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MATHEMATICAL MODELING OF THE IMPACT OF MALARIA VACCINES ON THE CLINICAL EPIDEMIOLOGY AND NATURAL HISTORY OF PLASMODIUM FALCIPARUM MALARIA: OVERVIEW

ABSTRACT

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
We report a major project to develop integrated mathematical models for predicting the epidemiologic and economic effects of malaria vaccines both at the individual and population level. The project has developed models of the within-host dynamics of Plasmodium falciparum that have been fitted to parasite density profiles from malariatherapy patients, and simulations of P. falciparum epidemiology fitted to field malariologic datasets from a large ensemble of settings across Africa. The models provide a unique platform for predicting both the short- and long-term effects of malaria vaccines on the burden of disease, allowing for the temporal dynamics of effects on immunity and transmission. We discuss how the models can be used to obtain robust cost-effectiveness estimates for a wide range of malaria vaccines and vaccination delivery strategies in different eco-epidemiologic settings. This paper outlines for a non-mathematical audience the approach we have taken and its underlying rationale.

INTRODUCTION

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Despite considerable efforts over the last three decades, and millions of dollars spent, there is still no registered vaccine against Plasmodium falciparum malaria. Several candidate malaria vaccines are now in phase I or phase II clinical trials or have entered pre-clinical testing. There is a need for rational approaches to decide how to prioritize different malaria vaccine development programs and to plan for the deployment of the vaccine(s) once efficacy has been established.

Mathematical models have been valuable decision-making tools for vaccination strategies against infectious diseases, in particular for those covered by the Expanded Program on Immunization (EPI).¹ Compared with other organisms that cause infectious diseases, P. falciparum has a complex life cycle, expressing many different potential targets for vaccines and various candidate vaccines targeting different stages of the parasite are in clinical development.² The history of ineffective or partially effective control of malaria and failed vaccination attempts has led to the assumption that the efficacy of a malaria vaccine is unlikely to approach 100%, but since P. falciparum is one of the most frequent causes of morbidity and mortality in areas where it is endemic,^3–5 even a partially protective vaccine may be highly cost-effective and a critically important public health tool. However, it is not obvious what minimum level of efficacy must be achieved before major investments in vaccine production can be justified. The issue arises that if a number of partially effective candidates with different profiles become available, how should their development be prioritized?

In this context, mathematical models of both the natural history and epidemiology of malaria are needed to guide the process of malaria vaccine development. Malaria models have several roles that transcend their obvious limitations in making precise predictions.⁶ They offer the possibility of systematically comparing the likely benefits of alternative types of vaccines and vaccine delivery scenarios, of predicting likely cost-effectiveness, and of identifying the role of vaccination within integrated control approaches. In addition, they provide a means of identifying current gaps in knowledge that need to be filled for rational planning of vaccine development strategies.

Plasmodium falciparum malaria was one of the first pathogens to be described by a mathematical model.⁷ Subsequent developments, i.e., the Ross-Macdonald models and the malaria model of the Garki project^8,9 have played seminal roles in the design of malaria control policies and the global malaria eradication campaign carried out in the 1950s and 1960s.^10,11 However, these malaria models were not designed to predict the likely impact of malaria vaccination.

We now describe the challenges that a model must address if it is to provide useful predictions of the potential impact and cost-effectiveness of malaria vaccines, and then outline our malaria modeling project that aims to meet this objective. The accompanying articles describe the different components of our models, and the conclusions we have so far been able to draw from them.

REQUIREMENTS OF A PREDICTIVE MODEL FOR THE EFFECTS OF MALARIA VACCINES

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Characteristics of individual P. falciparum infections. A model for use in predicting the population impact of a vaccine must embed within it a relevant description of the course of individual infections. For many infectious diseases, the necessary description is quite simple. For instance measles, pertussis, rubella, and varicella have well-defined latent periods, followed by acute episodes of morbidity and infectiousness, making it realistic to use a common modeling approach to address all of them. In contrast, the course of a single infection of P. falciparum, such as that shown in Figure 1, is far more complex, with infectiousness and the risks of acute morbidity and mortality varying as consequences of erratic temporal patterns of parasite densities. A number of recent models have analyzed how these patterns depend on clonal antigenic variation of the parasite^12–14 or on erythrocyte dynamics.^15,16

View larger version (23K): [in this window] [in a new window]       FIGURE 1. Course of a single infection of Plasmodium falciparum. Data of a characteristic malariatherapy patient (Patient S-1044).85 Dashed line = gametocytes; solid line = asexual parasites. (Reproduced with permission of the American Society of Tropical Medicine and Hygiene).

