Spatial heterogeneity and the design of immunization programs

Spatial Heterogeneity and the Design of Immunization Programs ROBERT M. MAY Department of Biology, Princeton University, Princeton, New Jersey 08544 AND

ROY M. ANDERSON Department of Pure & Applied Biology, imperial London University, London S W7 2BB, England

College,

Received I5 June 1984; revised 15 June 1984

ABSTRACT Conventional epidemiological models usually assume homogeneous mixing, with susceptible and infected individuals mingling like the molecules in an ideal gas. It has recently been noted, however, that variability in transmission rates-arising, for example, from some hosts being in dense aggregates while others are in small or remote groups-can result in the intrinsic reproductive rate, R,, of a microparasitic infection being greater than would be estimated under the usual assumption of homogeneous mixing; this implies the infection may be harder to eradicate under a homogeneously applied immunization programme (that is, a larger proportion of the population must be vaccinated) than simple estimates might suggest. In this paper we consider a spatially heterogeneous population arbitrarily subdivided into n groups, with one transmission rate among individuals within any one group, and another, lower transmission rate between groups. We define an optimum eradication program as that whichtreating different groups differently-achieves its aim by immunizing the smallest overall number in each cohort of newborns, and we show this optimum program requires fewer immunizations than would be estimated under the (false) assumption that the population is homogeneously mixed. We prove this result in general form, and illustrate it for some special examples (in particular, for a population subdivided into one large “city” and several small “villages”).

1.

INTRODUCTION

Our aim in this paper is to explore the way nonuniformities in the rates of transmission of an infectious disease that arise from spatial inhomogeneities in the distribution of the total population can affect the design of immunization programs. Even for the cognoscenti, however, it seems advisable to sketch in some background before focusing on the technical details.

MATHEMATICAL

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OElsevier Science Publishing Co., Inc., 1984 52 Vanderbilt Ave., New York, NY 10017

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ROBERT M. MAY AND ROY M. ANDERSON

Most work on mathematical epidemiology is based on the assumption of “homogeneous mixing,” with susceptible and infectious individuals mixing uniformly, without regard to age, location, or other such factors (see, for example, Bailey [l]). Under this assumption, the rate at which new infections appear is simply proportional to the number of susceptible individuals (X) times the number of infectious ones (Y); thus it is given by p XY, where p, the “transmission parameter,” is some proportionality constant. For such a homogeneously mixed population, we can define the basic reproductive rate of a microparasitic infection (sensu Anderson and May [2]), R,, as the average number of secondary infections produced when one infectious individual is introduced into a wholly susceptible population [3-51. This quantity R, characterizes the population biology of the association between the infection and its host. As the infection becomes established, so that only a fraction x of the population is susceptible, the effective reproductive rate of the infection (R , the number of secondary infections on average actually produced by each infectious individual) will be given as R, discounted by the fraction susceptible: R = R,x. At equilibrium, the effective reproductive rate will be unity (see, for example, [6]), whence R, and the fraction susceptible at equilibrium in a homogeneously mixed population, x*, are related by Rex* =l.

(1.1)

The same ideas lead directly to an estimate of the critical fraction, pc, of the population that must successfully immunized in order to eradicate the infection. The fraction remaining unimmunized, 1 - p,, must be insufficient to keep R above unity; that is, we require R,(l-p,) ~1 [7]. This, in combination with Equation (l.l), gives the eradication criterion p,=l-x*.

(1.2)

Here, as defined above, x* is the fraction susceptible in the preimmunization population, and p, is the fraction successfully immunized (or, equivalently, the fraction of each cohort of newborns immunized approximately at age zero-allowing for the effects of maternal antibodies-over many years). Estimates based on Equation (1.2), if treated with due caution, can provide a useful guide to preliminary thinking about the design of immunization programs against specific infections [5,8]. Equations (1.1) and (1.2) moreover, make testable predictions that appear to accord with some of the facts for the immunization programs against measles in the U.K. and U.S.A. [9,10]. For a more full discussion of these questions, see, for example, the papers in the Dahlem Conference volume [ll] or [10,12,13]. Simple estimates based on Equation (1.1) or (1.2) may, however, be seriously compromised by the fact that a variety of mechanisms can result in

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the host population not being homogeneously mixed. Some of these mechanisms are as follows. First, the “transmission parameter” p,, may vary with the age of the susceptible (i) and of the infectious (j) individuals. There is, indeed, evidence that for many childhood infections the transmission is weaker among preschool children and among adults than among those in the roughly 5-15 age range. Broadly speaking, relatively low values of p,, in the early years tend to make the critical fraction immunized, p,, larger than would be estimated from Equation (1.2) by assuming homogeneous mixing, while relatively low values of /3,, in laters years tend to make p, smaller than would be correspondingly estimated. These age-related inhomogeneities are discussed elsewhere [14- 181. Second, real populations are likely to be genetically heterogeneous with respect to such factors as susceptibility to infection or infectiousness once infected. It could, for example, be that the apparent decline in the transmissibility of infections such as measles among older (more than 15 years) age groups is simply because the more susceptible genotypes have largely been filtered out. If this is the case, then estimates of p, based on preimmunization values of age-specific transmissibility will be too optimistic: as overall infection rates decline under an immunization program, an increasing number of more susceptible genotypes will move into the older age classes of susceptibles, in effect causing the age-specific transmission parameter to increase in the older age classes over the value pertaining prior to vaccination. This seems to us to be an important source of inhomogeneity that is neglected in essentially all conventional studies, and we pursue it in detail elsewhere [ 181. Third, the rates of contact between susceptible and infectious individuals can exhibit a great deal of variability arising from the geographic or social setting. For example, Becker and Angulo [19] have shown that transmission rates for variola minor within households are typically much greater than between households; Yorke et al. [20] have shown that realistic models for the spread of gonorrhea need to distinguish a significant subpopulation of “superspreaders”; and Murray and Cliff [21] have used stochastic simulations in pursuit of an understanding of the observed epidemiology of measles in circumstances where several distinct regions are linked relatively weakly. Mathematical models embodying such spatial heterogeneity have been explored by several authors [8,22-251. These studies have obtained formal expressions for the conditions under which the infection may be maintained. Other authors have adopted a somewhat simpler approach, pointing out that the effects of spatial heterogeneity upon R, can sometimes be estimated as (1.3)

