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Dynamics of acquired immunity boosted by exposure to infection
Dynamics of Acquired Immunity Boosted by Exposure to Infection JOAN L. ARON Biology Department, Princeton University, Princeton, New Jersey 08544, ond Biometry Branch, L.andow Building SC - 19, National Cancer Institute, Bethesda, Maryland 20205 Received 29 December 1982; revised 23 February 1983
ABSTRACT A mathematical model is presented to describe the dynamics of immunity which can be boosted by reexposure to infection. Immunity is assumed to last until a specified interval of time elapses without an exposure. This assumption is incorporated into a compartmental model as a differential-delay equation. When the model is applied to malaria epidemiology, the prevalence of disease among adults is greatest at intermediate rates of infection. Observed
age-prevalence
curves
have
shapes
similar
to those
of the predicted
curves.
Immunity in an individual is formulated in terms of a stochastic process, and an expression for the average duration of immunity is obtained. The average duration of immunity increases with the rate of exposure, but the presence of mortality (or ether kinds of removal) risks.
I.
shortens
the average
duration
observed,
in analogy
with the theory
of competing
INTRODUCTION
Mathematical models of the epidemiology of infectious diseases have long incorporated some notion of immunity as refractoriness to infection acquired after an initial infection (Bailey [l]). In most models, the duration of immunity is prescribed independently of continued exposure to infection. However, epidemiological evidence suggests that in some cases, such as malaria (Boyd [2] and Molineaux and Gramiccia [3]), the duration of immunity is extended by continued exposure to infection. Recently, mathematicians have begun to describe the interdependence between exposure to infection and the duration of immunity for malaria. In a continuum model (Elderkin et al. [4]), levels of “resistance” depend dynamically on levels of parasites in the blood. A more epidemiological model (Dutertre [5]) contains a discrete class of immunes whose reversion to susceptible is retarded by reexposure calculated on a monthly basis. MATHEMATICAL
BIOSCIENCES
64:249-259
QElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017
(1983)
249 0025-5564/83/$03.00
JOAN L. ARON
250
In this paper, the model is a continuous-time process in which immunity lasts until the occurrence of a gap of T years without exposure. The underlying assumption is that acquired immunity gradually weakens in the absence of exposure to infection but quickly strengthens if additional exposure occurs before T years elapse from the previous exposure; on the other hand, if exposure occurs after T years, the individual, no longer immune, develops an infection. In addition, mortality may remove an immune individual at any time. This model in effect incorporates a locking time for a Type II counter (Karlin and Taylor [6, p. 1791) and standard assumptions on the competing risk of mortality (Gail [7]).
II.
FORMULATION
AS DIFFERENTIAL-DELAY
EQUATION
Striking features of the age-specific pattern of prevalence (proportion infected) in cross-sectional surveys may be due to the dynamics of immunity boosted by reexposure. Formulation of a differential-delay equation model permits the analysis of the effect of immunity in a model of prevalence. Linear models with transitions between “compartments” of susceptibles, infecteds, etc. are useful in the construction of age-prevalence curves (Muench [8]). It is straightforward to incorporate into this kind of model the number of immune individuals, z(t), whose dynamics obey the assumptions stated in the introduction. If conditions for transmission of infection have not changed dramatically over the lifespan of the oldest member of the population, then trends with increasing age are equivalent to changes of a birth cohort through time. Consequently, an equation is formulated for z(l), and then this equation is used in an age prevalence model where t is age. The rate of change of z(t) is the balance between the inflow of new immunes and the outflow due to mortality and loss of immunity. Lettingf( t) be the rate of nonimmune individuals entering the immune class at time r, h be the per capita rate of infection, p be the per capita rate of mortality, and T be the critical gap for the loss of immunity, we have
The third term on the right-hand side characterizes the loss of immunity. Those individuals who became immune exactly T years before, i.e. f( t - T), and those immune who were reexposed exactly 7 years before, i.e. hz( t - T), are those at risk of losing immunity at time t. Of those at risk, only a fraction will survive the interval without being reexposed, i.e. e--(h+P)7, and hence will revert to the susceptible class.
DYNAMICS OF ACQUIRED IMMUNITY
251
In calculating the equilibrium for Equation (2.1) by setting dz/dt note that if f(t) and z(t) are f and Z at equilibrium, then
It is illuminating
to compare equation
(2.2) with the equilibriaf
2 -f(r)-Pz(r)-Yz(r),
to zero,
and 5 of (2.3)
which obey the relationship
f=(y+/.L)z.
(2.4)
Choosing y so that ( y + p) Z matches the right-hand y=
side of equation (2.2), i.e.