Short-term effects on the vaccinated individual. The effects on morbidity and mortality of a partially protective vaccine are likely to be more complex than the effects on primary infections in the non-immune host.¹⁷ Even if the effect of the 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. Pre-existing immunity and heterogeneity in the efficacy of a vaccine both lead to reduction in the effectiveness in preventing infection.¹⁸ The efficacy of vaccination against post-infection outcomes such as morbidity and mortality may be very different from that against infection.¹⁹ A model to predict population impact of a vaccine needs to include these processes that modulate the impact of infection.

Long-term effects on the vaccinated individual. Field trials of malaria vaccines carried out thus far consider only impacts that can be measured during the 6–18 months follow-up periods.^20,21 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.

The introduction of insecticide-treated nets for malaria control has been accompanied by extensive debate about possible long-term effects. Related issues arise with regard to vaccines. Since reduction in exposure to the parasite will delay the acquisition of immunity, it has been conjectured that other factors being equal, long-term transmission control might only delay severe disease or even death.^22–24 Supported by data from long-term follow-up of transmission control projects,^25–28 others have contended that the benefits will outweigh any such potential effects.^29–32 Such possible delays in acquisition of immunity need to be considered in appraisals of the cost-effectiveness of malaria interventions, including vaccination.³³

Some of the long-term effects of malaria control are extremely difficult to predict. These include effects on children’s attendance rates and performance at school, higher education achievements, aspirations and forgone opportunities to enter competitive job markets, general well-being, and equity.³⁴ On the macroeconomic scale, malaria has measured effects on foreign direct investment, population mobility, tourism, and international trade, but the causal relationships of how malaria delays social and economic advancement of whole societies remain elusive.^35,36

Interdependence of hosts. An epidemiologic model for the effects of a vaccination program must consider the dependence between events in different individuals. All malaria vaccine field trials done so far have been designed with the objective of directly protecting the vaccinated individuals either from infection or from consequent morbidity and mortality, and have not considered broader effects on transmission. Evaluations of transmission effects do not form part of standard methods for evaluating vaccines against pre-erythrocytic or asexual blood stages of malaria.³⁷

The importance for mathematical models of the dependence between events in different individuals was already recognized by Ronald Ross some 90 years ago,³⁸ and has been the core of most subsequent malaria modeling exercises. This is the key element that distinguishes infectious disease modeling from that of non-infectious diseases.³⁹ The analysis of this dependence has been the objective of most previous models for vaccination against malaria,^40–43 which have concentrated on identifying the conditions for controlling or interrupting transmission.

The current burden of malaria morbidity and mortality, particularly in sub-Saharan Africa, is so large^3–5,44 that even a vaccine 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.

Cost-effectiveness analysis (CEA). This has become increasingly important for evidence-based decision-making in health care in resource-constrained settings. There is now consensus among economists about the main points of CEA methodology,^45,46 although there continues to be important advances in techniques related to CEA such as modeling uncertainty.⁴⁷ Little work has been done on the cost-effectiveness of malaria vaccination,⁴⁸ and this has not considered the potential savings in health care costs or productivity of workers. However, CEA of malaria interventions, even when based on careful costing,^33,49–51 have generally not taken into account either the transmission effects or the dynamics of the long-term impact. An adequate model for CEA of malaria vaccines needs to consider these elements.

STRUCTURE OF THIS PROJECT

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
None of the existing malaria models satisfies all of the requirements articulated above, so we have been able to make only limited use of them in developing new models to make quantitative predictions of the potential impact of vaccination against P. falciparum malaria.

