On the Spread of Epidemics by Carriers

On the Spread of Epidemics by CarriersAuthor(s): George H. WeissSource: Biometrics, Vol. 21, No. 2 (Jun., 1965), pp. 481-490Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2528105 .

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ON THE SPREAD OF EPIDEMICS BY CARRIERS

GEORGE H. WEIss'

Institutte for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., U.S.A.

1. INTRODUCTION

Quantitative problems in the phenomenon of epidemnics have been of interest for over fifty years. Specific applications of the theoretical developments have been made to malaria (Ross [1911]; Martini [1921]), to measles (Wilson et al. [1939]; Bartlett [1957]), and to other com- municable diseases. Specific references may be found in the excellent monograph by Bailey [1957]. All of the diseases whose theory has been developed so far are such that a population can be divided into sub- populations whose members are considered to be susceptible, infected, or imnmune. However, there are several diseases in which carriers are a significant factor in the spread of the epidenmic. The prime example of these is typhoid, although carriers may be important in the spread of bilharzia, amoebic dysentery, and typhus. A carrier is defined to be an individual who does not have overt disease symptoms but nevertheless is able to communicate the disease to others. Under this category we may include not only human carriers but also inanimate sources of disease such as polluted streams which may be used by a fairly large population.

Diseases involving carriers are still important notwithstanding mod- ern health controls-as witness the recent outbreak of typhoid in Zermatt, Switzerland. In more primitive societies the problem can obviously be more acute. To date, no theory seems to have been developed to make quantitative the factors involved in a carrier-borlne disease. It is the purpose of this paper to analyze a fairly simple and admittedly incomplete model of an epidemic involving carriers. I will assume in the present work, that only carriers are responsible for the spread of the disease. By implication this assunmes that public health measures are efficient enough to isolate infected individuals who may be able to transmit the disease to others. This may not be as un- realistic as it sounds. Consider, for example, the case of typhoid. It is estimated that about one or two percent of all those who have re-

'Present Address: National Cancer Institute, National Institutes of Health, Bethesda, Md., U.S.A.

481



482 BIOMETRICS, JUNE 1965

covered from typhoid may harbor pathogenic bacilli within them (Smadel [1963]). Most of these individuals will have had all of the symptoms of typhoid and can be checked by public health authorities. How- ever, some people may contract the disease in such a mild form that the symptoms are unnoticeable. It is these people who are a potential danger to the community. In this paper I will try to give a theoretical estimate of the degree of this danger.

There are generally two types of epidemic theory: the deterministic theory in which the numbers involved are treated as continuous var- iables, and the stochastic theory which takes into account random effects. It is felt that the stochastic theory is a much better model of any actual situation, but the deternministic equations are usually much easier to handle. We will begin the analysis with a brief account of the determninistic theory of epidemics with carriers, mainly because the analysis is quite simple. Following the treatment of the deterministic theory we shall discuss the statistics of a particular parameter, the ultimate size of the epidemic. A complete account of the time-dependent prop- erties of the stochastic theory is quite complicated and will not be given in the present paper. By considering the embedded Markov chain in the description of the epidemic, it is possible to derive a result in a simple, closed, form for the ultimate size of the epidemic in a fixed population, due to the introduction of n carriers into that population.

2. THE DETERMINISTIC THEORY

Let m(t) be the number of susceptible individuals at time t, and let n(t) be the number of carriers at time t. It will be assumed in both the deterministic and in the stochastic theory that the instantaneous elimination of carriers will talke place at a rate (3n(t), where a is a constant, and that the rate of infection spread at time t will be aen(t)n(t), where a is likewise a constant. In addition we will define a parameter cr as the ratio

o- = a/ (1)

Then, by our assumption as to the mechanism of the spread of the epidemic we have the equations

mi(t) = -atm(l)n(t) (2)

(t)= - On(t) whose solutions are

n(t) = n(O) exp (-fSt) (3)

m(t) = m(O) exp [-on(O)(1 - eCt)].



