Estimation of incidence and recovery rates of Plasmodium falciparum parasitaemia from longitudinal data

Estimation of incidence and recovery rates of Plasmodium falciparum parasitaemia from longitudinal data A. BEKESSY,' L. MOLINEAUX,2 & J. STOREY3 A method is described ofestimating the malaria incidence rate h and the recovery rate r from longitudinal data. The method is based on the assumption that the phenomenon of patent parasitaemia can be represented by a reversible two-state catalytic model; it is applicable to all problems that can be represented by such a model. The method was applied to data on falciparum malaria from the West African savanna and the findings suggested that immunity increases the rate of recovery from patent para- sitaemia by a factor of up to 10, and also reduces the number of episodes ofpatent para- sitaemia resulting from one inoculation. Under the effect of propoxur, h varies with the estimated man-biting rate of the vector while r increases, possibly owing to reduced super- infection. The longitudinal study of infants, who are assumed to be negative (i.e., free from malaria) at birth, is commonly used to estimate the incidence rate of malaria and the longitudinal study of patients removed from exposure to infection has been used to estimate the recovery rate. In both cases, one starts with a homogeneous group and it is assumed that only one type of transition-from negative to positive in the first case, from positive to negative in the second-is possible between two consecutive surveys. This paper describes the simultaneous estimation of incidence and recovery rates, from the longitudinal study of a mixture of negatives and positives; the estimation procedure allows for several transitions between consecutive surveys. Both inci- dence and recovery are based on patent parasitaemia as assessed by a standard method of blood exami- nation. The apparent incidence rate includes the inci- dence both of new infections and of parasitological 1 Statistician, Division of Health Statistics, World Health Organization, 1211 Geneva 27, Switzerland. 3 Medical Officer, Division of Malaria and Other Parasitic Diseases, World Health Organization, 1211 Geneva 27, Switzerland. 2 Laboratory Technician, Project MPD 012, World Health Organization, Kano, Nigeria. relapses; it is generally accepted that the incidence rate estimated in infants includes only new infections. The apparent recovery rate includes true recovery and latency; the recovery rate estimated in patients removed from exposure also includes both. METHOD OF ESTIMATION Identified persons are surveyed at defined intervals, and at each survey every identified person is classi- fied as positive or negative by a standard method of examination. Between two consecutive surveys and for a given population group (e.g., one defined by age and location) the following transition frequencies, which include only persons examined and classified in both surveys are computed: (a) the proportion of positives, at the second survey, in those negative at the first (a = N_+/N_); and (b) the proportion of negatives, at the second survey, in those positive at the first (,B = N+_/N+). It is assumed that, in a given population group and in a given time interval, negatives become positive at a constant rate h and positives become negative at a constant rate r. This defines a Markov process, in which the relationship between transition rates and transition frequencies is known. 3566 - 685 BULL. WORLD HEALTH ORGAN., Vol. 54, 1976 A. 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CC. ~0U)iiL0 '-') (5D LC 0) U-C U D 0 0-0 0 -H 6- 'a C- -C - 0 5- 0 U) c ,i 0 5. 00a U) r:- U CZ U) U) ca -H c 0 0n 0 U) -.