The main component of the project is a stochastic simulation model for the epidemiology of P. falciparum that incorporates insights from the within-host models, but is implemented independently of them.^49–55 We have used this epidemiologic model to simulate the results⁵⁶ from the recently completed trials of the malaria vaccine RTS,S/AS02 carried out in adult men in The Gambia⁵⁷ and in children 1–5 years of age in Mozambique.²¹ The model has also been used to predict the potential epidemiologic impact of such a vaccine,⁵⁸ and the cost-effectiveness of introducing it via the EPI.⁵⁹ To make these predictions, we incorporated costing data^59,60 and a model for the health system currently in place in a low-income country context, based largely on data from Tanzania.⁶¹

In addition, we have also made progress on developing models of within-host dynamics of malaria.⁶² This work is intended to complement earlier within-host models,^13,63 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.

STRATEGY OF EPIDEMIOLOGIC MODELING

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Processes modeled. To give reasonable predictions, our 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. Figure 2 gives a simplified illustration of how these processes and outcomes relate to the malaria transmission cycle. For our model, we use as input the seasonal pattern of transmission (measured by the entomologic inoculation rate [EIR]), and make predictions of the consequent infection rate of humans.⁵⁴ We then consider how this relationship may be modified by naturally acquired immunity⁵⁴ or by vaccination.⁵⁶

View larger version (16K): [in this window] [in a new window]       FIGURE 2. Simplified diagram of the malaria transmission cycle. Dashed arrows indicate the points at which vaccines are intended to act.

We analyze the relationship between asexual parasite densities and infectivity to the vector in malariatherapy patients to derive a model for the transmission to the mosquito vector.⁵³ 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.⁴⁹

Acute episodes of clinical malaria are predicted to occur as a consequence of high parasite densities.⁵⁵ A further stochastic sub-model is used to specify when these lead to severe disease or malaria-related mortality.^51,52

An important simplification in our strategy is to avoid predicting those intermediate variables whose quantitative relationships with epidemiologic outcomes are very uncertain. We do not dissect protection during the pre-erythrocytic stages of infection into that against sporozoites and that against liver stages because effects on these different pre-erythrocytic stages cannot be distinguished in large-scale field studies. We do not model levels of immune effector molecules, such as antibodies or cytokines. We consider levels of gametocytemia only as part of the validation of the sub-model for infectiousness because the quantitative relationships between gametocytemia and infectiousness to mosquitoes are problematic.^53,64 These simplifications do not compromise the ability of our models to make predictions of the effectiveness and cost-effectiveness of vaccines.

Stochastic simulation. We use individual-based simulations with five-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 state 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.⁶⁴ It provides estimates of distributions of outcomes, rather than only predicting averages. A disadvantage is that it is computationally more intensive than the deterministic alternatives. All modules shown in Figure 3 (except that to predict the prevalence of anemia) were implemented using the FORTRAN programming language using numerical and statistical libraries provided by the Numerical Algorithms Group (http://www.nag.co.uk/). These core components were wrapped in a application written in Java (http://sun.java.com) and accessed via the Java Native Interface (http://java.sun.com/j2se/1.4.2/docs/guide/jni)⁶⁵ for three reasons. First, implementation of data-holding components, which provide the input data for the core model and store the generated results, is easier with a programming language that supports object-oriented programming. Second, we developed a graphical user interface to simplify the process of defining simulation scenarios and to facilitate exploratory analysis of model predictions. Third, the use of the Java Remote Method Invocation (http://java.sun.com/products/jdk/rmi/)⁶⁶ allowed us to distribute the computation to a large number of computers and thereby cope with the considerable computational requirements posed by the data fitting process. The model for prediction of anemia was implemented as part of the analysis module in Java.

View larger version (29K): [in this window] [in a new window]       FIGURE 3. Key processes and relationships simulated by the dynamic models of Plasmodium falciparum transmission and morbidity.

In contrast, we have fitted different components of our model to a wealth of datasets from many different ecologic and epidemiologic settings. We then validated them by comparing our predictions with further field data. Stochastic simulations are more difficult to fit to data than are deterministic models. Our approach leads to implicit statistical models requiring many repeated simulations to make approximate parameter estimates.⁷⁰ We were able to fit these using a simulated annealing algorithm,^71,72 distributing simulations across our local computer network.

Modular structure. Since 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, different sub-models were fitted separately. The analyses described in subsequent reports^49–55,73 contributed sub-models to the overall model of malaria epidemiology (Figure 3).