SPREAD OF EPIDEMICS BY CARRIERS 483

Thus, the ultimate number of survivors of the epidemic is

m(cO) = m(O) exp [-un(O)]. (4)

It will be seen that the stochastic theory leads to essentially the same result for the mean when on(0) is small, but the deterministic theory overestimates the expected size of the epidemic when un(0) is not small.

3. THE STOCHASTIC THEORY

In what follows we will not discuss the complete time dependent stochastic theory since the equations are difficult to solve. However, when we restrict our attention to the statistics of the ultimate size of the epidemic, the treatment of the problem becomes much simpler. A similar analysis was given by Bailey [1953] for the statistics of the KIermack-McKendrick epidemic model. A complete time-dependent analysis can, however, be given rather simply for the case n(O) = 1. This will be discussed below.

Let us therefore define a function r,(in, n) to be the probability that a population of mn susceptibles are reduced to k survivors at the termination of an epidemic initiated by n carriers. Termination can occur either when the number of carriers is reduced to zero or when all members of the population are infected. We assume that the probability that a carrier is eliminated in (t, t + dt) is ir-dt where r is the number of carriers extant at time t. The probability of a carrier in- fecting a susceptible in (t, t + dt) will be axrsdt where s is the number of susceptibles at time t. It will be assumed that a susceptible who has become infected will not be a further source of infection but will be identified and removed from the population. Thus, we will be dealing with a random walk on a lattice specified by points (in, n) where mn is the initial number of susceptibles and n is the initial number of carriers. The permissible transitions allowed by the model are (in, n) -> (in, n -1) and (in, n) -> (m - 1, n) for m, n 5 0. The transition probabilities are

Pr {(m, n) (m - 1, n)} I = +l

Pr {(m,n) -(m,n--1)} =1+ I

in which o- has the same meaning as in Equation (4). With these transition probabilities we can write the equations satisfied by gr(in, n) in the form

7rk(mn,n) [=rk(rn,n - 1) ? nIr,k(m - 1,n)]/(l + orn), n - 0. (6)




A boun-dary condition is

Ik k(m, 0) = akm* (7)

The condition along the n axis (m = 0) is automatically taken account of by the transition probabilities of Equation (5) which allow only for the possibility (0, n) -> (0, n - 1). Certain of the "rk(rn ,n) can be derived directly. In particular we must have

lIk(k, n) = 1/(1 + k1o-), n # 0, (8)

since if there are no victims, all of the carriers have been eliminated independently and with the same probability.

In order to solve Equation (6) we introduce the generating functions co

17kmi(S) = Z Ik(r, n)s'. (9) n=1

It should be noted that nOkm(S) = 0 for m < k. When rm is greater than k equation (6) implies that the f7km(s) satisfy the recurrence relation

'7kmn(S) = f + - 7lk (m- 1) (S) (10) 1 + 0-mn - s

From Equation (8) it follows that f7kk (S) is given by

't7kk(S) -

1+ kr-s (11)

Equations (10) and (11) are easily solved to yield the expression

'7km(S) n s + 1

r(l - s + 1 -12 mr s CJ

r o s + + +

The next step in determining the 7rk(m ,n) is to decompose the product into partial fractions

i= kjJ?1 5 i-k,]0-?l -S (13)

where the Ai are

As - (i = )i 1 _ = A ) ) (T7-1'k1

- k) (14) (j- k)!(rn - 0-! - (rn- k)!j gk/ o-




This result implies that lkm(8) can be written

7km(s) (= n) E ( 1)jok (m k +) s (15)

or

rk(m, n) = ( Ek) E (-lI)i- (m_ k) (j 1 1) (16)

When m is large this sum may be difficult to evaluate numerically, consisting as it does of alternately positive and negative terms. Hence it is convenient to return to Equation (15) and cast it into a different form by means of the identity

1 - = f e(i+l-8) dt. (17)

jo-+ I - sO

When this is substituted into Equation (15) one finds

?Jk1Z,(S) = st(k) f e1t >? ) c(_1 -ei dt (18)

