Monitoring Short Term Changes of Malaria Incidence in Uganda with Gaussian Processes

Abstract

A method to monitor communicable diseases based on health records is proposed. The method is applied to health facility records of malaria incidence in Uganda. This disease represents a threat for approximately 3.3 billion people around the globe. We use Gaussian processes with vector-valued kernels to analyze time series components individually. This method allows not only removing the e ect of speci c components, but studying the components of interest with more detail. The short term variations of an infection are divided into four cyclical phases. Under this novel approach, the evolution of a disease incidence can be easily analyzed and compared between di erent districts. The graphical tool provided can help quick response planning and resources allocation.

Keywords

• Gaussian processes
• malaria
• kernel functions
• time series

Introduction

More than a century after discovering its transmission mechanism, malaria has been successfully eradicated from di erent regions of world [ 15 ]. However, it is still endemic in 100 countries and represents a threat for 3.3 billion people approximately [ 20 ]. In Uganda, malaria is among the leading causes of morbidity and mortality [ 19 ]. Di erent types of interventions can be carried on to prevent and treat malaria [ 20 ]. Their success depend on how well the disease can be anticipated and how fast the population reacts to it. In this regard, mathematical modelling can be a strong ally for decision-making and health services planning. Spatiotemporal modelling for mapping and prediction of infection dynamics is a challenging problem. First of all, because of the costs and di culties of gathering data. Second, because of the challenges of developing a sound theoretical model that agrees with the data observed.

The Health Management Information System (HMIS) operated by the Uganda Ministry of Health provides weekly records of the number of patients treated for malaria in di erent hospitals across the country. Unfortunately, the number of reporting hospitals is not consistent across time. This variation is prone to create arti cial trends in the observed data. Hence, the underreporting e ect has to be estimated to be removed.

A common approach for time series analysis is to decompose the observed variation into speci c patterns such as trends, cyclic e ects or irregular uctuations [ 4, 3, 7 ]. Gaussian process (GP) models are a natural approach for analyzing Copyright c 2015 for this paper by its authors. Copying permitted for private and academic purposes. functions that represent time series. GPs provide a robust framework for nonparametric probabilistic modelling [ 18 ]. The use of covariance kernels enable to analyse non-linear patterns by embedding an inference problem into an abstract space with a convenient structure[ 14 ]. By combining di erent covariance kernels (via additions, multiplications or convolutions) into a single one, a GP is able to describe more complex functions. Each of the individual kernels contributes by encoding a speci c set of properties or pattern of the resulting function [ 5 ].

We propose a monitoring system for communicable diseases based on Gaussian processes. This methodology is able to isolate the relevant components of the time series and study the short term variations of the disease. The output of this system is a graphical tool that discretizes the disease progress into four phases of simple interpretation. 2

Background

Say we are interested in learning the functional relation, between inputs and output, based on a set of observations f(xi; yi)gin=1. GP models introduce an additional latent variable fx, whose covariance kernel K is a function of the input values. Usually, yi is considered a distorted version of the latent variable.

To deal with multiple outputs, GP models resort to generalizations of kernel functions to the vector-valued case [ 1 ]. In time series literature, vector-valued functions are commonly treated in the family of VAR models [ 12 ], while in geostatistics literature co-Kriging generalizations are used [ 8, 11 ]. These approaches are equivalent. Let hx = (fx1; : : : ; fxd)> be a vector-valued GP, its corresponding covariance matrix is given by cov(hx; hz)ij = cov(fxi ; fzj ) : (1) The diagonal elements of the correlation matrix cov(hx; hz)ii are just the covariance functions of the real-valued GP elements. The non-diagonal elements represent the cross-covariance functions between components [ 9, 10, 2 ]. 3

Method Used

Suppose we have data generated from the combination of two independent signals (see Figure 1a). Usually, not only we are not able to observe the signals separately, but the combined signal they yield is corrupted by noise in the data collected (see Figure 1b). For the sake of this example, suppose that the two signals of the example represent a long term trend (the smooth signal) and a seasonal component (the sinusoidal signal). For an observer, the oscillations of the seasonal component masks the behaviour of the long term trend. At some point, however, the observer might want to know whether the trend is increasing or decreasing. Similarly, there might be interest in studying only the seasonal component isolated from the trend. For example, in economics and nance, business recession and expansion periods are determined by studying the cyclic component of a set of indicators [ 16 ]. The cyclic component tells if an indicator is above or below the trend, and its di erences tell if it is increasing or decreasing.

