How does work?

Mathematical models are important decision-making tools for the control of infectious diseases and malaria was one of the first infections to which this was applied. In the 19th century, a pioneer of malaria modeling, Ronald Ross, saw the value of mathematical analyses in planning malaria control, and developed the first mathematical model for malaria.

The likely impact of malaria control interventions has generally been inferred from clinical trial results, even though only short-term effects are assessed in trials. Few comparative analyses of the potential impact of different malaria control strategies have aimed to quantify the inevitable longer term impacts.

The inherent uncertainties make it difficult to optimize malaria control strategies or to prioritize areas of research.

Requirements of a predictive model for the effects of malaria interventions.

Characteristics of individual P. falciparum infections. A model for use in predicting the population impact of a malaria intervention strategy must embed within it a relevant description of the course of individual infections.

Short-term effects on individuals. Drug treatments and vaccines may have effects on outcomes that are more complex than the effects on primary infections in the non-immune host.

If the effect of a vaccine is simply to reduce the force of infection, the short-term consequences in terms of morbidity and mortality risks are not simply proportional to the reduction in infection rate.

Long-term effects individual impact. The introduction of insecticide-treated nets (ITNs) for malaria control has been accompanied by extensive debate about possible long-term effects. Related issues arise with regard to vaccines.

Long term effects of vaccination programs are even more difficult to predict, since field trials of malaria vaccines carried out thus far consider only impacts that can be measured during the 6-18 months follow-up periods. Unfortunately, the longer term consequences of a vaccination program cannot simply be extrapolated from the results of such trials. For example, some benefits of vaccination may take an extended period to become evident. This will be particularly the case if there is natural boosting or if there are effects on transmission dynamics. Conversely, vaccination may result merely in delay of morbidity and mortality in some individuals, in which case field trials may suggest a greater benefit than will be observed during implementation and scaling up of malaria vaccine programs.

Interdependence of hosts. An epidemiologic model for the effects of an intervention program must consider the dependence between events in different individuals. Field trials of interventions are generally designed with the objective of directly protecting individuals either from infection or from consequent morbidity and mortality, and do not consider broader effects on transmission.

Interdependence of hosts has been the core of most subsequent malaria modeling exercises.

Structure of the malaria models used by

Swiss Tropical Institute has developed new models to make quantitative predictions of the potential impact of vaccination against P. falciparum malaria.

The main component of these models is a stochastic simulation of the epidemiology of P. falciparum that incorporates insights from the within-host models, but is implemented independently of them. We have used this epidemiologic model to simulate the results from the recently completed trials of the malaria vaccine RTS,S/AS02 carried out in adult men in the Gambia and in children aged 1-5 years in Mozambique.

The model has also been employed to predict the potential epidemiologic impact of such a vaccine, and the cost-effectiveness. To make these predictions we incorporated costing data and a model for the health system currently in place in a low-income country context, based largely on data from Tanzania.

In addition Swiss Tropical Institute has also made progress on developing within-host dynamics of malaria. This work is intended to complement earlier within-host models, specifically with a view to providing insights relevant to modeling vaccination, useful for informing the epidemiologic models. The within-host models have been fitted to data from malariatherapy patients and lead to conclusions that are particularly relevant to the modeling of asexual blood-stage vaccination.

An important strength is that the epidemiologic model ties together an ensemble of interconnected sub-models validated against actual field data from various settings across AFRICA. In view of the complex malaria life cycle and gaps in our current knowledge there are inherent limitations attached to some of these components, which in turn influence the overall model outcomes.

To the best STI team's knowledge this is the most comprehensive population-based simulation of malaria yet developed. It represents an important new tool for rational planning of malaria control and vaccine development, and can readily be adapted to assess efficacy and cost-effectiveness of other malaria control interventions employed singly or in combination. This makes it possible to integrate epidemiologic and economic considerations in rational formulation of policy to reduce the intolerable burden of malaria.

The current burden of malaria morbidity and mortality, particularly in sub-Saharan Africa, is so large that even an intervention that modifies the course of infection in only a proportion of recipients without any effects on transmission may be worth pursuing. Transmission effects should not be ignored, but need to be just one part of a model that includes also the independent effects.

Strategy of epidemiologic modeling

The models need to simulate the processes that may be affected by vaccination, and also to capture the relationships between these processes and outcomes of public health importance. For our model we use as input the seasonal pattern of transmission, and make predictions of the consequent infection rate of humans. We then consider how this relationship may be modified by naturally acquired immunity, or by vaccination.

Swiss Tropical Institute embed an empirical description of within-host asexual parasite densities in the model for the infection process to give stochastic predictions of parasite densities as a function of the age of a malaria infection, and model the effect of immunity to asexual blood stages by considering how the distribution of parasite densities is modified in the semi-immune host. This model for immunity provides a straightforward basis for analyzing possible effects of asexual blood-stage vaccines, which can be simulated by a function that reduces parasite densities.

Swiss Tropical Institute analyze the relationship between asexual parasite densities and infectivity to the vector in malariatherapy patients in order to derive a model for the transmission to the mosquito vector. This relationship is used to simulate the transmission-blocking effects of vaccines. This makes use of the simulated population distribution of parasite densities to predict the human infectious reservoir for P. falciparum.

An important simplification in the strategy is to avoid predicting those intermediate variables whose quantitative relationships with epidemiologic outcomes are very uncertain.

Stochastic simulation. Swiss Tropical Institute use individual-based simulations with 5-day time steps to implement our models of P. falciparum epidemiology. This approach makes it possible to model populations of hosts and infections, each characterized by a set of continuous and static variables (parasite densities, infection durations, and immune status variables for individual hosts). This approach can allow more realistic consideration of the stochastic interactions between individual hosts and pathogens than the use of compartment models. A disadvantage is that it is computationally more intensive than the deterministic alternatives. All modules were implemented using the FORTRAN programming language.

Fitting to real data. The uncertainty inherent in complex models needs to be minimized by ensuring that all elements of the model conform as much as possible to reality.

Swiss Tropical Institute has fitted different components of the model to a wealth of datasets from many different ecologic and epidemiologic settings. Our approach leads to implicit statistical models requiring many repeated simulations in order to make approximate parameter estimates. Swiss Tropical Institute was able to fit these using a simulated annealing algorithm, distributing simulations across our local computer network.

Modular structure. The computational demands and complexity of the fitting process meant that it was not feasible to fit our overall model to all the relevant data simultaneously, so different sub-models were fitted separately.

These sub-models were fitted to field data quantifying the relationship between malaria transmission and the outcome of interest. Each sub-model was thus fitted conditionally on the parameter estimates made at earlier stages in the fitting process. This approach made it possible for us to allow for the dynamic effects of the treatment of clinical episodes, an important consideration when STI use the model to predict the impact of interventions.

Equations. In view of the modular structure of the project, the underlying equations of the epidemiologic model are grouped around six main components: (i) infection of the human host; (ii) characteristics of the simulated infections; (iii) infectivity to mosquitoes; (iv) acute morbidity; (v) mortality; (vi) anemia.