(h+p)e-(h+p)’ 1 -
e
-(h+P)T
’
(2.5)
results in the identical equilibrium. Thus, solving for the equilibrium of Equation (2.1) is formally equivalent to solving for the equilibrium of the simpler differential equation (2.3). It also turns out that y is directly related to the formulation of the model as a stochastic process. As shown in the next section, l/(y + cc) is the mean duration of time spent in the immune state and CL/(y + p) is the probability of dying before immunity is lost. Analysis of equation (2.1) by itself cannot generate an age-prevalence curve. The equation must be inserted into an epidemiology model which includes other groups of individuals such as susceptibles and infecteds. Switching from population sizes to population fractions (for prevalence) requires merely the removal of all mortality terms (setting k to equal zero) if mortality acts equally on all individuals (Cohen [9], Aron [ 10, Appendix G]). Although, in fact, mortality rates vary according to age and disease status, the simpler assumption is a reasonable approximation if a detailed study of mortality is not important (e.g. Dietz et al. [I 11). Even if variation in mortality is introduced, age prevalence is affected by rates of differential mortality, not rates of absolute mortality. The epidemiology of malaria provides a good example because the duration of immunity seems to depend on continued reexposure. The typical picture of immunity to malaria is based on data from tropical Africa, where Plasmodium falciparum is the dominant species of parasite causing malaria in humans (Molineaux and Gramiccia [3]; Boyd [2]). Immunity is acquired slowly after several infections under heavy exposure. An immune individual
252
JOAN L. ARON
is not refractory to infection; instead, immunity suppresses the parasites, thereby limiting the disease caused by the infection, while permitting transmission from human to mosquito. Most relevant to this discussion, the immunity gained may not be permanent. Not only are antibodies to malaria gradually lost in the absence of exposure (Kuvin and Voller [ 121; AmbroiseThomas et al. [13]), but short-term use of antimalarials causes a slight but noticeable loss of immunity (Molineaux and Gramiccia [3]; Pringle and Avery-Jones [ 141). A simplified model of the age prevalence of malaria demonstrates the effect of immunity which needs continual boosting. The population is divided into three groups: susceptibles (x), the proportion which is uninfected; infecteds (y), the proportion with severe infections; and immunes (z), the proportion with chronic, mild infections. Individuals are born susceptible to become infected at a rate of h infections per year. They subsequently recover at rate r, becoming susceptible again. Infected individuals may also acquire immunity at a slow rate, 6. Immune individuals remain so according to the dynamics of reexposure described by Equation (2.1) with f(t) replaced by 6y(t). If there is no differential mortality, the dynamics of age prevalence obey the following equations:
$=A(+T)+&)-h(t),
(2.6)
$=hx(t)-v(r)-&y(r),
(2.7)
$=By(l)-A(W),
(2.8)
A(r-r)=[8(y(t-r))+h(~(t-r))]e-~‘,
(2.9)
where x(t)+ r(t)+ z(r) = 1 and x(0) = 1. The prevalence at age t is defined to hey(r). Figure 1 illustrates the possible shapes of age-prevalence curves for different values of the infection rate, h: h = O.O5/yr, h = OS/yr, and h = 5/yr corresponding to low, medium, and high endemicity, respectively. The prevalence among children always increases with the rate of infection, but the prevalence among adults is highest at intermediate rates of infection. The crossover of the age-prevalence curve with increasing endemicity matches the pattern described by Boyd for tropical Africa in Figure 2 (Boyd [2, p. 5711). Thus, the model provides a first step for formalizing the intuition of some epidemiologists who fear that partial control could increase adult prevalence (Colboume [ 15, p. 231; Molineaux and Gramiccia [3, p. 111).
253
DYNAMICS OF ACQUIRED IMMUNITY
1.0
0
20 AGE
FIG. 1. Proportion infected, y, as a function of age in years for the model defined by Equations (2.6) through (2.9). (a) h = O.O5/yr; (b) h = 0.5/yr; and (c) h = 5/yr. The rest of the parameters are r = 0.8/yr, 6 = 0.2/yr, and 7 = 5 yr. See text for discussion.
The shapes of the age-prevalence curves do not depend strongly on the detailed model of acquired immunity. Indeed, a model based on Equation (2.3) rather than Equation (2.1) gives almost identical results (Aron and May [16]). Age-prevalence patterns seem to depend mainly on the average duration of immunity rather than the exact distribution of duration of immunity. Although the insensitivity to the details of the immune process is advantageous for producing robust conclusions, this very insensitivity poses a problem in the reverse analysis of using prevalence studies to understand acquired immunity in more detail. For this aim, it is necessary to turn to longitudinal studies which require a stochastic formulation geared to following individuals.
III.
FORMULATION
AS STOCHASTIC
PROCESS
The dynamics of immunity can be studied by following immune individuals who are subject to additional exposure and mortality (or other loss from
254
JOAN L. ARON
1.0
a
5
IO
15
20
25 25+
AGE FIG. 2. Prevalence of acute malaria infection versus age in years in stable indigenous populations in Africa for differing levels of endemicity: (a) low endemicity; (b) moderate endemicity; (c) high endemicity and (d) hyperendemicity. The curves are not specific data but rather illustrative patterns by Boyd; unfortunately the method of detecting acute malaria infection is not specified. See text for discussion.
