Infectious diseases pose an incessant threat to human health and welfare. Influenza, as one of the most prevalent diseases worldwide, is a typical recurrent seasonal epidemic disease. We shall seek to model the pattern of its behavior and its causes, based on the provided data and external data online. Qualitative patterns are proposed, supplemented a quantitative feature extraction along with its higher-order rectified version, to analyze the possible causes of influenza. Concrete policies are provided, both regarding its effectiveness and propagation viability, after a detailed regional validation of our models.
2 General Idea and Analysis
The organization of our model is mainly composed of five parts. First, we need to preprocess the data and carefully select the relevant information. Then we shall analyze the property of the available data qualitatively and build our model as per the indication and real-life scenarios. Thirdly, the principal features shall then be extracted and their validity shall be tested against a priori criteria. Now we can use such features to fit our models and derive the weights of each feature. A higher-order perturbed version of the model is also introduced to address the international behavior, and we shall test the effectiveness of both models. Finally, we design the corresponding policy to prevent the spread of influenza as suggested by our main model. The pipeline for our models is plotted in Figure 1.
Based on our formulation of the problem, we first augment the data related to the influenza mortality. Besides the provided data on consumption, health indicators, connectivity, immunization, sanitation, and water quality, we also resort to the World Bank Open Data  to collect other possible indicators, such as labor or agriculture. In total, we make a massive collection of indicators with dimensions. The rationale behind this resort to external data is due to the sparsity and insufficiency of our provided data. Although there are still missing values in the collected indicators, we can obtain a dense dataset via matrix completion . Finally, we shall use support vector machines (SVM)  to certify that our augmented dataset is indeed consistent and satisfies the basic criteria.
Qualitative Analysis of Periodic Behavior.
Since influenza, like all recurrent infectious diseases, follows a somewhat periodic behavior, we have sound reason to conjecture a periodic outbreak of influenza. Discrete Fourier transforms  on weekly influenza activity both regionally and globally would reveal such pattern indeed, and we find the period to be approximately one year. Thus we can safely use only the provided annual data, as it is the representative of the mean of influenza activity each year.
We shall use a combination of autoencoders  and principal component analysis (PCA)  to extract useful features from the augmented data. Autoencoders are used to introduce nonlinearity and thus robustness to our model. Then PCA is introduced for an orthogonal design. We shall use autoencoders to extract our features to a dimensional Riemann manifold, and use PCA to analyze its principal characteristics. Our method is in effect a kernel PCA method which enjoys both orthogonality and nonlinearity.
To analyze the causes of influenza based on extracted features, we propose a simple linear regression to fit the weights based on the influenza mortality data, as opposed to prevalent models  of PDE / integro-differential equations. The rationale behind the choice of the model is analyzed as follows. On the one hand, we do not possess a sufficiently fine resolution of data points in time at such a global scale, thus rendering a continuous model errant. On the other hand, nonlinearity is already introduced in our data pre-processing and feature extracting process. Therefore a stable linear model would appear more rational and can be easily modified by international higher-order perturbation terms.
Taken into account the global migration of population and trade of products, we can propose a rectified version of the regional model using the provided migration data, and trade data from Trade Map . Infectious diseases can indeed spread via migration and trade, so we model the international flow of migration and trade as a convolution on regional influenza activity. Furthermore, we use a second-order polynomial to model the region-wise dependence of the convolution kernel. We jointly optimize the coefficients of the deconvolution and the linear regression towards the correct prediction. Regional data shall be used to verify the correctness of our models.
Based on the analysis of features and their importance, we shall design corresponding policies to mitigate the negative influences of influenza. Cost-effectiveness and viability shall also be taken into consideration when concrete policies are designed for each region.
3 Main Models and Guiding Methods
3.1 Discrete Fourier Transform
Fourier analysis is used to analyze the behavior of a continuous function in the frequency domain. Specifically, Fourier transform is a linear bijective mapping of the space to itself. When given discrete data in a finite interval, Discrete Fourier Transform  as a variant of such methods can be used to analyze the periodic behavior quantitatively.
For any given region, we consider the available data on influenza activity. For the known data points of weekly influenza activity , we use the DFT technique to get coefficients:
We can infer the periodic behavior of influenza activity based on the peaks and periodic decaying property of the coefficients .
3.2 Matrix Completion
Matrix completion  is a conventional technique for filling out missing information in a matrix of a large scale, with the aim of maintaining a sparse and low-rank structure. The precise formulation of the problem is defined as follows:
Though as an NP-hard problem to find the optimal completion, we can introduce regularization and relaxation to address the problem, even with noisy input. We shall resort to the Python package fancyimpute for a numerical implementation of completing the missing data from collected indicators relevant to the influenza mortality.
