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ABSTRACT |
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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Different outcomes have different age patterns: the more severe the outcome, the younger the age group most affected. The age-pattern of each outcome also varies with the intensity of transmission.2 In stable endemic areas, the incidence of clinical malaria episodes is highest at intermediate levels of transmission.3 Hospital-diagnosed severe malaria in children also appears to be most frequent at intermediate transmission,3 but in infants shows an increase with transmission intensity,4 as does all cause mortality.5 Hospital case fatality rates are age-dependent, with the highest rates in young infants and older children and minimum rates in an intermediate age group.6,7 Malaria-specific mortality rates might therefore be expected to show different relationships with age and transmission intensity than do morbidity rates. Community-based estimates of malaria-specific mortality rates have been estimated using verbal autopsies for a number of endemic areas but the relationships with transmission intensity are unclear. One reason may be that verbal autopsies have poor sensitivity and specificity for malaria.8,9
The risk of malaria-diagnosable morbidity and mortality is thought to depend on other risk factors such as malnutrition and co-infections. It has been suggested that approximately 60% of malaria mortality is attributable to low weight, vitamin A deficiency and/or zinc deficiency.10 Eight percent of severe malaria cases in Kenya were found to be bacteremic.11
In addition to causing direct malaria mortality, P. falciparum is likely to be a contributory factor in many deaths that would not be diagnosed as malaria by a physician.12–14 Many malaria control or local elimination programs decreased all-cause mortality by more than the initial estimates of malaria specific mortality.15–19 The differential mortality required to explain frequencies of sickle cell hemoglobin (HbAS) is substantially greater than that generally attributed to malaria alone.14,16,20 However, the relative contribution of this indirect mortality has been debated.2,21
We propose a model to explain these patterns as consequences of two processes with different relationships to host age and the level of malaria transmission. The first of these is the level of immunity to asexual blood stages of the malaria parasite. The second is the chance that the host defenses are compromised by some co-morbidity or enhanced susceptibility around the time of the clinical malaria attack.
To predict the long-term impact of potential interventions on P. falciparum malaria, there is a need for dynamic models linking severe and fatal malaria to transmission.14 We now incorporate our proposal for the causes of severe malaria and malaria attributable mortality into a simulation model of malaria transmission, parasitemia, and acute morbidity.22,23 We fit the model to published data and show that the apparently conflicting observations relating morbidity and mortality rates to malaria transmission can be reconciled within a coherent framework that corresponds to current knowledge of malaria biology.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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) include a subset that are severe (B, Figure 1). We propose that severe malaria episodes can occur as a result of one or other of two distinct processes (B1 and B2). These categories do not necessarily correspond to any of the specific syndromes of severe malaria.
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) comprises those that occur when the host experiences an overwhelming parasite density. We define H(i,t) to be the clinical status of individual i at time t, PB1 (i,t) to be the probability that a clinical malaria episode in individual at time t is severe as a result of this process, and specify this probability using
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where 
indicates membership of a set, Pr indicates probability, and | is the symbol for conditional probability. Y*B1,(i,t) is then a critical value and Ymax(i,t) is the simulated maximum of daily parasite density measurements in individual i at five-day time interval t. We evaluate the fit of two possible algorithms for Y* B1,(i,t):
s, of that required to cause an uncomplicated episode in the same individual at the same timepoint, i.e. (i,t) Y* B1 = sY*(i,t), where Y*(i,t) is the previously defined pyrogenic threshold for individual i at time t.23 The second subset of severe malaria episodes (B2) occurs when an otherwise uncomplicated malaria episode happens to coincide with some other insult (e.g., a bacterial infection, malnutrition, or anemia2), which occurs with risk F(a(i,t)), a function of age a(i,t) of individual i at time (t). We considered three proposals for the age profile of these non-malaria insults (Appendix 1).
We assume that conditional on the age of the host, the risk of an acute malaria attack is independent of the risk of such an insult, but that the risk of severe malaria does depend on F(a(i,t)). The probability that an episode belonging to class B2 occurs at time t, conditional on there being a clinical episode at that time is PB2 (i,t)) defined as
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and calculated as
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The age and time specific risk of severe malaria morbidity conditional on a clinical episode is then given by
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The term PB1 (i,t) PB2 (i,t) is subtracted to avoid double- counting of that small proportion of episodes that qualify as severe by both definitions. Thus, the unconditional risk of a severe malaria episode is PB(i,t) Pm(i,t) where Pm(i,t) is the probability of a clinical episode.23
Direct malaria mortality.
