A Novel Honey Badger Algorithm with Multilayer Perceptron for Forecasting COVID-19 Time Series Data

: The COVID-19 pandemic was affected the health, economy, and all aspects of human lives around the world. Accurate prediction of the new daily cases of COVID-19 is critical for precise programming and needed measures to prevent the outbreak of it. Hence, in the present paper, we implement a new hybrid intelligent model, namely the artificial neural network (ANN) hybridized with the Honey Badger Algorithm (HBA-ANN) for accurately daily new cases COVID-19 prediction in Brazil, India, Russia, and the USA. The performance of the hybrid model was compared with the stand-alone ANN and Gene Expression Programming (GEP) model using statistical (R 2 , RMSE, SI, and NSE) and graphical (Taylor and scatter diagrams and box plot) criteria. Results showed that the HBA-ANN model with the high value of R 2 , law value of RMSE, and the least distance from actual values outperformed the ANN and GEP models in each country. Hence, it is recommended to implement the HBA algorithm to increase the prediction accuracy of the models in medicine field.

speed of transmission from one person to another and is easily transmitted through sneeze, cough, breathe, speak or sing.Symptoms of the disease include fever, dry cough, and sometimes breathing problems such as shortness of breath, shortness of breath, sore throat, and runny nose [2].
On 30 February 2020, the COVID-19 virus has killed more than 2,709 people and more than 73,332 people have been confirmed with this virus in 80 countries, including Thailand, South Korea, Japan, Taiwan, Australia, Singapore, Nepal, Vietnam, Indonesia, Germany, Russia, Fiji, France, Iran and the United States were infected.By May 27, 2020, the number of people infected and died in the world due to the Covid-19 virus was 5,790,103 and 357,432, respectively.Following the high epidemic and rising death rate due to the Covid-19 virus, the World Health Organization has announced a global health issue alert [3,4].By the March 16, 2021, worldwide, 120,512,041 infected cases and 2,665,742 deaths due to the corona virus have been reported.On March 6, about 291 million children and young people were forced out of school due to the widespread closure of schools by governments to slow the spread of the Corona virus.Nineteen countries closed all schools in their territory, which affected students.The grades were from preschool to high school.
To prevent the epidemic and deaths caused by the corona virus, governments around the world adopted various strategies, including lockdown, international travel ban, sanitizing, social distancing, work from home, and so on.
Despite these controls, in different countries of the world, people were getting infected with the virus, the death rate was high, and it was difficult for governments to control the existing conditions.In addition to health problems, the corona virus also created many economic problems.For the first time since World War II, in 2020, the global GDP decreased by 5.2% [5].At first, the death rate caused by the corona virus was unpredictable, especially for the children and young age groups.Most deaths occurred in people over 80 years with underlying diseases such as cancer, diabetes, and cardiovascular diseases [6].
The machine learning methods were showed better performance in predicting the new daily cases of COVID-19 [18].Some of the models which are used with researchers are including, LSTM [19,20]; Attention-based model using Bayesian Optimizer, and LSTM using Bayesian Optimizer [21]; MPA and ANFIS [22,23] Data-driven or parametric methods are widely used by researches for propagation and virus infection, and machine learning methods are utilized for forecasting virus spreading dynamics [39,40].In a study [41] to forecast COVID-19 pandemic in India have implemented linear regression, Vector Autoregressive model (VAR), and Multilayer perceptron (MLP) and concluded that the MLP outperformed the other implemented models.In the other study [42] implemented a Gompertz model and Artificial Neural Network (ANN) to forecast cumulative COVID-19 deaths in Mexico and indicated the superior performance of the ANN.Recently, [4] integrated Theta method and autoregressive neural network (ARNN) as a novel model (TARNN) to forecast daily new COVID-19 cases in the UK, the USA, Brazil, Canada, and India.In this study, the TARNN hybrid model outperformed the other traditional models.Also, authors in [1] used the ARIMA model to forecast the COVID-19 affected patients, and the results showed that in the future three months the affected patients will be 9 to 15 million people.
Due to the non-linear and complex nature of the difficulties and disorder of estimating the various parameters in the wide ranges of scientific fields (e. g. economic, finance, medicine, hydrology, environment, and so on) the swarm intelligence (SI) methods are developed and integrated with the artificial intelligence (AI) models.Using these methods, majority could increase the prediction performance of the AI models [9].The accurate prediction of COVID-19 is crucial in terms of health, economy, and social fields, so, it is critical to choose a precise model for prediction.
The complex feature of the COVID-19 data and high ability of the integrated of the AI models with SI algorithms was encouraged us to develop and compare the capability of the Honey Badger Algorithm (HBA) as a new hybrid model to predict COVID-19 around the world.To the best of our knowledge, there is not any research which has used so far on the implementing of hybridization of the HBA with the artificial neural network (HBA-ANN) to forecast COVID-19 daily new cases around the world.We compare the HBA-ANN model with the stand-alone ANN model and Gene Expression Programming (GEP) model.
The rest of the study is categorized as follow: Section 2 describes the material and methods and the data utilized.The results displayed in Section 3, and the conclusion is provided in Section 4.

