key: cord-0178996-brzv7z2n
authors: Rizk-Allah, Rizk M.; Hassanien, Aboul Ella
title: COVID-19 forecasting based on an improved interior search algorithm and multi-layer feed forward neural network
date: 2020-04-06
journal: nan
DOI: nan
sha: 5db0486bb05240fd38921178c42e1262c2ad9661
doc_id: 178996
cord_uid: brzv7z2n

COVID-19 is a novel coronavirus that was emerged in December 2019 within Wuhan, China. As the crisis of its serious increasing dynamic outbreak in all parts of the globe, the forecast maps and analysis of confirmed cases (CS) becomes a vital great changeling task. In this study, a new forecasting model is presented to analyze and forecast the CS of COVID-19 for the coming days based on the reported data since 22 Jan 2020. The proposed forecasting model, named ISACL-MFNN, integrates an improved interior search algorithm (ISA) based on chaotic learning (CL) strategy into a multi-layer feed-forward neural network (MFNN). The ISACL incorporates the CL strategy to enhance the performance of ISA and avoid the trapping in the local optima. By this methodology, it is intended to train the neural network by tuning its parameters to optimal values and thus achieving high-accuracy level regarding forecasted results. The ISACL-MFNN model is investigated on the official data of the COVID-19 reported by the World Health Organization (WHO) to analyze the confirmed cases for the upcoming days. The performance regarding the proposed forecasting model is validated and assessed by introducing some indices including the mean absolute error (MAE), root mean square error (RMSE) and mean absolute percentage error (MAPE) and the comparisons with other optimization algorithms are presented. The proposed model is investigated in the most affected countries (i.e., USA, Italy, and Spain). The experimental simulations illustrate that the proposed ISACL-MFNN provides promising performance rather than the other algorithms while forecasting task for the candidate countries.

A novel coronavirus, named COVID-19, emerged in December 2019 from Wuhan in central China. It causes an epidemic of pneumonia in humans and poses a serious threat to global public health. This virus appears through a range of symptoms involving shortness of breath, cough, and fever [1] . Despite the drastic containment measures taken by governments associated with different countries, it is swiftly spread to hit different parts of several countries. Besides China presents the mainland for the outbreak for this epidemic, the USA found itself at the top country with the worst outbreak for this epidemic. The exponential daily increase in the infected people, some governments implements a decree to lockdown entire parts of the country. As the confirmed cases are increased daily and dangerousness of this virus, drastic policies and plans must be explored. In this sense, developing a critical forecasting model to predict the upcoming days is vital for the officials in providing a drastic protection measure.

