assume that

( 8 ~ j ) --

'81 81 81 82 83 83 84 84 85 8S

83 84 85 83 84 85 83 84 85 84 84 85 85 85 85

NIELS G. BECKER AND VLADIMIR ROUDERFER

90

Transmission matrices of this form are labeled W A I F W 1 by Anderson and May [13, p. 177]. Estimates of t h e / 3 ' s are obtained, separately for measles and rubella, from estimates of the forces of infection for the corresponding age groups. We use the following estimates of the forces of infection obtained from different communities. Measles

A1 = 0.184, An :

A2 =

0.579,

A3 =

0.202,

A5 = 0 . 1 0 0

0.100,

(3)

(estimated from case notifications in England and Wales, 1966, see [13, Table 9.1]) and A1 =

0.100,

A2 =

0.240,

A3 =

0.362,

A5 = 0 . 0 8 6

A 4 = 0.086,

(4)

(estimated from a serological study in New Haven, Connecticut, 1955-1958, see [13, Table 9.1]). Rubella

A1 = 0.081,

h 2 = 0.115,

h 3 = 0.115,

A4 = 0.083,

A5 = 0.067,

)t 6 =

0.067

(5)

(estimated from a serological study in southeast England, 1980-1984, see [32]) and A1 = 0.089,

A2 =

0.134,

A3 =

0.151,

A4 = 0.148,

A5 = 0.126,

A6 =

0.126

(6)

(estimated from case notifications in l e e d s , 1978, see [13, Table 9.7]). 4.

TWO-DOSE INDISCRIMINATE VACCINATION

Consider vaccination strategies consisting of two doses administered indiscriminately at two different ages. In other words, on each occasion all available individuals are vaccinated, irrespective of their disease and vaccination history. Our aim is to find ages a 1 and a 2 at which to administer the first and second doses of M M R such that both the criterion I r for measles and the criterion C r for rubella are as low as possible. It is unlikely that there exists a pair of vaccination ages (al, a a) such that I r and C r are minimized simultaneously. The hope is that

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

91

there exists a pair of vaccination ages such that both I r and C r are near their minimum values. Unless otherwise specified, the value T = 20 years is used, as this seems an appropriate interval over which to plan a vaccination schedule. 4.1. V A C C I N A T I O N SCHEDULES FOR M E A S L E S For each of the two sets of forces of infection (3) and (4) for measles, and using other parameters as specified in Section 3, we constructed the surface of I20, the cumulative number of measles cases over 20 years, as a function of the vaccination ages a 1 and a 2. Figure 3 shows the surface for the cumulative number of cases with forces of infection (3), assuming that coverage is 80% at the first dose and as specified by Figure 2 for the second dose. It is seen that over the range of values displayed, low values for I20 occur for a very narrow range of values of a2, with steep rises occurring on both sides of a 2 = 5. In Figure 3, the range of values considered for a 1, the age at the first dose, is 0.7-1.3 years. When the graph is extended beyond those values for al, we find steep rises beginning in 120 at about 0.5 year, near the lower end of the interval, and at about 1.5 years at the upper end. This makes a case for using the ages a I = 1 and a 2 = 5 when the forces of infection (3) apply. The corresponding surface for 120 using the forces of infection (4) is similar in form to that in Figure 3. The main difference is that the trough located at a 2 = 5 is wider, extending from 5 to about 8 years, and

\ Oo

I9

7~

17

Y5

15

Age

14

Y3

at

72

77

70

second

9

8

dose

FIG. 3. Number of measles cases over 20 years (Iz0) for a two-dose vaccination schedule at ages (a 1, a2), when forces of infection are given by (3).

NIELS G. B E C K E R A N D V L A D I M I R R O U D E R F E R

92

the rise in the surface beyond a 2 = 8 is not quite as steep. In fact, the minimum value of 120 occurs at (al,a 2) =(0.9,10.0). This difference seems reasonable when we note that the forces of infection (4) differ from those in (3) by being lower for the 5 - 9 years age group and higher for the 10-14 years age group. 4.2.