the study). Formulation as a stochastic model predicts probabilistic features of the duration of immunity which can be compared with observations on individuals. Using the model of immunity given in the introduction, stochastic analysis proceeds under the following three assumptions: (1) The exposures follow a Poisson process with intensity h. (2) An individual’s risk of dying follows a Poisson process with rate CL, independent of exposure. (3) An immune individual remains immune until the occurrence of a time interval of length T without any exposure. The third assumption implies that only the latest exposure affects the duration. With each new exposure, the outcome is independent of the history of prior exposures. According to these assumptions, an immune individual receives N exposures, where N may be equal to zero. After the last exposure, the individual will either die after a time L or become susceptible after a time T, with
DYNAMICS OF ACQUIRED IMMUNITY
255
probabilities c and I- c, respectively. Letting Wi be the waiting time from the (i - 1)st to the ith exposure, the duration of immunity, T, can be expressed as
T-
5 Wi+cL+(l-c)~
(3.1)
i-l
It follows from the assumptions that the random variables (W}, N, and L and that the ( Wi} have a common distribution. Thus
are all mutually independent
= 2 Pr(N=i)iE(W,),
(3.2)
i-l
which leads to a simple form for the mean duration:
E(T)=E(N)E(W,)+cE(L)+(l-C)T.
(3.3)
In order to analyze the sequence of repeated exposures, consider the event of a first infectious exposure in (t, t + dt). The event consists of an exposure in (t, t + dt) preceded by (0, r) with neither exposure nor mortality. Thus, letting U be the waiting time for an infectious exposure conditional on the event that the exposure precedes mortality, we have
Pr(r
ra0.
The probabilityp of exposure occurring before mortality the probability that U is less than T, namely, p = /‘e -(h+c)fh 0
dr
=
w&l_
(3.4)
and before time T is
&h+W)
(3.5)
An individual remains immune as long as this event repeats sequentially. Since N is the number of repetitions without interruption, it has a geometric distribution with expectation E(N)=+
(3.6)
The distribution of the waiting time W, between the repeated that of U restricted to the range from zero to T, so that e-(h+P)rh
Pr(r
P
exposures
is
dr
’
06r67,
(3.7)
256
JOAN
L. ARON
and Te
1 _
-(h+a)r e
-(h+lr)T
(3.8)
’
In order to analyze removal from the immune state after interruption of the sequence of exposures, consider the alternative outcomes during the time interval of length T. Mortality, rather than exposure, may be the first event to occur. The probability, p’, that mortality occurs first is equivalent to p from Equation (3.5) with the processes reversed, i.e.
“=
L(lh+/t
e-(h+‘)‘).
(3.9)
The remaining possibility is that neither exposure nor mortality time T. The probability, p”, of the third outcome is
occurs by
p” = e -(h+t)T
(3.10)
It is not difficult to verify that p, p’, and p” sum to one. Since the interval of immunity after the last reexposure terminates either dying or exceeding length T, the probability of mortality, c, is P’
c=p,
by
(3.11)
[Notice that c is equal to y/(y + EL), where y is as in Equation (2.5), as discussed earlier.] If mortality occurs, the distribution of the waiting time to mortality is exactly like that of I+‘, with the processes reversed. However, the distribution of W, in Equation (3.7) depends only on the sum of the rates of the Poisson processes of exposure and mortality. Therefore, E(L) Substituting the results Equation (3.3) yields
from
E(T)=
Using Equations
equations
(3.12) (3.6), (3.1 l), and
(l-P”)E(W,)+P”T l-p
(3.12) into
.
(3.13)
(3.5), (3.8), and (3. lo), E(T)
[Notice that E(T) discussed earlier.]
= E(W,).
=
is equal to l/(y
1 -
e -_(h+P)T
w + he -(h+p)T ’
+ CL),where y is as in Equation
(3.14) (2.5), as
DYNAMICS OF ACQUIRED IMMUNITY
257
Thus, if the model holds, the mean residence time in the immune class, E(T), increases with the minimum duration of immunity, T, and the rate of exposure, h, but decreases with the rate of mortality, CL.The dependence on T is easily seen from Equation (3.14), since the numerator increases and the denominator decreases with increasing 7. Longer T hence requires longer study periods. The relationship with h holds because =sgn((h+~)~-(l-e-(h+P)T)}>O.
(3.15)
An observation showing that immunity lengthens with increasing infection rate is the key to demonstrating the existence of transmission-dependent immunity. Lastly, the relationship with ~1holds because = sgn{(h + p)7e-(“+‘)‘-(l-
e--(“+“)‘)}
-C 0.
(3.16)
The mean duration includes the contribution from those lost to mortality. Unfortunately, it is not correct to factor out the effect of mortality by ignoring deaths. The reason is that T,,the duration of immunity for those who eventually become susceptible again, also depends on the mortality rate p. The dependence is clearly seen in the following equation: E(C)
= E(N)E(I+‘,)+7,
(3.17)
since, as p increases, E(N),E(W,)and consequently E( T,) decrease. Even if the only observations are made on those who become susceptible, the analysis of the duration of immunity must take into account the rate of mortality or other processes which remove individuals from observation. The problem is analogous to estimating the risk of one cause of mortality when other causes of mortality are also acting on the population (Gail [7]). In designing an epidemiological study, it is desirable to minimize the chance of an individual dying before immunity is lost, c from equation (3.11). This chance increases not only with mortality ).L,but also with the rate of exposure, h, and the minimum duration of immunity, T. The relationships with /Aand T are straightforward, since p’ from Equation (3.9) increases with both andp” from Equation (3.10) decreases with both. The dependence on h follows from the observation that (3.18) Because 1/c decreases with increasing h , c increases with h . The results with h and T also make sense intuitively, because extending the duration of
JOAN L. ARON
258 TABLE 1 Sample Values ofp,p’,p”,
c, E(T)
and E(T,) Defined in the TexP
h
W-)
0.5
I
E(C)
P
PI
PII
c
(Yr)
(Yr)
.8414 .9638
.0337 .0193
.I249 .0169
.2122 S327
10.61 26.64
II.17 28.28
(Yr-‘)
aThe parameters used are c = O.O2/yr and 7 = 4 yr for two values of h: 0.5/yr and l/yr.