Artificial Neural Network (ANN)
Artificial Neural Network (ANN) is categorized among the supervised machine learning models.The basic building of the ANN is related to Warren Mc Calk and Walter Pitts (1940) which implemented a logic model for simulation [43].The main idea of ANN is to model the processing characteristics of the human brain to approximate conventional computational methods with biological processing methods.An ANN model consists of three layers, including input layer, hidden layer, and output layer.Each layer contains a group of neurons that are generally connected with all the neurons of other layers.Multilayer Perceptron (MLP) and Radial Basis Function (RBF) are two types of the ANN model.Bayesian regularization (BR), Levenberg-Marquardt (LM), scaled conjugate gradient (SCG), adaptive learning rate (GDX), and gradient descent with momentum are the algorithms of the ANN models.The relationship between input (x) and output (Y) in an ANN is shown as: where f is the action function; b is the bias; Wi is the weight of link. Figure . 1 presents the structure of a common ANN model.
Figure1.The structure of the artificial neural network (ANN)

Honey Badger Algorithm (HBA)
Honey badger is a type of mammal animal from the weasel family and native to Africa, Western Asia and the Indian subcontinent.This animal has been called "the most fearless animal on earth" and "unrivaled warrior".Honey badger has high intelligence and is one of the few non-primitive animals that uses tools.The favorite food of the HB is honey and their name was chosen for this reason.Recently, to solve many optimization problems, the metaheuristic methods were developed which have been stimulated from nature behaviors of living creatures.Honey Badger Algorithm (HBA) is a novel metaheuristic optimization algorithm which is inspired from the intelligent foraging behavior of honey badger [44].The HBA is determined by exploration/exploitation equilibrium, this competency increases the capability of the approach to solve difficult optimization issues with multi-local solutions.
The HB walks continuously and slowly and determines its prey by scenting and digging or by tracking the honeyguide bird.First stage is Digging mode and Recognition as honey mode are the first and second stages, respectively.HB implements sniffing skills to estimate prey's location, in the previous phase.In the final mode, HB as a guide to locate beehives directly implements the honey guide bird.
The population of candidate solutions (X) in HBA is shown as: j-th honey badger position The steps in running a HBA model is as follow: Step 1: initialization: For determining the relative positions of HB with n populations, the following expression is utilized: where, Ubi and LBi refer to the upper and lower bounds of the search space, respectively; xj shows the j-th honey badger position and in a population with size n determines a nominee solution.
Step 2: Define the intensity: The efficacy of concentration of the prey and the distance among it and the j-th honey badger affect the intensity (I).Ij presents the intensity of the smell of the prey; when the scent is low, slow motion occurs, and when it is high, fast motion occurs.It is equipped by inverse square law (ISL) (Kapner et al, 2007), as shown in Fig. 2 and presented by (5): =   −   (7) where S, an indicator of the concentration of a source (prey position), and dj, distances among j-the badgers and prey.
Step 3: Updating the density factor: With each iteration, the updated decreasing factor () decreases randomly using Eq.8: where (C) is a fixed number (the default value is 2), and Maxiter refers to the maximum number of iterations.
Step 4: Fleeing the local solution: The fourth, fifth and sixth steps are used to flee from the local solution area.HA implements a flag (F) to change the search direction and furnishes desirable situation for precise scanning of the search field.
Step 5: update the positions of the operators: Digging mode and honey mode are two modes in updating HBA positions (Xnew).
Detailed descriptions are provided below: • Digging mode: The honey badger's dig has a cardioid shape which Eq. ( 9) simulates this digging mode: where the best position that the prey has achieved to date is shown by Xprey.β is a fixed number that is bigger than 1 (default = 6).The r3, r4, and r5 generated within an interval of [0,1] and are random number.F represents a flag that exchanged the direction of the search, and identified by (10): Three factors majorly affect the behavior of HB in the digging mode: decreasing operator (), the distance among prey and HB (dj), and a measure of the intensity of the scent (I) of the prey (Xprey).
• Honey mode: The honeyguide bird exploring a beehive with the honeyguide bird using Eq. ( 11): Eq. ( 8) calculate the value of () and Eq. ( 10) calculate (F).Xnew shows the new position of the prey, and Xprey describes the location of the prey.From (11) it is assumed that the HB hunts near Xprey (the optimal prey position) based on the distance information (dj).In this step, the evolution of search behavior () affects search results.
Based on the theory, the HBA is considered a global optimization algorithm as the stages of exploration and exploitation.The advantages of the HBA algorithm are simple to implement, hybridization, reducing the number of agents that is necessary to be meliorated, broad applicability.Three variables determine the HBA algorithm, including the number of populations (n), the number of variables in a state (d), and the maximum number of iterations (Maxiter).
Figure .2, presents the flowchart of hybrid algorithm (HBA-ANN).Table 1 shows the parameters used in the modeling stage of ANN and HBA-ANN models.Gene expression programming model (GEP) is a meta-heuristic optimization method to achieve the correct estimation of the target variable.This model is inspired by nature towards evolution and the absolute optimum goes.This method is based on circular algorithms and Darwin's evolutionary theory, which was presented by Ferreira in 2001 [45].The GEP method is a combination of Genetics algorithm (GA) and genetic programming (GP) is that the genotype of chromosomes in the GEP model is similar to GA, having a linear structure with fixed length and the phenotype of chromosomes as a tree structure with different length and size is similar to GP [46].It is said that, the GEP is 10 thousand times more effective than GP.Because in the GEP model, all tree structures of different sizes and shapes are coded in linear chromosomes of fixed length, this causes the complete separation of phenotype and genotype, thus allowing the system to benefit from all evolutionary advantages.In this method, optimization including the processes of mutation, inversion, reproduction and selection of the best gene is done in a linear structure and then expressed as a tree structure.This issue causes only modified genomes to be passed to the next generation.Therefore, there will be no need for heavy structures for reproduction and mutation Genetic Programming (GEP) could be used for any specified time series.In the GEP, on the structure of the relation among independent and dependent variables, there is not any assumption.Two components of the GEP are a parse tree (a functional set of mathematical functions) and the terminal set (including parts of functions and their parameters).In the present study, these two components are chosen as: In the present paper, for implementing GEP model, the GenXpro software application [45,46 ] was used.Figure 1 presents the structure of a common GEP model.was low in all countries, but then we see an increasing trend, especially in India and Russia.In the period of the study, India had the highest (with 90802 cases) and Russia had the lowest (with 11656 cases) COVID-19 cases.The ups and downs of cases in Brazil have been more than the other three countries.India has the highest trend of infected cases and the highest decreasing trend of infected cases is related to Russia.

