Swarming morlet wavelet neural network procedures for the mathematical robot system

Abstract

The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet neural network (MWNN). The MRS is divided into two classes, infected $I\left(\theta \right)$ and Robots $R\left(\theta \right)$. The design of the fitness function is presented by using the differential MRS and then optimized by the hybrid of the global swarming computational particle swarm optimization (PSO) and local active set procedure (ASP). For the exactness of the AI based MWNN-PSOIPS, the comparison of the results is presented by using the proposed and reference solutions. The reliability of the MWNN-PSOASP is authenticated by extending the data into 20 trials to check the performance of the scheme by using the statistical operators with 10 hidden numbers of neurons to solve the MRS.

1. Introduction

The prevalence of communicable diseases has long been a problem over the world. To name only few viruses, dengue fever (DF), which affects 2.5 billion people worldwide, is the most dangerous, contagious, and pandemic diseases. Due to ignorance and insufficient knowledge, DF is generally observed in the warmest areas in the world, such as South Asia [1,2]. There are deadly, contagious illnesses on every region of the planet, including HIV. HIV spreads like some of the other infections, and during the past few decades, numerous civilian casualties have been observed [[3], [4], [5]]. Lassa disease is a very severe illness that is primarily found in underdeveloped areas [6]. Malaria is another prevalent condition that transmits through indirect contact between hosts [7,8]. According to world health organization statistics, there have been claimed to have been one million cases from malaria, which primarily affects pregnant women and children from the United States and South Africa [9]. An additional contagious, fatal virus called Ebola is spread by people to infected animals [10]. The world has just been plagued by a deadly coronavirus epidemic that has had a negative impact on global economies, education sectors, sport events, tourism, business and airline sectors [[11], [12], [13]]. One human can infect another with the coronavirus [14]. Since the beginning of the coronavirus, it has abruptly halted people from living busy, rapid lives and has caused uncertainty throughout the entire planet. More deadly than before, it has now appeared in a wide variety of shapes, variants, and phases. It had an influence on the economies of numerous developed and emerging nations. Currently, one, two or three doses of various vaccines have been used throughout the world to begin the immunization process [15].

The majority of infections lack effective treatments or vaccines. Consequently, only a few medical safeguards have been approved to prevent the propagation of these infections. Because of this, the globe adopted several other precautions to prevent the spread of such infections, such as quarantines, social distances, handwashing, avoiding crowded areas, etc. One of the main causes of disease is free healthcare, which is provided by medical personnel such as doctors and nurses. A serious issue for the entire world is being caused by the rising number of positive coronavirus cases [16,17]. Numerous analytical/numerical approaches to the problem of infectious illnesses have been developed. To mention some of them are Mickens employed a vaccination strategy based on distinct time to prevent the transmission of recurring viruses. Newton's embedded approach using the optimal control is applied by Ogren et al. [18] for the SIR dynamical system. The spatial measles outbreak was used on the fractional SIR epidemic by Goufo et al. [19]. Another few research projects based on infectious systems and theoretical advancements are given in these references [[20], [21], [22], [23]]. Consequently, using robots to identify coronavirus-positive patients is crucial. The benefit of using robots is that they can provide medical assistance for those who are ill and stop the coronavirus spread. The dynamics of the MRS is divided into two classes, infected $I\left(\theta \right)$ and Robots $R\left(\theta \right)$, mathematically presented as [24]:

$$\{\begin{array}{l}{I}^{\prime }\left(\theta \right)=\left(a-bR\left(\theta \right)-u\right)I\left(\theta \right)+C\left(I\right),I_0=i_1\text{,}\\ R^{\prime }\left(\theta \right)=c-dR\left(\theta \right),R_0=i_2\text{,}\end{array}$$ (1)

where $I\left(\theta \right)$ and $R\left(\theta \right)$ indicate the infected and Robot populations. a and c represent the newly infected individual and robot production rates. $C\left(I\right)$ is the migration factor using the human infected population, u shows the infected individual rate to death due to coronavirus, b is the detection rate based infected robots individual, d provides the stopping robot functioning rate, $\theta$ is the time, while $i_1$ and $i_2$ are the initial conditions (ICs).