We refer to deaths resulting from episodes of either class B1 or B2 as direct malaria mortality (classes C1 and C2 in Figure 1). We assume that 48% of severe malaria episodes present to hospital (Appendix 2) and that this applies equally to both class B1 and class B2 episodes. Age-specific hospital case fatality rates were taken from those reported from Tanzania.7 We assume that these hospital case fatality rates remain the same, even if the case mixture of type B1 and B2 severe malaria episodes varies between transmission settings.
The mortality risk for a severe episode at age a in the community Qc(a), is estimated with
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Qh(a) is the reported hospital case fatality rate at age a, and 
1 is the estimated odds ratio for death in the community compared with death in inpatients. Malaria mortality is then predicted by the sum of the hospital and community malaria deaths. We estimate 1 by fitting to malaria-specific mortality rates.
Indirect malaria mortality. In addition to the direct malaria mortality C1 and C2 we need to model additional, indirect, malaria deaths to replicate the association between all cause mortality and the entomologic inoculation rate (EIR), specifically in infants. We define as indirect malaria deaths those that would not have occurred in the absence of prior malaria exposure, but where the terminal illness would not have been diagnosed as malaria by a competent physician. We do not classify deaths in class C2 as indirect mortality because we consider a death in the same five-day interval as a precipitating clinical malaria episode to be diagnosable as malaria.
In the cases of indirect deaths, we propose that malaria exposure acts to enfeeble the individual, leading to subsequent mortality. These deaths would be prevented if malaria was removed, and so should be included in predictions of the potential impact of malaria interventions.
We consider two distinct classes of indirect mortality (Figure 1). D1 comprises neonatal mortality resulting from maternal infection during pregnancy. The model we use to predict the incidence of such deaths is considered in an accompanying paper.24 D2 comprises post-neonatal indirect mortality that is provoked by an acute attack of malaria, which together with other co-morbidity or enhanced susceptibility, leads to subsequent death.
The insults contributing to a death in class D2 could be sequential or they could occur together. Since this makes little difference to the predicted incidence, for mathematical convenience we use a model analogous to that for severe malaria in class B2. In this model, an event in class D2 is instigated at time t, conditional on there being a clinical episode at that time, with probability PD2 (i,t) defined as
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and calculated as
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where QD is the limiting value of PD2 (i,t) at birth.
The deaths in class D2 are simulated as occurring 30 days after time t. This allows for the possibility that the host dies of an event in class C1 or C2 before the indirect death occurs.
Data and fitting of the model.
Severe morbidity.
Data on the relative incidence of severe malaria in children less than nine years of age across different transmission intensities have been collated by Marsh and Snow.6 They summarize the relationship between severe malaria hospital admission rates and P. falciparum prevalence in children less than nine years of age. To obtain a continuous function relating hospital incidence to prevalence, we linearly interpolated between data points. To convert the hospital incidence rates to community severe malaria incidence, we divided the hospital admission rates by the assumed proportion (48%) of severe episodes presenting to hospital (Appendix 2). To fit our model to this relationship, we ran our simulation model of P. falciparum incidence, parasitology, and clinical episodes, and assumed one of the models for severe malaria described earlier in this paper, with the published transmission patterns for all the sites in Table 1 as input. We compared the predicted absolute incidence of severe malaria with the value on the interpolated curve corresponding to the predicted prevalence for the simulated site.
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For both sets of sites we simulated the incidence of severe malaria using a version of the model without effects of treatment of uncomplicated malaria episodes or any malaria mortality. The simulated population was stable in size, and the age distribution was fixed to be approximately the same as Ifakara, Tanzania using the algorithm reported in an accompanying paper.25
We simultaneously fitted our models to both the absolute incidence of severe malaria in children less than nine years of age and the age-specific relative risks by weighted least squares of the log-transformed rates, where the weights were chosen so that the two analyses were weighted approximately equally.
Simulated annealing26,27 was used to identify the parameter values that minimized the weighted residual sum of squares. Approximate confidence intervals were obtained by estimating the Fisher information for the parameters from a least squares fit of local quadratic approximations to the (stochastic) log likelihood. In addition to considering the formal model fit, we also assessed the biologic plausibility of the models and the predictions for the age groups for which we had no data.
Direct malaria mortality.
