Gradual acquisition of immunity to severe malaria with increasing exposure

Previous analyses have suggested that immunity to non-cerebral severe malaria due to Plasmodium falciparum is acquired after only a few infections, whereas longitudinal studies show that some children experience multiple episodes of severe disease, suggesting that immunity may not be acquired so quickly. We fitted a mathematical model for the acquisition and loss of immunity to severe disease to the age distribution of severe malaria cases stratified by symptoms from a range of transmission settings in Tanzania, combined with data from several African countries on the age distribution and overall incidence of severe malaria. We found that immunity to severe disease was acquired more gradually with exposure than previously thought. The model also suggests that physiological changes, rather than exposure, may alter the symptoms of disease with increasing age, suggesting that a later age at infection would be associated with a higher proportion of cases presenting with cerebral malaria regardless of exposure. This has consequences for the expected pattern of severe disease as transmission changes. Careful monitoring of the decline in immunity associated with reduced transmission will therefore be needed to ensure rebound epidemics of severe and fatal malaria are avoided.

The ward of each person presenting at hospital was recorded, and so we were able to include in the analysis only those severe malaria patients who lived in the included wards, as well as using the populations of the included wards from the Tanzania National Census (2002) as the denominator for calculating incidence. The population at risk used to estimate incidence was considered to be the population of the wards from which cases were reported plus a handful of wards which were close to the hospital but reported no cases in the study period. At the district hospital in the highest transmission setting, children were recruited only every second day, due to the very large number of admissions. The data were weighted accordingly.

Symptoms and mortality parameters
The parameters that determine the probability of experiencing each symptom, given that an episode of severe malaria occurs, and the case fatality ratio for each combination of symptoms were assumed to be independent of transmission intensity and past exposure. These parameters were estimated by maximum likelihood using the individual-level data from Tanzania.

Transmission model
The model-fitting for estimating the parameters relating to immunity against severe disease was done using Bayesian methods. The procedure for fitting the model to data was as follows:  find the equilibrium model solution for each location, conditional on a given EIR, using the model and parameters from [2];  for each location, find the predicted incidence of severe malaria by age;  calculate the likelihood that the model produced the observed data.
We grouped the individual level data from Tanzania into six areas, and treated each as a single transmission setting for the purpose of calculating the model prediction: Tanga and Kilimanjaro regions were each divided into low, mid and high altitude (below 600m, 600-1200m and above 1200m respectively). Wards were classified according to the mean altitude, weighted by the population in each square kilometre. In the description that follows, these data are referred to as dataset A.
We additionally used data from nine sites across Africa reported in [3,4], referred to as dataset B. The severe disease incidence data from Bakau and Sukuta, The Gambia, Kilifi North, Siaya and Kilifi South, Kenya were originally published in [5], while the incidence data from Kilifi Township, Kenya, Mponda, Malawi, Foni Kansala, The Gambia and Ifakara, Tanzania were first published in [3]. The data point in [3] from an area of Ethiopia where malaria is epidemic rather than endemic was excluded since our analysis focusses on endemic malaria transmission. The denominator population for calculating incidence was taken as "pre-defined communities located with 15km of essential clinical services" ( [3], figure 4).
The data from dataset B was grouped into 10 one year age groups, from birth to age 10. To model variation in incidence not explained by transmission intensity, we included a random effect u for each of the nine sites, having a normal distribution with standard deviation For a given set of parameters let the model-predicted incidence of severe malaria in site j and age group i be In dataset B, where the population at risk was defined within a fixed radius of the hospital, the incidence rate was much higher than in dataset A, where the population of a larger area was taken as the population at risk. It is likely that there were also differences in health care availability leading to different catchment areas and different local population densities.
These differences could not be resolved given the historic nature of the data. Instead of having random effects we therefore introduced scaling factors so that we could jointly fit the model to the two different types of data. We introduced a total of six scaling factors one for each of the transmission settings for the Tanzanian data. While this scaling was used to adjust overall incidence, it did not affect estimates of the age-profiles which were the main motivation for this analysis.
Dataset A was available at individual level. For each of the six areas into which the data were categorised, denote the scaling factor by k r for area k . Then there was an additional scaling factor for each ward where j x is the median travel time to hospital reported by patients living in ward j , and 0 x and  are parameters to be estimated. A small number of wards had no reported travel time. For these, the time was interpolated from neighbouring wards.
The data in each ward were put into 31 age groups. Starting at age 0, they were: 4 of width 3 months, 14 of width 1 year, and 13 of width 5 years. If the model-predicted incidence of severe malaria in ward j and age group i is ji  , and the person-time at risk is ji T , then the expected number of events is kj is the area that ward j belongs to. In order to give comparable statistical weight to that given to the data in dataset B, we assume that there is a gamma-distributed random effect  for each of 14 age groups, 10 groups up to age 10 (as for dataset B), and The parasite prevalence in 2 to 10 year-olds was also known for each site in dataset B. This was modelled using a beta-binomial likelihood with site-level normally distributed random effects for the logit prevalence, as in [2].
For each site, we fitted the EIR as parameters, with log-normal prior distributions. The log EIR for each site in dataset B was given a prior mean based on [7], and a standard deviation of 2, so that the prior distribution is relatively flat and information on the transmission intensity comes mainly from the parasite prevalence. The fitting to uncomplicated malaria, parasite prevalence and EIR data reported in [2] included parasite prevalence data from 24 villages from the Kilimanjaro and Tanga regions, which for model fitting were grouped into the same six areas by region and altitude as dataset A here. Hence we had an estimate of the log EIR for each area, which we took as the prior mean, and we took the standard deviation to be 0.3.