Our model for how infection rates are related to the EIR in the naive host⁵⁴ was fitted to data from The Gambia⁷⁴ and Kenya.⁷⁵ The core (parasitologic) model for infection and parasite densities^50,54 was jointly fitted to datasets from Ghana, Nigeria, and Tanzania. The sub-model to predict clinical episodes⁵⁵ was fitted to data from Senegal conditional on the parasitologic model and uses the same point estimates of the parameters of the parasitologic model. Similarly, the sub-model for severe malaria⁵² is conditional on both the parasitologic and clinical sub-models. Those for mortality⁵² depend on the parasitologic, clinical, and severe malaria sub-models (Figure 3) and, like the model for severe malaria, were fitted to rates from multiple African settings.

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 (i.e., on the sub-models higher up in Figure 3). This approach made it possible for us to allow for the dynamic effects of the treatment of clinical episodes, an important consideration when we use the model to predict the impact of interventions.

The sub-models for the infection of the vector^49,53 and for anemia⁷³ were fitted to independent datasets. To make predictions of vector infection rates and of anemia prevalence we apply the estimated functions to the outputs of the parasitologic sub-model.

Equations. The equations of the epidemiologic model are summarized in Appendix 1. In view of the modular structure of the project, they are grouped around six main components: infection of the human host, characteristics of the simulated infections, infectivity to mosquitoes, acute morbidity, mortality, and anemia.

STRENGTHS AND LIMITATIONS OF OUR MODELING APPROACH

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Strengths. It may be unrealistic to expect any model to deliver quantitative predictions of the cost-effectiveness of malaria vaccination with any degree of confidence. However our models can certainly suggest where to look for possible counter-intuitive impacts of vaccination. At each stage in our modeling approach, we have examined what are the main uncertainties that could impinge on the estimates of outcomes, and thus explored what are the important gaps in knowledge of malaria epidemiology. We also compare the fit of competing models, and thus choose between alternative model formulations. Although each component of the project is linked (Figure 3), each paper leads to its own conclusions. Many of these conclusions have bearing on vaccine development strategies independently of their consequences for quantitative predictions of vaccine impact.

Limitations. With all models of biology, there is a trade-off between parsimony and the fitting of details to what is know about the dynamics being modeled. A model is only useful if it represents a simplification, indicating which elements of the processes being analyzed are important. However, the better the fit to reality, the more likely are the predictions to be accurate. In the case of malaria the need for accurate models required simulation of many different processes. The requirement for a good fit to field data has thus committed us to developing a model with many different components and parameters. At present, some of the processes we modeled are ill-understood or lack relevant data, leading to uncertainties that cannot be captured by statistical measures of imprecision.

One role of modeling is to identify such gaps in knowledge. There are also other simplifications that limit the extent to which our models should be applied uncritically. The models developed so far do not address the issue of differences between ethnic groups in their response to infection, although such differences are known to exist within the savannah zone of West Africa,^76,77 the source of many of the data available to us, and are likely to be even more important in extensions of the model outside Africa. Our models do not consider effects of micro-heterogeneity in transmission within the human population.⁷⁸ This limits their applicability as a tool for estimating the basic reproductive number, and thus for predicting the conditions for elimination.

Our models do not capture all the epidemiologic phenomena that are relevant to immunity to malaria. In endemic areas, chronic asymptomatic infection appears to play a role in effective clinical immunity⁷⁹ and may be necessary for long-term maintenance of immune memory.⁸⁰ These phenomena, sometimes referred to as premunition, very likely involve several distinct immunologic mechanisms. In our models concomitant infections induce clinical immunity mainly by increasing the threshold level of parasitemia necessary for an acute malaria episode.⁵⁵ A further element is innate immunity to hepatic stages which could be stimulated by either hepatic or erythrocytic stages of P. falciparum. We allow for this implicitly by including density-dependent regulation of the infection process, but have so far not been able to explicitly model effects of erythrocytic infections on the control of hepatic stages.⁵⁴ We do not make any allowance for decay of either pre-erythrocytic or blood stage immunity because we have no good quantitative data from which to estimate rates of decay. The limited field data that do exist suggest that even exposure many years in the past provides important clinical protection.⁸¹