-a -C C PI 0 (AU) UI- co 0) N 0) U) II ' Xi' 10 &7 m ea cr ESTIMATION OF MALARIA INCIDENCE AND RECOVERY RATES It is further assumed that the probability of making one transition-from negative to positive or from positive to negative-in one time unit (e.g., in one day) is small and that the probability of making two or more transitions in one time unit is negligible. The transition rates h and r can be estimated from the transition frequencies a and ,B by the following formulae: a 1 h = t(a + I) I -(a±+f) t(a + ln) 1 -(a +/) where t is the time interval (in days for the estimation of daily rates). The derivation of these formulae is given in Annex 1, with a method for calculating the variance of the estimates. Annex 2 provides a programme for the computation of h, r, Sh, and Sr, given N_, N_+, N+, N+_, and t, using the Hewlett-Packard 65 pocket calculator. APPLICATION The method was applied to data from the malaria research project conducted by the Government of the Federal Republic of Nigeria and the World Health Organization in Garki, Kano State, Nigeria. Surveys were conducted at intervals of 10 weeks. At each survey, 200 fields of a standardized Giemsa- stained thick film were examined, and every person was classified as positive or negative for Plasmodium falciparum infection. The baseline period extended from survey 1 to survey 8 (70 weeks) and the inter- vention period from survey 8 to survey 16 (80 weeks). The main transmission seasons of 1971, 1972, and 1973 corresponded to the intervals between surveys 4 and 6, 9 and 11, and 14 and 16, respectively. For the baseline period, transition frequencies were calculated for the 16 villages observed in surveys 3-8. For the intervention period, the cal- culations refer to the four villages (part of the 16) treated exclusively with the residual insecticide propoxur and examined throughout the 3 con- secutive years of baseline and intervention periods, i.e., surveys 1-16. One day (24 h) was chosen as the unit of time. RESULTS Table 1 and Fig. 1 and 2 show the observed tran- sition frequencies and estimated transition rates in relation to age and season during the baseline period. The yearly average of the daily incidence rate h was 0.0134 in infants below 1 year of age; this rose to a maximum of 0.0195 in those aged 5-8 years and decreased to approximately 0.0070 in persons aged 19 years and over. This decrease was statistically significant but the earlier increase was not. The yearly average of the daily recovery rate r was 0.0045 in infants below 1 year of age; this fell to a minimum of 0.0016 in those aged 1-4 years and rose to a maximum of 0.0194 in the oldest age group. This increase was statistically significant, but the earlier decrease was not. The inverse of r, or the expected duration of a positive episode, increased from 222 days in infants below 1 year of age to 625 days in those aged 1-4 years and later decreased to a minimum of 52 days in the oldest age group. The expected equilibrium parasite rate hl(h + r) was very close to the one actually observed, except in infants since they had not had time to reach equili- brium (see Table 1, last two rows). In the same population, the infant conversion rate was also estimated from the proportion of infants found positive for the first time after each interval, using the formula: A I1h =-- In (1 -p) t where h is the daily incidence, t the length of the interval in days, and p the proportion found positive for the first time. The yearly average, i.e., the mean of the five relevant intervals, was 0.0074 + 0.0008. This was not significantly different from the estimate derived from the transition frequencies in the infants, given the large error in the latter estimate. Fig. 2 shows the variation of h by age and season. Since there were no significant differences among the three oldest age groups, these have been com- bined. The relative difference between seasons ofhigh and low transmission was large below 1 year of age, decreased to a minimum in those aged 5-8 years and increased in the older age groups. 687 A. BEKESSY ET AL. 0.005 0 -c1 1-4 5-8 9-18 19-28 29-43 44+ A g e (years) Fig. 1. Yearly average h and r by age in the baseline period, in 16 villages combined. Table 2 and Fig. 3 show the change in h and r in those aged 19 years and over under the impact of propoxur. Age groups and intervals were combined after it had been shown that they did not differ significantly with respect to h and r. Table 2 and Fig. 3 also show the corresponding man-biting rates estimated by night-bait collection in two of the four villages. The continuation of malaria trans- mission under propoxur was proved both by the presence of sporozoite-positive vectors and by the parasitological conversion of infants. There was no rise in h in the wet season of 1972; it decreased