Applications of artificial neural network to solve the nonlinear COVID-19 mathematical model based on the dynamics of SIQ

Abstract

The purpose of this research work is to present a numerical study through artificial neural networks (ANNs) to solve a SIQ-based COVID-19 mathematical model using the effects of lockdown. The effects of lockdown are considered to the three-dimensional mathematical model, “susceptible”, “infective” and “quarantined”, i.e. the SIQ system. ANNs and Levenberg–Marquardt backpropagation (LMB) are used to present the numerical analyses of the SIQ system-based COVID-19. Three different types of authentications, testing and training as sample data are applied to solve the SIQ system. Statistical ratios for the SIQ-based COVID-19 mathematical model are selected – 80% for training and 10% for both testing and authentication. The obtained numerical results of the SIQ mathematical system have been compared with the reference dataset, which is constructed through Adams solutions. The obtained numerical performances of the nonlinear SIQ dynamical model are testified with a reduction of the error in mean square sense in the range of 10⁻⁹ to 10⁻¹².

KEYWORDS:

• COVID-19
• nonlinear SIQ system
• artificial neural networks
• Levenberg–Marquardt backpropagation
• reference dataset

1. Introduction

There are many deathly diseases that can be transmitted from human to human and animals to humans, like HIV, Ebola and dengue [1–4]. One of the serious diseases is COVID-19, which was first revealed in the Chinese city, Wuhan, at the end of 2019. This virus spread across each country of the world and rapidly in humans [5,6]. In a short period of time, COVID-19 spread throughout the whole world with a large number of casualties. The fundamental reason for the propagation of COVID-19 was the transportation or travelling of masses from China to different regions of the world. China started the policy of lockdown and controlled the disease in Wuhan [7,8]. The option available to control COVID is vaccines and targeted drugs. COVID-19 badly hit industries, education sectors, businesses and major economies of the world like the United States of America, India, European countries, Brazil, etc. It is reported in literature that the migration of people plays a substantial role in the spread of infections like flu. It has been noted that immigration not only introduces infections in inherent people, but also alters the occurrence of infections and grows the scenarios of local spread [9]. The migration of individuals from COVID-19 areas has postured a danger to the local populace and increased environmental assortment of the virus rapidly [10].

Until now, a variety of COVID-19–based mathematical models with varied characteristics have been designed and various methodologies have been used on these systems. To mention a few of them, Rhodes et al. [11] presented the ODEs based model for public distress due to COVID-19. Mustafa et al. [12] proposed a mathematical model for analyzing and forecasting the transmission of COVID-19, Benvenuto et al. [13] designed the ARIMA model based on COVID-2019, Sivakumar [14] reviewed the predictive control analysis for COVID-19 in India, Thompson [15] proposed epidemiological models using important tools for guiding COVID-19 interventions, and Nesteruk [16] estimated the COVID-19 pandemic dynamics in Ukraine based on two data sets. Also, Libotte [17] introduced a vaccine administration strategy for COVID-19, Gumel et al. [18] introduced a mathematical model for COVID-19 pandemic, Sadiq et al. [19] discussed the role and impact of nanomaterial in tackling COVID-19 pandemic, and Ortenzi et al. [20] provided transdisciplinary studies of COVID-19 in Italy. Umar [21] implemented theoretical studies for a drug to treat COVID-19, Moore et al. [22] provided a mathematical model for COVID-19 to study the impact of vaccination as well as non-pharmaceutical interventions, Baba et al. [23] introduced rigorous analysis to curtail the spread of COVID-19, and Anirudh [24] narrated the prediction of transmission dynamics of COVID-19. In addition, Zhang et al. [25] described COVID-19 with stochastic perturbation analysis, Chen et al. [26] introduced the impact of social distance in COVID-19 using mathematical modeling and Soumia et al. [27] provided the potential inhibitors of COVID-19.