VACCINATION SCHEDULES FOR RUBELLA

Using each of the two sets of forces of infection (5) and (6) for rubella, with other parameters as specified in Section 3, we constructed the surface of C2o, the cumulative number of cases of CRS over 20 years, as a function of the vaccination ages a I and a 2. Figure 4 shows the surface for forces of infection (6) assuming again that coverage is 80% at the first dose and as depicted in Figure 2 for the second dose. From a contour plot corresponding to this surface (not shown here), we found that C20 is minimized at (a 1, a 2) = (1.0,15.0). Note that there is a steep rise in Ce0 just below a 2 = 5 years and after a 2 = 15 years. For a 2 between 5 and 15 there is a more gentle slope. This is encouraging because it offers some scope for choosing an age for the second dose that is somewhat removed from the optimal age but predicts a number of cases of CRS that is only a little higher than its minimum. The corresponding surface for C20 using the forces of infection (5) is similar in form to that shown in Figure 4. One difference is that the

f

i/

t I~

I8

¢ 17

i 16

¢ 15

Age

f 74

I 73

ot

I 72

7 ~

I 70

second

I 9

I 8

I 7

I 6

I 5

I 4

~

2

dose

FIG. 4. Number of cases of CRS over 20 years (Cz0) for a t~o-dose vaccination schedule at ages (al,a2), when forces of infection are given by (6).

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

93

gentler slope for a 2 in the range 5-15 years is in the opposite direction so the minimum value for C20 occurs at (a1,a2)=(0.9,5). This is explained by the fact that estimates of the forces of infection (6) are relatively higher for older age groups. Another difference is that the surface of Czo for forces of infection (6) is uniformly higher than that for (5), as is expected since the estimates (6) are uniformly higher than those for (5). The more gradual slopes in the rubella surfaces for 5 ~
o

Cases of Ueasels • 0 0 1 . Cases of Ivleosles • 0 0 1 . Cases of C R 5 , 5 E E n g l a n d

•

UK

forces

of

infection

(3}

Iofces of i n f e c l i o n

USA focee

of

infection

'

'

'

(4)

(5)

A

~o

do

0

o

L I I

I

I

2

,5

4

5

I

i

1

I

r

I

I

6

7

8

9

I0

11

12

Age eL second

i

1.3

I

I

I

t

i

I

14

15

16

17

18

19

20

dose

FIG. 5. Numbers of measles and rubella cases as functions of age at second dose, when the first dose is given at age I year, with different forces of infection and using a two-dose schedule only.

94

NIELS G. BECKER AND VLADIMIR ROUDERFER

For the two sets of forces of infection for measles, the optimal age for the second dose is age 5 years, as is the case for rubella with forces of infection (5). With such forces of infection, the vaccination ages ( a l , a 2 ) = ( 1 , 5 ) simultaneously minimizes 120 for measles and C20 for rubella. This finding gives support to the two-dose vaccination schedules adopted by the United States and is a desirable policy in terms of cost of vaccine and its delivery. However, this vaccination schedule is not optimal for rubella if the forces of infection (6) apply. In that case, the value of C2o can be reduced by about 15% if the vaccination schedule (al, a 2) = (1,5) is used for rubella instead. This observation points to the need for reliable estimates of the age-specific forces of infection for the community of interest. 4.4. PERMANENT VACCINE-INDUCED IMMUNITY In the above discussion it was assumed that vaccine-induced immunity waned at a rate such that 5% of successfully vaccinated individuals become susceptible again within 10 years. The available data seem to suggest that this is an appropriate assumption. We now report on corresponding results under the alternative assumption that immunity induced by vaccination is permanent for both measles and rubella. This is of interest for two reasons. It is of interest to see how the assumption that vaccine-induced immunity is permanent may mislead us about effective vaccination schedules, particularly since the assumption of permanent immunity is often made. Further, vaccines continue to improve, so the assumption of permanent immunity may become reasonable. We computed surfaces of 120 and C20 after replacing only the assumption that 0 = 0.005 by 0 = 0. The surfaces were similar in form, with two main differences in detail. First, the values of 120 and C20 are uniformly lower, as is expected, and therefore predict more optimistic outcomes than are warranted when 0 is in fact 0.005. Second, the range of values of a 2 for which the surface remains near its minimum is extended in all cases. For example, the very narrow trough in Figure 3 near a 2 = 5 is extended to 5 ~
SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