Parameter estimates
Tables S1 and S2 contain the maximum likelihood estimates and 95% confidence intervals for the parameters determining which symptoms occur and the case fatality of each combination of symptoms. These were estimated from dataset A.   Figure S2a shows the distribution of travel times to hospital in dataset A for all severe disease patients included in the analysis for whom this data was available, up to 10 hours.

Travel time to hospital
Each bar includes the upper limit: for example reported times of exactly 1 hour are included in the first bar, not the second bar. Most patients (79%) travelled less than two hours to hospital (47% 1 hour or less, 32% 1 to 2 hours), but a substantial number travelled for many hours, with the longest journey times being 10, 12 and 20 hours.
We estimate that the proportion of severe malaria patients reaching hospital decreases sharply as their travel time to hospital increases from two to four hours ( Figure S2b), which is consistent with previously published work reviewed in [8]. This suggests that the overall incidence of severe malaria and malaria mortality in this study could be an underestimate due to the high probability of missing cases amongst populations more than 2 hours away. Figure S2 (a) Reported travel times to hospital among severe disease patients. (b) Probability that a case of severe malaria will present to hospital by travel time to the hospital for that ward in hours, relative to someone with zero travel time.

Observed and fitted symptoms by age
We assume in the fitted model that the probability of each type of symptom given that there is an episode of severe malaria is independent of transmission intensity, depending only on age. Figure S3 shows the observed probability of each of the three symptoms, or of none of them in the whole Tanzanian dataset and also in two subsets of the data, low altitude Tanga and mid altitude Kilimanjaro: 89% of the patients in the analysis were from these areas. The former has much higher transmission than the latter. For plotting, the data are aggregated into the age groups 0-2, 2-5, 5-10 and 10-20 years.
Generally the model fits the overall dataset well, although at older ages the model may slightly under-estimate the probability of developing respiratory distress. However, in each age group the proportion of severe cases with severe anaemia is lower in mid altitude Kilimanjaro than in low altitude Tanga, and the proportion with none of the three symptoms and the proportion with cerebral malaria are both higher. Figure S3 Observed probability of each symptom among those meeting the criteria for severe malaria in [1], with 95% binomial confidence intervals, and fitted model. Figure 3D in the main text shows the observed and model predicted case fatality in the Tanzanian dataset. We again plot the observed data for mid altitude Kilimanjaro and low altitude Tanga, in Figure S4. In the fitted model case fatality is assumed to depend only on the symptoms suffered, and so for a given age is assumed to be independent of transmission intensity. There possibly a lower case fatality in the youngest children in mid altitude

Case fatality by age and transmission intensity
Kilimanjaro. Figure S4 Fitted case fatality, and observed data in two areas of Tanzania.