In the long-term, vaccines are likely to exert selective effects on parasite populations, and selection in favor of non-vaccine parasite genotypes has already been demonstrated in one phase IIb trial carried out in Papua New Guinea.⁸² Selection of other parasite traits, such as virulence⁸³ is also possible but we contend that an adequate epidemiologic model is a pre-requisite for convincing models of such effects.⁸⁴

CONCLUSION

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
In this introductory paper, we have provided a succinct overview of our approach for developing a dynamic mathematical model for prediction of the epidemiologic and economic impact of a malaria vaccine. An important strength of this framework is that it 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. However we are confident that the material presented in the remaining 14 papers provides a sound foundation on which improved models can be built. To the best of our knowledge, this is the most comprehensive population-based simulation of malaria yet developed. It represents a major new tool for rational planning of malaria vaccine development, and can readily be adapted to assess efficacy and cost-effectiveness of other malaria control interventions used 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.

APPENDIX 1 EQUATIONS OF THE EPIDEMIOLOGIC MODEL INFECTION OF THE HUMAN HOST54

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
E_a(i,t), the age-adjusted entomologic inoculation rate (EIR) for individual i at time t, is given by

where, A(a(i,t)) is the average body surface area estimated for an individual of age a(i,t) and A_max is the average surface area of people 20 years of age in the same population. E_max (t) refers to the usual measure of the EIR computed from human bait collections. The force of infection is then

where S_imm, X_p*, E*, _p, S are constants (Table 1) and:

The number of infections h(i,t) introduced in time step t, is distributed as

CHARACTERISTICS OF THE SIMULATED INFECTIONS50

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Each new infection j, initiated in individual i at time t₀ is assigned a duration of t_max, sampled from

The log density in the absence of previous exposure at each time point, = 0,1,...,_max(i,j) of the infection j in host i is then normally distributed with expectation

where, y_G(, _max) is an empirical description of malariatherapy patients from the Georgia hospital and d(i) represents between-host variation drawn from a log-normal distribution with variance _i².

We measure exposure to asexual blood stages with

where Y(i,) is the total parasite density of individual i at time and y(i,j,,) is the density in individual i for infection j at time , and

the expected log density for each concurrent infection is then

where M(t) is the total multiplicity of infection and

and X_y*, X_h*, D_x, a_m*, and _m, are further constants.

Variation within individual hosts is quantified by a term _y²(i,j,), where

and ₀² and X_v* are constants (Table 1). The simulated densities are specified using:

The total density at time t in host i is then the sum of the densities of the various co-infections j i.e.

MODEL FOR INFECTIVITY OF THE HUMAN HOST49,53

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Let

where t is in 5-day units, and

where, ß₁,ß₂,ß₃,,_g² are constants (Table 1). Define

where is the cumulative normal distribution, y_g* is the density of female gametocytes necessary for infection of the mosquito, and * = (ln() – ln(y_g*))/_g. Then the proportion of mosquitoes that are infected feeding on individual i at time t is

and the probability that a mosquito becomes infected at any feed is:

where is a constant scale factor.

Define _u⁽⁰⁾(t) as the value of _u (t) in the simulation of an equilibrium scenario to which an intervention has been applied. Let E_max⁽⁰⁾ (t + l_v) be the corresponding entomologic inoculation rate. _u⁽¹⁾(t) and E_max⁽¹⁾ (t + l_v) are the corresponding values for the intervention scenario. Then

where l_v corresponds to the duration of the sporogonic cycle in the vector, which we approximate with two time steps (10 days). E_max⁽⁰⁾ (t + l_v)/_u⁽⁰⁾ (t) is the total vectorial capacity).

ACUTE MORBIDITY52,55

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
An episode of acute morbidity occurs in individual i, at time t, with probability

where Y* is the pyrogenic threshold and Y_max is the maximum density of five daily densities sampled during the five-day time interval t. The pyrogenic threshold evolves over time via:

with the initial condition Y* (i, 0) = Y₀* at the birth of the host and , , Y₂* are constants.

We consider two different classes of severe episodes, B₁ and B₂. P_B1 (i,t) is the probability that an acute episode (A) is a class B₁ severe episode and is specified using

where Y_B1* is a constant and H(i,t) is the clinical status.