Recently some efforts have been presented to address the COVID-19. Zhao et al. [2] developed some statistical analysis based on the Poisson concept to expect the real number of COVID-19 cases that had not been reported in the first half of January 2020. They estimated that the unreported cases reach 469 from 1 to 15 January 2020 and those after 17 January 2020 had increased 21-fold. Nishiura et al. [3] presented a statistical estimation model to determine the infection rate regarding the COVID-19 on 565 Japanese citizens (i.e., from 29 to 31 January 2020) which are evacuated from Wuhan, China, on three chartered flights. They estimate the infection rate which is 9.5% and the rate for death which is 0.3% to 0.6%. Tang et al. [4] developed a likelihood-based estimation model to estimate the risk of transmission regarding COVID-19. They concluded the reproduction number can be effectively reduced by the isolation and avoiding the intensive contact tracing. In [5] , the transmission risk of COVID-19 through human-to-human is studied on 47 patients. Duccio Fanelli et al. [6] develop a differential equations model to analyze the exponential growth of the COVID-19 on three countries including China, Italy, and France in the time window from 22 / 01 to 15 / 03 / 2020. Accordingly, the literature involves some models that were developed for forecasting some epidemics. These models include the compartmental model proposed by De. Felice et al. [7] to forecast transmission and spillover risk of the human West Nile (WN) virus. They applied their model on the historical data reported from the mainland of this virus, Long Island, New York, form 2001 to 2014. In [8] , forecasting pattern via time series models based on time-delay neural networks, multi-layer perceptron (MLP), auto-regressive, and radial basis function, is proposed to gauge and forecast the hepatitis A virus infection, where these models are investigated on thirteen years of reported data from Turkey country. They affirmed that the MLP outcomes the other models. In [9] , a forecasting model using an ensemble adjustment Kalman filter is developed to address the outbreaks of seasonal influenza, where they are employed the seasonal data of New York City from 2003 to 2008. In [10] , a dynamic model based on the Bayesian inference concept is presented to forecast the outbreaks of Ebola in some African countries including Liberia, Guinea, and Sierra Leone. Massad et al. [11] developed a mathematical model to forecast and analyze the SARS epidemic while Ong et al. [12] presented a forecasting model for influenza A (H1N1-2009). Moreover, a probability-based model is proposed by Nah et al. [13] to predict the spreading of the MERS. The feed-forward neural network (FNN) [14] presents one of the most commonly used artificial neural networks (ANNs) that has been applied in a wide range of forecasting applications with a high level of accuracy. The FNN possesses some distinguishing features that it makes valuable and attractive for forecasting problems. The first feature is a data-driven self-adaptive technique with few prior assumptions about the model. The second is that it can generalize. The third is that a universal functional approximation that provides a high degree of performance while approximating a large class of functions. Finally, it possesses nonlinear features. Due to these advantages of FNN, it has drawn overwhelming attention in several felids of prediction or forecasting tasks. For example, FNN was presented with two layers for approximating functions [14] . Isa et al. [15] presented the FNN based on multilayer perception (MLP) that is conducted on the data set from the University of California Irvine (UCI) repository. Lin . Although many related studies seem to be elegant for the forecasting tasks, they may deteriorate the diversity of solutions and may get trapped in local optima. Furthermore, these methods may be problem-dependent. Therefore these limitations can deteriorate the performance of the forecasting output and may achieve the unsatisfactory and imprecise quality of the final outcomes. This motivated us to presents a promising alternative model for forecasting tasks with the aim to achieve more accurate outcomes and avoid the previous limitations by integrating the strengths of a chaotic-based parallelization scheme and interior search method to attain better results.

Interior Search Algorithm (ISA) is a novel meta-heuristic algorithm that is presented based on the beautification of objects and mirrors. It was proposed by Gandomi 2014 [33] for solving global optimization problems. It contains two groups, namely mirror and composition to attain optimum placement of the mirrors and the objects for a more attractive view. The prominent feature of ISA is contained in involving only one control parameter. ISA has been applied to solving many engineering optimization fields [34, 35, 36, 37, 38] . However, ISA may face the dilemma of the sucking in the local optimum while implemented for complicated and/or high dimensional optimization problems. Therefore, to alleviate these shortages, ISA needs more improvement strategies to acquire great impacts on its performance. In this paper, an improved interior search algorithm (ISA) based on chaotic learning (CL), a strategy named ISACL is proposed. The ISACL is started with the historical COVID-19 dataset, and then this dataset is sent to the MFNN model to perform the configuration process based on parameters of the weight and biases. In this context, the ISACL is invoked to improve these parameters as the solutions by starting with ISA to explore the search space and CL strategy to enhance the local exploitation capabilities. By this methodology, it is intended to enhance the quality and alleviate the falling in local optima. The quality of solutions is assessed according to the fitness value. The process of algorithm is continued for updating the solutions (parameters) iteratively until the stop condition is reached, and then the achieved best parameters are invoked for configuring the structure of the MFNN model to perform the forecasting and analyzing the number of confirmed cases of COVID-19. The contribution points for this work can be summarized are as follows:

 A brilliant forecasting model is proposed to deal with the COVID-19 and the analysis for the upcoming days is performed on the basis of the previous cases. The rest sections of this paper are organized as follows. Section 2 introduces the preliminaries for the MFNN model and the original ISA. Subsequently, the proposed ISACL-MFNN framework is provided for introduced in Section 3. Section 4 shows the experiments and simulation results. Finally, conclusions and remarks are provided in Section 5.

This section provides the basic concepts of the multilayer feed-forward neural network and the original interior search algorithm.

