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Random Variable Models

Compartmental Models for Malaria

RandomVariables.Rmd

We present the computational algorithms that support a probabilistic approach to malaria epidemiology. We start with a semi-Markovian model of malaria exposure and infection, whose states are represented by random variables that describe the multiplicity of infection (MoI) in a host and the age of infection (AoI). Assuming that parasite densities can be predicted by the AoI in a statistical sense, we can compute probability distribution functions describing parasite densities, parasite counts, and detection in an individual chosen at random from the population. From this, we present a model for parasite detection and parasite counts. This same approach has been extended to predict disease, immunity, treatment with anti-malarial drugs, and a brief period of chemo-protection.

The probabilistic approach is both highly realistic and descriptive, but our goal was a synthesis. This synthesis involves a few steps:

  1. We develop formula and functions to compute the mean MoI, the mean AoI and all its moments, and the probability of detection.

  2. Hybrid models for the mean MoI for malaria superinfection were developed by Nåsell (1985)1, We extend this approach, developing systems of differential equations that track the mean and higher order moments of the distribution of the AoI.

  3. We a new random variable describing the age of the youngest infection (AoY). We show how the variable serves as a basis for computing parasite density distributions in complex infections.

  4. We derive a hybrid variable for the mean AoY.

  5. We demonstrate that a simple system of ordinary differential equations can be used in place of the random variables for most applications.

To put it another way, we can reduce the behavior of these highly complex probabilistic systems to a simple system of equations that has a high degree of accuracy. The computational and conceptual simplicity of hybrid models have some simplicity over compartmental models and stochastic individual-based models, and with the supporting probabilistic framework, provide a sound basis for a synthesis of observational malaria epidemiology.