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Here R,, is the value of the basic reproductive rate obtained by using the average value of the transmission parameter p, and VAR(/~) represents the variance exhibited by p. In detail, most of the studies that lead to Equation (1.3)-or something essentially equivalent-are for schistosomiasis [26], malaria [27], or other parasites whose transmission cycle involves an intermediate vector [28]; in this case, VAR( /I)more specifically corresponds to the covariance between rates of transmission from primary to intermediate hosts and from intermediate to primary hosts. In general, however, Equation (1.3) can represent any circumstances where spatial heterogeneity leads to variability in the transmission parameter [29,30]. Although sometimes acknowledging that the covariance effects could in principle be negative (for instance, patterns of human settlement could lead to patches of high vector density being associated with low host density, or dense urban aggregates could have higher standards of hygiene that effectively reduce transmission rates), most of the above authors have emphasized that the effects encapsulated in Equation (1.3) are likely to result in the actual value of R, being larger than that estimated from average contact rates, &,. If this is the case, it further implies that pC (the minimum fraction of the population to be successfully immunized in order to eradicate the infection) will be larger than estimated from Equation (1.2) using the simple average value of x* (the overall fraction of the population that are susceptible, before immunization). That is, it has usually been suggested that Equation (1.2) gives too optimistic an estimate of pC, if the effects of spatial heterogeneity are significant [29]. At last we come to the present paper. We consider a microparasitic infection in a host population that is divided in some way into n distinct subpopulations or groups. Within any one group we assume homogeneous mixing, with some common value j? for the intragroup transmission parameter. Contacts between individuals in different groups are assumed to be weaker, with the intergroup transmission parameter having a value EP (e Q 1). We further assume that individuals who have recovered from infection are immune for life. Within the framework of this simple yet natural and fairly general model, we show that the value of p, is indeed larger than would be estimated from Equation (1.2)-that is, estimated by assuming the population is homogeneously mixed-provided we assume the proportion immunized is the same in all groups. Alternatively, however, we investigate the “optimal” eradication program, defined as that which employs the smallest number of immunizations consistent with overall eradicution of the infection; in general, this optimal program will require immunization of different fractions in different groups. We show that the overall proportion of the total population immunized under this optimal eradication scheme is smaller than would be estimated by treating the population as homogeneously mixed [Equation (1.2)]. (Note, incidentally, that this definition of “optimal” is essentially a static one, referring to the fraction of newborns immunized each

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year in each group, This should not be confused with other studies of “optimal control,” such as those by Sethi [31] or Wickwire [32], which ask different questions about the dynamics of changing rates of immunization within homogeneously mixed populations.) In short, treating this spatially inhomogeneous population as if it were homogeneously mixed will, in general, lead to an assessment of p, that is too optimistic if immunization is applied uniformly, immunizing the same fraction in each group. But this assessment of p, will be too pessimistic for an optimally designed program of immunization. The former observation accords with those made by earlier authors, as noted above; the latter observation is a new result. This paper is organized as follows. Section 2 sets out the general mathematical model, and indicates the criteria for eradication of infection under various assumptions about the immunization program. Section 3 obtains explicit results for x* and pc under an optimal eradication program, assuming the transmission parameter has one value within groups, and another between groups (as specified above); the distribution of the population among the n groups, however, remains arbitrary. Section 4 explores a particular example in detail, giving a variety of graphical results: in this example a fraction (f) of the population is in one large “city,” while the remaining fraction (1 - f) is divided equally among m “villages.” Section 5 briefly considers another explicit example, in which the population is divided among the n groups according to a negative binomial distribution (with a “clumping parameter” k describing the extent to which the groups show more aggregation than they would if individuals were distributed among groups independently and randomly). Section 6 briefly summarizes the salient conclusions and shortcomings of this work.

2.

THE PROBLEM

DEFINED

We follow Hethcote [El, Post et al. [24], and others in considering n groups, with the population of the i th group being N, (i = 1,2,. . , n). Births and deaths are assumed to balance in each group, so that all N, are constant. The population in each group is made up of three classes: susceptibles, X,; infectious, x:; and recovered and immune individuals, Z,. New infections appear in the i th group at a rate equal to the number of susceptibles times the “force of infection” A,; in this spatially inhomogeneous situation A, is a weighted sum over all infected individuals: n

4= Here

the transmission

parameter

c 4,‘:.

(2.1)

J=t p,,

represents

the probability

that

an

ROBERT M. MAY AND ROY M. ANDERSON

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infectious individual in group j will infect a susceptible individual in group i, per unit time. The dynamics of this system then obeys the standard set of differential equations, dX

2 =pN,-(A, dt

+p)x,,

dY dt

-=A,X,-(o+p)x. Here p is the per capita birth rate, assumed equal to the death rate and the same in all patches. This assumption of a constant, age-independent death rate p (corresponding to the ecologists’ “Type II survivorship”) is made in most epidemiological models, on grounds of mathematical simplicity rather than realism. The assumption is discussed in detail elsewhere [lo], where its outcome is shown to give results closely approximating those obtained with realistic, age-specific vital rates. Infectious individuals recover to the immune class at a rate u, and immunity is deemed lifelong. Although a third set of differential equations can be written for 2, (t), the three equations are not independent, but rather are related by X,(t)+I:(t)+Z,(t)=N,=constant.

(2.4)

The model can be extended to include classes of newborn hosts protected against infection by maternal antibodies, or of latent (infected but not yet infectious) hosts. For more full discussion of this general model, and its possible extensions and refinements, see Hethcote [8] and Anderson and May [lo, 111. EQUILIBRIUM

The equilibrium values of the numbers of susceptible and infectious individuals in the ith patch, X,* and y*, are obtained simply by putting all time derivatives equal to zero in Equations (2.2) and (2.3). For X,* this gives

(2.5) Using Equation (2.5) in the equilibrium version of Equation (2.3) now gives an expression for q* in terms of ATT,which can be substituted into Equation (2.1) to give a set of n simultaneous equations for the n quantities AT (i=1,2 ,..., n): xy =

(2.6)

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Thus if the distribution of the host population among the n patches, N, , and the spatially inhomogeneous transmission parameters, /I,,, are specified, we can evaluate XT from Equation (2.6). Substituting these quantities into Equation (2.3, and summing over all patches, we can find the value of the total fraction of the population that is susceptible, at equilibrium: x*=

2 JT. i=l

Here N is, of course, the total population

ERADICATION

(2.7)

size: N = IX,N,.