Alongside many different mathematical models of COVID-19, artificial intelligence (AI) based computing infrastructure is used to exploit the neural networks, and optimization techniques are also widely introduced by the research community for perdition and control of COVID-19 [28–33]. AI-based computing stochastic numerical procedures have been implemented in reported studies of diversified areas [34–37]. A few recent applications of stochastic solvers are the systems of third order governed with Emden–Fowler equations [38,39], heat conduction model [40], periodic differential model [41], a multi-singular system of equations [42,43], delayed, pantograph and prediction differential system [44], functional differential models [45] and differential fractional order models [46], food chain models [47,48], atomic physics [49], dusty plasma [50], electric circuits [51], thermal explosion model [52], financial models [53] and environment economic models [54]. Additionally, these stochastic solvers are extensively exploited for providing an accurate, robust, reliable and consistent platform for numerical treatment of the solution dynamics of COVID-19 mathematical models [55–58]. These are potential motivational facts for authors to exploit the knack of AI-based numerical solvers for the analysis of COVID-19 mathematical models. The computing stochastic numerical procedures have been implemented in diverse domains of applied sciences; however, the nonlinear SIQ dynamical system based COVID-19 propagation studies involving lockdown impacts [59] look promising to solve with AI-based numerical procedures.

The aim of this work is to provide the solution dynamics of the nonlinear SIQ dynamical system based COVID-19 using the knack of AI via artificial neural networks (ANNs) and Levenberg–Marquardt backpropagation (LMB). The dynamical models have various applications in different sectors [60–63]. The contribution and innovative insights of the computing stochastic numerical procedures are provided as follows:

• Design of novel computing stochastic numerical procedure inspired by knacks of AI via ANNs-LMB is introduced viably for the numerical treatment of the nonlinear SIQ dynamical system of COVID-19.

• Competency of Adams numerical solver is exploited for creating data samples used for testing, validating and training ANNs-LMB to construct networks for approximate solution dynamics of the COVID-19 mathematical model.

• The precise and consistent overlapping of the numerical outcomes in good agreement with standard solutions with slight absolute error (AE) verifies the value of the computing stochastic procedure for the SIQ dynamical system of COVID-19.

• Analysis through error histograms (EHs), MSE-based convergence curves, correlation and regression substantiate the computing stochastic procedure.

The rest of the paper is organized as follows: The numerical measures to solve the nonlinear SIQ model using the ANNs-LMB are labeled in Section 2. The numerical performances through ANNs-LMB are described in Section 3. The conclusions along with future guidance are provided in Section 4.

2. System model: SIQ-based nonlinear COVID-19

This study emphasizes the lockdown effects as precautionary measures based on mathematical models. The COVID-19 based SIQ system has three classes: susceptible S(t), infective I(t) and quarantined Q(t). The general form of the SIQ mathematical system is presented as [59] (1) $\begin{array}{r}\{\begin{array}{ll}{\displaystyle S^{\mathrm{\prime }}(t)=a-\mu S(t)}& \\ -{\displaystyle \frac{\beta I(t)S(t)}{\eta I(t)+\alpha }-(\theta -1)m,}& S(0)=y_1,\\ {\displaystyle I^{\mathrm{\prime }}(t)=\theta m-(\mu +{\delta }_1+\sigma +{\alpha }_1)I(t)}& \\ -{\displaystyle \frac{(k-1)\beta I(t)S(t)}{\eta I(t)+\alpha },}& I(0)=y_2,\\ {\displaystyle Q^{\mathrm{\prime }}(t)=\frac{k\beta I(t)S(t)}{\eta I(t)+\alpha }}& \\ {\displaystyle -(\mu +{\delta }_2+{\alpha }_2)Q(t)+\sigma I(t),}& Q(0)=y_3.\end{array}\end{array}$(1) The comprehensive details of nonlinear SIQ models are shown in Table 1. A further description of the selection of appropriate values or intervals of these parameters can be seen in reference [59] with their local and global stabilities as well as optimal control sensitivity. Whereas in the present research work, we confined our studies to the numerical treatment of systems of ODEs in set (1) by exploiting knacks of AI via design ANNs-LMB.

Table 1. Comprehensive details of each parameter of the nonlinear SIQ system.

