article: Infectious diseases are one of the primary healthcare problems worldwide, leading to millions of deaths annually. To develop effective control and prevention strategies, we need reliable computational tools to understand disease dynamics and to predict future cases. These computational tools can be used by policy makers to make more informed decisions. In this study, we developed a computational framework based on Gaussian processes to perform spatiotemporal prediction of infectious diseases and exploited the special structure of similarity matrices in our formulation to obtain a very efficient implementation. We then tested our framework on the problem of modeling Crimean–Congo hemorrhagic fever cases between years 2004 and 2015 in Turkey. We showed that our Gaussian process formulation obtained better results than two frequently used standard machine learning algorithms (i. e. , random forests and boosted regression trees) under temporal, spatial, and spatiotemporal prediction scenarios. These results showed that our framework has the potential to make an important contribution to public health policy makers.
  Infectious diseases constitute a major part of healthcare burden worldwide, leading to millions of deaths annually, which are especially seen among poor and young populations in low and middle income countries [1]. In addition to pandemic infectious diseases such as influenza and tuberculosis, there are also emerging infectious diseases such as Ebola virus disease and Zika fever, which require a worldwide effort to combat. Thus, predicting the case counts of infectious diseases is of great importance in developing control and prevention strategies. In particular, there might be spatial dependencies (e. g. , humid conditions for malaria) and temporal dependencies (e. g. , seasonal effects for influenza) that control the emergence and spread of such diseases [2]. To be able to develop protective measures against infectious diseases, it is very important (i) to clearly identify the disease spread and (ii) to make reliable predictions for future cases. When the disease spread is known, policy makers can develop preventive strategies against, for instance, environmental factors that promote the disease. Once we have reliable predictions for future cases, policy makers can make informed decisions on, for example, vaccine purchases, public awareness campaigns and training programs for healthcare workers. Machine learning algorithms can contribute to the control of infectious diseases by addressing aforementioned two aims. In the literature, standard machine learning algorithms such as random forests [3] and boosted regression trees [4,5] were frequently used in ecological and epidemiological applications [6–10]. These algorithms have been picked by the applied researchers mainly because they have a relatively simple interface for nonspecialists. However, they might fail to capture highly complex dependencies in disease modeling scenarios. Thus, we used Gaussian processes [11] to be able to identify highly nonlinear dependencies and to make more reliable predictions. We proposed a computational framework that uses Gaussian processes as the basic building block to perform spatiotemporal prediction of infectious diseases. We first noted that the kernel matrices have a special structure owing to their dependencies on both spatial and temporal covariates and then exploited this special structure to obtain a very efficient inference algorithm. We tested our proposed framework on Turkey’s country-wide surveillance data set of a vector-borne infectious disease Crimean–Congo hemorrhagic fever, which is a widespread endemic infectious disease seen in Africa, the Balkans, the Middle East, and Asia with a case fatality rate of 5–40% [12]. We present the overview of our proposed computational framework with three possible prediction scenarios in Fig 1. We assume that the reported case counts of location and time period pairs have been recorded with additional information about their spatial and temporal properties. We first extract spatial and temporal features for each location and time period, respectively, from these properties. We then calculate two similarity matrices among locations and time periods, respectively, using the extracted features. These two similarity matrices are combined to obtain a larger similarity matrix between location and time period pairs. Using the combined similarity matrix and reported cases counts, we train a Gaussian process regression model to be able to make predictions under three different scenarios: (i) temporal prediction (i. e. , predicting case counts for future time periods, leading to predicting disease prevalence for each location in the future), (ii) spatial prediction (i. e. , predicting case counts for unseen locations, leading to predicting disease spread within the same time frame in other locations), which can be used to complete missing case counts for the locations that we could not obtain historical data, and (iii) spatiotemporal prediction (i. e. , predicting case counts for unseen location and future time period pairs, leading to predicting disease spread to new locations in the future), which is especially important to be able to prepare against emerging infectious diseases since there will be no historical data for the locations that experience the disease for the first time.
  We used an unpublished surveillance data set of 9,636 CCHF infection cases reported in Turkey between years 2004 and 2015, which was collected by the Ministry of Health of Turkey (S1 File). The reported cases were mainly because of infected tick bites, and they were diagnosed with clinical symptoms such as fever, myalgia, and bleeding from various sites. These infected cases were also confirmed with blood tests. The Ministry of Health of Turkey provided us with spatial information (province, district, and town names) and temporal information (year and month) for each case, which made this data set suitable for studying spatiotemporal characteristics of CCHF. The data set does not include clinical covariates of infected cases, which forces our study to investigate only spatial and temporal covariates. Infectious disease spread is usually driven by both location and time, which means nearby locations and time periods have similar characteristics. The disease spreads to adjacent province much more easily than distant provinces due to spatial dependency. Case counts in consecutive time periods or in time periods within the same season are usually heavily correlated due to temporal dependency. We suggest using Gaussian process regression (GPR), which is suitable to capture highly complex dependencies between input and output variables thanks to its nonlinear nature brought by kernel functions. We propose a computational strategy based on GPR that enables us to perform predictions under spatial (i. e. , predicting case counts for unseen locations), temporal (i. e. , predicting case counts for future time periods) and spatiotemporal scenarios (i. e. , predicting counts for unseen location and future time period pairs) for infectious diseases. We first give a brief description of GPR. We then show how GPR can be modified for infectious disease modeling by introducing a structured kernel function based on two separate kernel functions over spatial and temporal covariates, respectively, and how this modified GPR formulation can be implemented very efficiently. We describe three different prediction scenarios encountered in spatiotemporal modeling of infectious diseases. We lastly discuss two baseline algorithms from the literature that will be used to benchmark against.