CRITERION

Suppose a constant fraction, p,, of newly born individuals in group i are effectively immunized, essentially at birth. This corresponds to replacing the term p N, in Equation (2.2) by PN, (1 - pi); a compensating term pN,p, will appear on the RHS in the equation for dZ,/dt, corresponding to newborns appearing directly in the immune class. At equilibrium under this immunization schedule, the force of infection in i th group, I:, will be given by the appropriately modified version of Equation (2.6):

(2.8) As discussed by Hethcote [8], Post et al. [24], Schenzle [16], and others, the eradication criterion corresponds to all h: + 0. That is, a set of p,-values corresponding to eradication obeys the set of relations (i = 1,2,. . . , n)

(2.9)

Here a,, = 1 if i = j, 6,, = 0 if i + j. Equation (2.9) represents a set of homogeneous linear equations in the variables AT, corresponding to the minimal set of p,-values that eradicate infection. For a consistent solution of Equation (2.9) we require det]A,,] = 0.

(2.10)

Here A,, is the n X n matrix whose elements are given by the expression inside the square brackets in Equation (2.9). Equation (2.10) gives a constraining relationship among the immunized fractions p, in the various groups that are consistent with eradicating the infection.

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90 UNIFORM

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M. MAY AND ROY M. ANDERSON

TION PROGRAM

If the fraction immunized is the same in each group, regardless of its relative size, we have p, = p for all i. This critical value of p = pc is then uniquely determined by Equation (2.10); this corresponds to determining l/(1 - p,) from the eigenvalues of the n x n matrix whose elements are p,,N,/( u + p). This is essentially the formal result obtained by Hethcote [8]. The result is also broadly similar to those obtained for age-related (rather than spatial) inhomogeneities by Schenzle [16].

OPTIMAL

IMMUNIZATION

PROGRAM

As defined above, the optimal schedule is that which minimizes the total number of immunizations delivered per unit time. We denote the fraction of newborns in the i th patch that are immunized under this optimal schedule as $i. The overall optimal fraction of the total population then immunized is (2.11)

Here f, is the fraction of hosts in the i th group, f, = N,/N. It is also useful to define Q as the total fraction optimal program:

(2.12) not immunized

P.

Q=l-

under the

(2.13)

Determination of the optimal schedule is now a standard problem in maximization subject to constraints. We require to minimize P (that is, maximize Q), subject to the constraints that the set of j, obeys Equation (2.10), and that all pi obey 12 a, 2 0.

3.

OPTIMAL

ERADICATION

The problem to this transmission parameters, Introduction, where the tragroup contacts, and contacts:

FOR “COUPLED

GROUPS”

point is formulated in terms of a general set of pi,. We now specialize to the case described in the transmission parameter has one value, /?, for inanother value, I$ (where F< l), for intergroup

P,, = fib,,

3

b,,=E+(l-E)??,,.

(3.la) (3.lb)

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Here a,, is the Kronecker delta defined following Equation (2.9). E + 1 we will clearly recover the conventionally assumed case neous mixing. We now proceed to evaluate Q,, the maximum fraction not under the optimal eradication program, for this system. First, tionally convenient to define the quantities qi (the unimmunized the i th patch), s,, and p as

In the limit of homogeimmunized it is notafraction in

(3.2) (3.3) (3.4) The quantity p essentially sets the scale of the basic reproductive rate of the infection: in the limit e + 1, p + R,. With these definitions, the task is to maximize the quantity Q of Equations (2.11) and (2.13),

Q=

i

s,,

(3.5)

1=1

subject to the constraint

(2.10), which now reads detlpbiiS, - a,,1 = 0.

(3.6)

This determinant is evaluated explicitly in Appendix Equation (3.6) reduces to the condition

I, where it is shown that

(3.7) We also require all 1 > q, > 0, which corresponds

(i =1,2 >..,, n). This problem can be solved by standard For each i, we require

aQ YJF jpt=o. I

to

Lagrange multiplier

techniques.

(3.9)

I

Here F( sl, s2,. . , s,,) is the function defined by the LHS of Equation (3.7) and y is a Lagrange multiplier. Using Equations (3.5) and (3.7) in Equation

ROBERT M. MAY AND ROY M. ANDERSON

92 (3.9), we have

l-

&y&

That is, choosing the negative subsequently be satisfied,

square

root so that Equation

1-(Y4’*

ps =

Finally, the Lagrange multiplier y may be determined s,-values of Equation (3.11) into Equation (3.7), to get

( YW’)1’2 = 1

_

(3.7) may

(3.11)

1-E

I

(3.10)

=O.

I’+nE .

by substituting

the

(3.12)

We thus arrive at the result for the critical fraction remaining unimmunized in the i th patch, q,,c, under an optimal immunization program:

l

ff9r,c=s,= p(l-E+H&)' The total fraction not immunized n patches [Equation (3.5)]:

is obtained

PQ,= l-~+nE' ’

by summing

(3.13)

this result over all

(3.14)

Some aspects of this optimal result should be noted. The first comment is primarily a mathematical one. The basic result, Equation (3.14), is obtained under the assumption that all the s, of Equation (3.13) obey Equation (3.8). Substituting from Equation (3.13) into Equation (3.8), we see that Equation (3.14) will indeed be the solution provided 1 Pfl a

(3.15)

l-E+Pl&’

for all i. This is likely to be the case if p >> 1, so long as none of the f, are very small. The second, related comment is primarily biological. Equation (3.13) corresponds to the simple condition that, under the optimal immunization policy, q,f, = constant. That is, the fraction remaining unimmunized in small groups is greater (in inverse proportion to the group size) than in large groups (provided p >> 1, as discussed in the preceding paragraph). This

