Identifying children with excess malaria episodes after adjusting for variation in exposure: identification from a longitudinal study using statistical count models

As expected from previous studies [2], the number of clinical episodes varied with increasing age showing an initial increase and subsequent decrease in risk, consistent with initially increased mosquito biting rates as the child grows and loss of protective maternal antibodies, and subsequent acquired natural immunity to clinical malaria (Fig. 1a). This increase and decline was most evident among the children living with the highest exposure to P. falciparum parasites in the microenvironment (i.e. with the highest exposure index), followed by those in the medium, and then the lowest of the three exposure index tertiles.

In order to demonstrate the degree of over dispersion in an unadjusted analysis, we first restricted analysis to 5 years of surveillance for each individual. The mean number of clinical malaria episodes for children that had accumulated within the 5 years of surveillance was 0.6 per child per year. A Poisson distribution based around this mean would predict that very few children would have a sum of more than eight episodes of febrile malaria (i.e. <1 %) from at least 5 years of follow-up. However, this was not the case and over 16 % of the children had more than eight episodes (Fig. 1b). The Poisson distribution is therefore a poor fit for the data.

We then included the full data set with adjusting for time at risk for each individual in order to compare how well different statistical models fit the data. We compared Poisson, zero-inflated Poisson, negative binomial, and zero-inflated negative binomial models adjusting for measures of P. falciparum exposure in the microenvironment: exposure index, calendar year, and age. We included significant (i.e. P <0.05) non-linear effects using multiple fractional polynomials as described previously [22] and used child-years of observation as the unit of analysis in these models, and therefore clustered individual observations by child using the robust-sandwich estimator to account for linked observations. The zero-inflation models used exposure index as a logistic function to predict the risk of zero counts. The negative binomial model was a significantly better fit for the data than the Poisson model as judged by the AIC test (AIC = 16,811). Similarly, the zero-inflated binomial model was a significantly better fit than the negative binomial model when the two were compared by Voung test (i.e. Z = 3.17, P = 0.001). This is not surprising, considering that 47 % (1,167 of 2,463) of the children included in the model accumulated zero episodes and the distribution clearly did not fit a Poisson distribution. The relationships between age, EI, and calendar year with malaria, the building blocks of the model, are shown by plotting the partial predictors and residuals in Fig. 2. The final model with non-linear transformations is shown in Additional file 1.

Having determined the best fitting model out of those tested, we then ran 100 simulations applying the parameters returned from the model to the covariates observed in our population and selected children above the 95^th percentile for observed minus expected numbers of clinical cases as an EM group (Fig. 1c). We noted a positive skew with an excess of children with extreme positive observed–expected values. We restricted this analysis to children with 5 or more years of follow-up to reduce uncertainty. We then compared this EM group with a group of children matched for similar EI but who had only experienced an average number of clinical episodes (AM), defined as children with the expected number of cases of malaria or fewer, as determined by the zero-inflated negative binomial model. We aimed to match each EM child with the closest two AM controls, one with higher and one with lower EI. There were 212 out of 2,463 children that fulfilled our criteria for EM and we identified 319 exposure-matched AM children. The average time in follow-up and age at entry were similar between the AM and EM groups: 8.03 (confidence interval (95 % CI), 7.8–8.3) vs 7.70 (95 % CI, 7.6–8.2), P = 0.5, and 1.88 (95 % CI, 1.64–2.11) vs 1.75 (95 % CI, 1.55–1.95) years, P = 0.4, respectively. Exposure indices were 0.51 (95 % CI, 0.48–0.53) vs 0.49 (95 % CI, 0.47–0.51), P = 0.2, for EM and AM children, respectively. Insecticide-treated net usage was similar between the EM and AM children (P = 0.9, Fisher’s exact; 84 % vs 83 %, respectively).

Compared to AM, the EM children had (1) a higher geometric mean parasitaemia density during asymptomatic infections determined from cross-sectional surveys of blood smears in the dry season when P. falciparum transmission is minimal [43,891 (95 % CI, 8,500– 79,282) vs 4,597 (95 % CI, 1447– 7746) parasites per μ L], P = 0.0001; Fig. 3 panel c), (2) a higher geometric mean parasitaemia density during clinical malaria diagnosed from active weekly surveillance [80,230 (95 % CI, 67,718–92,742] vs 72,056 (95 % CI, 65,804–78,307) parasites per μL], P = 0.0001; Fig. 3 panel a), and (3) were less likely to be of the sickle cell (HbAS) trait genotype that is known to protect from malaria (Table 1). Although the proportion of children admitted to hospital with malaria was higher amongst EM than AM children, the differences were not significant (Table 2). There were no differences in the levels of axillary body temperatures during clinical malaria (Fig. 3b). Finally, there was no significant difference in the prevalence of asymptomatic P. falciparum positive blood smears at pre-transmission cross-sectional surveys (0.22 (CI, 0.16–0.28) vs 0.16 (CI, 0.12–0.21), P = 0.2, for EM and AM, respectively; Fig. 3d).

Both EM and the matched AM children were frequently found in the same geographical locations, further supporting the idea that having excess malaria is not always explained by environmental factors like increased exposure to malaria in the microenvironment (Fig. 4). For example, 79 out of 173 EM children in this study area (Junju) shared homesteads with at least one or more AM children (Fig. 4).

All these epidemiological markers point to EM as a group of children, who, over a sustained period of time, fail to acquire the ability to control parasite growth. Together, our analyses suggest that the factors responsible for increased susceptibility in the EM group are host dependent and not environmental, with sickle-cell trait being an obvious example tested.

We compared the outputs of classification of EM based on the zero-inflated binomial regression model and the output of a simple cut-off, varying the cut-off from >5 to >10 clinical episodes irrespective of exposure and other variables (Additional file 2). There was no cut-off that closely replicated the results of the model, the best compromise was a cut-off of >7 episodes, for which 27 children were classified as EM by the model but not by the cut-off, and 11 children were classified as EM by the cut-off but not by the model.