immunity by increasing the rate of reexposure, or minimum duration of immunity, makes it more difficult to observe the loss of immunity. In order to demonstrate how pronounced this effect can be, Table 1 shows an example of a twofold increase in the rate of exposure. In this example, the proportion of people dying before losing immunity increases from 21% to 53%, while the mean duration of immunity more than doubles. IV.
SUMMARY
A mathematical model has been presented to describe the dynamics of immunity which can be boosted by reexposure to infection. The key assumption is that immunity is lost after a specified interval of time passes without exposure to infection. Consequently, the duration of acquired immunity depends on the rate of exposure. In Section II, the model has been formulated as a differential-delay equation which can be embedded in compartmental epidemiological models in order to examine its impact on prevalence. As an example, analysis of a simple model of the epidemiology of malaria shows that the prevalence of disease among adults will be greatest at intermediate rates of infection, in accord with some reports. In Section III, the model has been formulated as a stochastic process which is more appropriate for longitudinal studies of the duration of immunity in individuals. The main result, not surprisingly, is that the average duration of immunity increases with increasing rates of exposure. However, analysis of the model has also demonstrated the effect of concomitant mortality (or other processes of removal), which is not quite as obvious. The probability that a person dies before immunity is lost may be high, especially when mortality is high, exposure is heavy, or the unboosted duration of immunity is long. Further, amongst those who do survive long enough to lose immunity, its average duration is shorter than would be expected in the absence of any mortality. The booster effect of infection on immunity has thus been formally analyzed at the level of the population and at the level of the individual. The
259
DYNAMICS OF ACQUIRED IMMUNITY
model, although simplified, establishes dynamics of acquired immunity.
a guideline for an examination
of the
I thank R. M. May for technical guidance, and R. M. Anderson, J. E. Cohen, M. Gail, I.. Molineaux, I.. Nunney, P. Prorok, D. I. Rubenstein, D. W. Tonkyn, D. A. J. Tyrrell, G. Weiss, D. C. Wiernasz, J. A. Yorke, and an anonymous reviewer for helpful discussion and criticism. I also thank C. Ball for her careful typing of the manuscript. This work was supported in part by an NSF graduate fellowship and the Princeton University Harold Willis Dada% Fellowship. REFERENCES N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, (2nd ed.), Macmillan, New York, 1975. 2 M. F. Boyd (Ed.), Malariologv, Saunders, Philadelphia, 1949. 3 L. Molineaux and G. Gramiccia, The Garki Project: Research on the Epidemiology and Control of Malaria in the Sudan Savannah of West Africa, World Health Organization, Geneva, 1980. 4 R. H. Elderkin, D. P. Berkowitz, F. A. Fart%, C. F. Gunn, F. J. Hickemell, S. N. Kass, F. I. Mansfield, and R. G. Taranto, On the steady state of an age dependent model for malaria, in Nonlinear Systems and Applications (V. Lakshmikantham, Ed.), Academic, New York, 1977, pp. 491-512. 5 J. Dutertre, Etude dun modele (rpidemiologique applique au paludisme, Ann. Sot. I
6
Beige Medecine Tropicale 56:127-141 (1976). S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic,
New York, 1975. M. Gail, Competing risks, in Encyclopedia of Sfatistical Sciences, Vol. 2 (S. Katz and N. Johnson, Eds.), Wiley, New York, 1982, pp. 75-81. 8 H. Muench, Catabtic Models in Epidemiology, Harvard U. P., Cambridge, Mass., 1959. 9 J. E. Cohen, When does a leaky compartment model appear to have no leaks?, Theoret. 7
Population Biol. 3:404-405 (1972). IO J. L. Aron, Population Dynamics of Immunity to Malaria, Ph.D. Thesis, Biology Dept.,
Princeton Univ., Princeton, N.J., 1981. 11 K. Diem, L. Molineaux, and A. Thomas, A. malaria model tested in the African Savannah, Bull. World Health Organization 501347-357 (1974). 12 S. Kuvin and A. Voller, Malarial antibody titres of West Africans in Britain, British Medical J. 2~477-479 (1963). 13
P. Ambroise-Thomas, W. H. Wemsdorder, B. Grab, J. Cullen, and P. Bertagna, Etude sero-epidemiologique longitudinale sur le paludisme en Tunisie, Bull. World Health
14
G. Pringle and S. Avery-Jones, Observations on the early course of untreated falciparum malaria in semi-immune African children following a short period of protection, Bull. World Health Organization 34~269-272 (1966). M. Colboume, Malaria in Africa, Oxford U. P., London, 1966.
Organization 54:355-367
(1976).
