Introduction

At present, the epidemic COVID-19 s and the short period of time spread rapidly around the world, which had a destroying impact on the welfare and health of people different countries, as well as on the global economy. At 30.05. 2021, approximately 170 million have contracted COVID-19, of which approximately 3.5 million have officially died due to the disease. Fast, accurate and automated diagnosis of COVID-19 is essential for patient care and pandemic control. Although COVID-19 medicine features are known (respiratory symptoms, cough, fever, shortness of pneumatic and breath), but they do not always indicate COVID-19 and may be at ordinary pneumonia, which complicates the diagnosis of the problem.

Currently, COVID-19 diagnostics uses the following methods:

1. The RT-PCR test [1][2][3] is the standard for the diagnosis of COVID-19. The disadvantages of RT-PCR provides that it is painful, difficult, time consuming, costly, non-automated and insufficiently accurate [4][5][6].

2. Computed tomography (CT) [7][8][9] allows to visualize the chest cell (to receive CCT image) and is a quick and simple procedure. Compared to RT-PCR, the image CT less accurate in the ordinary pneumonia case, but more accurate in case COVID-19. The disadvantages of CT include the risk of disease transmission when using a CT scanning device and its high cost [10][11][12].

3. Radiography [13][14][15] allows visualization of the chest cell (receive CXR image) and is fast and simple procedure. Compared to CT, it is much faster and more economical, since it requires less scarce and expensive equipment. The disadvantages of radiography include the inability to differentiate COVID-19 from other types of pneumonia and is less accurate than CT [16,17]. Currently, intelligent diagnosis of COVID-19 is usually based on deep and shallow machine learning techniques. At the same time, the most popular are artificial neural networks [18].

The artificial neural networks advantages are:

• The adaptation and training networks;

•

The ability to recognition of patterns, their generalization, extracting knowledge from data, i.e., knowledge about the entity (parametric model for object) is not required; • Data parallel processing, which increases computational complexity. The artificial neural networks disadvantages are:

•

The difficulty of determining the artificial network structure, there are no methods for determining the layers count and neurons in its for different applications;

•

The difficulty of forming a representative set of pattern; • High probability of getting the adaptation and learning method into a local extremum; • Inaccessibility for knowledge accumulated human understanding by the neural network (it is impossible to represent the relationship between input and output in the form of rules), knowledge are distributed among all neural network items in the form of synaptic weight. In [19], a deep training model was proposed based on a combination of CNN (for feature extraction) and long short-term memory (LSTM) (for classification) using CXR images, which provided a diagnostic probability of 99.4%.

In [20], the Dark Net deep learning model was proposed, which consists of 17 CNN and YOLO. The disadvantage of this method is the limitation of CXR images.

In [21] used SVM with four kinds of cores (linear, polynomial, sigmoidal, radial basis functions) using CXR images, and for extracting features in the used 4 different models CNN: Google Net, ResNet18, ResNet50 and ResNet101. A diagnostic probability of 100% was achieved with 2 classes (with COVID-19 and no disease) and 97.3% with 3 classes (with COVID-19, with common pneumonia and without disease), but the processing rate was low.

In [22], a deep learning model was proposed through a huge set of CXR images. The disadvantage is that the dataset is unbalanced: 358 CXR images with COVID-19 and 13,000 CXR images with common pneumonia and no disease.

In [23,24], a combination of a deep learning model and a simple CNN using CXR and CCT images was proposed, which provided a diagnostic probability of 94.4%. The disadvantage is that the dataset was small.

In [25], a deep learning model COVIDX-Net was proposed, which consists of 7 different CNN models: InceptionV3, VGG19, ResNe tV2, DenseNet121, Inception-ResNet-V2, MobileNetV2, and Xception using CXR images. Moreover, the probability of diagnosis varied from 60% to 90%, i.e. was insufficient.

In [26], an ensemble of classifiers was used (neural network, decision tree, support vector machine (SVM), naive Bayes, k-nearest neighbours) using CXR images, which provided a 98% diagnostic probability. The downside is that the dataset was small.