The second subset of severe malaria episodes (B₂) occur when an otherwise uncomplicated malaria episode happens to coincide with some other insult, which occurs with risk

where F₀ is the limiting value of F(a(i,t)) at birth, and a_F* is the age at which it is halved.

The probability that an episode belonging to class B₂ occurs at time t, conditional on there being a clinical episode at that time is P_B2 (i,t) where

The age and time specific risk of severe malaria morbidity conditional on a clinical episode is then given by

MORTALITY52

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
Malaria deaths in hospital are a random sample of those severe malaria cases deemed to be admitted, with age-dependent sampling fraction Q_h(a), the hospital case fatality rate, derived from the data of Reyburn and others.⁸⁶

We estimate the severe malaria case fatality in the community, Q_c(a) for age group a with

Where _l, the estimated odds ratio for death in the community compared to death in in-patients, is an age-independent constant and Q_h(a) is the hospital case fatality rate. Malaria mortality is the sum of the hospital and community malaria deaths.

The risk of neonatal mortality attributable to malaria (death in class D₁) in first pregnancies is set equal to 0.3µ_PG where µ_PG is given by

where x_PG is related to x_MG, the prevalence in simulated individuals 20–24 years of age via

and x_MG* and x_PG * are constants (Table 1).

An indirect death in class D₂ is provoked at time t, conditional on there being a clinical episode at that time, with probability P_D2 (i,t) where

where Q_D is limiting value of P_D2 (i,t) at birth and a_F * is a constant. Deaths in class D₂ occur 30 days (six time steps) after the provoking episodes.

ANEMIA73

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 
The prevalence of anemia, p_A(a,t), in age group with mid-age a, at time t is specified by

where p_P(a,t) is the prevalence of patent parasitemia in the age group and ß₀, ß_P,p*,ß_al,a*,ß_I are constants.

Received September 18, 2005. Accepted for publication November 25, 2005.

Acknowledgments: We thank Dan Anderegg for editorial assistance, and 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) Malaria Vaccine Initiative, and GlaxoSmithKline 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 Insitute, Socinstrasse 57, PO Box CH-4002, Basel, Switzerland. E-mail: Thomas-A.Smith{at}unibas.ch

Authors’ addresses: Thomas Smith, Nicolas Maire, Amanda Ross, Fabrizio Tediosi, Guy Hutton, Jürg Utzinger, and Marcel Tanner, Swiss Tropical Institute, Socinstrasse 57, PO Box, CH-4002, Basel, Switzerland, Telephone: 41-61-284-8273, Fax: 41-61-284-8105, E-mails: Thomas-A.Smith{at}unibas.ch, nicolas.maire{at}unibas.ch, amanda.ross{at}unibas.ch, fabrizio.tediosi{at}unibas.ch, guy.hutton{at}unibas.ch, juerg.utzinger{at}unibas.ch, and marcel.tanner{at}unibas.ch. Gerry F. Killeen, Ifakara Health Research and Development Centre, Ifakara, Kilombero District, Tanzania, Telephone: 255-748-477-118, Fax: 255-23-262-5312, E-mail: gkilleen{at}ihrdc.or.tz. Louis Molineaux, Peney-Dessus, CH-1242 Satigny, Geneva, Switzerland. Klaus Dietz, Department of Medical Biometry, University of Tübingen, West-bahnhofstrasse 55, 72070 Tübingen, Germany, Telephone: 49-7071-29-78253, Fax: 49-7071-29-5075, E-mail: klaus.dietz{at}unituebingen.de.

Reprint requests: Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, PO Box, CH-4002, Basel, Switzerland.

REFERENCES

TOP ABSTRACT INTRODUCTION REQUIREMENTS OF A PREDICTIVE... STRUCTURE OF THIS PROJECT STRATEGY OF EPIDEMIOLOGIC... STRENGTHS AND LIMITATIONS OF... CONCLUSION APPENDIX 1 EQUATIONS OF... CHARACTERISTICS OF THE SIMULATED... MODEL FOR INFECTIVITY OF... ACUTE MORBIDITY52,55 MORTALITY52 ANEMIA73 REFERENCES

 

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