to a minimum in the dry season of 1973 and increased again in the wet season of 1973, but not beyond its prespraying dry season level. There was a significant increase in r under propoxur. DISCUSSION The model used in estimating h and r from longitudinal observations is the reversible catalytic model used by Muench (1) to estimate incidence and recovery rates from age-specific prevalence curves. The basic assumptions certainly constitute a simplification of the situation. However, such a simplification is probably acceptable for studies pertaining to a relatively short interval and a popula- tion which, in relation to age and location, is apparently quite homogeneous with respect to both exposure and immunity. An increase of the sample size augments the precision of the estimate, but at the same time it may decrease the homogeneity of the sample. The formulae given for h and r do not permit an estimate if (a + fi) > 1. This may be due to random fluctuations or to the inadequacy of the model, e.g., when the rates vary too much in the interval. In our study this happened only with small samples, which suggests the first explanation. The estimations of h and r in Garki provide some interesting insights into the dynamics of infection U.UD, 1; 688 ESTIMATION OF MALARIA INCIDENCE AND RECOVERY RATES A h 0.03 0.02 0.01 0 3-4 4-5 5-6 6-7 7-8 Interval Fig. 2. Estimated daily incidence of falciparum para- sitaemia by age and season in the baseline period. with P. falciparum in an area with a very high level of transmission and a large seasonal variation. The effect of age on h (Fig. 1) may be interpreted as follows: h may increase in early life either because of increased exposure, as suggested by the biting behaviour of anophelines, or because an increasing proportion of the negatives are really latent positives and subject to relapse; later on, h decreases either because of reducing exposure (not suggested by the biting behaviour of anophelines) or because of increasing immunity, which expresses itself as decreasing susceptibility to new infection or as fewer positive episodes resulting from one inocula- tion. The effect of age on r (Fig. 1) may be interpreted as follows: from the infant group to the group aged 1-4 years, r^ may decrease either because of loss of maternal immunity or because of superinfection; later it increases, by a factor of more than 10, because of increasing immunity. The combined effect of age and season on hA (Fig. 2) may be interpreted as follows: from birth to the age of 5-8 years, the relative seasonal varia- tion decreases since new infections are progressively diluted by relapses, which are less dependent on Table 2. The observed transition frequencies and the estimated transition rates (incidence and recovery rates of P. falciparum parasitaemia) at ages 19 years and over, by season, in the baseline period and under propoxur, in 4 villages combined Interval Transition frequencies Man-biting rate eween - Incidence rate Recovery rate (A. gambiae Year Season surveys Treatment N_ N+..r pu A. fnestusaSUN8YS ____ -a =+ ±sh r sr (bites/man/(average t) N..+ iht 1971 dry 1-4 - 289/1570 438/734 0.0054 ± 0.0004 0.0175 i 0.0011 1.88(t = 66.3) wet 4-6 - 405/1153 271/525 0.0100 ± 0.0010 0.0148 ± 0.0015 51.20(t = 81.5) 1972 dry 6-9 - 282/1675 434/742 0.0044 ± 0.0003 0.0152 ± 0.0009 2.15(t= 71.7) wet 9-11 propoxur 212/1389 227/326 0.0049 ± 0.0005 0.0221 + 0.0022 1.70(t = 70.0) 1973 dry 11-14 - 188/1952 242/331 0.0028 + 0.0003 0.0213 i 0.0018 0.06(t = 73.0) wet 14-16 propoxur 234/1369 142/200 0.0054 ± 0.0007 0.0226 i 0.0031 7.24(t = 76.0) a Estimated by night-bait collection; mean of 2 villages and of indoor and outdoor collections. a l -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 689 A. BEKESSY ET AL. U h, IfA 10010 o 0.01 Season dry wet dry wet dry wet Year 1971 1972 1973 Propoxur + + Fig. 3. h and r in persons aged 19+ years, both in the baseline period and under propoxur, in four villages combined. season. In older age groups, the relative seasonal variation decreases further as a consequence of immunity, since there are fewer relapses per new infection. The effect of propoxur on h (Fig. 2) parallels its effect on the vector man-biting rate. Its moderate size reflects the persistence