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makes good sense; it is intuitively reasonable that attention should be differentially focused on high levels of immunization in large urban aggregations, where transmission is high. This optimal program, however, contrasts markedly with the uniform immunization coverage assumed in most previous analyses: the optimal program has q, = constant/f, in contrast with the usually-assumed q, = constant. The differences between these two policies will be increasingly pronounced as the variation in size among the groups increases. For a homogeneously mixed population, Q, would [from Equation (1.2)] be equal to x*, the total fraction susceptible in the preimmunization population. For the present model, x* is found from Equations (2.5) and (2.7) with AT determined from Equation (2.6). Although an explicit solution of this system of equations is not in general possible, we can obtain an excellent approximation in most situations by noting that XT B p (that is, the average age at infection, l/AT, is typically much less than the average life span, l/p). Replacing the factor X:/(X: + p) by unity in Equation (2.6) gives us an approximate value, x,, for the preimmunization force of infection in the i th patch: (3.16)

That is, using Equation

(3.1) for bi, and remembering x:=/.&+(1-e)f,].

that Zif, = 1, (3.17)

Similarly replacing XT+ p by x, in Equation (2.5) for X,, we obtain approximation to the equilibrium value of the fraction susceptible, x’:

x’=pC n F.fl from Equation

(3.17) into Equation px’=

t i-1

(3.18)

I

i=l

Substituting

an

(3.18) gives the final result:

ft

&+(I- &If,

Notice that the essential approximation underlying XT B p, which, from Equation (3.17), corresponds p s+ l/[& +(l~)f,] for all i. This is much the same for there to be a finite level of immunization in all constrained optimal solution is an interior solution), In short, both Equations (3.14) and (3.19) are usually

(3.19)

this result was simply to the approximation as the condition (3.15) n patches (that is, the giving the result (3.14). valid so long as p B 1,

ROBERT M. MAY AND ROY M. ANDERSON

94

corresponding to an infection with a reasonably Equation (3.4), which defines p]. We conclude this section by proving that

high reproductive

Q,. > x‘.

rate [see

(3.20)

That is, we prove the total fraction of the population not immunized under the optimal eradication program is greater than would be estimated (x’ = x*) assuming the population to be homogeneously mixed. (As just emphasized, both the expression for Q, and the approximation x’= x* will in general hold if p > 1.) To prove this inequality, we first observe that the average value of the fraction of the population in any one of the n patches is f = l/n. Equation (3.14) can thus be rewritten as (3.21) Then, from Equations

(3.19) and (3.21), we have

P(Qc-x')=E+(lE-E)/~=~

f-f! 2 ,-t(l-,)~~

(3.22)

Were it not for the appearance of f, in the denominator of the expression being summed, this summation would clearly give zero. As it is, the quantity ~+(l~)f, is smaller when f, f; that is, the coefficient of f - f, is larger when i - f, is positive than when it is negative. Therefore, the sum in Equation (3.22) is positive, or at very least zero, for all f,, whence the inequality of Equation (3.20) follows. The equality pertains either when E = 1 (which corresponds to homogeneous mixing) or, less trivially, when f, =f for all i (all groups of equal size). [We note, parenthetically, that if all groups are of equal size, we effectively have a homogeneously mixed population. Although p,, is not constant, having different values within groups and between groups, the essential property of homogeneity-namely, each individual has the same average experience as any other individual-is fulfilled. It is a simple exercise to show that, if all f, = l/n, then we have the exact result Q, = x* = x’ = l/R,. Here R, system,

is the basic reproductive

rate of the infection

R,=p(r++i_

(3.23) in this symmetrical

(3.24)

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With such symmetry, the optimal immunization program is identical with the uniform one, and it can also be seen that the value 4c = 1 - p, obtained from Equations (2.9) and (2.10) is just q, = Q,.] Having obtained the very general results (3.14), (3.19), and (3.20) for Q,, x’, and their relative magnitudes in the limit p B 1, we now turn to a more detailed study of some explicit examples.

4.

EXPLICIT

EXAMPLES:

“CITY AND VILLAGES”

For an explicit example with a manageable number of parameters, we consider the situation where a fraction f of the total population live in one big “city,” while the remaining fraction, 1 - f, are equally divided among m

1 8

Z8 .N

Q FIG. 1,

Fraction Curve A shows the critical

living fraction

in

city,

not immunized

f (multiplied

by p ),

pQ,,

under the optimal eradication program: curve C shows the corresponding quantity, py,, under a program that immunizes the same fraction in every patch: and curve B shows the total fraction susceptible before immunization (multiplied by p). p.x*. All quantities are shown as functions of f9 the fraction of the population living in the one “city,” as distinct to from the m “ villages.” These curves are all for p --* cc [where p is roughly proportional the reproductive rate of the infection, Equation (3.4)], in the case where m = 10 and P = 0.1. The features of these results are discussed in the main text. Note in particular that the optimal program (Q, , curve A) requires fewer immunizations, and the uniform program (4,. curve C) more immunizations, than would be estimated by assuming the population to be homogeneously mixed (that is, estimated from x’, curve B).

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ROBERT

small “villages” [with a fraction (1- f)/m total number of patches is now n = m + 1.

OPTIMAL

M. MAY AND ROY M. ANDERSON

in each village]. Note that the

ERA DICA TION PROGRAM

From our exact general result, Equation (3.14), the critical fraction remaining not immunized under an optimal eradication program, QC, is m+l

Q, = P(rn&Sl) Under the optimal program, from Equation in the city, ql, is

(4.1)

(3.13), the fraction unimmunized

1

41 =

(4.2)

Pf(1-t me> . -8

FIG. 2. As for Figure 1, except now p = 40. Note the “edge effects” that are manifested by curve A for the optimal eradication program when p is finite: curve A drops down to join curves B and C at the right, corresponding to no immunization in villages for large f; conversely, the drop at the left corresponds to no immunization in the city for small f. The dashed curve B’ illustrates the approximation to the fraction susceptible before immunization, px’, as given by Equation (3.19). For fuller discussion, see the text.

