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Mathematical Malaria Epidemiology

An Overview of Concepts and Models

Overview.Rmd

Mathematical models of malaria can serve an indispensible role in all of this, in part, because they make it possible to convey quantitative information accurately. The models can also be useful for understanding malaria, and they have played an important role in malaria research and analytics. The first mathematical models of malaria were published in 1908 and 1911 by Ronald Ross, who wrote:

… all epidemiology, concerned as it is with the variation of disease from time to time or from place to place, must be considered mathematically, however many variables are implicated, if it is to be considered scientifically at all. To say that a disease depends upon certain factors is not to say much, until we can also form an estimate as to how largely each factor influences the whole result. And the mathematical method of treatment is really nothing but the application of careful reasoning to the problems at issue.

In 1957, the preface to George Macdonald’s book The Epidemiology and Control of Malaria, he argues that the models had, so far, fallen short:

The mathematical studies of Ross appeared an attractive approach to new explanation, but experiment showed they did not complete the picture or provide explanation…

Macdonald noted that all of the work that had been done on the mathematical epidemiology of malaria — by Ross, Lotka, and McKendrick — had all been done on a single, simple model.

Innovation in malaria mathematical epidemiology was slow, but it started to accelerate after the Global Malaria Eradication Programme (GMEP) ended in 1969. In 1950, Macdonald had published a mathematical model of superinfection, but (as explained beautifully by Paul Fine), the mathematics were flawed. Macdonald’s superinfection math can be understood through a branch of mathematics called queuing theory, but even if he had got it right, it was one feature among many. The first model of malaria with immunity was published in 1974 as part of the Garki Project. The first within-host model was published by Hellriegel in 1992. Severe disease first appeared in 1999.

Over the past century, mathematical models have been developed to explore various aspects of malaria. Depending on the needs of a study, a model may not need to include every feature of malaria, but it should be able to build models that have all of the following features:

  • Exposure — the number of infectious bites, per person, per day varies geographically, seasonally, across years, and within a population by age and other factors.

  • Time Course of Infection — simple malaria infections have a complex time course: an acute growth phase (lasting up to a month) is followed by a chronic phase that lasts for months (on average). Over the time-course of infection, parasite densities fluctuate enormously. The toolbox for handling infections includes stage-of-infection, age-of-infection, within-host models, and probability theory.

  • Superinfection — the rate of exposure to malaria often exceeds the slow rate that infections clear, so superinfection is common. Queuing theory has been the main toolbox: these are compartmental models that sub-divide infection into an infinite number of states – the multiplicity of infection (MoI) – and that explore the dynamics of the MoI. Here, we will also develop hybrid variables that model MoI distributions.

  • Antimalarial Drugs — Infections can be treated and cured with anti-malarial drugs, and treatment is often followed by a short period of chemo-protection. This raises questions about the adherence to the prescribed drug-regimens, and the evolution of drug resistance.

  • Infectiousness – the probability a mosquito becomes infected after blood feeding on a human – is related to gametocyte densities and development of transmission-blocking immunity.

  • Disease — malaria is a disease of humans with an enormous health burden; while fever is an important symptom, but the primary concerns are severe disease and anemia

  • Immunity — immunity to malaria develops with age and cumulative exposure; it modifies infection, infectiousness, and disease;

  • Diagnostics and Detection — to understand malaria in populations, we must measure it. To be useful, it must be possible to relate the true prevalence of malaria to the prevalence that would be observed, the probability of detection by various diagnostics.

  • Demography — If these models are to be used for policy, then it is essential to account for human demography and to consider age.

All these are quantitative phenomena are interrelated, and there is probably no sensible way of describing them that does not involve mathematics. Each one of these facets of malaria has been addressed in mathematical models, but it has proven difficult to formulate a synthesis.

This website is focused on the mathematical epidemiology of malaria. The goal is to provide a single repository for mathematical models, and to develop some models of malaria that are useful for research and policy. In some cases, the documentation includes some mathematical derivations. The basic building blocks for malaria include:

  • Compartmental models describing disease dynamics, and with states to describe:

    • stage structured infection to handle the complex time course;

    • superinfection and the multiplicity of infection using queuing theory;

    • stage structured immunity;

    • disease states;

    • chemoprotection after treatment;

    • human population age structure;

    • parasite population structure;

    • states to describe infectiousness.

  • Formulations that use random variables for some part of the process;

  • Hybrid variables describing the means and higher order moments of the MoI, the age of infection, cumulative exposure, or parasite populations;

  • Within-host models and individual-based simulation;

  • Functions that describe the observational process to translate the state space into observable outputs;

  • Models for human demography, including vital dynamics.

  • Ordinary differential equations, delay differential equations, partial differential equations, discrete time systems, and stochastic processes.

Malaria epidemiology is now more than a dozen dozen years old, if we start counting from Laveran. Even if we define malaria epidemiology narrowly, we can identify these core issues:

  1. exposure;

  2. the complex time course of infection and

  3. superinfection;

  4. heterogeneous exposure;

  5. disease, including fever, anemia, and severe disease;

  6. the causes of death from severe disease;

  7. gametocytes and infectiousenss;

  8. treatment with anti-malarial drugs and its effects on infection, infectiousness, and disease;

  9. immunity and its effects on infection, infectiousness, and disease;

  10. malaria in pregnancy;

  11. vaccines

  12. the problem of diagnostics and detection by light microscopy, rapid diagnostic tests (RDTs), and various alternatives;

  13. human age and sex and its effects on malaria;

  14. human genetic blood disorders associated with malaria and their effects on malaria;

  15. malaria spatial dynamics;

  16. other kinds of epidemiologically relevant heterogeneity, including nutritional status and its effects on malaria, coinfection with other pathogens, propensity to seek care, and adherance to antimalarial drug regimens;

  17. genetic diversity in the parasite population as an underlying explanation for malaria epidemiology; and

  18. the evolution of resistance to drugs, diagnostics, and vaccines.