The purpose of this study is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet neural network (MWNN) under the optimization of the hybrid global swarming computational particle swarm optimization (PSO) and local active set procedure (ASP). The AI based stochastic solvers have been used in many applications, some of them are singular models [25,26], periodic differential models [27], food chain models [[28], [29], [30]] and economic/environmental models [31]. These presented stochastic based applications motivated the authors to explore the AI based MWNN-PSOIPS for solving the nonlinear MRS to achieve the stable and reliable numerical performances. Few of the novel features related to MWNN-PSOIPS are presented as:

• •The solutions of the nonlinear MRS to examine the positive coronavirus cases using the AI based stochastic computing PSOIPS are presented.
• •The stochastic AI based MWNN-PSOIPS is efficiently implemented to present the solutions of the nonlinear MRS.
• •The exactness of the AI based MWNN-PSOIPS is obtained through the comparison of the proposed and reference results.
• •The corroboration of AI based MWNN-PSOIPS is obtained using the statistical Theil inequality coefficient (TIC) and mean square error (MSE) to validate the dependability of the nonlinear MRS to examine the positive coronavirus cases.
• •The absolute error (AE) is performed in good measures, which authenticates the accuracy of the MWNN-PSOIPS.

The remaining structure of the paper is presented as: Sect 2 presents the structure of the MWNN-PSOIPS. Sect 3 provides the comprehensive performances of the solutions. Sect 5 provides the conclusions along with upcoming reports.

2. Methodology

The current section presents the AI based MWNN-PSOIPS formulation to solve the MRS. Fig. 1 shows the graphical performances of the AI based MWNN-PSOIPS.

Fig. 1.

Workflow illustration of MWNN-PSOASP for the MRS.

2.1. MWNN-PSOIPS design

The AI based MWNN-PSOIPS structure is presented to solve the MRS, which is shown in Eq. (1). The proposed solutions are represented as $\hat{I}$ and $\hat{R}$ with the derivatives of 1st kind is shown as:

$$\left[\hat{I}\left(\theta \right),\hat{R}\left(\theta \right)\right]=\left[\sum _{q=1}^s{y}_{I,q}N\left({\Psi }_{I,q}\theta +z_{I,q}\right),\sum _{q=1}^s{y}_{R,q}N\left({\Psi }_{R,q}\theta +z_{R,q}\right)\right]\text{,}$$ (2)

$$\left[{\hat{I}}^{\prime }\left(\theta \right),{\hat{R}}^{\prime }\left(\theta \right)\right]=\left[\sum _{q=1}^s{y}_{I,q}{N}^{\prime }\left({\Psi }_{I,q}\theta +z_{I,q}\right),\sum _{q=1}^s{y}_{R,q}{N}^{\prime }\left({\Psi }_{R,q}\theta +z_{R,q}\right)\right]\text{,}$$

where s presents the neurons and the unknown vectors $\left[\mathit{y},\mathbf{\Psi },\mathit{z}\right]$ are given as:

$\mathit{W}=\left[{\mathit{W}}_I,{\mathit{W}}_R\right]$, for ${\mathit{W}}_R=\left[{\mathit{y}}_R,{\mathbf{\Psi }}_R,{\mathit{z}}_R\right]$ and ${\mathit{W}}_I=\left[{\mathit{y}}_I,{\mathbf{\Psi }}_I,{\mathit{z}}_I\right]$, where

$$\begin{array}{l}{\mathit{y}}_I=\left[y_{I,1},y_{I,2},...,y_{I,s}\right],{\mathit{y}}_R=\left[y_{R,1},y_{R,2},...,y_{R,s}\right],{\mathbf{\Psi }}_I=\left[{\Psi }_{I,1},{\Psi }_{I,2},...,{\Psi }_{I,s}\right],\\ {\mathbf{\Psi }}_R=\left[{\Psi }_{R,1},{\Psi }_{R,2},...,{\Psi }_{R,s}\right],{\mathit{z}}_I=\left[z_{I,1},z_{I,2},...,z_{I,s}\right],{\mathit{z}}_R=\left[z_{R,1},z_{R,2},...,z_{R,s}\right]\text{.}\end{array}$$