15 16 J. L. Aron and R. M. May, The population dynamics of malaria, in Population Dynamics of Infectious Diseases (R. M. Anderson, Ed.), Chapman and Hall, London, 1982, pp. 139-179.
ABSTRACT A mathematical model is presented to describe the dynamics of immunity which can be boosted by reexposure to infection. Immunity is assumed to last until a specified interval of time elapses without an exposure. This assumption is incorporated into a compartmental model as a differential-delay equation. When the model is applied to malaria epidemiology, the prevalence of disease among adults is greatest at intermediate rates of infection. Observed
age-prevalence
curves
have
shapes
similar
to those
of the predicted
curves.
Immunity in an individual is formulated in terms of a stochastic process, and an expression for the average duration of immunity is obtained. The average duration of immunity increases with the rate of exposure, but the presence of mortality (or ether kinds of removal) risks.
I.
shortens
the average
duration
observed,
in analogy
with the theory
of competing
INTRODUCTION
Mathematical models of the epidemiology of infectious diseases have long incorporated some notion of immunity as refractoriness to infection acquired after an initial infection (Bailey [l]). In most models, the duration of immunity is prescribed independently of continued exposure to infection. However, epidemiological evidence suggests that in some cases, such as malaria (Boyd [2] and Molineaux and Gramiccia [3]), the duration of immunity is extended by continued exposure to infection. Recently, mathematicians have begun to describe the interdependence between exposure to infection and the duration of immunity for malaria. In a continuum model (Elderkin et al. [4]), levels of “resistance” depend dynamically on levels of parasites in the blood. A more epidemiological model (Dutertre [5]) contains a discrete class of immunes whose reversion to susceptible is retarded by reexposure calculated on a monthly basis. MATHEMATICAL
BIOSCIENCES
64:249-259
QElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017
(1983)
249 0025-5564/83/$03.00
JOAN L. ARON
250
In this paper, the model is a continuous-time process in which immunity lasts until the occurrence of a gap of T years without exposure. The underlying assumption is that acquired immunity gradually weakens in the absence of exposure to infection but quickly strengthens if additional exposure occurs before T years elapse from the previous exposure; on the other hand, if exposure occurs after T years, the individual, no longer immune, develops an infection. In addition, mortality may remove an immune individual at any time. This model in effect incorporates a locking time for a Type II counter (Karlin and Taylor [6, p. 1791) and standard assumptions on the competing risk of mortality (Gail [7]).
II.
FORMULATION
AS DIFFERENTIAL-DELAY
EQUATION
Striking features of the age-specific pattern of prevalence (proportion infected) in cross-sectional surveys may be due to the dynamics of immunity boosted by reexposure. Formulation of a differential-delay equation model permits the analysis of the effect of immunity in a model of prevalence. Linear models with transitions between “compartments” of susceptibles, infecteds, etc. are useful in the construction of age-prevalence curves (Muench [8]). It is straightforward to incorporate into this kind of model the number of immune individuals, z(t), whose dynamics obey the assumptions stated in the introduction. If conditions for transmission of infection have not changed dramatically over the lifespan of the oldest member of the population, then trends with increasing age are equivalent to changes of a birth cohort through time. Consequently, an equation is formulated for z(l), and then this equation is used in an age prevalence model where t is age. The rate of change of z(t) is the balance between the inflow of new immunes and the outflow due to mortality and loss of immunity. Lettingf( t) be the rate of nonimmune individuals entering the immune class at time r, h be the per capita rate of infection, p be the per capita rate of mortality, and T be the critical gap for the loss of immunity, we have
The third term on the right-hand side characterizes the loss of immunity. Those individuals who became immune exactly T years before, i.e. f( t - T), and those immune who were reexposed exactly 7 years before, i.e. hz( t - T), are those at risk of losing immunity at time t. Of those at risk, only a fraction will survive the interval without being reexposed, i.e. e--(h+P)7, and hence will revert to the susceptible class.
DYNAMICS OF ACQUIRED IMMUNITY
251
In calculating the equilibrium for Equation (2.1) by setting dz/dt note that if f(t) and z(t) are f and Z at equilibrium, then
It is illuminating
to compare equation
(2.2) with the equilibriaf
2 -f(r)-Pz(r)-Yz(r),
to zero,
and 5 of (2.3)
which obey the relationship
f=(y+/.L)z.
(2.4)
Choosing y so that ( y + p) Z matches the right-hand y=
side of equation (2.2), i.e.