The odds ratio for death of a case in the community relative to that in hospital, 1, was estimated by fitting the malaria-specific mortality rates in children less than five years of age predicted by the severe malaria models above assuming the published hospital case fatality rate. The data were derived from verbal autopsy (VA) studies in sites with prospective demographic surveillance and were adjusted for the effect of malaria transmission intensity on the sensitivity and specificity of the cause of death determination.28 Sites with both VA data and seasonal patterns of the EIR (Table 2) were used for estimating 1. The fitting algorithm was the same as for the severe malaria model and we assume that there was no effective treatment of uncomplicated malaria episodes.
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). We use the entomologic data as input and estimate QD and the infant mortality that is independent of malaria Qn by the same fitting algorithm that was used for the severe malaria and direct mortality components. The model for indirect mortality is conditional on our models for severe malaria and direct malaria mortality, and assumes no effective treatment of uncomplicated malaria episodes. Since a study found no clear relationship between all-cause mortality for children 1–4 years of age and transmission intensity,5 we did not use data for children more than one year of age to estimate the parameters of the model for indirect mortality.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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). Both models produced similar predictions (Figure 2). Both gave a good fit to most of the data, and in particular they reproduced the decrease in incidence with transmission intensity in highly endemic areas.
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), and this led to estimates of malaria mortality rates that exceed recorded all-cause mortality rates in some age groups.30 Predicted mortality rates in older age groups were lower with model 2. The assumption of a constant parasitemia threshold for severe malaria in model 2 is also more attractive because of the evidence that total parasite biomass is critical in precipitating severe malaria episodes.31 The estimate of this threshold of 784,000 parasites/µL is high, but within the range observed in severe malaria patients. We do not attach much credibility to the precise value of this threshold because our simulation model only reproduces distributions of parasite densities very approximately.25
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). The proportion of predicted severe malaria cases that belong to class in this model increases with transmission B2 intensity because the infections tend to occur at younger ages (Figure 5).
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). The predicted age-specific community case fatality is shown in Figure 6.
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, together with the predictions for the same sites using the severe malaria model that we have adopted (model 2). Both the observed data and predictions show no obvious trend with transmission intensity, and there is a large variation between the sites in the verbal autopsy-based rates.
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). Using the severe malaria model with a multiplier for the pyrogenic threshold (model 1), adults 20–39 years of age had a rather high predicted malaria mortality rate (Figure 8c). This is the result of the high predictions for incidence of severe malaria with this model.
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). However, this quantity was estimated very imprecisely because the parameters QD and Qn are highly correlated.
There was an association between the observed all-cause IMR and transmission intensity, as previously reported using broader inclusion criteria5 (Figure 9). The predicted IMR for these sites using model 2 (incorporating the effects of severe malaria and malaria mortality models as above) reproduces this apparent trend.
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). Although the deaths in infants tend to increase, this is not the case for either direct or indirect malaria mortality for older age groups. Taking all age groups together, the ratio of indirect: direct malaria deaths was 0.6 for an EIR of 5. This increased to 1.4 for an EIR of approximately 100 and did not increase further for higher transmission intensities.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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) represent malaria- specific morbidity. These are more frequent at moderate levels of transmission and account for the peak in the incidence of pediatric severe malaria at intermediate levels of transmission. Within our model, this is mainly because maternal immunity helps to control the first infections at very high levels of transmission, so that the initial infections are less well controlled if they occur later in life.25 The patterns of events in classes B1 and B2 with age and with transmission have similarities to those described for severe malaria anemia, and for patterns for cerebral malaria,32 respectively. However, our simple structure for the different classes of events does not aim to map onto the pathophysiology of these syndromes. Recent work has suggested that the different syndromes of severe malaria are overlapping.33 A major uncertainty lies in the choice between models 1 and 2 for the relationship between parasite density and severity of disease. This is likely to have an important effect on our predictions of the impact on interventions that affect blood stage densities, and points to a gap in our knowledge of pathogenesis.
The fitting of these models suggests that a substantial proportion of severe malaria episodes involve age-dependent cofactors that are concentrated in the youngest children. This is consistent with the fact that these children have the least immunity to other infections and are also at highest risk of nutritional problems.