The AI based MWNN have been applied first time to solve the MRS. The Morlet function is mathematically depicted as:

$$N\left(\theta \right)=\cos \left(1.75\theta \right)\exp \left(-\frac{1}{2}{\theta }^2\right)\text{,}$$ (3)

Eq. (2) is updated as:

$$\begin{array}{l}\left[\hat{I}\left(\theta \right),\hat{R}\left(\theta \right)\right]=\left[\begin{array}{l}\sum _{q=1}^s{y}_{I,q}\cos \left(1.75\left({\Psi }_{I,q}\theta +z_{I,q}\right)\right)e^{-0.5{\left({\Psi }_{I,q}\theta +z_{I,q}\right)}^2},\\ \sum _{q=1}^s{y}_{R,q}\cos \left(1.75\left({\Psi }_{R,q}\theta +z_{R,q}\right)\right)e^{-0.5{\left({\Psi }_{R,q}\theta +z_{IRq}\right)}^2}\end{array}\right]\text{,}\\ \left[{\hat{I}}^{\prime }\left(\theta \right),{\hat{R}}^{\prime }\left(\theta \right)\right]=\left[\sum _{q=1}^{s}2y_{I,q}{\Psi }_{I,q}\left(\frac{e^{\left({\Psi }_{I,q}\theta +z_{I,q}\right)}}{1+{\left(e^{\left({\Psi }_{I,q}\theta +z_{I,q}\right)}\right)}^2}\right),\sum _{q=1}^{s}2y_{R,q}{\Psi }_{R,q}\left(\frac{e^{\left({\Psi }_{R,q}\theta +z_{R,q}\right)}}{1+{\left(e^{\left({\Psi }_{R,q}\theta +z_{R,q}\right)}\right)}^2}\right)\right]\text{,}\end{array}$$ (4)

To present the solution of the MRS, a merit function is presented as:

$$M_F=M_{F-1}+M_{F-2}+M_{F-3}\text{,}$$ (5)

where, $M_{F-1}$ and $M_{F-2}$ are the merit functions using the infected and Robot population, $I\left(\theta \right)$ and $R\left(\theta \right)$ that are constructed using the different MRS, while the construction of $M_{F-3}$ is based on the ICs of the MRS, shown as:

$$M_{F-1}=\frac{1}{N}\sum _{q=1}^K{\left({\hat{I}}_q^{\prime }-\left(a-b{\hat{R}}_q-u\right){\hat{I}}_q-C\left({\hat{I}}_q\right)\right)}^2\text{,}$$ (6)

$$M_{F-2}=\frac{1}{N}\sum _{q=1}^K{\left({\hat{R}}_q^{\prime }+d{\hat{R}}_q-c\right)}^2\text{,}$$ (7)

$$M_{F-3}=\frac{1}{2}\left({\left({\hat{I}}_0-i_1\right)}^2+{\left({\hat{R}}_0-i_2\right)}^2\right)\text{,}$$ (8)

$$Kh=1,{\hat{I}}_q=\hat{I}\left({\theta }_q\right),{\hat{R}}_q=\hat{R}\left({\theta }_q\right),{\theta }_q=qh\text{.}$$

2.2. Optimization: PSOASP

The current section shows the procedure of optimization using the PSOASP to solve the MRS. Fig. 1 shows the workflow illustrations based on the MWNN-PSOASP for the MRS.