(h+p)e-(h+p)’ 1 -
e
-(h+P)T
’
(2.5)
results in the identical equilibrium. Thus, solving for the equilibrium of Equation (2.1) is formally equivalent to solving for the equilibrium of the simpler differential equation (2.3). It also turns out that y is directly related to the formulation of the model as a stochastic process. As shown in the next section, l/(y + cc) is the mean duration of time spent in the immune state and CL/(y + p) is the probability of dying before immunity is lost. Analysis of equation (2.1) by itself cannot generate an age-prevalence curve. The equation must be inserted into an epidemiology model which includes other groups of individuals such as susceptibles and infecteds. Switching from population sizes to population fractions (for prevalence) requires merely the removal of all mortality terms (setting k to equal zero) if mortality acts equally on all individuals (Cohen [9], Aron [ 10, Appendix G]). Although, in fact, mortality rates vary according to age and disease status, the simpler assumption is a reasonable approximation if a detailed study of mortality is not important (e.g. Dietz et al. [I 11). Even if variation in mortality is introduced, age prevalence is affected by rates of differential mortality, not rates of absolute mortality. The epidemiology of malaria provides a good example because the duration of immunity seems to depend on continued reexposure. The typical picture of immunity to malaria is based on data from tropical Africa, where Plasmodium falciparum is the dominant species of parasite causing malaria in humans (Molineaux and Gramiccia [3]; Boyd [2]). Immunity is acquired slowly after several infections under heavy exposure. An immune individual
252
JOAN L. ARON
is not refractory to infection; instead, immunity suppresses the parasites, thereby limiting the disease caused by the infection, while permitting transmission from human to mosquito. Most relevant to this discussion, the immunity gained may not be permanent. Not only are antibodies to malaria gradually lost in the absence of exposure (Kuvin and Voller [ 121; AmbroiseThomas et al. [13]), but short-term use of antimalarials causes a slight but noticeable loss of immunity (Molineaux and Gramiccia [3]; Pringle and Avery-Jones [ 141). A simplified model of the age prevalence of malaria demonstrates the effect of immunity which needs continual boosting. The population is divided into three groups: susceptibles (x), the proportion which is uninfected; infecteds (y), the proportion with severe infections; and immunes (z), the proportion with chronic, mild infections. Individuals are born susceptible to become infected at a rate of h infections per year. They subsequently recover at rate r, becoming susceptible again. Infected individuals may also acquire immunity at a slow rate, 6. Immune individuals remain so according to the dynamics of reexposure described by Equation (2.1) with f(t) replaced by 6y(t). If there is no differential mortality, the dynamics of age prevalence obey the following equations:
$=A(+T)+&)-h(t),
(2.6)
$=hx(t)-v(r)-&y(r),
(2.7)
$=By(l)-A(W),
(2.8)
A(r-r)=[8(y(t-r))+h(~(t-r))]e-~‘,
(2.9)
where x(t)+ r(t)+ z(r) = 1 and x(0) = 1. The prevalence at age t is defined to hey(r). Figure 1 illustrates the possible shapes of age-prevalence curves for different values of the infection rate, h: h = O.O5/yr, h = OS/yr, and h = 5/yr corresponding to low, medium, and high endemicity, respectively. The prevalence among children always increases with the rate of infection, but the prevalence among adults is highest at intermediate rates of infection. The crossover of the age-prevalence curve with increasing endemicity matches the pattern described by Boyd for tropical Africa in Figure 2 (Boyd [2, p. 5711). Thus, the model provides a first step for formalizing the intuition of some epidemiologists who fear that partial control could increase adult prevalence (Colboume [ 15, p. 231; Molineaux and Gramiccia [3, p. 111).
253
DYNAMICS OF ACQUIRED IMMUNITY
1.0
0
20 AGE
FIG. 1. Proportion infected, y, as a function of age in years for the model defined by Equations (2.6) through (2.9). (a) h = O.O5/yr; (b) h = 0.5/yr; and (c) h = 5/yr. The rest of the parameters are r = 0.8/yr, 6 = 0.2/yr, and 7 = 5 yr. See text for discussion.
The shapes of the age-prevalence curves do not depend strongly on the detailed model of acquired immunity. Indeed, a model based on Equation (2.3) rather than Equation (2.1) gives almost identical results (Aron and May [16]). Age-prevalence patterns seem to depend mainly on the average duration of immunity rather than the exact distribution of duration of immunity. Although the insensitivity to the details of the immune process is advantageous for producing robust conclusions, this very insensitivity poses a problem in the reverse analysis of using prevalence studies to understand acquired immunity in more detail. For this aim, it is necessary to turn to longitudinal studies which require a stochastic formulation geared to following individuals.
III.
FORMULATION
AS STOCHASTIC
PROCESS
The dynamics of immunity can be studied by following immune individuals who are subject to additional exposure and mortality (or other loss from
254
JOAN L. ARON
1.0
a
5
IO
15
20
25 25+
AGE FIG. 2. Prevalence of acute malaria infection versus age in years in stable indigenous populations in Africa for differing levels of endemicity: (a) low endemicity; (b) moderate endemicity; (c) high endemicity and (d) hyperendemicity. The curves are not specific data but rather illustrative patterns by Boyd; unfortunately the method of detecting acute malaria infection is not specified. See text for discussion.
the study). Formulation as a stochastic model predicts probabilistic features of the duration of immunity which can be compared with observations on individuals. Using the model of immunity given in the introduction, stochastic analysis proceeds under the following three assumptions: (1) The exposures follow a Poisson process with intensity h. (2) An individual’s risk of dying follows a Poisson process with rate CL, independent of exposure. (3) An immune individual remains immune until the occurrence of a time interval of length T without any exposure. The third assumption implies that only the latest exposure affects the duration. With each new exposure, the outcome is independent of the history of prior exposures. According to these assumptions, an immune individual receives N exposures, where N may be equal to zero. After the last exposure, the individual will either die after a time L or become susceptible after a time T, with
DYNAMICS OF ACQUIRED IMMUNITY
255
probabilities c and I- c, respectively. Letting Wi be the waiting time from the (i - 1)st to the ith exposure, the duration of immunity, T, can be expressed as
T-
5 Wi+cL+(l-c)~
(3.1)
i-l
It follows from the assumptions that the random variables (W}, N, and L and that the ( Wi} have a common distribution. Thus
are all mutually independent
= 2 Pr(N=i)iE(W,),
(3.2)
i-l
which leads to a simple form for the mean duration:
E(T)=E(N)E(W,)+cE(L)+(l-C)T.