Error Analysis
In the present paper, to explore the performance of the implemented models for predicting the COVID-19 cases in the Brazil, India, Russia, and USA we utilized four performance criteria, including correlation coefficient (R), Root Mean Square Error (RMSE), (SI), and Nash sufficient Error (NSE) which is shown in Table 2.In this Besides, we also utilized graphical criteria, including Taylor, scatter, and error diagrams to evaluate the performance of the implemented models.Taylor diagram considers three performance criteria (standard deviation, R, and RMSE) to evaluate the performance of the models.Based on those three criteria, in the space, a specific point is appeared for any model, and the model which has the lowest distance from observation point will be the most accurate model for prediction.

Result
In the present study, we utilized three model, including ANN, HBA-ANN, and GEP to predict the COVID-19 cases in Brazil, India, Russia, and the USA.The model's inputs were determined with Average Mutual Information (AMI) method.The AMI measures the general dependence of two variables, whereas the ACF and PACF show the dependence from the perspective of linearity [47].This delayed mutual information is based on the Shnanon's entropy and is able to properly determine the optimal time lag in comparison with the ACF.These recognizes that the most important quantities included are entropy (i.e. the amount of information that is contained in a random variable) and mutual information (i.e. the amount of information in common between two random variables [48].The results of the AMI method are shown in Fig. 6.The first local minimum is chosen for the number of inputs.The results of the AMI showed that up to 3, 4, 13, and 4 time-delays can be implemented as the optimal time delays for the daily COVID-19 cases in Brazil, India, Russia, and the USA, respectively.Figure 6.The AMI results for selecting optimal time lags To evaluate and compare the performance of the ANN, GEP, and HBA-ANN models in prediction the COVID-19 cases in the tasting phase in Brazil, India, Russia, and USA, we utilized four performance criteria, including R, RMSE, SI, and NSE which the results are shown in Table 5.It can be seen that the HBA-ANN model has the highest R among the models in each four countries.The value of R for the Brazil, India, Russia, and USA is 0.999, 0.899, 0.853, and 0.993, respectively.Besides, the HBA-ANN model could decrease the RMSE by 97.22%, 43.68, 85.41%, and 93.83% for the Brazil, India, Russia, and USA, respectively.The SI for the HBA-ANN model in each four countries was less than 0.1 indicating the superior performance of this model.Hence, the HBA-ANN model is the most accurate model for predicting the COVID-19 in those countries.The other result of this Table 3 is that, the performance of the ANN model was better than the GEP.The Taylor diagram for the implemented models is shown in Figure 7.For the ANN, GEP, and HBA-ANN models in each country, a point is determined on Taylor diagram.Also, there is a point that shows the observation point.In the right, for each model, it is shown a numerical measure which is the distance among the predicted values with that model and the observation point.It can be seen that this measure for the HBA-ANN model is 228.709,2113.18,64.54, and 1047.72,respectively which is the lowest distance among the implemented models.This indicates the high accuracy of the HBA-ANN model in prediction the COVID-19 cases in Brazil, India, Russia, and the USA.We utilized the time series analysis for visual comparison of the temporal variation of the COVID-19 (Fig. 10).
According to Fig. 10, it can be seen that the predicted values with the HBA-ANN model are fitted well with the observed values for all countries.This may be related to the better generalization ability of the HBA, which could best optimize the internal parameters of the ANN model.

