key: cord-0925390-gstj22kt
authors: Cheema, Tahir Nawaz; Raja, Muhammad Asif Zahoor; Ahmad, Iftikhar; Naz, Shafaq; Ilyas, Hira; Shoaib, Muhammad
title: Intelligent computing with Levenberg–Marquardt artificial neural networks for nonlinear system of COVID-19 epidemic model for future generation disease control
date: 2020-11-23
journal: Eur Phys J Plus
DOI: 10.1140/epjp/s13360-020-00910-x
sha: e0d5dd14ad4fa73319f7a58c02e38d277087082d
doc_id: 925390
cord_uid: gstj22kt

The aim of this work is to design an intelligent computing paradigm through Levenberg–Marquardt artificial neural networks (LMANNs) for solving the mathematical model of Corona virus disease 19 (COVID-19) propagation via human to human interaction. The model is represented with systems of nonlinear ordinary differential equations represented with susceptible, exposed, symptomatic and infectious, super spreaders, infection but asymptomatic, hospitalized, recovery and fatality classes, and reference dataset of the COVID-19 model is generated by exploiting the strength of explicit Runge–Kutta numerical method for metropolitans of China and Pakistan including Wuhan, Karachi, Lahore, Rawalpindi and Faisalabad. The created dataset is arbitrary used for training, validation and testing processes for each cyclic update in Levenberg–Marquardt backpropagation for numerical treatment of the dynamics of COVID-19 model. The effectiveness and reliable performance of the design LMANNs are endorsed on the basis of assessments of achieved accuracy in terms of mean squared error based merit functions, error histograms and regression studies.

Transmission coefficient (infection) β 1 Transmission coefficient (super spreaders) k

Exposed to infectious rate ρ 1 Exposed to infected rate ρ 2 Exposed to super spreaders rate γ a Rate of being hospitalized class γ i Recovery rate without hospitalized γ r Recovery rate of hospitalized patients δ i

Death rate due to infected people δ p Death rate due to super spreaders

In December 2019, a new disease known as coronavirus was declared as a viral infection with high rate of transmission in Wuhan city of China. Corona virus is originated by the acute respiratory syndrome 2 (SARS-Covid-2) declared by the Group of International Committee (GIC) on Taxonomy of virus on February 11, 2020. It was identified as the causative virus by Chinese authorities on January 1, 2020 [1] . A chain of analysis reported on Bats are key reservoir in this research [2, 3] .

The COVID-19 epidemic mainly effects on people's health, economy daily life routine [4] . Due to these major causes, the governments of several countries have made public policy about both highlighted aspects. 19) originally started on December 31, 2019, from Wuhan city of China, which is now the capital of Hubei territory. In the previous medical history of viruses, spreading of viruses always have some logical reasoning for which the accessible medications are found for the treatment. Further, it has been verified through reliable data that the transmission of the infection is only possible from humans to humans [5] . During the reported time, many cases were spread in Wuhan city as well as to different urban communities of China rapidly. Besides this, the infection spreads to other parts of the world, for example, Europe, North America and Asia within short span of time. Meanwhile it is reported that the appearance of the symptoms based on cough, breathing troubles and high fever of the corona (COVID-19) within 2 to 10 or 2 to 14 days approximately.

Nowadays, the dynamics of COVID-19 models have been growing interest in the research community and may mathematical models are designed for the better interest of people around the world, such as the model of eight classes based on susceptible, infected, diagnosed, ailing, recognized, threatened, healed and extinct (SIDARTHE) [6] , five classes based on SEIAR represented with 5 number of ordinary differential equations [7] , a new θ -SEIHRD model represented with nine classes [8] , modified SEIRS model system with five classes [9] , four class modified SIR model [10] , SAIR system based COVID-19 model for complex networks [11] . Beside, these variety of COVID-19 model are introduced by the researchers [8, [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] . However, in the current scenarios, we have taken a complex 8 classes model based on Susceptible (S), exposed (E), symptomatic and infectious (I), super propagation (P), infection but asymptomatic (A), hospitalized (H), recovery (R) and fatality (F) classes, i.e., SEIPAHRF for numerical investigations [23] .

Mathematical relations of Covid-19 dynamics with SEIPAHRF model are represented with following initial value problem (IVP) as [23] :

where definitions of each parameter of COVID-19 on SEIPAHRF model (1-9) is provided in nomenclature table. The graphical representation of SEIPAHRF model for COVID-19 dynamics is shown in Fig. 1 to decipher the information more evidently.