In [27], deep learning model was proposed based on the use of 3 different CNN models: Inception-ResNetV2, InceptionV3, ResNet50 using CXR images, and the best diagnostic probability was shown by ResNet50 -98%.

In [28], a shallow CNN was proposed, which provided high accuracy. The disadvantage is that the set of CXR images was small.

Recently, neural networks have been combined with fuzzy inference systems [29].

The advantages of fuzzy inference systems are:

• Presentation of knowledge in the form of rules, easily accessible for human understanding;

•

No need for an accurate assessment of variable objects (incomplete and inaccurate data); The disadvantages of fuzzy inference systems are: • Inability of their training and adaptation (the parameters of the membership functions cannot be automatically adjusted);

•

Inability of parallel processing of information, which increases the computing power. In [30] has been proposed adaptive neuro-fuzzy inference system (ANFIS), which provided 98.67% probability of diagnosis at high speed training. The disadvantage of the proposed system was the non-automated determination of the number of values of linguistic variables and the number of fuzzy rules.

In this regard, it is relevant to create a method for intelligent diagnosis of COVID-19, which will eliminate these disadvantages.

The aim is to increase the efficiency intelligent diagnostic COVID-19 due to an artificial neural network with generalized bell-shaped function, which is trained by back propagation method allows to automate the process of extraction of knowledge.

To achieve this goal, it is necessary to solve the following tasks:

1. The choice of a method for the formation of diagnostic features.

Formation of diagnostic features CXR image

Formation of diagnostic features CXR image based on the method of gray-level co-occurrence matrix (GLCM), which allows to reduce feature space and performed as follows:

1. To each color image of a set of data is converted into a grey Picture X, i.e. based on matrices, the matrix is calculated in the form

1 2 1 2 1 2 [ ], [ ], [ ] l l l l l l R r G g B b = = = 12 [] ll Ss =

in the Y'UV and Y'IQ models,

1 2 1 2 1 2 1 2

0.299 0.587 0.114

l l l l l l l l s r g b = + + , 12 , 1, l l L  , or in the HDTV model (ITU-R BT.709 standard) 1 2 1 2 1 2 1 2 0.2126 0.7152 0.0722 l l l l l l l l s r g b = + + , 12 , 1, l l L  , where 1 2 1 2 1 2

,, l l l l l l r g b are red, green and blue components of the pixel.

Each grey image is used method of gray-level co-occurrence matrix (GLCM). Preparation matrices denotes as

Creating Mathematical models and neural network bell-shaped function diagnostics COVID-19

For the diagnosis of COVID-19, the work has further improved the artificial neural network models through the use of generalized bell-shaped functions (they are a modification of the Cauchy distribution density), which makes it possible to reduce the number of hidden layers, which simplifies the identification of the parameters of the artificial neural network.

The structure of a neural network model of generalized bell-shaped functions (GBFNN) in the graph form is shown in Fig. 3. The functioning of the neural network of generalized bell-shaped functions is presented as follows. The hidden layers are calculated multidimensional generalized bell-shaped functions (corresponding to the aggregation of sub-conditions of fuzzy rules, connected in conjunction)

1 ( ) ( ) Z h j j zj z z y f gbell x = ==  x , 1 2 ( ) 1 zj b z zj zj z zj xc gbell x a −  −  =+   , 1, jJ  .

In the output layer, the sums of weighted multidimensional generalized bell-shaped functions are calculated (corresponds to the aggregation of activated fuzzy rules with the same conclusions, i.e. diagnoses)

1 J out h k jk j j y w y = =  , 1, kK  .

Thus, the mathematical model is a neural network generalized bell-shaped functions represented as

1 2 1 1 1 zj b Z J z zj out k jk j z zj xc yw a − = =  −  =+     , 1, kK  ,(1)

To decide on the choice of action for the model and (1), the following rule is used

* arg max out k k ky = , 1, kK  .