The artificial neural network (ANN) presents one of widely artificial intelligence methods that are employed for forecasting tasks. Its structure involves the input layer, hidden layers and the output layer. The ANN simulates or look alike the human brain and it contains a number of neurons, where it performs the training and testing scenarios on the input data [39] . Input data in the present work include day and the output data present the daily number of cases. The ANN keeps updating iteratively its network weights that connect the input and output layers to minimize the error among the input data. The ANN has advantages include, it can easily learn along with making decisions and also has the ability to depict a relationship among inputs and 5 the output data without obtaining mathematical formulation. Furthermore, the ANN is easy to implement and the flexibility when it is employed for modeling. However the ANN includes some disadvantages which are, it may generate error while forecasting process, training process may reach unstable results, and involves high dimensions parameters (weights) that are need to be found in optimal manner. Furthermore, low convergence and small sample size issues are two common shortages of ANNs.

To construct the neural network model, the input and output data are determined, then the number of neurons among the number of hidden layers must be carefully chosen because they influence on the training accuracy.

The ANN-based forecasting pattern in the present work uses the MFNN along with two hidden layers. The input layer involves several neurons equal to network inputs (days), where N and M neurons are adopted for the first and second hidden layers, respectively and one neuron has assigned for the output layer [40, 41] . In this context, the hidden neurons transfer function is the sigmoid function, and transfer function for the output is a linear activation function. The associated output for a certain hidden neuron ( jth ) is determined as follows:

where ji  represents the weight among the th i input neuron and the th j first hidden neuron, i x denotes the th i input, j y defines the first hidden layer output. Here, j b defines the base of the first hidden layer. Also, the output induced by second hidden layer that is denoted by the symbol  k is calculated as follows. where lk v denotes the weight for second hidden neuron ( th k ) and the output neuron ( th l ). The assessment of the algorithm performance while training process is determined by the means of the error that equals the difference between the output of MFNN and the target. Here, the mean square error (MSE) is considered: 

In this sense, gradient-based back-propagation algorithm is presented to update weights and then minimizes the MSE value among the target output and computed output from the MFNN. Also, the MSE is provided to update the biases through the output layer towards the hidden layers [40] .

Therefore, the weights and biases can be updated as follows:

Where  defines the learning rate,  ji represents the change in the weight which hooks up the inputs of the first hidden neurons, and  

(1) ISA involves two main stages which simulate the architecture of decoration and interior design. The first stage defines the composition stage in which the composition of elements (solutions in terms of optimization viewpoint) is altered to attain a more beautiful environment, which represents the better fitness in terms of optimization viewpoint. The final one represents a mirror search that aims to explore better views between elements of this stage and the fittest one. More details about ISA are described as follows.

(2) Generate a population of elements randomly between the search bounds, upper ( upper  ) and

lower bounds ( lower  ), and record the fitness value for each element. 

where k i  defines the th i elements of the th k iteration and 2 r dedicates a random value ranged from 0 to 1.

(6) For the mirror group, the mirror is invoked among each element and the better one (global best). Therefore, the position for th i element at th k iteration of a mirror is expressed as follows. 1 , 3 3 .

(1 ).

where 3 r defines a random value within the interval 0 and 1. The position of the image or virtual position of the element depends on the mirror location, and this can performed as follows.

For enhancing the position of the global best, the random walk is implemented as a local search to make slight change for the global best position. This strategy is formulated as follows. 

(9) The procedures are stopped, if assessment criteria are satisfied, repeat from step 2.

The pseudo code for the traditional ISA is portrayed in Fig. 1 9 Fig. 2 . The framework of the ISA

As the series epidemic of the coronavirus, COVID-19, that is outbreak to hit several countries of the globe and causing a considerable turmoil among the peoples, thus the practical intent of the proposed work is to assist the officials with estimating a realistic picture regarding the time and the epidemic peak (i.e., estimating and forecasting the max. no. of infected individuals by the means of forecasting model) and thus can help in developing drastic containment measures for the officials to avoid the epidemic spreading of this virus. In respect of the proposed methodology, an improved interior search algorithm (ISA) is enhanced with chaotic learning (CL) strategy to improve the seeking ability and avoid trapping in the local optima, named ISACL. The ISACL provides an optimization role in achieving an optimal configuration of the MFNN by training process in terms of tuning its parameters. Accordingly, the simulation results 

The proposed ISACL algorithm is developed via two improvements which are the composition group based on individuals' experience to emphasize the diversity of the population, and chaotic learning strategy is carried out on the best solution to improve its quality during the optimization process. The detail behind the ISACL is elucidated as follows.