  Table 1 reports PCC values of RFR, BRT, and GPR algorithms on our CCHF data set for three prediction scenarios. We see that GPR algorithm obtained the best PCC values by improving the results of temporal, spatial, and spatiotemporal prediction scenarios by 1. 05%, 26. 31%, and 16. 45%, respectively. Note that RFR and BRT algorithms failed to capture the spatial spread of CCHF when predicting case counts for unseen provinces (i. e. , in spatial and spatiotemporal scenarios), whereas GPR algorithm was able to capture this spread by obtaining more than 70% PCC for these two scenarios. All algorithms achieved PCC values around 75% and 85% for temporal scenario since capturing temporal dynamics is easier owing to annual periodicity of CCHF cases. Table 2 shows NRMSE values of RFR, BRT, and GPR algorithms on our CCHF data set for temporal, spatial, and spatiotemporal prediction scenarios. We see that GPR algorithm again obtained the best NRMSE values by improving the results of temporal, spatial, and spatiotemporal prediction scenarios by 21. 39%, 20. 38% and 15. 65%, respectively. Even though BRT algorithm obtained a PCC value comparable to that of GPR algorithm for temporal scenario, GPR algorithm obtained considerably better NRMSE values than both RFR and BRT algorithms. This shows that GPR algorithm is better than the other two algorithms in terms of capturing the range of CCHF cases in the test sets as discussed below. Fig 5 shows the total observed and predicted case counts by RFR, BRT and GPR algorithms for years 2014 and 2015 over the five provinces with the highest case counts among 40 common test provinces of all scenarios. We see that all three algorithms captured the annual periodicity of CCHF cases, whereas GPR algorithm performed the best in terms of predicting the observed case counts. RFR algorithm was not able to predict the observed case counts owing to its lack of high order interactions between covariates, whereas BRT algorithm performed better owing to its second order interactions. The same results were also valid if we took the first 10,15, and 20 provinces from 40 common test provinces (S14, S15 and S16 Figs). S17 Fig gives a detailed comparison between observed and predicted case counts of RFR, BRT, and GPR algorithms for the same five provinces reported in Fig 5. We see that GPR algorithm produced predictions mostly in agreement with the range of observed CCHF case counts, whereas RFR and BRT algorithms underestimated CCHF case counts in most of the time periods. BRT algorithm obtained NRMSE value comparable to that of GPR algorithm for temporal scenario, whereas GPR algorithm reduced NRMSE values by 0. 277 and 0. 170 for spatial and spatiotemporal scenarios, respectively. The results of the computational experiments reported in this study can be analyzed from different perspectives. We analyzed the results with respect to prediction scenarios, machine learning algorithms, computational complexity, dependency on training set size, and dependency on sampling over provinces. We performed computational experiments under three different scenarios. As we can see from Tables 1,2, Fig 5 and S17 Fig, making temporal predictions (i. e. , predicting future time periods by looking at the historical data) is strikingly easier than making spatial and spatiotemporal predictions (i. e. , generalizing to unseen locations). Most infectious disease outbreaks occur in cycles (i. e. , ascending, plateau, and descending phases), and this structure makes temporal prediction easier. The disease we addressed is a vector-borne infectious disease mainly transmitted by infected tick bites, leading to a strong temporal dependency owing to the sleep cycles of ticks. We used three machine learning algorithms for predicting case counts. As we discussed before, GPR algorithm was able to capture the range of CCHF case counts better than RFR and BRT algorithms. We think that this was mainly due to the capability of GPR algorithm to model highly complex dependencies between input and output covariates thanks to nonlinear kernel functions such as the Gaussian kernel we used. We also noted from Fig 5 and S17 Fig that the main improvement of GPR algorithm over the others was the ability to better capture the range of case counts in the time periods with nonzero observed case counts. In the literature, RFR and BRT algorithms were frequently used as classification algorithms to predict whether there will be cases. In terms of classification performance, we would not expect major differences between three algorithms. Instead of using a naive version of GPR algorithm, we implemented an efficient variant that exploits the special structure of the kernel matrix to make inference very fast. We decomposed the kernel matrix into a Kronecker product of two smaller kernel matrices calculated on spatial and temporal covariates, respectively. By doing so, we were able to perform inference for our structured GPR formulation in the order of milliseconds, whereas RFR and BRT algorithms took several minutes to complete using drastically higher physical memory. To show the dependency of GPR on training set size, we performed an additional set of experiments by changing the number of years used for training. We used CCHF case counts of the last two, four, six, eight, and ten years between 2004 and 2013, respectively. Table 3 shows PCC and NRMSE values of GPR algorithm for this new set of experiments. We can see that there was an increasing trend in predictive performance as we increased the training set size. Up to this point, we performed our experiments on a fixed training and test set split (S13 Fig), which was designed to make training and test sets as similar as possible, to better illustrate the differences between machine learning algorithms. We also compared the predictive performances of RFR, BRT, and GPR on 100 different training and set set splits constructed by random sampling on 81 provinces. Fig 6 shows PCC and NRMSE values of the algorithms for spatial and spatiotemporal modeling scenarios. We see that our algorithm GPR was statistically significantly better (i. e. , p < 0. 001) than other two algorithms for both scenarios in terms of PCC values. In spatial prediction scenario, GPR achieved statistically significantly better NRMSE values than RFR (i. e. , p = 0. 023), but it obtained NRMSE values comparable to BRT (i. e. , p = 0. 052). In spatiotemporal prediction scenario, NRMSE values of GPR were statistically significantly better than those of BRT (i. e. , p < 0. 001), whereas NRMSE values were comparable between GPR and RFR (i. e. , p = 0. 932).