We assumed the same age dependence in co-morbidity in estimating the contribution of malaria to indirect deaths (D2 in Figure 1) and thus the effects of co-morbidity dominate those with high parasite densities in determining the impact of P. falciparum on all-cause mortality in the youngest children. The strong age dependence is supported by ecologic comparisons of all-cause mortality rates and malaria transmission intensity, where there is no clear association after the first year of life.5 It is also in agreement with analyses of HbAS frequencies that have suggested that indirect malaria mortality is likely to be concentrated in the youngest children.20
Clinical malaria episodes are also more concentrated in younger children as the transmission intensity increases.23 Therefore, within our model, the probability that these risks coincide to cause either severe malaria episodes (B2) or sub-sequent indirect mortality (D2) increases with transmission level. We used clinical malaria episodes for the predisposing factor for indirect deaths, but it is also possible that symptomless parasitemia plays this role.21
The model points to other important areas of uncertainty. Malaria in adults is an example of this. It is generally thought that severe malaria occurs only infrequently in adults in the stable endemic conditions prevailing in much of Africa,6,29 and although severe malaria is commonly diagnosed in African adults, many of these represent misdiagnoses.34,35 In a randomized trial, insecticide-impregnated nets did not reduce mortality in Ghanaian adults, suggesting that malaria is not a major cause of death in this age group.36 However, immunologically naive adult visitors to endemic areas are highly susceptible and major epidemics with high case fatality may occur in areas of initially low transmission to which malaria returns after having been nearly eliminated.37,38 A recent observational study in an endemic area of Papua New Guinea suggested that mosquito nets have a substantial effect in reducing all-cause mortality in adults in an area of moderate transmission.39 These results suggest that malaria may be an important cause of adult mortality in areas of low endemicity.
We expect that severe malaria is infrequent in those adults with a substantial history of exposure to P. falciparum because they control parasite densities and thus rarely develop any acute clinical episodes. Major epidemics should not occur as a rebound if malaria control is abandoned in areas of very high previous exposure because of persistence of immunity against asexual stages of the parasite. In contrast, people who become infected with P. falciparum after spending most of their lives without being exposed are highly susceptible to severe episodes. Current efforts to control malaria may lead to sustained reductions in malaria transmission without eliminating the parasite, and this could place many older children and adults in this position. The shape of our function for co-morbidity is critical in our predictions of the public health burden that this implies. If co-morbidity follows the strong increase in infectious disease mortality with age that is observed in adults, then we would predict that in low and unstable transmission settings where most adults never acquire much immunity P. falciparum may be an important cause of mortality in elderly people. There is a need to test whether this is the case.
There are many other factors influencing the risk and outcome of severe malaria that we have not been able to consider explicitly. These include effects of host genetic markers and of seasonality.12 In addition, field estimates of malaria morbidity and mortality rates are unavoidably plagued by effects of attendance bias and diagnostic uncertainties. The empirical basis for estimating the effect of in-patient care on case fatality rates is particularly weak. There are estimates of four relevant quantities, the hospital case fatality rate,6,34 the overall malaria mortality rate,28 the proportion of malaria deaths that seek care in health facilities,40 and the per capita admission rates for severe malaria.6 However, these do not provide a basis for convincing estimation of the case fatality rate in the community. This adds considerable uncertainty when our model is used to estimate the likely public health impact of improving curative services.41
In the context of recent developments in malaria control42–44 there is a need for comparisons of the likely epidemiologic impact of different intervention strategies. Randomized controlled trials provide a solid basis for predictions of the short-term impact but these cannot necessarily be extrapolated beyond the time horizon of the trial, which is rarely more than 1–2 years. Adverse consequences resulting from interference with the acquisition of natural immunity may take much longer than this to become apparent, and the full impact of malaria interventions on human-vector transmission is also only likely to be seen over longer periods. The model we propose represents a first step towards making predictions of longer term effects that can allow for these factors.
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APPENDIX 1 |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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). Infectious disease morbidity in rural African sites decreases strongly over the first few years of life so we require that, at least over this period, F(a(i,t)) should be a decreasing function of the age a(i,t) of individual i at time t.
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F(a(i,t)) = ß1exp(-ß2a(i,t)) where ß1 and ß2 are constants.
A second proposal is a hyperbolic curve.
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where a*F and F0 are constants.
An alternative is to use an empirical function. We explored a function based on the first principal component of the life tables for demographic surveillance sites in predominantly rural communities in Africa.30 This curve decreases with age in very young children but increases with age in adults. We expect this to represent mainly the age-pattern of infectious disease mortality (excluding that due to human immunodeficiency virus), but it is not necessarily an appropriate curve to represent the age-pattern of relevant co-morbidity. We scale the risk of an insult that would convert an uncomplicated episode to a severe attack by assuming in our model that
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The age patterns were best reproduced using the hyperbolic curve (option ii), and adopt this proposal as part of our model. The estimates for a*F and F0 are given in Table 3. We assumed the same function for co-morbidity contributing to indirect deaths (equation 7). For this, we estimate the prevalence at birth of co-morbidity QD, but the same value for a*F was used because we fit the indirect model only to infant data that does not give information about the decrease of the function with age (Table 3).