The strength of artificial intelligent (AI) based computing solvers has been exploited by the research community on large scale to obtain the approximated solutions of many problems arises in broad fields of applied science and technology. Some potential, recent reported studies having paramount significance including Van-der-Pol oscillatory systems, optics, electrically conducting solids, reactive transport system, nanofluidics, nanotechnology, fluid dynamics, astrophysics, circuit theory, plasma, atomic physics, bioinformatics, energy, power and functional mathematics see [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] and references cited therein. The said information is the motivational affinities to investigate in AI base numerical computing solver for the COVID-19 model. As per our literature survey no one yet implemented AI based computational procedure through Levenberg-Marquardt artificial neural networks (LMANNs) to solve initial value problems (IVBs) of nonlinear systems of ordinary differential equations (ODEs) represented COVID dynamics as given in (1) (2) (3) (4) (5) (6) (7) (8) (9) . We present the design of intelligent computing paradigm through LMANNs for numerical treatment of Covid-19 based SEIPAHRF model for five different cities of China and Pakistan including Wuhan, Karachi, Lahore, Rawalpindi and Faisalabad. Research related Covid-19 model and its applications will be useful to different models of diseases emerging in science, particularly, bio-mathematicians for design and development of alternate computing solver to study the dynamics of the systems numerically.

The innovative contributions of the presented study for Levenberg-Marquardt artificial neural networks (LMANNs) for COVID-19 models are highlighted as follows.

• A novel design based on two-layers structure of Levenberg-Marquardt artificial neural networks (LMANNs) is presented to examine the dynamics of COVID-19 model represented with initial value problems of eight systems of ODEs. 

The mathematical models for the development for the COVID-19 systems for one big city of China and 4 cities of Pakistan are presented in Sect. 2, methodology of LMANNs is provided in Sect. 3, the numerical simulation and analysis are presented for different cases COVID-19 dynamics in Sect. 4, while concluding inferences are given in the last Section.

Mathematical development of COVID-19 for different cities of China and Pakistan is provided in this section. Fixed setting of parameters as tabulated in Table 1 reported recently in [23] for SEIPAHRF model of COVID-19 is used throughout in the presented study. 

The essential information related to our proposed mathematical modeling together with performance metrics are presented in this section. The implemented mathematical modeling based on three phases: in phase one COVID-19 model for five different cities of China-Pakistan is evaluated that are considered as input reference dataset for FFNNs, phase two, layer structure formulation of NN-BPML models and training of NN-BPML is performed with Levenberg-Marquart solver in phase three. The graphical abstract of presented study is shown in Fig. 2 .

The Adams predictor corrector method procedure [60-61] is presented to the system (9-10). By using Adams method formulation, first we used predictor solution then corrected in whole numerical procedure to improve the accuracy level of results with provided information of predicted results. The Eqs. (9-10) of predictor corrector method can be given as: 

Eur. Phys. J. Plus (2020) 135:932 The relation for predictor 2-step formula in case of first equation of set (15) is given:

while 2-step corrector relation formula in case of first equation of set (15) is written as:

Accordingly, the formulae of Adam predictor and corrector method for rest of equations in set (15) are formulated. The dataset of FFNN can be created with Adams method as summarized in Eqs. (11) (12) (13) for solving the PLFMs. However, the presented study, we have generated the dataset of FFNN using 'NDSolve' routine of Mathematica with algorithm 'Adams' for each scenario of PLFMs.

The layer structure of FFNN models with log-sigmoid activation function and 10 number of neurons in the hidden layer are exploited for solving each scenario of PLFMs. The constructed architecture of FFNN is presented in Fig. 3 The training the FFNNs is conducted with backpropagation of Levenberg-Marquardt method (LMM), i.e., FFNN-LMM by defining an error base merit function. The objective function is constructed of mean square error (MSE) metric and optimization of the objective function is performed with LMM for each case.

The mathematical notations of the performance metrics through absolute error (AE), figure of merit, i.e., mean square error (MSE) and regression coefficient are given below:

here S j ,Ŝ j andS j stand for reference, approximate and mean of solution of jth input, while k represent total number of input grids. The unit value of R, i.e., square root of R 2 , is the desire parameter for perfect modeling, while AE and MSE are equal to zero for perfect modeling scenarios. The overall process flow diagram of proposed LMANNs is described in Fig. 2 . The proposed LMANNs are implemented through 'nftool' (neural network fitting tool) in neural network toolbox in Matlab environment, while Levenberg-Marquardt (L-M) is used to train the weights of neural networks. The designed LMANNs are conducted for five different cases where first four cases are constructed on real data of big cities of Pakistan: Karachi, Lahore, Faisalabad and Rawalpindi, and last case is on real data of Wuhan city with fixed parameters as tabulated in Table 1 . The papulation survey of 2017 of Pakistan is used for related parameters.

The reference data of SEIPAHRF model for COVID-19 are generated for 60 days as inputs with step size of 0.5 through the solutions of Adams numerical approach by using Mathematica environment 'NDSolve' built-in function for numerical results of ODEs for each case of SEIPAHRF model for COVID-19. The dataset values for S, E, I, P, A, H, R and F classes for 121 input points that are arbitrarily distributed to produce a set for train, validation and test with 90%, 5% and 5%, respectively. The two layered structure LMANNs based computing paradigm of neural networks with backpropagation of L-M along ten hidden layers are contracted for the results of SEIPAHRF mode for COVID-19 classes that shown in Fig. 3 .