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The corresponding

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fraction unimmunized

in each village, q2, is

q2=p(l--f;;l+m)

(4.3)

This “interior solution,” with a finite fraction immunized in both city and villages, will pertain so long as both q1 and q2 are less than unity. If f is sufficiently close to unity that the RHS of Equation (4.3) exceeds unity (most people living in the one “city”), we put q2 =l. That is, no “village” people are immunized and the optimal program consists of immunizing only city people. The constraining relation (3.7) now takes the form of a simple equation for the proportion remaining unimmunized in the city, ql, and gives Pf41 = The quantities

l-4-f)P

(4.4)

l-b(l-f)p'

a and b are defined for notational

convenience

as

l--E

a=&--

(4.5)

112 ’

(4.6)

b=(++;).

-08

18

0)

N .C z E .C 1

zL OL

I

I

I

I

I

I

0.5

cc

f FIG. 3.

Exactly

as for Figure 2, except

p = 20.

I

1 1

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ROBERT

The overall critical fraction remaining

Q,.=h

M. MAY AND ROY M. ANDERSON

not immunized

in this “urban”

limit is

+(1-f)-

(4.7)

with q1 given by Equation (4.4). In the extreme f + 1, we recover the case of homogeneous mixing (with everyone living in the city and Q< -+ l/p). Conversely, if f is sufficiently small for the RHS of Equation (4.2) to exceed unity (most of the population living outside the city in villages), we put qi = 1. That is, no city people are immunized, and the optimal policy consists of immunizing only village people. This does not make much sense biologically, as our “city” is now smaller than our “ villages” unless m is very large, but we need to consider this limit for mathematical completeness. The constraining relation (3.7) in this opposite extreme gives an equation for the fraction remaining unimmunized in the villages, q2, which leads to the result

P(l-“0% = Again,

a and b are defined by Equations

PO: Q

yg$.

0.5

(4.8)

(4.5) and (4.6), respectively.

The

i

f FIG. 4. As for Figures 2 and 3, except here p = 10. Notice that curve A generally lies less Par above, and curve C less far below, curve B as p becomes smaller. Curve B’ also becomes a poorer approximation to curve B as p decreases.

IMMUNIZATION

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99

of the total population unimmunized in this “rural” limit is

Q,=f+(l-.f>qz,

(4.9)

with q2 given by Equation (4.8). These results, Equations (4.1), (4.7), and (4.9), for Q,. as a function of f are plotted in Figures 1-6, for various values of E (the ratio of transmission rates between groups to those within groups),

m (the number of separate

villages) and p (a measure of the reproductive rate of the infection). The features of these figures will be discussed below, after we have dealt with x* and q, as functions off. FRACTION

SUSCEPTIBLE

The preimmunization villages,

BEFORE

IMMUNIZA

TION

values of the force of infection

AT and A;, follow from two equations

obtained

in the city and in from the general

f FIG. 5. The effect of changing E, the ratio of intergroup to intragroup transmission. upon the results shown in Figures 1-4. Specifically, we compare the critical fraction remaining unimmunized under an optimal eradication program (multiplied by p). pQ, (shown as solid lines), with the estimate based on treating the population as if it were homogeneously mixed, px’ (shown as dashed lines), as functions of f for p = 30, m = 10, and E =1,0.3,0.1,0.03, as indicated. The essential features shown in Figures l-4 remain, unless e is close to unity.

100

ROBERT

expression

M. MAY AND ROY M. ANDERSON

(2.6):

(4.10)

(4.11)

As above, a is defined by Equation (4.5). The equilibrium fraction susceptible, x*, in turn then follows from Equations (2.5) and (2.7):

*-lL!!l+

px-q+p

PL(l-f) x;+j.l

(4.12)

For specified values of f, E, m, and p, Equations (4.10) and (4.11) can be solved for the dimensionless variables ATT/p and X;/p, and x* then com-0 al N .-

do- -\

\ \ \

c

E’

.-E C 3

m-m

\

-

\

\

\

\\

\

\\ \\

\

FIG. 6. Similar to Figure 5, showing the effects of changing the number of villages. nz. among which the rural population is subdivided. The solid curves show pQ, and the dashed curves show px*, as functions of f, for p = 30, e = 0.1. and m = 2.10.30, as indicated. The broad features of Figures l-4 are maintained in all cases, being least pronounced for nz = 2 and most pronounced for m -t 03.

IMMUNIZATION

IN HETEROGENEOUS

POPULATIONS

101

puted from Equation (4.12); the quantity ).L does not really enter the calculation, serving only to set the scale of A. These numerical computations can be easily executed, as an iterative scheme based on the approximate results discussed below (and in Section 3) converges rapidly. The results are shown in Figures l-6. The approximate values discussed in Section 3 for the city (Xi) and country (X;) forces of infection, and for the fraction susceptible at equilibrium before any immunization (x’), here take the explicit form [see Equations (3.17) and (3.19)]: (4.13) x;

=~p

(l- E)(l-f)

E+

[

(l-f)

f px’=

(4.14)

m

C+(l-,)f

(4.15)

+ &+(l-E)(l-f)/WZ’

This expression for x’ is also shown in Figures 1-4, in order to indicate the accuracy of the approximation. It will be seen that x’ gives a good approximation to the exact fraction susceptible, x*, even for quite modest values of p. Note also that x’ > x* for all values of f and p: in Section 3 we proved that Q, > x’, and thus, a fortiori, Q, > x*. UNIFORM

IMMUNIZATION

PROGRAM

In this case, as discussed earlier, the fraction q = 1 - p, in all groups. For the city-villages appropriately specific form of Equation (3.7):

not

immunized is the same, model, q is given by the

E(l_f)W 1+

(l-:;;q-l

+

(l-e)(l-f)pq/m-1

=O.

(4.16)

[This is equivalent to finding the eigenvalues of the matrix in Equations (2.9) and (2.10).] Equation (4.16) reduces to a quadratic equation for q, of the form (4.17) Here the coefficients

with

CI and

A and B are defined as A=bf(l-f),

(4.18)

B=f

(4.19)

+a(1-f),

b given by Equations

(4.5) and (4.6) above.