95

4.5. ASSESSMENT OVER A IO-YEAR PERIOD In order to see how sensitive conclusions are to the choice of T = 20 years, we compared the performance of alternative vaccination schedules over a 10-year period. The values of I10 and C10 are obviously lower than I20 and C20, respectively, but the shapes of the corresponding surfaces are very similar, particularly when 0 = 0. In the case of 0 = 0.005 there are small differences, because there will be less waning of immunity over a 10-year period. For example, the narrow trough in Figure 3 is extended from 5 ~
TWO-DOSE VACCINATION WITH CATCH-UP AT SCHOOL ENTRY

It seems very wasteful to revaccinate children a few years after their first dose when vaccine-induced immunity is long-lasting. Consequently, there is considerable interest in vaccination policies that require evidence of vaccination at school entry, with strong encouragement to have previously unvaccinated children vaccinated at that stage. This seems to be a cost-effective way of preventing disease transmission, because it avoids a large number of unnecessary vaccinations compared with indiscriminate vaccination at school entry, takes advantage of the greater availability for vaccination arising from school attendance, and increases the overall level of immunity at an age when children begin a period of intense mixing with other children. We therefore consider a change in vaccination strategy from a single dose to a two-dose strategy with a catch-up vaccination at school entry, that is, at age 5 years. This is of particular interest in Australia, where a policy of this type has been proposed. With the same assumptions about achievable coverage, the addition of a catch-up vaccination at school entry increases overall coverage and is therefore expected to reduce incidence. 5.1. A CATCH-UP DOSE MAKES SIMULTANEOUS CONTROL MORE FEASIBLE

We assume that all available individuals, of those not previously vaccinated, are vaccinated at school entry. With this catch-up vaccination included, we constructed the surfaces of 120 and C2o as functions of a 1 and a2, the ages at which the first and second indiscriminate doses of MMR are administered, where a I < 5 and a 2 >~5. Figure 6 shows the surface 120 under the same parameter values as for the surface shown in Figure 3 except that catch-up vaccination has

96

NIELS G. BECKER AND VLADIMIR ROUDERFER

//

o j

vOo"~C.~

~.~,~

FIG. 6. Number of measles cases in 20 years (120) for a two-dose vaccination schedule at ages (a 1,a 2) and a catch-up vaccination at school entry, when forces of infection are given by (3).

been added at school entry. This means that 80% of unvaccinated children, that is, 16% of all children, were vaccinated at school entry. By comparing the surfaces in Figures 3 and 6, we deduce that with the addition of the catch-up vaccination 120 remains near its minimum value for a much larger range of a 2 values. In particular, whereas a two-dose schedule with a I = 1 and 0 ~< a 2 ~< 14 is very inferior to the schedule ( a l , a 2) = (1,5) without a catch-up vaccination, it is seen that both schedules do very well when catch-up vaccination at school entry is in place. Similar conclusions apply for /2o under the alternative forces of infection (4) and for C20 under each of the forces of infection (5) and (6). In fact, the forces of infection (3), for which surfaces are shown in Figures 3 and 6, give the worst-case scenario in this regard, as is seen by comparing the graphs of Figure 7 with the corresponding graphs of Figure 5. The graphs of Figures 5 and 7 differ only in that those in Figure 7 are computed under the assumption that catch-up vaccination is in place at school entry. 5.2. A SPECIFIC NUMERICAL COMPARISON Consider the two vaccination schedules (al, a 2) = (1, 5.1) and (a l, a 2) = (1,14). The age 5.1 is chosen in the first schedule to ensure that the

SIMULTANEOUS CONTROL OF MEASLES AND RUBELLA

97

Cose$ Of Meosle| x 0.05. UK force of infection ( 3 ) C o r n of Measles . O.OS. USA force of infection ( 4 ) Coses of CRS. S ( [nglond force of infection ( 5 ) [

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FIG. 7. Numbers of measles and rubella cases as functions of age at second dose, when the first dose is given at age 1 year, with different forces of infection and using a two-dose schedule plus catch-up at school entry.