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APPENDIX 2 |
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ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
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). For simplicity, they ignore age and season dependence and consider very approximate average rates for children.
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). Pediatric hospital admission rates for the studies reported by Marsh and Snow4,6 (Figure 2) average approximately 30/1,000 person-years, and overall malaria mortality rates (from VA studies) is approximately 10 per 1,00028 (Figure 7). Combining Qh=1 with the ratio of inpatient admissions to overall malaria deaths, Ph/Pc = 30/10 = 3, gives an estimate of QhPc/Ph, the proportion of deaths that occur in hospital, of 0.3. This is similar to the results of a retrospective study of VAs in Tanzania40 that found that about 33% of children less than five years of age who died of malaria had attended the hospital at some time during their terminal illness, though the proportion who died there was lower.
In rural sub-Saharan Africa, since many hospitals are difficult to reach and often provide poor standards of care, attendance is likely to be even less frequent than in the Tanzanian study where public health services have a relatively high ratio to population. However in the research settings that contributed most of the VA and hospital data (many of them the same sites) hospital attendance rates may have been higher. In view of this, we assume that proportion of cases treated is Ph/PB = 0.48, in agreement with the proportion of severe episodes receiving inpatient treatment in the model of Goodman and others.45 Using the formulae in Table 4, this gives an estimate of 31% for the case fatality rate in the community, corresponding to this level of treatment (arrows in Figure 12) and 21% for the overall case fatality rate. This implies that the health system prevents approximately 33% of malaria deaths. This compares with an estimate of 44% for the proportion of (all cause) deaths prevented by a good Kenyan district hospital.46
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1 = 2.09 for the ratio of odds of community death to inpatient death by fitting our stochastic model to verbal autopsy data adjusted for sensitivity and specificity (see Materials and Methods).
Irrespective of the proportion of episodes resulting in admission, the low values of 1 = 1 that we propose at first sight appear to indicate that inpatient treatment has little benefit. The reality is undoubtedly more complex than this simple model. We dichotomized clinical malaria into severe and uncomplicated classes and assumed each class to be homogeneous in prognosis. In practice, there is a continuous range of severity and inpatients are likely to disproportionately represent the most severe cases, many of whom arrive at health facilities when it is too late for treatment to be effective. This selection bias leads to an underestimate the benefit of seeking treatment. Treatment may be life-saving even when administered less than optimally or based on imperfect diagnoses. Contact is made with formal health facilities at some stage during the terminal illness in many more cases than those who die in hospital.40 For every case that dies despite making contact with the health services, many more may be saved.
Received September 18, 2005. Accepted for publication November 20, 2005.
Acknowledgments: We thank Dan Anderegg for translating papers from French to English and Professor Klaus Dietz for helpful discussions. We also thank the members of the Technical Advisory Group (Michael Alpers, Paul Coleman, David Evans, Brian Greenwood, Carol Levin, Kevin Marsh, F. Ellis McKenzie, Mark Miller, and Brian Sharp), the Project Management Team at the Program for Appropriate Technology in Health (PATH) Vaccine Initiative, and Glaxo-SmithKline Biologicals S.A. for their assistance.
Financial support: The mathematical modeling study was supported by the PATH Malaria Vaccine Initiative and GlaxoSmithKline Biologicals S.A.
Disclaimer: Publication of this report and the contents hereof do not necessarily reflect the endorsement, opinion, or viewpoints of the PATH Malaria Vaccine Initiative or GlaxoSmithKline Biologicals S.A.
* Address correspondence to Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, Postfach, CH 4002 Basel, Switzerland. E-mail: Thomas-A.Smith{at}unibas.ch 
Authors’ addresses: Amanda Ross, Nicolas Maire, and Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, Postfach, CH 4002 Basel, Switzerland, Telephone: 41-61-284-8273, Fax: 41-61-284-8105, E-mails: amanda.ross{at}unibas.ch, nicolas.maire{at}unibas.ch, and Thomas-A.Smith{at}unibas.ch. Louis Molineaux, Peney-Dessus, CH-1242 Satigny, Geneva, Switzerland.
Reprint requests: Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, Postfach, CH 4002, Basel, Switzerland.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX 1
APPENDIX 2
REFERENCES |
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= observed malaria-specific mortality in children less than five years of age 
= model predictions. Predictions and observed data for the same sites are vertically aligned because they have the same transmission intensity.
= infant mortality rates from field data; 
= predictions using model 2.