Parameters	Description
$\alpha$	Half saturation constant
$a$	Recruitment rate
$\eta$	Positive value
$\mu$	Natural death rate
$\beta$	Transmission infection rate
M	Migrants number
${\delta }_1$	Recovery rate of infective population
$\theta$	Infected migrants’ rate
K	Contact tracing rate
${\delta }_2$	Recovery rate of quarantined population
$\sigma$	0.59 per day
${\alpha }_1$	Disease-associated infective population’s death rate
${\alpha }_2$	Disease-associated quarantine’s population death rate
y1, y2 and y3	Contents for the initial conditions (ICs)

3. Methodology: artificial neural networks-Levenberg–Marquardt backpropagation

The proposed computing stochastic numerical procedure is delivered in two steps for the SIQ dynamical COVID-19 system.

• Critical and detailed explanations are provided for the computing stochastic numerical procedure based ANNs-LMB.

• Implementation procedures support the computing stochastic numerical procedure for the SIQ dynamical COVID-19 system.

An appropriate design process flow structure using the computing stochastic numerical procedure of the SIQ dynamical COVID-19 model is plotted in Figure 1. ANNs-LMB operates through the multi-layers developed in Figure 2. The procedure is applied in “MATLAB” using layer structure, i.e. single input, output and hidden layers, 10 hidden neurons, n-fold cross validation, log-sigmoid activation function, 1000 epochs, and Levenberg–Marquardt optimization algorithm with default values of iteration, stoppage limits, tolerances and step size that are used for training. Also, label data for input and targets are obtained from standard numerical solution, while 80% of arbitrary data are selected for training and 10% for both testing and validation, respectively.

Figure 1. Workflow structure of the proposed stochastic structure for the SIQ dynamical model of COVID-19.

Figure 2. Proposed framework based on the single neuron.

Inspired by the related recent studies [64–70], the proposed neural network models with 10 neurons are taken on the basis of the best compromise between overfitting and underfitting of neural networks in the process of training and validation with epochs 1000. If we take a smaller number of neurons, we may encounter the issues of underfitting, i.e. results in premature convergence, while taking more neurons, i.e. greater than 10, may encounter the overfitting problem, i.e. similar accuracy with higher complexity. The training, testing and validation samples for the ANNs-LMB procedure are chosen arbitrarily with 80%, 10% and 10%, respectively. If we have a smaller number of samples for training i.e. 40%–60%, then the MSE for testing increases from 20% to 30% and results in overall poor performance of the algorithm. On the other hand, if we have more samples for training like 90%, and 5% testing and validation, the accuracy of the algorithm slightly increases for the label data, i.e. bias input and targets, but robustness of the scheme is not rigorously verified for unbiased data without prior knowledge of the targets.

4. Results of numerical simulations with interpretation

Three variants of the nonlinear SIQ dynamical system of COVID-19 through the ANNs-LMB have been presented in this section. The mathematical representations of each variant are provided as

Case 1: Consider the nonlinear SIQ dynamical system of COVID-19 with $a=67447,\beta =2.1\times {10}^{-8},\alpha =5,\eta =1,{\delta }_1=0.4,{\delta }_2=0.4,{\alpha }_1=1.78\times {10}^{-5},m=25000,\theta =0.9\text{and}{\alpha }_2=1.78\times {10}^{-5}$ given as (2) $\begin{array}{r}\{\begin{array}{ll}{\displaystyle S^{\mathrm{\prime }}(t)=67447-5.285\times {10}^{-05}S(t)}& \\ -{\displaystyle \frac{2.1\times {10}^{-08}I(t)S(t)}{I(t)+5}+2500,}& S(0)=1.32,\\ {\displaystyle I^{\mathrm{\prime }}(t)=22500-0.99I(t)}& \\ +{\displaystyle \frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5},}& I(0)=2.29,\\ {\displaystyle Q^{\mathrm{\prime }}(t)=\frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5}}& \\ -0.40007Q(t)+0.59I(t),& Q(0)=\mathrm{3.5.}\end{array}\end{array}$(2) Case 2: Consider the nonlinear SIQ dynamical system of COVID-19 with $a=67447,\beta =2.1\times {10}^{-8},\alpha =5,\eta =1,{\delta }_1=0.4,{\delta }_2=0.4,{\alpha }_1=1.78\times {10}^{-5},m=30000,\theta =0.9\text{and}{\alpha }_2=1.78\times {10}^{-5}$ written as (3) $\begin{array}{r}\{\begin{array}{ll}{\displaystyle S^{\mathrm{\prime }}(t)=67447-5.285\times {10}^{-05}S(t)}& \\ -{\displaystyle \frac{2.1\times {10}^{-08}I(t)S(t)}{I(t)+5}+3000,}& S(0)=1.32,\\ {\displaystyle I^{\mathrm{\prime }}(t)=27000-0.99I(t)}& \\ +{\displaystyle \frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5},}& I(0)=2.29,\\ {\displaystyle Q^{\mathrm{\prime }}(t)=\frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5}}& \\ -0.40007Q(t)+0.59I(t),& Q(0)=\mathrm{3.5.}\end{array}\end{array}$(3) Case 3: Consider the nonlinear SIQ dynamical system of COVID-19 with $a=67447,\beta =2.1\times {10}^{-8},\alpha =5,\eta =1,{\delta }_1=0.4,{\delta }_2=0.4,{\alpha }_1=1.78\times {10}^{-5},m=35000,\theta =0.9\text{and}{\alpha }_2=1.78\times {10}^{-5}$ shown as