second indiscriminate dose is administered after the catch-up vaccination, when the latter is offered. Age 14 is chosen in the second schedule because Figure 7 shows it to be a good schedule. The following table gives 120 and C20 for both vaccination schedules and the forces of infection (h) given by (3)-(6). Measles (120) No catch-up

Rubella (C20)

With catch-up

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8,750 76,200

3,100 4,400

6,100 4,800

130 295

875 760

100 80

760 415

The schedule (1,5.1) is seen to be better for three of the four forces of infection when there is no catch-up, whereas schedule (1,14) is better for three of the four when there is catch-up vaccination. Furthermore, with catch-up vaccination there is a very substantial reduction in incidences for both diseases. 5.3. SENSITIVITY OF THE COMPARISON TO A V A I L A B I L I T Y FOR VACCINATION

We now check how sensitive the comparison of the U.S. policy (indiscriminate vaccination at ages 1 and 5 years) and the Australian

98

NIELS G. B E C K E R A N D V L A D I M I R R O U D E R F E R

policy (indiscriminate vaccination at ages 1 and in the range 10-16 years, plus catch-up at 5 years) is to the availability for vaccination during the years of school attendance. To make the comparison more specific we take the second dose in the Australian policy to be at 11 years. This lies in the range specified and is convenient in that it represents the age of entry into secondary school. It is generally recognized that availability is high during the years of school attendance, and we only change the availability of those who were not vaccinated with the first dose. The achievable coverage for these children is doubtful because it includes children who cannot be vaccinated for health, religious, or other reasons. Figures 8 and 9 give the number of cases 120 and C20 , respectively, as functions of c01, the availability of previously unvaccinated children during years of school attendance, for both policies. The Australian policy has the advantage of extra vaccinations at school entry, while the U.S. policy has the apparent advantage that the indiscriminate second-dose vaccination occurs at an earlier age. In making these comparisons we must remember that our results for measles indicate that without the catch-up vaccination the Australian two-dose schedule in vastly inferior to the American two-dose schedule. However, with the catch-up vaccination we see from Figures 8 and 9 that the Australian policy is better for all values of coverage c01 for

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FIG. 8. Numbers of measles cases (•20) for the schedule with two doses at ages l and 5 years (U.S. policy) and the schedule with doses at ages 1 and 11 years plus catch-up at age 5 years (Australian policy) as a function of c01, the school-years availability of children not vaccinated at age 1 year.

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FIG. 9. Numbers of C R S cases (C20) for thc schedule with two doses at agcs 1 and 5 years (U.S. policy) and thc schedule with doses at ages I and II years plus catch-up at agc 5 ycars (Australian policy) as a function of c01, the school-ycars

availability of children not vaccinated at age 1 year.

rubella and also for measles if forces of infection (4) apply. For measles with the forces of infection (3) we see that the U.S. policy is better for all coverages c01. 6.

DISCUSSION

The results of this study confirm the commonly held belief that the best age at which to administer the first dose of MMR is at about 12 months. In essence this is as early as possible, when we allow for the period of protection by maternal antibodies. In contrast, the best age for a second dose of MMR, offered indiscriminately to all available, depends more on the age-specific forces of infection acting in the community for each of the diseases. Using two existing sets of estimated forces of infection, for each of measles and rubella, and allowing for availability for vaccination, the results suggest that a two-dose schedule offered at ages 1 year and 5 years gives near optimal control of measles cases over a 20-year period. For cases of CRS the answer is less clear; the ages 1 and 5 are near optimal for set (5) of estimated forces of infection, but for set (6) of estimated forces of infection this schedule is estimated to lead to about 18% more cases of CRS than the schedules with doses at ages 1 and 15 years.