The critical,

ROBERT

102 maximum eradication

fraction remaining not immunized program, q, , is thus given by

P4, =

VERSUS OPTIMAL

under

a uniformly

B-(B2-4A)L’2 2A

Figures l-4 also show q, as a function in all cases. UNIFORM

M. MAY AND ROY M. ANDERSON

applied

(4.20)

of f, and it can be seen that x* 2 q,

IMMUNIZATION

PROGRAMS

Figure 1 compares the critical fraction not immunized under an optimal eradication program (Q,., curve A) and under a uniform immunization program (q,,, curve C) with the estimate of this critical fraction obtained by treating the population as if it were homogeneously mixed (x*, curve B), in the limit p B- 1. Specifically, Figure 1 compares the relative values of pQ,., living in the city, px*, and pq, as functions of the fraction of the population f, for m = 10 (ten villages) and E= 0.1 (transmission between groups one-tenth that within groups), in the limit p ---)00. We see that the three curves merge at f=l,atf=O,andatf=&,all o f whihc correspond to situations where all groups are equivalent, effectively giving homogeneous mixing (in the limit p -+ cc, curve A is a horizontal line, plunging to join the other curves at f = 1 and f = 0). For moderate-to-large values of f in this example, however, it is apparent that Q,. can substantially exceed x*, and that qc can lie substantially below x*. That is, for this particular example, estimates of how close one must come to 100% immunization (p + 1) in order to eradicate infection that are based on assuming homogeneous mixing (that is, based on x*) can be too optimistic by factors of around 2 under a uniformly applied immunization program, and too pessimistic by factors also of around 2 under an optimal program. Figures 2-4 similarly show pQ,, (curves A), px* (curves B), and pq, (curves C) as functions of f for m = 10 and E= 0.1, but now for finite values of p. Specifically, p = 40 in Figure 2, p = 20 in Figure 3, and p = 10 in Figure 4. The dashed curves B’ in Figures 2-4 illustrate the approximation to the total fraction susceptible, x’, that was obtained above in Equations (3.19) and (4.15); it can be seen that x’ is a reasonable approximation to x* even when p is as small as 10, and becomes an excellent approximation as p increases. The essential features of Figures 2-4 remain as discussed for Figure 1. The main difference between Figures 2-4 and Figure 1 is that the critical fraction remaining unimmunized under the optimal eradication program, Q(, falls steeply as f-1 [or as f + 01, corresponding to the situation where no villagers are immunized [or no city dwellers are immunized] in a population that is predominantly urban [or rural]. Notice, incidentally, that the critical value p, of p (as a function of f for m = 10 and e = 0.1) below

IMMUNIZATION

IN HETEROGENEOUS

POPULATIONS

103

which the infection cannot maintain itself even in the absence of immunization, is given simply by putting qc = 1 in Figures 1-4; that is, curve C (which is the same in all four figures) can be taken to express p, as a function of f. Figure 5 compares the critical fraction Q, that need not be immunized under an optimal eradication program (the solid curves) with the estimates x* obtained assuming homogeneous mixing (the dashed curves), as functions of f, for m=lO, p=30, and various values of E (~=1,0.3,0.1,0.03, as indicated). Likewise, Figure 6 compares Q, (solid curves) with x* (dashed curves) for ~‘0.1, p=30, and various values of m (m=2,10,co, as indicated). The essential features remain as before, with estimates based on x* leading to underestimates of Q< that can be substantial for some values of f, particularly if m is not too small. In brief, the patterns exhibited by Figures l-4 for m = 10 and E= 0.1 remain broadly true for other values of m and e. 5.

SPECIFIC

EXAMPLE:

“DISPERSED

AGGREGATIONS’

The “city-villages” example of Section 4 has the advantage that it permits an explicit and detailed analysis, and the accompanying disadvantage that it is too artificial to be useful for other than illustrative purposes. In this section we explore the somewhat more realistic situation where the distribution of hosts among the n patches, {f, }, is described by a negative binomial distribution, with a “clumping parameter” k describing the degree of aggregation (small k corresponds to high aggregation, with a few patches containing most of the host population; k + co corresponds to a Poisson distribution, with hosts distributed independently and randomly). Our analysis proceeds along the lines laid down to Section 3, comparing Q,. and x* in the limit p > 1. The generating function for the negative binomial distribution with mean M and clumping parameter k is g(z)=[l+(l-z)M/‘k]~‘.

(5.1)

That is, the probability of finding j individuals in a given patch, p(j), is given by the coefficient of ZJ in the power series expansion of g(z). The mean number of individuals in a patch is M, and the meaning of the parameter k is seen by noting that the coefficient of variation, CV, from this distribution is given by cv2

E

variance =1+1 M mean2

k’

(5.4

A fuller account of the properties of the negative binomial are given from a mathematical standpoint by Anscombe [33], and from a biological standpoint by Southwood [34] and May [35].

ROBERT

104

M. MAY AND ROY M. ANDERSON

For p z+ 1, the overall fraction of the population who are susceptible before immunization is implemented, x*( p + co) = x’, is given by Equation (3.19), which we repeat: px*=

2 z+(l-fl

,=I

(5.3)

&If,

Here f, is the fraction of the population found in the i th patch; f, = j/N if the patch contains j of the total population of N individuals. For a population that is negatively binomially distributed among patches, the average value of the summand in Equation (5.3) for any one patch is

Here p(j) is the negative binomial probability of finding j individuals in the patch; the mean of the distribution is M = N/n. If the number of patches, n, is reasonably large, we may approximate px* by multiplying this average value of the summand in Equation (5.3) by the number of patches: px*

_@(j>

nc

3

N j e+(l-e)(

(5.5)

j/N)

(This approximation becomes less reliable as n becomes smaller: for n = 2, the number of hosts in the second patch will be N - j if there are j in the first patch, and averaging separately over the two patches is clearly invalid.) The summation in Equation (5.5) is performed in Appendix II, and for N B 1 we get the result

px*=“e”E,+,(K).