  Infectious diseases cause important health problems worldwide and create difficult challenges for public health policy makers. To be able to make correct and effective decisions, it is quite important to understand the characteristics of each infectious disease, which includes environmental factors such as climate and animal population in addition to molecular evolution of disease sources such as bacteria and viruses. In this study, we addressed to capture the effect of environmental factors on infectious diseases by modeling their spatial and temporal dependencies on these factors. For this purpose, several computational methods have been proposed in the literature, whereas we focused only on machine learning algorithms applied to this problem. Easy-to-use machine learning algorithms such as random forests and boosted regression trees were frequently used in infectious disease modeling studies. However, Gaussian processes might capture highly complex dependencies better than these tree-based algorithms. Thus, we formulated a computational framework based on Gaussian processes that can be used to perform spatial, temporal, or spatiotemporal prediction of infectious diseases. We integrated spatial features (such as geographical coordinates) and temporal features (such as seasonal conditions) for location and time period pairs that were used as data instances in our Gaussian process formulation. However, a naive implementation of Gaussian processes would become computationally infeasible owing to very high numbers of pairs being modeled. We exploited the special structure (i. e. , Kronecker) of similarity matrices in our formulation to obtain a very efficient implementation, which enabled us to train models for around 10,000 data instances in the order of milliseconds. We applied our framework to the problem of predicting the case counts of a vector-borne infectious disease Crimean–Congo hemorrhagic fever using the data set of infected case counts between years 2004 and 2015 collected by the Ministry of Health of Turkey. We performed predictions under three different scenarios (Fig 1), which correspond to making predictions for unseen provinces (i. e. , spatial prediction), future time periods (i. e. , temporal prediction), or unseen province and time period pairs (i. e. , spatiotemporal prediction) to show the suitability of our approach to distinct problems. Predicting future cases of infectious diseases is very important for the control and prevention of the disease. The predicted case counts can be used to develop new public health policies and intervention mechanisms. It is more useful for public health policy makers to be able to predict the possible number of infected cases for a region and a time period pair rather than predicting whether there will be cases or not. Policy makers can make use of predicted number of infected cases to purchase vaccines around the right amount, to raise public awareness in the region, to educate healthcare workers, etc. From that perspective, GPR algorithm did a better job than RFR and BRT algorithms by predicting CCHF case counts more accurately (i. e. , lower NRMSE values). We tested our proposed formulation on a single disease, but the same framework can be extended towards other vector-borne infectious diseases (e. g. , dengue fever, malaria, Zika fever) and as well as other infectious diseases (e. g. , influenza, measles, tuberculosis). We also made the source code publicly available to enable other computational and applied researchers to make such extensions easily.
summary: Infectious diseases cause important health problems worldwide and create difficult challenges for public health policy makers. That is why they need reliable computational tools to better understand disease and to predict case counts. They will benefit from such computational tools to make more informed decisions in developing control and prevention strategies. We formulated a computational framework that can be used to model spatial, temporal, or spatiotemporal dynamics of infectious diseases. We showed the utility of our framework on the problem of modeling Crimean–Congo hemorrhagic fever in Turkey.
keywords: turkey (country)
  machine learning algorithms
  medicine and health sciences
  applied mathematics
  crimean-congo hemorrhagic fever
  tropical diseases
  geographical locations
  health care
  simulation and modeling
  algorithms
  kernel functions
  mathematics
  artificial intelligence
  neglected tropical diseases
  infectious disease control
  research and analysis methods
  viral hemorrhagic fevers
  infectious diseases
  computer and information sciences
  operator theory
  people and places
  asia
  physical sciences
  viral diseases
  europe
  health care policy
  machine learning
title: Spatiotemporal prediction of infectious diseases using structured Gaussian processes with application to Crimean–Congo hemorrhagic fever