(4) $\begin{array}{r}\{\begin{array}{ll}{\displaystyle S^{\mathrm{\prime }}(t)=67447-5.285\times {10}^{-05}S(t)}& \\ -{\displaystyle \frac{2.1\times {10}^{-08}I(t)S(t)}{I(t)+5}+3500,}& S(0)=1.32,\\ {\displaystyle I^{\mathrm{\prime }}(t)=31500-0.99I(t)}& \\ {\displaystyle +\frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5},}& I(0)=2.29,\\ {\displaystyle Q^{\mathrm{\prime }}(t)=\frac{1.05\times {10}^{-08}I(t)S(t)}{I(t)+5}}& \\ -0.40007Q(t)+0.59I(t),& Q(0)=\mathrm{3.5.}\end{array}\end{array}$(4)

The numerical representations are provided for the nonlinear SIQ dynamical model of COVID-19 in input span [0, 1]. The command “nftool” in Matlab is applied to solve the SIQ dynamical system of COVID-19 using 10 neurons with the selected 80% for training and 10% for both authentication and testing. The obtained performances for the SIQ dynamical system of COVID-19 are plotted in Figure 3.

Figure 3. Stochastic structure for the SIQ dynamical COVID-19 model.

The graphs through the calculated results for the SIQ dynamical model based COVID-19 are plotted in Figures 4–8. The proficient procedures of all three variants of the SIQ dynamical nonlinear system based COVID-19 are derived in Figure 4 using the states of performances and transitions. The calculated MSE measures for best curves, training, authentication and testing are provided in Figure 4(a–c), while the state transitions are plotted in Figure 4(d–f) to solve the SIQ dynamical nonlinear system based COVID-19. The best results of the performances for the SIQ dynamical model are drawn at epoch 1000, which lie around 4.002 × 10⁻⁰⁵, 3.454 × 10⁻⁰⁷ and 1.449 × 10⁻⁰⁵, respectively. The gradient values for the SIQ dynamical system based COVID-19 are found as 2.89, 1.87 and 2.95, respectively. These graphical presentations designate the convergence and exactness of the SIQ dynamical system based COVID-19. The fitting curves are plotted in Figure 5(a–c) for each SIQ dynamical system based COVID-19, which shows the comparison of obtained and reference results. Figure 5(d–f) is drawn on the basis of EHs to solve the SIQ dynamical system based COVID-19 using the ANNs-LMB. It is observed that the EHs values for case 1 lie around 1.457 × 10⁻⁰³, −2.1 × 10⁻⁰⁵ and 4.92 × 10⁻⁰⁴, respectively.

Figure 4. MSE and state transitions for the SIQ dynamical COVID-19 model. (a) Case 1: MSE, (b) Case 2: MSE, (c) Case 3: MSE, (d) Case 1: Values of the state transition, (e) Case 2: Values of the state transition, (f) Values of the state transition.