(5.6)

E

Here E,,(z) is the exponential

integral of order n [36], and

K

is defined

as

nke

K=l--E.

(5.7)

The result (5.6) for the fraction susceptible, x*, is to be contrasted with the critical fraction not immunized under an optimal eradication program, Q, , which as before is given by Equation (3.14): pQ< = n/(1 - E + no). Remember, both results are for the limiting case p x=-1. Figure 7 compares pQc (the solid horizontal line) with px* (the dashed curve) for a range of values of the clumping parameter k, in the case where n = 10 and E = 0.1. Figure 7 for the negative binomial corresponds to Figure

IMMUNIZATION

IN HETEROGENEOUS

105

POPULATIONS

“,s N .-c

;

PQC

.

__---

5,/-

E .-c 3

,

4-

/

/’

,/;x*

: w 0 .a : fn 3 ln

3-

2-

C

4-

/ / /

,

/

,

/

/

,

/

,

/

/

/’ .-0

=: 5

.’ _cH

.’

0.01

clumping

Q FIG. 7.

Assuming

II

0.4I

the total population

40I

parameter,

k

to the apportioned

among

,z patches

according

to a negative binomial distribution, we show pQ, (solid horizontal line) and pi* (dashed curve) as functions of the “clumping parameter” k, in the limit p + cc (and with n = 10 and E = 0.1). The fraction susceptible before immunization, x*. provides a good estimate of the fraction that need not be immunized under an optimal eradication program. Q, , for

large k (corresponding to hosts distributed independently randomly among patches), but can provide a significantly pessimistic estimate for small k (corresponding to a clumped or overdispersed distribution). As discussed in the text, the basic tendencies exhibited here are similar to those shown in Figure 1 for the city-villages model.

1 for the city-villages example. Again it is apparent that the fraction remaining unimmunized under an optimal eradication program, Qc, can be substantially larger than would be estimated by assuming the population to be homogeneously mixed (that is, estimated from x*), if the distribution of hosts among patches is significantly clumped (k sufficiently small). In the limit k-+oo, corresponding to hosts distributed independently randomly among patches, all hosts have the same experience on average and the population is effectively homogeneously mixed; hence Q, + x* as k + 00, as shown in Figure 7. 6.

CONCLUSIONS

Several people have recently observed that spatial heterogeneities in the distribution of a population of hosts will usually (though not necessarily)

106

M. MAY

ROY M.

result in the immunization coverage required to eradicate an infection being higher than would be estimated [from Equation (1.2)] by treating the population as if it were homogeneously mixed. We have shown, however, that for a patchily distributed population in which intragroup transmission rates systematically exceed intergroup transmission rates, the coverage required under an optimal immunization program (defined as that needing the fewest total immunizations each year) is lower than would be estimated assuming homogeneous mixing. This result has been established in general, and elaborated in explicit detail for a “city-villages” model and for a population distributed negative-binomially among subpopulations. Figures 1-7 make the results plain, and further show that the differences between optimal and uniform immunization programs and estimates based on assuming homogeneous mixing are not necessarily picayune: treating the population as homogeneously mixed can underestimate the closeness to 100% coverage required for a uniformly applied immunization program by factors of 2 or more, and can overestimate the closeness to 100% required for an optimal program also by factors or 2 or more. The essential difference between uniform and optimal immunization programs is that the former aim to immunize the same fraction in each group (by definition), while the latter aim to leave unimmunized roughly the same number of individuals in each group [ q,f, = constant, Equation (3.13) except insofar as “edge effects” of the kinds illustrated in Figures 2-6 can leave some groups entirely unimmunized under the optimal procedure]. In practice, there will be obvious difficulties in estimating exactly what constitutes the optimal program, and in implementing it systematically, even if the basic model is taken as gospel. But the principal message emerging from the analysis of the optimal eradication program remains useful, telling us that-all other things being equal-attention should be differentially focused on the larger and higher density groups. This result is of some practical significance, given the recent implementation of the World Health Organization’s Expanded Program on Immunization (EPI). The aims of this program are substantially to reduce, by mass immunization, the incidence of certain common childhood infections-such as measles and pertussis-in the developing world; in these regions, such infections are often a significant cause of child mortality. In major cities, with dense populations and high birth rates, very high levels of vaccine coverage (probably greater than 95%) will be required to interrupt measles and pertussis transmission. It has therefore been argued that measles eradication programs cannot be as focused as was the smallpox eradication scheme, where success resulted from outbreak and case containment with general immunity results of often less than 50% in Africa and Asia. One view is that very high levels of measles immunization would have to reach virtually all regions of a given country [36,37]. Irrespective of the practical problems

IMMUNIZATION

IN HETEROGENEOUS

POPULATIONS

107

inherent in such an approach, we suggest that this is unlikely to be the case, since the rural-urban differences in population density and birth rates are often substantial in developing countries. Indeed, it is likely that in many low density rural areas, infections such as measles could not persist endemically (i.e., R, < 1) without repeated introductions from nearby urban centers. Under such circumstances, our optimal immunization program suggests that effort should be focused on high density urban centers, an approach with many practical advantages. This insight could be confounded if the intragroup transmission parameter /3 varied among groups, instead of being constant as we assume; in particular, if /3 had a propensity to be larger in smaller groups (for example because of lower hygiene in rural areas), we could have the VAR( /?) term negative in Equation (1.3) with the consequences discussed in Section 1. This is unlikely to be the case, however, for infections such as measles and pertussis. Beyond its special assumptions about p,, , our model has many other gross simplifications. The spatial heterogeneity we consider remains one in which each patch usually contains a large number of people. Thus the model may be suited to discussing heterogeneity on a geographical scale, but not to the fine-scale heterogeneities inherent in family units or social groupings (such as schools). Such microheterogeneity leads naturally to the larger questions of epidemiological differences among age classes and genotypes; as mentioned in Section 1, these complications are ignored here. In short, we are aware of the many shortcomings in the way our mathematical model oversimplifies or omits epidemiological details. The present discussion is, moreover, mathematically abstract, and detailed application to actual immunization programs would not be easy. But we do believe the model represents a useful first step in discussing the practical implications for immunization programs of large-scale spatial heterogeneity. Results like Figures l-7 help to clear the ground by sweeping away the misconception that spatial heterogeneity must necessarily make eradication more difficult than would be estimated assuming homogeneous mixing [that is, from Equation (1.2)]. APPENDIX

I

In this appendix we derive Equation (3.7) for the general equation (2.10), for the case where p,, is given by Equation (3.1). That is, we evaluate the determinant of the matrix A, where

(A.11 Here p and s, are as defined by Equations (3.4) and (3.3) respectively. If we subtract the final row of this matrix from each other row, we obtain a matrix in which the only nonzero elements are in the final row, the final

108

ROBERT

column,

and the diagonal. The determinant

0 .

