The Age of Infection: A Semi-Markov Framework for Developing Mechanistic Models of Malaria Epidemiology

Abstract

Malaria is an epidemiologically complex disease which poses a significant burden onhumanity, contributing an estimated 643,000 deaths in 2019 alone [1]. Infection with one cohort of parasites does not prevent concurrent infection with others [2], making prevalence alone an incomplete measure of infection in a population. Further, immunity is slow to develop in response to exposure and is only partially protective [3]. Therefore, it acts to both suppress disease in individuals and to mask infections from detection and treatment, which allows for longer periods of transmission to mosquitoes. In this way past exposure modifies the bias in the observed prevalence and incidence data collected from each cohort of hosts. Natural variability in the system further obfuscates the connection between data and the process which generates it. Determining the transmission and burden of malaria becomes incredibly difficult without the use of theoretical models to connect the observed data to the latent states and parameters which guide our basic understanding and policy decisions, such as the distribution of the number of infections in each host and the detectable fraction of infections. Currently existing mechanistic models typically fall into one of two categories: simple and transparent, but deterministic and not descriptive enough to connect to available data [4]; or incredibly detailed individual-based simulation frameworks which are realistic but difficult to implement or calibrate [5,6]. Here, we propose a different approach of intermediate complexity which embraces the probabilistic structure of the system. In chapter 1, we start by demonstrating how the distribution of the multiplicity of infection modeled above is impacted by access to treatment, a major factor for the state of the system. In chapter 2, using data from controlled human malaria infections, we show how infection age is statistically predictive of parasitemia, and therefore detection, fever rates, and transmission rates. These relationships imply that infection age, a relatively simple quantity to model, can help us build predictive models for the latent and highly variable parasite densities. Chapter 3 explores this through simulation, and develops a theoretical framework for tracking the probability densities using simple ODEs. Chapter 4 expands on this, and by treating the outcomes of interest as generalized linear models of our semi-Markov model of infection age, we obtain simple ODE models for these probability density function tracking variables, with closed form equations relating them to observables such as theoretical true prevalence, detection, and fever-prompted treatment (using the results from chapter 1 along the way), all while incorporating immunity as a covariate. Finally in chapter 5 we turn our attention to fitting these models to time series data of observables, and ultimately develop an algorithm for fitting without any probabilistic simulation of the underlying stochastic process. It is our hope that this series of papers inspires others to build upon and push the new tools developed here, and use them to better improve future efforts toward malaria eradication everywhere.