In ISA, the element or the individual of the composition group is updated by a random manner that may deteriorate the acceleration of the algorithm and the population diversity. Hence, an experience strategy is introduced to improve the acceleration and the population diversity. In this sense two individuals  k l and  k r are chosen from the population, randomly. Therefore, the likelihood search direction is illustrated in the updating the current element as follows.

The main merit of the chaos behavior is lies behind the sensitivity to initial conditions, which can potentially perform the iterative search with higher speeds than the conventional stochastic search caused by its ergodicity and mixing properties. To effectively increase the superiority and robustness of the algorithm, a parallelized chaotic learning (CL) strategy is introduced. The CL strategy starts the search with different initial points and thus enhances the convergence rate and overall speed of the proposed algorithm. The steps of CL strategy can be described as follows.

Step 1. Generation of chaotic values: In this step, a  ND matrix k C of chaotic values is generated according to N maps as follows: 

where D is the number of dimensions for the decision (control) variable, k jd  denotes the generated chaotic number within the range of (0, 1) for the th j chaotic map of the th k iteration on the th d dimension. The chaotic numbers in (13) are generated using the functions that are introduced in [ is considered as the best so for solution.

Step 3. Updating the best solution: If

Step 4. Stopping chaotic search: If the maximum iteration for the chaotic search phase is satisfied, stop this phase.

Based on the abovementioned improvements, the framework of ISACL is described by the pseudo-code as in Fig.3 and the flowchart in Fig. 4 , where the ISACL starts its optimization process by initializing a population of random solutions. Thereafter these solutions are updated by the ISACL and the best solution is refined and updated using the CL phase. Then the superior solution will go to feed the next iteration. The procedures of the framework are continued until any stopping criterion is met. .

(1 ).

Update the global best with random walk, 

In this section, the description regarding the COVID-19 dataset, the parameter settings for the presented algorithms, the performance measures, the results and modeling analysis, and discussions are presented.

This subsection provides the COVID-19 dataset that is presented in this study. 

To assess the proposed ISACL-MFNN while forecasting the COVID-19, it is compared with different algorithms including GA [29] , PSO [44] , GWO [44] , SCA [44] and the standard ISA.

Because of the high degree of haphazardness associated with the meta-heuristic algorithms and to ensure fairness, each algorithm is carried out with different independent runs and the best result is reported along with performance measures. To attain fair comparisons, the operation parameters such as a maximum number of iterations, populations' size are set to 500 and 10 which are common parameters iterations where the other related parameters associated with each algorithm are provided as reported in its corresponding literature where the overall parameters are tabulated in Table 1 . Besides, all algorithms are coded with Matlab 2014b, Windows 7 (64bit) -CPU Core i5 and 4GB RAM. Thus the closer the value of 2 R is to 1, the superior result for the candidate method. Table 2 The results of error analysis regarding the training set ( 

To forecast the COVID-19 of the confirmed cases of three countries (i.e., USA, Spain, Italy) that are most influenced at 3/4/2020, six optimization algorithms are implemented, where the ISACL presents the proposed one. In this sense, the results of the ISACL are compared with other competitors. The assessments regarding these algorithms are performed by some indices that are reported in Table 2 . Based on the reported results, it can be observed that the proposed ISACL outperforms the other models, where it can provide that the lower values for the overall indices including RMSRE, RMSE, MAPE, and MAE and can perceive a higher value for R 2 that indicate Table 3 . For further validation, the forecasting results of the confirmed cases of the test set , 31/3/2020-3/4/2020, are depicted in Fig. 8 , where training set is considered from 22/1/2020 to 30/3/2020. From Fig. 8 , it

can be noted that the proposed ISACL is very close to the actual historical data than the other algorithms.

Finally, based on the obtained results, it can be observed that the proposed ISACL-MFNN provides accurate results and has high ability to forecast the COVID-19 dataset regarding the studied cases. In this sense, the limitations of traditional MFNN are avoided due to the integration with the ISACL methodology through the PL strategy to enhance the local exploitation capabilities and obtain high quality of solutions.

This paper presented an improved interior search algorithm based on chaotic learning (CL) strategy, named ISACL, which is implemented to improve the performance of the MFNN by finding its optimal structure regarding the weights and biases. 

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