Introduction

Author

Affiliation

University of Washington

To understand and manage malaria, we measure it: mathematics and statistics have the grammar and vocabulary to deal with quantitative aspects of malaria, but we need to learn enough about the quantitative logic and the mathematical language to be conversant. Here, malaria is explored through mathematical models – usually dynamical systems. We start simple and then add complexity and realism.

The following gives an overview of the website’s structure. Click on the callout boxes to see an expanded discussion of the content.

In the Introduction, we start simple. The essays cover material that might be taught in advanced undergraduate or graduate college courses.

Introduction

The Parasite Life-Cycle – The parasite life cycle is the natural framework for understanding malaria. This describes the life-cycle with some commentary for malaria analysts.
Ross’s Quantitative Logic – After making his famous discovery, Ronald Ross turned his attention to the prevention of malaria. In the process, he pioneered the use of mathematical models to understand malaria. A lot of his ideas were before their time.
The SIS Model – we present SIS model, a very simple model for malaria transmisison.
Macdonald and Malaria Eradication – George Macdonald published 14 paper from 1950-1956 and a book in 1957. He was rapporteur for the sixth report of the WHO Expert Committee on Malaria, which laid out the plan for Global Malaria Eradication. Among Macdonald’s durable contributions to malaria are his analysis demonstrating that malaria transmission would be highly sensitive to mosquito survival, and a formula for \(R_0.\) This essay puts Macdonald in context.
A Ross-Macdonald Model – This presents a system of differential equations that uses Ross’s model for human infection dynamics (the SIS model), and Macdonald’s model for the sporozoite rate.
Solving Dynamical Systems – What does it mean to solve a system of differential equations? What are the parts of a model? What do malaria analysts need to know about numerical methods?
Analyzing Dynamical Systems – A discussion of qualitative analysis for differential equations, including stability analysis. What do malaria analysts need to know about the mathematical theory?
Stochasticity – A discussion of stochaticity? Do models need to be stochastic to be realistic or useful?
Model Building – What does it mean to build a model? What makes a model good?
Theory for Malaria Analysts – What is simulation-based analytics? Why should malaria analysts model malaria? What good is malaria theory?

After the introduction, we’re ready to delve into some of the topics we’ll need to do simulation-based analytics. For this, we need to address some of the complexity that was missing from the simple models that are reviewed in the introduction.

Malaria can be understood as a set of loosely coupled and locally peculiar, complex adaptive systems: each local system is made up of non-linear interactions among four different kinds of agents: local populations of mosquitoes, malaria parasites, humans as hosts for the parasites, and humans as malaria managers. We have developed vignettes that explore different aspects of each one of these four agents in focused discussions: 1) human malaria epidemiology; 2) parasite transmission dynamics among populations of humans and mosquitoes through blood feeding; 3) mosquito ecology; and 4) malaria control.

Agents

Human Malaria Epidemiology – An overview of models of malaria exposure, infection, disease, & infectiousness with discussions of treatment and chemoprotection, diagnostics and detection, and malaria vaccines.
Parasite Transmission – A look at blood feeding by adult mosquitoes and parasite transmission in populations. We set up a framework for understanding malaria transmission that can be extended to analyze models of malaria that are realistic enough for malaria policy.
Mosquito Ecology – Mosquito behavior and ecology, including population dynamics, are important for understanding malaria transmission dynamics and control. This section looks at mosquito ecology in detail.
Malaria Control – To be useful, theory of malaria transmission needs theory for malaria control, including concepts like effects, effect sizes, adjusted reproductive numbers, coverage, and response timelines.

If we want to use models to guide policy in a place, we need the models to represent the features of malaria epidemiology, transmission dynamics and control in that place. What are those features? We can use abstract models to give generic advice, but if we want to tailor interventions to context or target interventions, we highly granular information about mosquitoes, humans, parasites, and the environment. To build these models, we need a bigger toolbox. We might need to stratify the human population to deal with various kinds of heterogeneity, to consider spatial dynamics, and to evaluate malaria as a changing baseline (e.g. forced by weather) that has been modified by vector control.

Realism

Space – We describe malaria spatial dynamics, with a focus on microsimulation and metapopulation dynamic. We define malaria connectivity, and we discuss the computation of threshold conditions for spatial models.
Forcing – We set up a framework for understanding malaria as a changing baseline (e.g. modified by weather and resource availability) that has been modified by control.
Relevant Detail – We discuss the problem of relevant detail in malaria analytics: what features of a system affect malaria policy?
Thresholds – We present an extended discussion of thresholds for malaria in heterogeneous populations.

We are making a leap of faith that the world can be understood at all. None of us is omniscient, so we use the evidence to understand reality. We often get it wrong. We should expect to be often wrong, and we should be honest about the uncertainty that keeps us from developing an accurate understanding or making perfect decisions. The models are merely approximations of reality. They are always approximations, and therefore wrong in some sense.

The relevant question is whether the models are useful: do they help us understand the world and make decisions about it? It’s important to get the gist of the problem and make good decisions, but it is equally important to understand how we might be wrong. How robust are our conclusions?

Policy

The Burden of Malaria – We define burden in a general sense and then delve into the challenges of computing the burden and averted burden of malaria
Evolution – We look at the evolution of resistance from a mathematical perspective, including parasite resistance to anti-malarial drugs and diagnostics, and mosquito resistance to insecticides.
Malaria Elimination – We look at some problems arising in the context of malaria elimination from a mathematical perspective.

Finally, we note that this website is part of a cluster of websites (see Related Content).