The method for computationally optimizing global search swarming approach called PSO is used to substitute the genetic algorithms. The PSO technique was first established during the last decade by Kennedy and Eberhart [32]. PSO displays the outcomes of multiple intricate systems that manage a specific population utilizing the technique of optimum training. PSO can be completed easily due to the minimal storage capacity [33]. In recent decades, PSO is used in various submissions, e.g., engineering systems [34], multi-objective multimodal methods [35], solar energy models [36], cataloguing the photovoltaic based single/double/triple parameter diode [37], plant diseases [38], image organization [39], particle filter noise reduction based on the analysis of mechanical accountability [30] and green coal production networks [40]. These submissions stimulated the authors to perform the swarming approaches for the MRS.

A local search optimization technique known as active set programming is used to improve the performance of unconstrained and convex models. Recently, ASP is applied in various applications, some of them are realizable safety critical control [41], linearly non-lipschitz constrained nonconvex optimization [42], lung tumor detection and classification [43], atrial fibrillation detection based on transfer learning [44], characterizations of discrete-time descriptor systems [45], pressure-dependent models of water distribution systems [46] and embedded model predictive control [47]. The AI based MWNN-PSOASP is implemented to present the numerical solutions of the MRS. Fig. 1 shows the workflow illustration of MWNN-PSOASP for the MRS.

The pseudocode of the current study based on the optimization procedure to solve the MRS is presented below as:

Pseudocode using the MWNN-PSOASP for the MRS.

2.3. Statistical performances

The current section shows the mathematical performances based on the TIC and MSE for solving the MRS, which is shown as:

$${[\text{TIC}}_I{,\text{TIC}}_R]=\left(\frac{\sqrt{\frac{1}{n}\sum _{q=1}^n{\left(I_q-{\hat{I}}_q\right)}^2}}{\left(\sqrt{\frac{1}{n}\sum _{q=1}^n{I}_q^2}+\sqrt{\frac{1}{n}\sum _{q=1}^n{\hat{I}}_q^2}\right)},\frac{\sqrt{\frac{1}{n}{\sum _{q=1}^n\left(R_q-{\hat{R}}_q\right)}^2}}{\left(\sqrt{\frac{1}{n}\sum _{q=1}^n{R}_q^2}+\sqrt{\frac{1}{n}\sum _{q=1}^n{\hat{R}}_q^2}\right)}\right)\text{,}$$ (9)

$${[\text{MSE}}_I,{\text{MSE}}_R]=\left[\sum _{q=1}^n{\left(I_q-{\hat{I}}_q\right)}^2,\sum _{q=1}^n{\left(R_q-{\hat{R}}_q\right)}^2\right]\text{.}$$ (10)

3. Simulation and results

This section indicated the numerical simulations based on the AI based MWNN-PSOASP for the MRS. The comparison performances based on the reference and obtained results are presented to authenticate the consistency of the procedure. Consider $d=0.2$, $c=0.1$, $C\left(I\right)=C=2$, $b=0.1$, $a=0.5$, $u=0.3$, $i_1=0.6$ and $i_2=0.5$ are presented as:

$$\{\begin{array}{l}{I}^{\prime }\left(\theta \right)=\left(0.5-0.1R\left(\theta \right)-0.3\right)I\left(\theta \right)+2,I_0=0.5\text{,}\\ R^{\prime }\left(\theta \right)=0.1-0.2R\left(\theta \right),R_0=0.6\text{.}\end{array}$$

A $M_F$ takes the form as:

$$M_F=\frac{1}{N}\sum _{q=1}^K\left({\left({\hat{I}}_q^{\prime }-0.5{\hat{I}}_q+0.1{\hat{R}}_q{\hat{I}}_q+0.3{\hat{I}}_q+2\right)}^2+{\left({\hat{R}}_q^{\prime }-0.1+0.2{\hat{R}}_q\right)}^2\right)+\frac{1}{2}\left({\left({\hat{I}}_0-0.5\right)}^2+{\left({\hat{R}}_0-0.6\right)}^2\right)$$ (12)