= m (m ) f e-(+ls)t(l - e-t)mdk dt

which leads to the representation

7rk(M, n) e( 11)! (k) fn t7-le(k+l)t(1 _ e-t)mk dt (19)

which is manifestly positive and clearly satisfies the normalization condition

? 7k(Mi n) - 1. (20) k=O

The moments of the distribution of the number of ultimate survivors are easily derived from the expression of Equation (18). The mean and variance are found to be

k = M/(l + oa)n

k2_2i=(inl)+ in in2 (21) (2cr?1) (0cr?1)n (0r?1)2

When on is small the exact mean, given by this equation, is to be com- pared with the number of survivors predicted by the deterministic theory, m exp (-an). The agreement in this limit is close. For larger on the expected ultimate number of survivors according to the stochastic model is greater than that predicted by the deterministic theory.




The result for ir(in, 1) follows directly froii Equation (12);

7rk(m, 1)

1 lm! rk +?)

_ m! 1

( m + , (ml-?1 + -(m 2 + 1) k +

In particular when cr-1= all of the 7r (in, 1) are equal:

7((M, 1) = m/(m + 1). (23)

This comapletely randoml distribution renders any determninistic theory wholly inadequate. When n = 2 one can find ani explicit expression for 7r(in, 2) by differentiating -q,,,(s)1s and setting s =0,

( ) 1 in! 0 {( )-( )

= km 2 rk(m, 1 _ + 1 + k +-(2

{r 4 m k+l? m} +

where A,(x) is the logarithmic derivative of the gamma function (see, e.g. Davis [1935]),

{t(x) -dyln r(x). (25)

Expressions for the rk (m, n) for greater values of n can be written in terms of polygamma functions, which are also tabulated by Davis [op. cit.]. Values of lrk(10, 1) and lrk(10, 2) are given in Tables I and II for cr - 0.2 and cr = 1.0. As expected, the introduction of a second carrier significantly affects the results for so smuall a population.

An important case is that of large numbers of susceptibles, and a large epidemic. Specifically we will assumne

m?1>,-= 0=0(1)x. (26)

dx




TABLE I

PROBABILITY OF k SURVIVORS FOR n = 1, m = 10, a = 0.2

k 0 1 2 3 4 5 6 7 8 9 10

wk(1O,l) .00033 .002 .005 .012 .023 .042 .077 .110 .165 .238 .333

TABLE II

PROBABILITY OF k SURVIVORS FOR n = 2, in = 10, o- = 0.2: o- 1.0

k 0 1 2 3 4 5 6 7 8 9 10

7Wk(10,2) a- = 0.2 .002 .011 .022 .043 .069 .103 .150 .164 .177 .164 .111 0f = 1.0 .275 .184 .138 .108 .085 .067 .052 .039 .027 .017 .008

where 0 is the percentage of the initial population which falls prey to the disease. The application of Stirling's formula to Equation (12) yields the result

7r, (m, n) Inl- - (27)

When n - 1 this expression is a monotone function of 0 (increasing if o- < 1 and decreasing if o- > 1). When n is greater than 1 and o- is less than 1 this expression has a maximum at

= exp n ). (28)

4. TIME DEPENDENCE FOR n = 1.

Although the analysis of the time-dependent evolution of a carrier borne epidemic is somewhat complicated, the case n = 1 can be discussed fairly simply. In this situation the epidemic can be terminated in one of two ways; either the carrier is eliminated or else the entire population contracts the disease before the carrier is eliminated. Let us define pi (t) to be the probability that the population is reduced from m to j in time t conditional on the presence of the carrier at t. The probability that there are j susceptibles in the population at time t is pj(t) exp (-,Ot), the last term being the probability that the carrier has not been climinated by time t. The pi (t) then satisfy the equations




Pm(t) = -mcpm(t)

pm-l(t) = magpm(t) - (m - 1)apmil(t) (29)

po(t) = ap1(t).