Figure 5. Comparison and EHs values using the ANNs-LMB to solve the SIQ dynamical system based COVID-19. (a) Case 1: Comparison, (b) Case 2: Comparison, (c) Case 3: Comparison, (d) Case 1: EHs, (e) Case 2: EHs, (f) Case 3: EHs.

Figure 6. Regression measures through the ANNs-LMB for case 1.

Figure 7. Regression measures through the ANNs-LMB for case 2.

Figure 8. Regression measures through the ANNs-LMB for case 3.

The correlation measures are plotted in Figures 6–8 that indicate the regression performances. The correlation performances in terms of the coefficient of determination, i.e. values of R², lie around 1 to solve the SIQ dynamical system based COVID-19. These best plots of testing, training and verification indicate the precision of the scheme. The MSE-based convergence based on the training, epochs, authentication, testing, backpropagation measures and complexity are provided in Table 2 to solve the SIQ dynamical system based COVID-19. The computational complexity of the algorithm is determined by calculating the time spent for the training of networks, i.e. execution of all 1000 epochs, for the solution dynamics of SIQ dynamical system based COVID-19. The outcomes of the complexity presented in Table 2 show that the ANNs-LMB algorithm takes around 5 ± 1 s for all three cases of SIQ dynamical system based COVID-19.

Table 2. Proposed ANNs-LMB for the SIQ dynamical COVID-19 model.

	MSE
Case	[Training]	[Verification]	[Testing]	Performance	Gradient	Mu	Epoch	Time
I	2.708 × 10^−05	4.002 × 10^−05	2.811 × 10^−05	2.71 × 10^−05	2.89	1.00 × 10^−02	1000	6
2	2.825× 10^−07	3.454 × 10^−07	2.612 × 10^−07	2.83 × 10^−07	1.87	1.00 × 10^−04	1000	4
3	2.932 × 10^−06	1.449 × 10^−05	1.392 × 10^−15	2.93 × 10^−06	2.95	1.00 × 10^−03	1000	4

Figures 9 and 10 indicate the comparison plots and AE of the obtained and reference results to solve the nonlinear dynamical SIQ model. It is indicated that the reference and obtained results exactly overlapped for each variant of the nonlinear dynamical SIQ model. This perfect overlapping of the results indicates the exactness and perfection of the designed ANNs-LMB. The AE values for each case of the nonlinear dynamical SIQ model are provided in Figure 10. It is observed that the AE is found in good measures for each case of the nonlinear dynamical SIQ model using the ANNs-LMB. One can observe that the designed scheme is accurate and precise to solve the nonlinear SIQ model.

Figure 9. Results comparisons to solve each case of the nonlinear dynamical SIQ-based COVID-19 model. (a) Results for case 1, (b) Results for case 2, (c) Results for case 3.

Figure 10. AE for each case of the nonlinear dynamical SIQ-based COVID-19 model, (a) AE: case 1, (b) AE: case 2, (c) AE: case 3.

5. Concluding remarks

The aim of the present study is to present the numerical solutions of the nonlinear dynamical SIQ mathematical model based on COVID-19 using the ANNs-LMB. The SIQ model is dependent on three dimensions, “susceptible”, “infective” and “quarantined”, i.e. the SIQ system. The authentication, testing and training as sample data are applied in ANNs-LMB for the SIQ system. The data based on the statistics to solve the SIQ mathematical model are selected as 80% for training and 10% for both testing and validation. The obtained numerical results of the SIQ mathematical system have been compared with the reference dataset, which is constructed through the Adams solutions. One can see that the obtained results overlapped with the reference solution, which indicates the exactness of the scheme. The obtained results based on the nonlinear SIQ dynamical model are testified to reduce the MSE. To observe the exactness, trustworthiness, and efficiency of the scheme, the procedures are accomplished using the MSE, correlation, EHs and regression. Moreover, the plots of AE designate a better performance of the scheme for the SIQ model.

In future, the designed procedures through the ANNs-LMB can be implemented to present the numerical solutions of the computational fluidic systems, fractional order models, computer virus models and bioinformatics studies.

Data availability statement

All data generated or analyzed during this study are included in this article.

Additional information

Funding

This study was supported by the Deanship of Scientific Research at Umm Al-Qura University [grant number 22UQU4282396DSR16].