The optimization performances using the AI based MWNN-PSOASP for the MRS are provided for twenty independent trials to perform the parameter of the model. The values of the numerical outputs have been achieved in the [0, 1] with the step size 0.05. The proposed values of the $I\left(\theta \right)$ and $R\left(\theta \right)$ are shown as:

$$\hat{I}\left(\theta \right)=3.41\cos \left(\frac{4}{3}\left(0.61\theta -2.48\right)\right)e^{-\frac{1}{2}{\left(0.61\theta -2.48\right)}^2}+0.56\cos \left(\frac{4}{3}\left(1.58\theta +0.97\right)\right)e^{-\frac{1}{2}{\left(1.58\theta +0.97\right)}^2}-0.57\cos \left(\frac{4}{3}\left(1.42\theta +2.34\right)\right)e^{-\frac{1}{2}{\left(1.42\theta +2.34\right)}^2}+1.16\cos \left(\frac{4}{3}\left(0.61\theta +0.68\right)\right)e^{-\frac{1}{2}{\left(0.61\theta +0.68\right)}^2}+2.24\cos \left(\frac{4}{3}\left(1.12\theta +2.75\right)\right)e^{-\frac{1}{2}{\left(1.12\theta +2.75\right)}^2}-0.31\cos \left(\frac{4}{3}\left(0.98\theta +1.89\right)\right)e^{-\frac{1}{2}{\left(0.98\theta +1.89\right)}^2}-1.29\cos \left(\frac{4}{3}\left(1.48\theta +1.34\right)\right)e^{-\frac{1}{2}{\left(1.48\theta +1.34\right)}^2}-2.83\cos \left(\frac{4}{3}\left(-0.1\theta +0.25\right)\right)e^{-\frac{1}{2}{\left(-0.1\theta +0.25\right)}^2}-2.27\cos \left(\frac{4}{3}\left(-0.4\theta +0.71\right)\right)e^{-\frac{1}{2}{\left(-0.4\theta +0.71\right)}^2}+3.61\cos \left(\frac{4}{3}\left(2.50\theta +4.80\right)\right)e^{-\frac{1}{2}{\left(2.50\theta +4.80\right)}^2}\text{,}$$ (13)

$$\hat{R}\left(\theta \right)=-0.1\cos \left(\frac{4}{3}\left(0.18\theta +0.55\right)\right)e^{-\frac{1}{2}{\left(0.18\theta +0.55\right)}^2}+0.26\cos \left(\frac{4}{3}\left(0.18\theta +1.24\right)\right)e^{-\frac{1}{2}{\left(0.18\theta +1.24\right)}^2}+0.14\cos \left(\frac{4}{3}\left(-0.08\theta -0.5\right)\right)e^{-\frac{1}{2}{\left(-0.08\theta -0.5\right)}^2}+0.61\cos \left(\frac{4}{3}\left(0.20\theta +2.01\right)\right)e^{-\frac{1}{2}{\left(0.20\theta +2.01\right)}^2}+0.05\cos \left(\frac{4}{3}\left(-0.2\theta +0.09\right)\right)e^{-\frac{1}{2}{\left(-0.2\theta +0.09\right)}^2}+0.03\cos \left(\frac{4}{3}\left(-0.7\theta -1.70\right)\right)e^{-\frac{1}{2}{\left(-0.7\theta -1.70\right)}^2}-0.008\cos \left(\frac{4}{3}\left(0.3\theta +0.030\right)\right)e^{-\frac{1}{2}{\left(0.3\theta +0.03\right)}^2}+0.34\cos \left(\frac{4}{3}\left(-0.1\theta -0.78\right)\right)e^{-\frac{1}{2}{\left(-0.1\theta -0.78\right)}^2}+0.2\cos \left(\frac{4}{3}\left(0.02\theta -0.098\right)\right)e^{-\frac{1}{2}{\left(0.02\theta -0.09\right)}^2}-0.15\cos \left(\frac{4}{3}\left(-0.1\theta +0.06\right)\right)e^{-\frac{1}{2}{\left(-0.1\theta +0.06\right)}^2}\text{,}$$ (14)