(3.3)
In order to analyze the sequence of repeated exposures, consider the event of a first infectious exposure in (t, t + dt). The event consists of an exposure in (t, t + dt) preceded by (0, r) with neither exposure nor mortality. Thus, letting U be the waiting time for an infectious exposure conditional on the event that the exposure precedes mortality, we have
Pr(r
ra0.
The probabilityp of exposure occurring before mortality the probability that U is less than T, namely, p = /‘e -(h+c)fh 0
dr
=
w&l_
(3.4)
and before time T is
&h+W)
(3.5)
An individual remains immune as long as this event repeats sequentially. Since N is the number of repetitions without interruption, it has a geometric distribution with expectation E(N)=+
(3.6)
The distribution of the waiting time W, between the repeated that of U restricted to the range from zero to T, so that e-(h+P)rh
Pr(r
P
exposures
is
dr
’
06r67,
(3.7)
256
JOAN
L. ARON
and Te
1 _
-(h+a)r e
-(h+lr)T
(3.8)
’
In order to analyze removal from the immune state after interruption of the sequence of exposures, consider the alternative outcomes during the time interval of length T. Mortality, rather than exposure, may be the first event to occur. The probability, p’, that mortality occurs first is equivalent to p from Equation (3.5) with the processes reversed, i.e.
“=
L(lh+/t
e-(h+‘)‘).
(3.9)
The remaining possibility is that neither exposure nor mortality time T. The probability, p”, of the third outcome is
occurs by
p” = e -(h+t)T
(3.10)
It is not difficult to verify that p, p’, and p” sum to one. Since the interval of immunity after the last reexposure terminates either dying or exceeding length T, the probability of mortality, c, is P’
c=p,
by
(3.11)
[Notice that c is equal to y/(y + EL), where y is as in Equation (2.5), as discussed earlier.] If mortality occurs, the distribution of the waiting time to mortality is exactly like that of I+‘, with the processes reversed. However, the distribution of W, in Equation (3.7) depends only on the sum of the rates of the Poisson processes of exposure and mortality. Therefore, E(L) Substituting the results Equation (3.3) yields
from
E(T)=
Using Equations
equations
(3.12) (3.6), (3.1 l), and
(l-P”)E(W,)+P”T l-p
(3.12) into
.
(3.13)
(3.5), (3.8), and (3. lo), E(T)
[Notice that E(T) discussed earlier.]
= E(W,).
=
is equal to l/(y
1 -
e -_(h+P)T
w + he -(h+p)T ’
+ CL),where y is as in Equation
(3.14) (2.5), as
DYNAMICS OF ACQUIRED IMMUNITY
257
Thus, if the model holds, the mean residence time in the immune class, E(T), increases with the minimum duration of immunity, T, and the rate of exposure, h, but decreases with the rate of mortality, CL.The dependence on T is easily seen from Equation (3.14), since the numerator increases and the denominator decreases with increasing 7. Longer T hence requires longer study periods. The relationship with h holds because =sgn((h+~)~-(l-e-(h+P)T)}>O.
(3.15)
An observation showing that immunity lengthens with increasing infection rate is the key to demonstrating the existence of transmission-dependent immunity. Lastly, the relationship with ~1holds because = sgn{(h + p)7e-(“+‘)‘-(l-
e--(“+“)‘)}
-C 0.
(3.16)
The mean duration includes the contribution from those lost to mortality. Unfortunately, it is not correct to factor out the effect of mortality by ignoring deaths. The reason is that T,,the duration of immunity for those who eventually become susceptible again, also depends on the mortality rate p. The dependence is clearly seen in the following equation: E(C)
= E(N)E(I+‘,)+7,
(3.17)
since, as p increases, E(N),E(W,)and consequently E( T,) decrease. Even if the only observations are made on those who become susceptible, the analysis of the duration of immunity must take into account the rate of mortality or other processes which remove individuals from observation. The problem is analogous to estimating the risk of one cause of mortality when other causes of mortality are also acting on the population (Gail [7]). In designing an epidemiological study, it is desirable to minimize the chance of an individual dying before immunity is lost, c from equation (3.11). This chance increases not only with mortality ).L,but also with the rate of exposure, h, and the minimum duration of immunity, T. The relationships with /Aand T are straightforward, since p’ from Equation (3.9) increases with both andp” from Equation (3.10) decreases with both. The dependence on h follows from the observation that (3.18) Because 1/c decreases with increasing h , c increases with h . The results with h and T also make sense intuitively, because extending the duration of
JOAN L. ARON
258 TABLE 1 Sample Values ofp,p’,p”,
c, E(T)
and E(T,) Defined in the TexP
h
W-)
0.5
I
E(C)
P
PI
PII
c
(Yr)
(Yr)
.8414 .9638
.0337 .0193
.I249 .0169
.2122 S327
10.61 26.64
II.17 28.28
(Yr-‘)
aThe parameters used are c = O.O2/yr and 7 = 4 yr for two values of h: 0.5/yr and l/yr.
immunity by increasing the rate of reexposure, or minimum duration of immunity, makes it more difficult to observe the loss of immunity. In order to demonstrate how pronounced this effect can be, Table 1 shows an example of a twofold increase in the rate of exposure. In this example, the proportion of people dying before losing immunity increases from 21% to 53%, while the mean duration of immunity more than doubles. IV.