The results of LMANNs for SEIPAHRF model for COVID-19 in terms of state transition dynamics are graphically described in Fig. 4 In the Figs. 5a, 6a, 7a, 8a and 9a, convergence through MSE for validation, train and test processes are illustrated for case 1-5 of SEIPAHRF model for COVID-19. The best network performance achieved at 90, 22, 74, 62 and 1000 epochs with MSE around 10 −04 to 10 −03 , 10 −04 to 10 −02 , 10 −04 , 10 −05 and 10 −04 to 10 −03 for case 1-5, respectively. The performance of LMANNs generated outcomes is examined with reference results of Adams numerical method for case 1-5 and respective results are shown in Figs. 5b, 6b, 7b, 8b and The analysis of regression studies is calculated through co-relation studies where the results are graphically shown in Figs. 10a-e for each case. Correlation R values are steadily around unity, i.e., desired value for perfect modeling, for training, testing and validation, which established the accurate working of LMANNs for solving SEIPAHRF model. Therefore, the numerical and graphical results of LMANNs are determined for the susceptible class (S), export class (E), symptomatic and infectious class (I), infectious but asymptomatic class (A), super spreaders class (P), hospitalized (H), recovery class (R), fatality class (F) to explain the behavior corresponding to 60 days for each five case. Numerical outcomes are portrayed in Figs. 11, 12, 13, 14, 15, 16, 17, 18 and 19 . The susceptible class (S) is graphically explains in subfigure 11a of case 1-4, the result values of S are lies in different ranges that is why subfigures for first four cases are shown that explains as more population higher susceptible class. The Figures 12a, 13a, 14a, 15a, 16a, 17a and 18a , describes graphi- Tables 3, 4 Tables 8, 9 , 10 and 11. AEs also satisfied the results of case 5 that is illustrated in Fig. 19d and tabular in Table 12 . AEs of class S ranges between 10 −02 and 10 −04 for cases 1, 3, 4, 5 and 10 −02 to 10 −03 for case 2. Range of AEs for class A are 10 −03 to 10 −04 for cases 1, 3, 4, 10 −02 to 10 −04 and 10 −02 to 10 −06 for case 2 and 5, respectively. AEs of class R are 10 −03 to 10 −05 for case 1, 10 −03 to 10 −04 for case 2 to 4, and 10 −02 to 10 −06 and for class F are 10 −03 to 10 −04 , 10 −02 to 10 −03 , 10 −03 to 10 −04 , 10 −03 to 10 −04 , and 10 −02 to 10 −04 of case 1-5, respectively. AEs are 10 −03 to 10 −05 of case 1, 3, 5 and, 10 −02 to 10 −04 of case 2 and 10 −03 to 10 −07 of case 4 for class E. The range of AEs for class I, P and H are 10 −02 to 10 −05 , 10 −03 to 10 −07 , 10 −03 to 10 −06 of case 1, 10 −02 to 10 −05 , 10 −03 to 10 −08 , 10 −02 to 10 −05 of case 2, 10 −02 to 10 −05 , 10 −03 to 10 −06 , 10 −03 to 10 −06 of case 3, 10 −02 to 10 −06 , 10 −03 to 10 −08 , 10 −03 to 10 −05 of case 4, 10 −03 to 10 −06 , 10 −04 to 10 −07 , 10 −04 to 10 −06 of case 5, respectively. These ranges of AEs for all classes of each case illustrates the accuracy of the proposed method that is up to 8 decimal places.

Artificial intellect based integrated computing intelligent platform is presented by means of neural networks with backpropagation of Levenberg-Marquard to find the solution of mathematical model SEIPAHRF for COVID-19 representing the spreading of Corona virus through different classes in the major cities of Pakistan and China for different cases that Eur. Phys. J. Plus (2020) 135:932 are constructed on the basis of real data. Dataset for SEIPAHRF model for COVID-19 is generated through Adams numerical solver for different classes. The 90%, 5% and 5% of the reference dataset is used as training, validation and testing for LMANNs. On the basis of above numerical study and investigation, following key findings of SEIPAHRF model for COVID-19 can be observed.

• Governing system of ODEs representing the radiative spread of COVID-19 are solved with the help of. LMANNs. • Comparison of proposed results with reference numerical solution obtained through Adams method upto 8 decimal places which shows the accuracy and convergence of the proposed LMANNs. In future, one may implement proposed LMANN for solving the systems representing computer virus models [35, 36] , prediction studies [37] [38] [39] [40] [41] , nonlinear fractional differential equation [42, 43] , bioinformatics models [44] [45] [46] and financial modeling [30, 47] .

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Conflict of interest The authors declare that they have no competing interests.