Selection index for evaluating the neural network generalized bell-shaped functions diagnostics COVID-19 model performance

The paper for evaluation of parametric identification of the mathematical model of neural network of generalized bell-shaped functions (1) is selected:

• The criterion of accuracy, which means selection of such parameters

I out ik ik k K k K i F y d I   =  =  →    , 1, 1, 1, 1, 1, 1,

1, arg max arg max , arg max arg max 0, arg max arg max .

out ik ik k K k K out ik ik out k K k K ik ik k K k K yd yd yd        =    =   (3)

•

The criterion speed, which means selection of such parameters, which deliver the least computational complexity of proposed model min FT  =→ .

(4)

Adaptation of the structure and parameters of a neural network generalized bell-shaped functions diagnosis COVID-19 model based on the method of backpropagation in batch mode

To adaptation the structure and parameters of a neural network generalized bell-shaped functions diagnostics COVID-19 model (1) in the work achieved further improvements in the procedure for determining these parameters based on the method of back propagation and batch training mode to speed up the training, which involves the following steps:

  + = −  , 1, zZ  , 1, jJ  , () ( 1) ( ) () zj zj zj En b n b n bn   + = −  , 1, zZ  , 1, jJ  , () ( 1) ( ) () zj zj zj En c n c n cn   + = −  , 1, zZ  , 1, jJ 

, where  -factor that determines the speed of learning,

01   , 1 1 ( )( ) I out j i ik ik i jk E f y d wI =  =−   x , ()2 11 2 ( ) 1 ( )1

3. Calculation of the output data of layer using KI threads that are grouped into K blocks. Each thread computes out ik y .

4. Calculation of the error energy using strands that are grouped into K blocks. In each block, on the basis of parallel reduction, a partial sum is calculated from I elements of the form ZJI threads which are grouped into ZJ blocks. In each block, on the basis of parallel reduction, the sum of the I elements of the form is calculated

( )2 1 2 1 ( ) 1 ( ) ( )( )K zj out j i zj z jk ik ik k zj b f gbell x w n y d Ia =  −−     x .

7. Setting parameters zj b , using ZJI threads which are grouped into ZJI blocks. In each block, on the basis of parallel reduction, the sum of I elements of the form is calculated ( ) ZJI threads which are grouped into ZJ blocks. In each block, based on parallel reduction, the sum of the I elements of the form is calculated

( )2 1 2 1 ( ) 1 ( ) ( )( ) so y is * k  ( j F ), 1, jJ 

, where j F -coefficient of fuzzy rules j R ,

Numerical research

Numerical research of the offered artificial neural network models, multilayer perceptron, neural network radial basis functions held in the package Matlab using the Deep-Learning Toolbox.

Table 1 shows the root mean square errors (RMSE) and computational complexity, the false diagnostic decisions making probabilities, obtained on the basis of data sets COVID-19 chest X-ray [31], COVID-19 database|SIRM [32], COVID-19 image data collection [33], Radiopaedia COVID-19 [34], Mendeley data -augmented COVID-19 X-ray images dataset [35] using an artificial neural network of the type multilayer perceptron (MLP) and a radial basis function neural network (RBFNN) with backpropagation (BP), and the proposed model (1) with backpropagation (BP). At the same time, MLP had 2 hidden layers (input layer and hidden layer contains of 4 neurons) with logistic activation function, RBFNN had one hidden layer of 8 neurons (twice the neurons count in the hidden layer) with Gauss activation function. P -is the training set power, N -is the number of iterations performed. According to experiments performed, the next conclusions can be done.

The number of neurons in the hidden layer MLP and RBFNN is not automated and is determined empirically, which reduces the classification accuracy and the speed and identification of model parameters.

The generalized bell-shaped activation function is more efficient than the logistic and Gaussian. The proposed models make it possible to eliminate these disadvantages.

Conclusions

1. To decide the problem of augmentation the efficiency of intelligent diagnostics of COVID-19, the corresponding methods of artificial intelligence were investigated. These studies have shown the use of fuzzy logic in combination with artificial neural networks for analysis CXR image is the most effective method in these days.