3.3 Support Vector Machine
Given data points , where is the input feature and is the output pattern, assumed to be or , support vector method approach 
aims at a construction of the classifier in the following form:
where is a tuning regularization parameter. We shall use SVM and a priori knowledge to verify the results after matrix completion.
Recent years have witnessed the success of variants of autoencoders , mainly because autoencoders have the power to express and represent nonlinear manifold in a high dimensional space. We use autoencoders to reduce the dimension of collected indicators, which can be modeled by the ‘bottleneck’ structure. To be specific, the architecture is plotted in Figure 2. The encoder part havecells () is reached in the middle. The decoder part is a mirror of the encoder, with output size . The features in the ‘bottleneck’ layer are recognized as the ‘code’ which most likely represent the most important information hidden in data, thus the name ‘autoencoder’.
In order to choose the optimal number of layers and number of cells in the ‘bottleneck’ layer, therefore balance between the reconstruction error and the model simplicity, we use the Bayesian information criterion (BIC)  to evaluate the overall performance of our model. Here, the BIC factor is defined as
where is the number of countries and regions in context, is the number of cells in the ‘bottleneck’ layer, and is the reconstruction error per country.
3.5 Principal Component Analysis
Principal Component Analysis 
is a prevalent method for data compression and feature extraction. It uses orthogonal transformation to extract the high-dimensional data into a sequence of uncorrelated features.
The algorithm is defined as follows. For a data matrix , we extract the first component by
Then we subsequently extract the -th component given the first ones by a subtracted matrix
and a similar maximization of Rayleigh quotient
The procedure could also be explained by the truncation of the largest singular values in the SVD decomposition. We use PCA to achieve an orthogonal design in the extracted feature space, in preparation for the regression.
3.6 Linear Regression
Since infectious diseases follow an exponential increase pattern, we take the logarithm of the death rate and find the data is of a Gaussian distribution. We shall perform linear regression
where is the normalized log death rate, is the data of the features derived by autoencoders and PCA, augmented by a column of as the interception, and is our target of the weight of each feature.
3.7 Higher-Order Rectification
After the derivation of our regional model , where the weights are optimized by linear regression, we shall rectify the model by a higher-order perturbation,
The matrix represents international flow of migration and trade, and is assumed small.
4 Concrete Analysis
4.1 Analysis of the Data After Completion
We use the technique of matrix completion, for a merged version world_bank_merge.csv of the data world_bank on World Bank . The processed data world_bank_impute.csv is tested by a support vector machine. We would like to check the low-rank and sparse property of our data.
To be precise, we divide the countries into developed ones and developing ones, which is an essential feature indicative of the general condition of the countries. We use SVM for classification based on the completed features, and plot the results in Figure 3 and 4. We find that the countries are successfully classified after completion, whereas the classification fails for the raw data, thus consolidating the effectiveness of our matrix completion.
4.2 Feature Extraction
We use autoencoders to extract features from the originally -dimensional information world_bank_impute.csv. We use a four-layer autoencoder, with a number of selected features following a pattern of geometric series to capture the information better. Thus the number of features in each layer is , , , , and finally on the final output layer.
The advantages of the autoencoder method over PCA lie in its nonlinearity and expressibility, which can be seen by comparing the reconstruction error of PCA and autoencoders in Figure 7. The error of autoencoders is far less than that of PCA given the same number of features. Thus autoencoders can indeed capture the nonlinearity of data features, which is beyond the abilities of PCA. However, taking the subsequent regression procedure into consideration, we shall invoke a PCA orthogonalization after the autoencoder.
Finally, we shall make some observations on our extracted features. The features are derived by a back-propagation of the results outputted by PCA, and they are linear combinations of the provided information. The specific data and corresponding information is stored in the zip file Feature_selection, with each .csv file corresponding to one extracted feature, listed as per importance.
4.3 Linear Regression and Higher-Order Rectification
We take the influenza mortality data in death_rate_ghe2016.csv from WHO , and form the normalized log death rate , stored in z_normal.txt. We use a linear regression to obtain the weights , where is the feature matrix obtained by autoencoders and PCA.
As a higher-order rectification, we use the model
where is assumed small by the intuition of perturbation. Each entry represents the normalized mortality transferred from region to region , which is composed of the flow amount multiplied by a region-dependent factor. The factor is further approximated by a second-order polynomial of , inspired by the local rectangular basis in finite element method:
where , are the coefficients to be optimized, and , are normalized migration and trade amounts from region to region .
The coefficients in the rectified version of our model can be jointly optimized with the linear regression, which essentially solves the approximated deconvolution problem. We store both of the outputs in out.txt.