of relapses and transmission; the related increase of r^. i.e., the shortening of positive episodes, may be due to reduced super- infection. In infants, h estimated from the transition fre- quencies may exceed the incidence rate conven- tionally estimated from first parasitological conver- sions because the former includes relapses in addition to new infections. However, if infants recover from parasitaemia at an appreciable rate-as demonstrated here-the conventional incidence rate constitutes an underestimate, the extent of which increases with the interval between surveys. In malaria caused by P. falciparum or other malarial parasites, h and rP may be useful indicators of the intensity of transmission and of the immune status of a population. The required transition fre- quencies are easy to measure, the assumptions made-rates constant over relatively short intervals and a priori relatively homogenous population groups-are easy to accept, and the actual computa- tion is simple. The method can be applied to other epidemio- logical problems, such as other parasitic diseases, that lend themselves to description by the same model, using one reversible step and constant rates. ex I ESTIMATION OF DAILY TRANSITION RATES AND THEIR VARIANCES: MATHEMATICAL DISCUSSION It is assumed that the underlying stochastic process is Markovian in character, continuous in time, and with two states only. State 1 is called the " nega- tive", i.e., the parasite-negative state, and state 2 the " positive " one. The forward Kolmogorov differential equations describing the process are: P'11 = - hPi'1 + rP12 Pi21 = - hP21 + rP22 | (1) P'12 = - rP12 + hPjj P'22 = -rP22 + hP21 (See, for example, Bharucha-Reid (2), formula 2.21; substitute i,j, k = 1,2; Qll =Q22 = 0, Q12 = Q21 = 1, q1 = h, q2 = r.) There are two parameters in these equations, the infinitesimal transition probabilities. If the unit of time chosen is sufficiently small, h and r can be inter- preted as transition rates per time unit. The time unit is sufficiently small if the probability of more than one transition during this interval is negligible com- 690 ESTIMATION OF MALARIA INCIDENCE AND RECOVERY RATES pared to the probability of a single transition. For malaria, one day seems to be short enough in this sense; h is then the measure of the daily transition rate from the negative to the positive state and r the daily transition rate in the reverse direction. Further, it is supposed that h and r are constants between successive observations. Without loss of P12(t) = h/(r + h)(I - e - (r+ h P21(t) = r/(r+ h)-(I e-(r+h The statistical problem is how to estimate the model parameters h and r, i.e., the daily transition rates, from observed data. The method will depend, of course, on the arrangement of the observations. Here we assume that one complete observation, from the statistical point of view, consists of two sub- sequent observations of the study population. At the time of the first observation, the number N_ of negatives and N+ of positives is established. Then, at the time (t) of the second, follow-up observations, the number N_+ of persons found positive among those negative at the first time, as well as N+_ (the h = a g) In (1 -a--),t(a+P) In theory, these estimators are asymptotically unbiased and asymptotically normally distributed. Their variance-covariance matrix is the inverse of the matrix Z2 In L b2n L] aZh2 abhar 72in L -2In L _bhbr ar2 h = h, r = r (See, for example, Kendall & Stuart (3).) A somewhat lengthy but otherwise straightforward algebraic calculation leads then to the following set of formulae (used as algorithm for the calculator programme): a = N+I/N_, , = N+_/N+ 'y = a + ,B U = (- In (l - y))/t, V= Y/[(l - y)t] (4) S1 = a(l - a)/N_, S2 = f(1 - P)/N+ generality, the time point of the first observation can be chosen as zero; the time of the second sub- sequent observation will be denoted by t and it should be measured in days. Then, with the initial conditions P1l(O) = P22(0)= 1, P12(0) = P21(0) = 0 the solu- tions of (1) are: Pll(t) = 1 - P12(t), P22(t) = 1 - P21(t). (2) corresponding number of negatives among those previously positive), should be registered. The ratio N_+/N_ = a is an unbiased estimator of P12 and similarly N+_/N+ = ,B estimates P21. These two statistics are jointly sufficient for estimating h and r. Hence the standard maximum-likelihood method can be applied. The logarithm of the likelihood is (apart from additive constants not depending on the parameters): In L = N+_ * In P21 + (N+ - N+) - In P22 + N_+ * In P.2 + (N-N_+) - In P1l. The equations ULfZh = 0, ZL/Zr= 0 can be solved and result in: r -p t(a±fl9) l(-- (3) the variances are then = (S1 (aV + gU)2 + S2 (aU - aV)2)/y4 (5) S= (S2 (flV + aU)2 + SI (lU-- V)2)/y4 According to (3), estimation is possible only if a + < 1. This follows from the assumptions made because it can be seen from (2) that P12 4+ P21 < 1 must hold. Thus if a + , > 1 it could be suspected that the model did not fit well: either the process was not Markovian or the parameters were not constant between subsequent observations. Even if the model is correct, the case a + P> 1 might occur, though not too often, because of random errors; even if P12 + P21 < 1 their estimators, a and ,B respectively, may differ from them considerably, an event to be expected especially with small sample size. Observa- tion bias can also influence the value of a and ,B so that a +± > 1. 691 A. BEKESSY ET AL. Annex 2 CALCULATOR PROGRAMME The formulae (3), (4) and (5) of Annex 1 were programmed for the Hewlett-Packard 65 pocket calculator. Programme: *, STO 3, gLST x, +, 1, RCL 3,-, x, STO 1, R/S, STO 4, gLST x, .,1, RCL 4,-, x, STO 2, R/S, STO 8, 1, RCL 3, RCL 4, +, STO 5,-, STO 6, f, LN, CHS, gx = y, ,STO 7, RCL 3, x, RCL 5, +, R/S, RCL 5, RCL 6, ., RCL 8, -, STO 6, RCL 7, -,STO, 9, RCL 1, RCL 4, RCL 7, x, RCL 3, D, RCL 2, RCL 3, E, R/S, RCL 4, RCL 7, x,RCL 5, *, R/S, RCL 2, RCL 3, RCL 7, x, RCL 4, D, RCL 1, RCL 4, E, RTN, LBL, E, RCL, 9, X, f-1, Vx, X, +, f, Vx, RCL 5, ., RCL 5, .,RTN, LBL, D, RCL 6, x, ,f, x,x, RTN. Stored in registers 1-9: S1, S2, a, 8, y, V, U, t, V - U Remarks: 1. t should be in days. 2. If a + f > 1, blinking display, no estimation is possible; verify by recalling a, , from registers 3, 4 and adding them. Example: N_+ = 160; N_ = 433; N+_ = 157; N+= 356; t =81 days; h = 0.0094; Sh= 0.0011; r = 0.0112; Sr = 0.0013. User instructions: Step 1 2 3 4 5 6 7 8 9 10 Instructions Enter programme Set display conveniently, e.g. . ...... . .. *** **. ................. . . . . . .. .. .. **. . . . . . ......... ***.*** . **. ** **.*******. InputN_+N_N+_N+t KeysDSP, ., 4tAtR/SR/S. R/S. R/S.. . R/S... Output I;h Sh r Sr RIISUMt ESTIMATION DES TAUX D'INCIDENCE ET DE GUERISON DE LA PARASITE-MIE A PLASMODIUM FALCIPARUrM, A PARTIR DE DONNEES LoNGITuDiNALES Une m6thode d'estimation du taux d'incidence h et du taux de gu6rison , de la parasitemie patente, A partir de donnees longitudinales, est decrite. La methode postule que le phenomene de parasit6mie patente peut etre repre- sente par un modele catalytique reversible A deux etats, et est applicable A tous les problemes qui peuvent etre representes par un tel modele. La methode a 6td appliquee A des donnees, concemant P. falciparum, de la savane de l'Afrique de l'Ouest. Avant intervention, le taux d'incidence quotidien h augmente de 0,013 chez les nourrissons jusqu'a 0,02 dans le groupe d'age de 5 A 8 ans, puis decroit jusqu'A 0,007 A 19 ans et plus; le taux de guerison quotidien r diminue de 0,005 chez les nourrissons jusqu'A 0,002 chez les enfants de 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESTIMATION OF MALARIA INCIDENCE AND RECOVERY RATES 693 1 a 4 ans, puis croit jusqu'a 0,02 A 44 ans et plus. h varie selon la saison; la variation saisonniere augmente, en valeur relative, avec I'age. Ces observations suggerent que l'immunite accroit le taux de guerison par un facteur de 10, et reduit le nombre d'episodes de parasitemie patente resultant d'une inoculation. Sous l'action du propoxur, h varie comme le taux d'aggressivite des vec- teurs pour l'homme, tandis que r augmente, peut-etre A cause d'une reduction de la surinfection. REFERENCES 1. MUENCH, H. Catalytic models in epidemiology. Harvard University Press, 1959. 2. BHARUCHA-REID, A. T. Elements of the theory of Markov processes and their applications. New York, McGraw-Hill,1960, p. 61. 3. KENDALL, M. G. & STUART, A. The advanced theory of statistics, 2nd ed. London, Griffin, 1967, Vol. 2.