100

NIELS G. BECKER AND VLADIMIR ROUDERFER

The extra coverage achieved by adding to a two-dose schedule a catch-up dose for all unvaccinated children available at school entry substantially decreases the incidences of measles and CRS over a 20-year period for some vaccination schedules (al,a2). Furthermore, when a. catch-up dose for unvaccinated children is in place at school entry, then a second dose given indiscriminately to all available for vaccination seems just as effective when given in the age group 10-14 years as one given just after school entry at age 5.1 years. The results support the vaccination schedule, with catch-up, proposed in Australia, provided the coverage achieved with the catch-up dose is high. On the other hand, as illustrated by the table in Section 5, if the catch-up component of the Australian policy is ineffective, then this policy is greatly inferior to the American policy. There is, however, another reason why the Australian schedule was proposed. The limited amount of data available on age-specific measles incidence in Australia [33, 34] indicate that a substantial number of infections occurred in the older age groups 5-9 and 10-14 years. When there is a change in a vaccination schedule with higher coverage, it is beneficial to offer vaccination to older age groups for an interim period. This has been confirmed with calculations. The sensitivity of our conclusions to various underlying assumptions has been checked. As mentioned in Sections 4.4 and 4.5, we have considered schedules under the assumption that vaccine-induced immunity is permanent and considered performances of schedules over a 10-year period. At the end of Section 5 we checked the sensitivity of results to cm, the coverage achieved for the second dose and catch-up vaccination of those not vaccinated at age 1. We have also considered schedules under the assumption that primary vaccination failures are higher, namely f = 0.08 instead of f = 0.03. With the higher primary failure rate, the surfaces were raised, indicating higher incidence, but the shapes of the surfaces, and therefore the relative merits of different vaccination schedules, remained essentially the same. The results are sensitive to the shape of the age-specific forces of infection, and informed decisions about vaccination schedules require reliable estimates of age-specific forces of infection for both diseases that are relevant to the community under consideration. In the absence of reliable estimates of the forces of infection it seems safer to have a two-dose schedule with a catch-up dose at school entry as such a policy makes a common two-dose schedule that is near optimal for measles and rubella more plausible. We have used 120, the number of measles cases in the next 20 years, and C20, the number of CRS cases in the next 20 years, as our objective function, although 110 and C m were also used. This is in contrast to

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101

m o s t o t h e r work, which c o n c e n t r a t e s on the e v e n t u a l o u t c o m e . T h e e v e n t u a l s t e a d y state usually t a k e s l o n g e r t h a n 50 y e a r s to reach. N o t e that this is l o n g e r t h a n v a c c i n a t i o n p r o g r a m s have b e e n in place. It s e e m s quite i n a p p r o p r i a t e to us to m a k e p r e d i c t i o n s o v e r such a long t i m e s p a n with a m o d e l w h o s e p a r a m e t e r s a r e very likely to c h a n g e o v e r t i m e as a results o f c h a n g e s in vaccines as well as in social a n d political changes.

Support by a grant f r o m the Victorian Health Promotion Foundation is gratefully acknowledged. REFERENCES 1 National Vaccine Advisory Committee, The measles epidemic: the problems, barriers, and recommendations, J. Am. Med. Assoc. 66:1547-1552 (1991), 2 T.R. Shum, D. B. Nelson, M. A. Duma et al., Increasing rubella seronegativity despite a compulsory school law, Am. J. Public Health 80:66-69 (1990). 3 P.W. Kelley, B. P. Petruccelli, P. Stehr-Green et al., The susceptibility of young Americans to vaccine-preventable infections, JAMA 226:2724-2729 (1991). 4 M. L. Lindegren, L. J. Fehrs, S. C. Handler et al., Update: rubella and congenital rubella syndrome, 1980-1990, Epidemiol. Rev. 13:341-348 (1991). 5 V. Fraser, E. Spitznagel, G. Medoff and W. Dunagan, Results of a rubella screening program for hospital employees: a five-year review (1986-1990), Am. J. Public Heatlh 138:758-764 (1993). 6 National Notifiable Diseases Surveillance, Communicable Disease Intelligence 18:346-348, Australian Gov. Publishing Service, Canberra, 1994. 7 Centers for Disease Control, Measles prevention: recommendations of the Immunization practices advisory committee (ACIP), MMWR 38(S-9):1-18 (1989). 8 Consensus Conference on Measles, Can. Communicable Dis. Rep. 19(10):72-79 (1993). 9 National Health and Medical Research Council, Report of the ll4th Session, AGPS, Canberra, 1992, pp. 7-8. 10 E . G . Knox, Strategy for rubella vaccination, Int. J. Epidemiol. 9:13-23 (1980). 11 R. M. Anderson and R. M. May, Vaccination against rubella and measles: quantitative investigations of different policies, J. Hyg. Camb. 90:259-325 (1983). 12 R. M. Anderson and R. M. May, Age-related changes in the rate of disease transmission: implication for the design of vaccination programmes, J. Hyg. Camb. 94:365-436 (1985). 13 R.M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Univ. Press, Oxford 1991. 14 A. R. McLean and R. M. Anderson, Measles in developing countries. Part I. Epidemiological parameters and patterns, Epidemiol. Infect. 100:111-133 (1988). 15 A . R . McLean and R. M. Anderson, Measles in developing countries. Part II. The predicted impact of mass vaccination, Epidemiol. Infect. 100:419-442 (1988). 16 H.W. Hethcote, Optimal ages of vaccination for measles, Math. Biosci. 89:29-52 (1988).