. .

0 .

. .

EPS,

X n

matrix A is then

0 0

[(l-E)Pwl]

0 . .

D,,of the n

0

[(l-E)Pvll D,, = det

M. MAY AND ROY M. ANDERSON

-[(l-~)w,!-11 -[(l-&)w,,-ll

[(l-e)ps-,-l]

. .

.

.

EPSZ

“’ . .

-[(lb&)pS,,yl]

.

.

. . P s,, -1

EPS,

(A.4

Expanding

about the first column, we get

D,,-,+~ps~ fi [(l-&~-l].

D,,= [(l-E)PS,-l]

(A.3)

/=2

of the corresponding Here D,,- 1 is the determinant involving s, with j = 2,3,. . . , n. Applying Equation the result

Q, =

(n ~ 1) X (n - 1) matrix (A.3) to D,,_ I leads to

[(I- E)P%-11[(I- E)P$-11Q-2 +

(1-r,“;r:,-1 +(l-:p;5,-1 Iiii[(I,=I

t

4w, -11

i (A.41

Recursive

application

of Equation

(A.3) thus leads eventually

)I

l+ ,51(I$;;,

-1

fI [(l-e)PS,-l]

to

(A.5)

/=1

Equation (2.10) says D,, = 0.If we assume none of the factors in the grand product inside the curly braces to the right in Equation (AS) are zero (which will in general be true), then Equation (2.10) reduces to the requirement that the term inside the curly braces to the left in Equation (AS) must vanish. This gives Equation (3.7).

APPENDIX

II

This appendix derives the analytic result (5.6) for px* from the general formula (5.5), in the limit N B 1. To perform the summation in Equation (5.5) we first reexpress the denominator by using the identity 1 =Jomexp{ -[ &+(I-

E)(_i/N)

(A4

IMMUNIZATION

Equation

IN HETEROGENEOUS

POPULATIONS

109

(5.5) can then be written as PX

1 * *=-- M o 1

dre?‘&p( j)z’.

(A.7)

i

Here z is defined as z = exp ,[

-!y].

(A.81

The sum in Equation (A.7) can immediately be evaluated in terms of the generating function for the negative binomial, g(z), which is given by Equation (5.1). Specifically, the sum is equal to zdg/dz, whence m

px* =

1+

/0

1++l)&

Ml-z(t)1 k

(A.9)

Here z(t) is defined by Equation (A.8). In the limit when N is very large, as will usually be the case in real applications, Equation (A.8) gives z 4 1 and 1 - z --) t(1 - E)/N. Thus for N >> 1 we have (A.lO) The quantity K is as defined in Equation (5.7). By writing Equation (A.lO) can be brought to the form

slWY

px*+

-(k+ue-“’

1+

(jy.

et/~

=

y,

(All)

The integral here is the exponential integral Ek+ I(~), as defined and discussed, for example, in Abramowitz and Stegun [38]. The expressions for px* in the opposite limits of large and small k may be noted, remembering that K = kne/(l - E). For k B 1, we have (see [38, formula 5.1.521)

px* +

n l-E+Tl&

l-

n&(1-E)

+O(k-‘)

k(1

-

E+

TIE)*

1

(A.12)

For k + 0 we have (see [38, formula 5.1.111)

px*-+ [&][ln(s)-v--O(k)].

(A.13)

110

ROBERT

M. MAY AND ROY M. ANDERSON

Here y is Euler’s constant, y = 0.577.. . More generally, illustrated for n = 10 and E = 0.1 in Figure 7.

px* has the form

This work was supported in part by the NSF under grant BSR83 -03772, by the Rockefeller Foundation, and by the Department of Health and Social Security (Chief Scientist’s Office), U.K. REFERENCES 1 2

N. .I. T. Bailey, The Muthemutad Theov of Infectious New York, 1975. R. M. Anderson and R. M. May, Population biology Nature 280:361-367 (1979).

Discuses. 2nd ed., Macmillan. of infectious

diseases:

Part

I.

3

G. Macdonald, (1952).

4

K. Dietz, Transmission and control of arbovirus diseases, in Epidemiolom (D. Ludwig and K. L. Cooke, Eds.), Society for Industrial and Applied Mathematics, Philadelphia,

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The analysis

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in malaria,

Trap. Dis. BUN. 49:813-X29

1975, pp. 104-121. R. M. Anderson and R. M. May, Directly transmitted infectious diseases: Control by vaccination, Science 215:1053-1060 (1982). A. Nold, The infectee number at equilibrium for a communicable disease, M&h.

11

Biosci. 46:131-138 (1979). C. E. G. Smith, Prospects for the control of infectious disease, Proc. Rqv. Sot. Med. 63:1181-1190 (1970). H. W. Hethcote, An immunization model for a heterogeneous population Theoret. Population Biol. 14:338-349 (1978). P. E. M. Fine and J. A. Clarkson, Measles in England and Wales. II: The impact of the measles vaccination programme on the distribution of immunity in the population. Internut. J. Epidemiol. 11:15-25 (1982). R. M. Anderson and R. M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hrg. 90:259-325 (19X3). R. M. Anderson and R. M. May (Eds.), Population Biology of Ilfefrctrous Diseases.

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Springer, New York, 1982. H. W. Hethcote, Measles and rubella in the United States, Amer. J. Epidemiol. (1983). K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctua-

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of measles, Rer. Infectlour Dis. 4:933-939 (1982). Handbook of Mrrthemuticul Functions, Dover, New