SUMMARY
A mathematical model has been presented to describe the dynamics of immunity which can be boosted by reexposure to infection. The key assumption is that immunity is lost after a specified interval of time passes without exposure to infection. Consequently, the duration of acquired immunity depends on the rate of exposure. In Section II, the model has been formulated as a differential-delay equation which can be embedded in compartmental epidemiological models in order to examine its impact on prevalence. As an example, analysis of a simple model of the epidemiology of malaria shows that the prevalence of disease among adults will be greatest at intermediate rates of infection, in accord with some reports. In Section III, the model has been formulated as a stochastic process which is more appropriate for longitudinal studies of the duration of immunity in individuals. The main result, not surprisingly, is that the average duration of immunity increases with increasing rates of exposure. However, analysis of the model has also demonstrated the effect of concomitant mortality (or other processes of removal), which is not quite as obvious. The probability that a person dies before immunity is lost may be high, especially when mortality is high, exposure is heavy, or the unboosted duration of immunity is long. Further, amongst those who do survive long enough to lose immunity, its average duration is shorter than would be expected in the absence of any mortality. The booster effect of infection on immunity has thus been formally analyzed at the level of the population and at the level of the individual. The
259
DYNAMICS OF ACQUIRED IMMUNITY
model, although simplified, establishes dynamics of acquired immunity.
a guideline for an examination
of the
I thank R. M. May for technical guidance, and R. M. Anderson, J. E. Cohen, M. Gail, I.. Molineaux, I.. Nunney, P. Prorok, D. I. Rubenstein, D. W. Tonkyn, D. A. J. Tyrrell, G. Weiss, D. C. Wiernasz, J. A. Yorke, and an anonymous reviewer for helpful discussion and criticism. I also thank C. Ball for her careful typing of the manuscript. This work was supported in part by an NSF graduate fellowship and the Princeton University Harold Willis Dada% Fellowship. REFERENCES N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, (2nd ed.), Macmillan, New York, 1975. 2 M. F. Boyd (Ed.), Malariologv, Saunders, Philadelphia, 1949. 3 L. Molineaux and G. Gramiccia, The Garki Project: Research on the Epidemiology and Control of Malaria in the Sudan Savannah of West Africa, World Health Organization, Geneva, 1980. 4 R. H. Elderkin, D. P. Berkowitz, F. A. Fart%, C. F. Gunn, F. J. Hickemell, S. N. Kass, F. I. Mansfield, and R. G. Taranto, On the steady state of an age dependent model for malaria, in Nonlinear Systems and Applications (V. Lakshmikantham, Ed.), Academic, New York, 1977, pp. 491-512. 5 J. Dutertre, Etude dun modele (rpidemiologique applique au paludisme, Ann. Sot. I
6
Beige Medecine Tropicale 56:127-141 (1976). S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic,
New York, 1975. M. Gail, Competing risks, in Encyclopedia of Sfatistical Sciences, Vol. 2 (S. Katz and N. Johnson, Eds.), Wiley, New York, 1982, pp. 75-81. 8 H. Muench, Catabtic Models in Epidemiology, Harvard U. P., Cambridge, Mass., 1959. 9 J. E. Cohen, When does a leaky compartment model appear to have no leaks?, Theoret. 7
Population Biol. 3:404-405 (1972). IO J. L. Aron, Population Dynamics of Immunity to Malaria, Ph.D. Thesis, Biology Dept.,
Princeton Univ., Princeton, N.J., 1981. 11 K. Diem, L. Molineaux, and A. Thomas, A. malaria model tested in the African Savannah, Bull. World Health Organization 501347-357 (1974). 12 S. Kuvin and A. Voller, Malarial antibody titres of West Africans in Britain, British Medical J. 2~477-479 (1963). 13
P. Ambroise-Thomas, W. H. Wemsdorder, B. Grab, J. Cullen, and P. Bertagna, Etude sero-epidemiologique longitudinale sur le paludisme en Tunisie, Bull. World Health
14
G. Pringle and S. Avery-Jones, Observations on the early course of untreated falciparum malaria in semi-immune African children following a short period of protection, Bull. World Health Organization 34~269-272 (1966). M. Colboume, Malaria in Africa, Oxford U. P., London, 1966.
Organization 54:355-367
(1976).
15 16 J. L. Aron and R. M. May, The population dynamics of malaria, in Population Dynamics of Infectious Diseases (R. M. Anderson, Ed.), Chapman and Hall, London, 1982, pp. 139-179.