This is a simple death process with the solution (Bharucha-Reid [1960])

p1(t) = ( ")eiat(i - eat)tn (30)

If we denote the probability density for the duration of the epidemic by sp(t) then this function is given by

,p(t) = O3e-t[l - (1 - e-a t)m] + mae( a+)t (1 -e- a t)mtl (31)

The first term is the probability density assuming termination by elimination of the carrier and the second term represents termination by involvement of the entire population before the carrier is eliminated, since p(t) is the probability density for the event 'population number reaches zero at time t'. Moments of the time to cessation of the epidemic may be derived from the Laplace transform of sp(t). This function is calculated to be

f e"'tto(t) dt - _ + _ rn!Q ?t (32) 3 ?u ar +1?m 1)

In particular, the expected time to the cessation of the epidemic is

- I1 1 M !1r(

r(l + m + 1)

o- 1 _ m( ( 1)).(3

For large mn Stirling's approximation can be applied to the factorials with the resulting expression

Ti: 1- (1 1/ )1. (34)




5. DISCUSSION

It is rather difficult to estimate a froin any presently available data, but we imay get a rough order of maagnitude estimate from the results of the recent outbreak of typhoid in Zerna-tt. The normal population of Zermatt is of the order of 1,000 people, although at the outbreak of the epideimic people had arrived for a ski festival. Let us therefore put the population at about 1,500. Roughly 100 cases of typhoid were reported. We can then get a crude estimate of o- by equating expected numbers to actual numbers. In this way we find

m %-1

5 = (1 + n) (35)

If we put n -1 then o- = .072, for n = 2, o- = .034 and for n = 3, CT = .023. We imay conclude that the appropriate value of o- for the Zermatt epidemic was somewhere between 0.005 and 0.1, allowing for the uncertainties of our calculation. If we assume that it takes on the order of magnitude of a week to eliminate a carrier, i.e., A = 1 week-', then a ranges between 0.005 and 0.1 week-'. It may be presumed that in more primitive societies the range of CT might be as high as 1-10.

The present theory clearly omits several factors which deserve further study. In particular, it is imaportanat, but much mlore difficult, to treat the case of an epidemic in which the disease can be spread both by carriers who have had overt symptoms and by those who have not. The problem of estimation of parameters requires analysis even for the present, rather simple, model. Here one needs the complete time-dependent solution of the equations in order to develop experimentally feasible estimators. Another factor which may have been important in the Zermatt epidemic but which is not treated by the present theory is that of immigration and emigration. In any epidemics which occur in a non-primitive society, such factors will be of growing importance as transportation becomes easier.

Several of these topics will be the subject of future research in elucidating the role of carriers in the spread of a disease.

REFERENCES

Bailey, N. T. J. [1953]. The Total Size of a General Stochastic Epidemic. Bio- metrika 40, 177-85.

Bailey, N. T. J. [1957]. The Mathematical Theory of Epidemtics. London: Griffin. Bartlett, M. [1957]. Measles Periodicity and Community Size, J. R. Stat. Soc.

A120, 48-60. Bharucha-Reid, A. T. [1960]. Elements of the Theory of Markov Processes and their

Applications. New York: McGraw-Hill. Davis, H. G. [1935]. Tables of the Higher Mathematical Functions, Vol. II. Indiana:

Principia.




Martini, E. [1921]. Berechnungen und Beobachtungen zur Epidemiologie und Bekamp- fung der Malaria. Hamburg: Gente.

Ross, R. [1911]. The Prevention of Malaria. London: Murray. Smadel, J. [1963]. Intracellular Infection and the Carrier State. Science 140,

153-7 (also personal communication). Wilson, E. B., Bennett, C., Allen, M., and Worcester, J. [1939]. Measles and

Scarlet Fever in Providence, R. I., 1929-35 with Respect to Age and Size of Family. Proc. Am. Phil. Soc. 80, 357-476.



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