Fig 2 .
Fig 2. Flowchart of the HBA-ANN model

Figure 4 .
Figure 4. Confirmed death per 100000 population in the world To modeling and prediction the daily COVID-19 patients in Brazil, India, Russia, and the USA, we used daily time series from 20-01-2020 to 15-09-2020.The data was divided into training and testing phases.Daily statistical characteristics of COVID-19 variable and coordinates of the studied countries is shown in Tabl.2, and the time series graph of the implemented data is presented in Figure5.It can be seen that in the first fifty days, the COVID-19 case

Figure 5 .
Figure 5.Time series of the daily COVID-19 cases in Brazil, India, Russia, and the USA

Figure 7 .
Figure 7.The Taylor diagram for the ANN, HBA-ANN, and GEP in predicting the COVID-19 cases in Brazil, India, Russia, and the USA

Figure 8 .
Figure 8.The predicted errors for the ANN, HBA-ANN, and GEP model

Figure 9 .
Figure 9. Box-plot of the ANN GEP and HBA-ANN models in Brazil, India, Russia, and the USA.

Figure 10 .Figure 11 .
Figure 10.Time series plots of the observed and predicted COVID-19 of the ANN GEP, and HBA-ANN models in the Brazil, India, Russia, and the USA In a study[35]used exponential smoothing model to predict ten days ahead corona virus cases, but this model had considerable forecast errors.A study is also was done for Italy, South Korea, and Iran by [36] using SIQRK model.For short term cumulative confirmed cases forecasting, [34] implemented integrated moving average (ARIMA), random forest, Cubist regression, support vector regression (SVR), stacking-ensemble learning (SEL), and ridge model.The studies of the[37,38]confirmed the high ability of the nonlinear incidence rates and building nonlinear models in forecasting the COVID-19 in European countries.

Table 1 .
Parameters used for ANN and HBA-ANN Brazil is the largest and most populous country in South America.The area of Brazil is 8,515,767 square kilometers and its population is 212,732,000 people.Brazil has vast agricultural lands and tropical forests, and with extensive natural resources and a rich labor force, it is the most powerful economy in South America.India is located in South Asia and is the most populous country in the world with a population of 1.43 billion people and ranks seventh in the world with an area of 3,402,873 square kilometers.The first corona infection cases occurred on 30th January in Kerala.India reached its first 1 lakh infection on18 th May 2020, and as of 11 th of July crossed 8.5 lakhs.The USA is located in North America, and with 333 million people and with 9,833,520 km 2 area is the most populous country in the Americas, the third most populous and largest country in the world.The first confirmed case of the novel coronavirus (COVID-19) in the USA was announced on January 21, 2020.Russia is the ninth most populous country in the world with about 145,926,000 people, and with 17,098,242 square kilometers has the title of the largest country in the world.The population density of Russia is 8.56 people per square kilometer, which is one of the lowest population density rates in the world.The first corona virus infection is occurred in Russia on 25 Feb. 2020, and on 19 June 2020, one million case and 49 thousand deaths are reported which was 22.62% of total COVID-19 deaths in the world.

Table 2 .
Daily statistical characteristics of COVID-19 variable and coordinates of the studied countries

Table ,
() and () refer to the forecasted and observed COVID-19 values, respectively; ̅ and ̅ show the average of the forecasted and observed COVID-19 values, respectively; and  is the numbers of observations (Shabani et al., 2021).

Table 3 .
The evaluation metrics of the models < 1If a model has the lowest value for RMSE, and the highest value for R between implemented models, it will be the most accurate model for prediction.For the NSE and SI performance metrics the various ranges that indicate the accuracy of a model are presented in Table4.