As a final refinement of our model, we compute the -value of each feature and shall reject the features with -value more than . The refined weights are listed in out_selected.txt and presented in Table 1.
A bootstrap cross-validation test is conducted to examine the behavior of our models. In Figure 8 and 9, we study the behavior of our model depending on whether the higher-order rectification is present. We can easily conclude that the modified model outperforms the original way by a considerable margin.
We now present some of our significant features and analyze their interpretation in real life scenarios, as in Table 2.
|Bad||Other greenhouse gas emissions||Fertilizer consumption||Rural poverty gap|
|Medium||Electricity production||Employment to population ratio||Imports of goods and services|
|Good||Average precipitation in depth||Community health workers||Health care|
5 Policy Design
5.1 Global Policies
Based on table 2, we can design the following policies worldwide.
We can control the use of environmental-harmful products, such as fertilizers and electronic devices prone to greenhouse emission.
We should design policies to diminish the poverty gap, so as to provide the rural population with a better environment against influenza.
We can enhance our health welfare, spending more of government expenditure in health insurance and community health care to conquer the influenza virus.
5.2 Regional Policies
In order to specifically design regional policies, we calculate the significant features for some typical countries, namely Nigeria, Japan and USA, as in Table 3.
We analyze the data of Nigeria, where influenza is of the most mortality. We conclude that feature is of the most significance in the region, and we thus should impose restrictions on environmental-harmful activities such as the emission of greenhouse gases.
We carry out similar analysis on Japan, where influenza is of great significance in periodicity and the -th least mortality. We conclude that feature should be enhanced in the region, and we thus should improve the number of community health workers. Japan is an isolated and highly populated country. Better community health care could lower the risk of spread of infectious diseases by intense contact.
Finally we study the cause for the spread of influenza in the United States, which has the -nd lowest influenza mortality rate. We conclude that feature should be taken special notice in the region. Namely we should heed the poverty gap and provide better health care for the poor.
The policies are indeed viable and can be propagated easily on internet. For example, we can put up slogans in marches and parades, to raise the general awareness of the public on environmental issues. The government should allocate more of its funding on community health care by adjusting its budgets. To address the problem of poverty gap, the government could increase the standard of minimal wages and impose relatively more taxes on the riches.
6 Robustness and Regional Validation
We introduce an additional column of information in world_bank_merge.csv with randomly generated data, to find that our results of feature extraction and weights regression remains similar, by a perturbation less than . Similarly, an additional row of region is introduced in world_bank_merge.csv with randomly generated data, and the output remains a perturbation less than as well. Therefore, we can deduce that our model is of great robustness and stability.
Also, we compute the residual of our models in residual.txt, as regional validations of the models. We find that our model is of satisfactory accuracy, with moderate errors as model errors and algorithm errors.
7 Assessment of Our Models
7.1 Strength of the Model
The model consists of a linear regression of important features and a higher-order perturbation. Thus it enjoys the property of generalization and combines regional as well as international information.
Exterior datasets are introduced to help reconstruct missing information and features. Matrix completion ensures sparsity as well as low-rank property, matching the case in real-life scenarios. A high-dimensional data space ensures the comprehensiveness of our features, and model reduction methods are deployed to capture the crucial information.
DFT method is invoked to verify the periodic behavior of influenza.
Casual inference, cross-validation, and tests of robustness are executed for the sake of stability.
The model is transferable to model other types of infectious diseases due to its interpretability and its comprehensiveness of all potential contributing factors to influenza behavior.
7.2 Deficiencies of the Model
We neglect the time evolution of influenza for lack of data.
A simplified version of the migration model is introduced, due to the complexity of modeling the dynamic system of a Markov process.
Policies are designed without taking into account the correlation between important features. Combined policies could be proposed as a future improvement.
We propose in this article a model of influenza spreading by a combination of massive feature engineering and international flow deconvolution. We detect a periodical behavior in influenza activity and use yearly normalized mortality data for regression. Features are extracted from the augmented dataset and weights are computed by higher-order rectification of graph deconvolution. We reach the conclusion that the spread of influenza is affected by its local environment and national economies. Policies are designed both regionally and globally, to mitigate the adverse effects of influenza.
Our model is of high interpretability. Bridging nonlinearity of the feature extraction into the linear regression model, supplemented by a higher-order graph deconvolution, can increase the robustness and consistency of the model. Furthermore, our methods of feature extraction and the main model can be easily transferred to analyze the spread of other infectious diseases.
We thank Citadel and Correlation One for all the dedicated efforts to hold such wonderful competition. This work is impossible without their help and support.
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