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17 H. W. Hethcote, Rubella, in Applied Mathematical Ecology, L. Gross, T. G. Hallam, and S. A. Levin, Eds., Springer-Verlag, Berlin, 1989, pp. 212-234. 18 D. Schenzle, An age-structured model of pre- and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol. 1:169-191 (1984). 19 D. Schenzle, Control of virus transmission in age-structured populations, in Proceedings of an International Conference on Mathematics in Biology and Medicine (Lect. Notes Biomath. 57), V. Capasso, E. Grosso, and S. L. Pavert-Fontana, Eds., Springer-Verlag, Berlin, 1985, pp. 171-178. 20 W. Katzmann and K. Dietz, Evaluation of age-specific vaccination strategies, Theoret. Popul. Biol. 25:125-137 (1984). 21 V. Rouderfer, N. G. Becker, and H. W. Hethcote, Waning immunity and its effects of vaccination schedules, Math. Biosci. 124:59-82 (1994). 22 V. Rouderfer and N. G. Becker, Assessment of two-dose vaccination schedules: availability for vaccination and catch-up strategies, Math. Biosci. 1994, to appear. 23 S.R. Preblud, M. K. Serdula, J. A. Frank, Jr. et al., Rubella vaccination in the United States: a ten-year review, Epidemiol. Rev. 2:171-194 (1980). 24 R. M. Anderson and B. T. Grenfell, Quantitative investigations of different vaccination policies for the control of congenital rubella syndrome (CRS) in the United Kingdom, J. Hyg. Camb. 96:305-333 (1986). 25 J.P. Grant, The State of the World's Children, United Nations Children's Fund, Oxford Univ. Press, New York, 1993. 26 D. J. Nokes, A. R. McLean, R. M. Anderson et al., Measles immunisation strategies for countries with high transmission rates: interim guidelines predicted using a mathematical model, Int. J. Epidemiol. 19:703-710 (1990). 27 B.S. Hersh, L. E. Markowitz, R. E. Hoffmann et al., A measles outbreak at a college with prematriculation immunization requirement, Am. J. Public Health 81:360-364 (1991). 28 A. Vaisberg, J. O. Alvarez, H. Hernandez et al., Loss of maternally acquired measles antibodies in well-nourished infants and response to measles vaccination: Peru, Am. J. Public Health 80:736-738 (1990). 29 W. Orenstein, K. Bart, A. Hinmam et al., The opportunity and obligation to eliminate rubella from the United States, JAMA 251:1988-1994 (1984). 30 R.R. Wittier, B. C. Veit, S. McIntyre et al., Measles revaccination response in a school-age population, Pediatrics 88:1024-1030 (1991 ). 31 R . G . Mathias, W. G. Meekison, T. A. Arcand et al., The role of secondary vaccine failures in measles outbreaks, Am. J. Public Heatlh 79:475-478 (1989). 32 D . J . Nokes, R. M. Anderson, and M. J. Anderson, Rubella epidemiology in south east England, J. Hygiene 96:291-304 (1986). 33 Infectious Diseases Unit, Health Department Victoria, Surveillance of Notifiable Infectious Diseases in Victoria-- 1991, 1992, pp. 47-49. 34 NSW Health Department, Public Health Bull. 5(8):91 (1994).