We propose a similar approach for monitoring disease incidence time series, but in our case, we will use a non-parametric approach. To extract the original signals, the observed data can be modelled using a GP with a combination of kernels, say exponentiated quadratics, one having a shorter lengthscale than the other. Figures 1c and 1d shows a model of the combined and independent signals. We also use a vector-valued GP to model directly the derivative of the time series, rather than using simple di erences of the observed trend. As a result, we are able to provide uncertainty estimates about the speed of the changes around the trend. Our approach is based on modelling linear functionals of an underlying GP [ 13 ]. If hx = (fx; @fx=@xi)>, its corresponding kernel is de ned as (xi; xj) = " K(xi; xj) : (2)

In most multi-output problems, observations of the di erent outputs are needed to learn their relation. Here, the relation between fx and its derivative is known beforehand through the derivative of K. Thus @fx=@xi can be learnt by relying entirely on fx. For the signals described above, Figures 1e and 1f show the corresponding derivatives computed using a kernel of the form of (2). The derivatives of the long term trend are computed with high con dence, while the derivatives of the seasonal component have more uncertainty. The last is due to the magnitude of the seasonal component relative to the noise magnitude. 4

Uganda Case

In this exposition we focus on Kabarole district, but provide a snapshot of the monitoring system for all the country. Our base assumption about the infection process of malaria is that it evolves with some degree of smoothness across time. Smooth functions can be represented by a kernel such that the closer the observations in the input space, the more similar values of the output. The Matern kernel family satis es this condition, as it de nes dependence through the distance between points with some exponential decay [ 18 ]. Di erent members of this family encode di erent degrees of smoothness, being the limit case the exponentiated quadratic kernel or RBF, which is in nitely di erentiable. To illustrate our method we will use an RBF kernel. Results with (rougher) Matern kernels do not di er much when used instead.

Despite malaria is a disease in uenced by environmental factors like temperature or water availability, we could not observe a seasonal e ect in HMIS data [ 6 ]. If that was the case, the model could be improved incorporating a periodic kernel in the covariance structure. Yet, the model t can be improved if a second RBF kernel is added. In this case, one kernel has a short lengthscale and 16 14 12 10 8 6 4 2 00 20 18 16 14 12 10 8 6 40 therefore represents short term variations, while the other represents long term changes.

An important factor to consider about HMIS data is that the number of health facilities is highly variable. See Figure 2a. This variation is prone to create arti cial trends in the incidence of malaria reported. Such trends can be removed by incorporating a linear kernel that describes the relation between reporting facilities and malaria cases. Unlike the RBF kernels mentioned above, which take time as input, the linear kernel takes the number of health facilities as input. In Table 1, we present a comparison of the model predictive performance, when using di erent kernels, based on the leave-one-out predictive probabilities [ 17 ]. The best predictive performance is achieved when considering short and long term changes and a correction for misreporting facilities. We have proposed a disease monitor based on vector-valued Gaussian processes. Our approach is able to account for uncertainty in both the level of each component and the direction of change. The simplicity for doing inference with this 2005 2007 2009 2011 2013

Kabarole (a) HMIS records from Kabarole (b) Phases on 2014/04/20 model is not compromised by the use of a vector-valued approach. The model can be bene ted if spatial information is available and encoded in the kernel function. Further research is needed to explore the bene ts of this model in practice. We expect that an analysis from this perspective can add situational awareness and contribute to interventions planning and resources allocation when facing infectious diseases.

Acknowledgements

Ricardo Andrade-Pacheco is supported by CONACYT and SEP scholarships.

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