Fig. 3, Fig. 4 indicates the statistical performances of the $I\left(\theta \right)$ and $R\left(\theta \right)$ for the MRS. The Mean, Maximum (Max), standard deviation (STD), Median (MD), Minimum (Min), and Semi-interquartile range (SIR) for the MRS are presented in Table 1 and Table 2 . The mathematical performance of the SIR is the difference of ½ times of quartile 3rd and 1st. The Mean, STD, MD and SIR performances have been performed for both $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS are 10⁻⁰⁶ to 10⁻⁰⁷. The Max values shows the poor results but found as 10⁻⁰⁵ to 10⁻⁰⁶ for both $I\left(\theta \right)$ and $R\left(\theta \right)$. The Min performances shows the good results, which are found for both classes are 10⁻⁰⁷ to 10⁻¹¹. These calculated operator small measures represent the dependability of the AI based on the MWNN-PSOASP for the MRS.

Fig. 3.

Performances of the statistical operators for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS.

Fig. 4.

TIC operator convergence measures performances for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS.

Table 1.

Statistical operator performances for.$I\left(\theta \right)$

$\theta$	$I\left(\alpha \right)$
	Mean	Max	STD	MD	Min	SIR
0	4.54674E-07	4.90402E-06	1.07541E-06	1.75826E-07	7.10649E-11	2.39063E-07
0.05	8.50202E-07	6.91587E-06	1.50625E-06	4.09634E-07	5.03369E-09	2.34732E-07
0.1	1.43577E-06	6.75494E-06	1.59467E-06	9.45636E-07	9.66264E-08	6.37282E-07
0.15	2.50968E-06	1.03194E-05	2.36168E-06	1.92842E-06	1.04942E-07	1.25302E-06
0.2	3.21751E-06	1.37793E-05	3.14077E-06	2.85585E-06	1.52113E-07	2.02217E-06
0.25	3.31684E-06	1.42339E-05	3.39703E-06	2.66063E-06	1.81681E-08	2.22107E-06
0.3	2.85017E-06	1.21896E-05	3.13704E-06	2.27382E-06	5.16728E-08	1.86375E-06
0.35	2.15496E-06	9.03015E-06	2.47878E-06	1.52121E-06	1.43166E-08	9.39160E-07
0.4	1.31191E-06	7.96198E-06	1.90939E-06	6.86554E-07	1.83868E-08	3.92892E-07
0.45	1.03345E-06	5.92670E-06	1.41285E-06	4.92309E-07	4.13522E-08	5.37661E-07
0.5	1.24158E-06	4.48179E-06	1.12813E-06	8.96362E-07	4.58559E-08	6.35915E-07
0.55	1.19577E-06	6.83674E-06	1.41559E-06	8.39599E-07	2.50308E-07	4.55329E-07
0.6	1.17749E-06	9.10337E-06	1.95084E-06	7.05512E-07	8.88368E-09	5.74924E-07
0.65	1.91972E-06	1.08727E-05	2.35191E-06	1.36164E-06	5.64084E-08	7.46043E-07
0.7	2.77014E-06	1.17677E-05	2.78416E-06	2.18574E-06	1.41055E-07	1.38304E-06
0.75	3.40234E-06	1.15413E-05	3.05703E-06	2.72830E-06	3.46908E-08	2.05816E-06
0.8	3.48297E-06	1.08901E-05	3.06579E-06	2.72537E-06	3.80029E-07	2.16012E-06
0.85	3.02570E-06	1.06894E-05	2.65767E-06	2.49113E-06	3.94092E-07	1.68071E-06
0.9	2.14565E-06	9.15875E-06	2.30348E-06	1.42499E-06	8.90565E-08	8.20232E-07
0.95	1.61955E-06	7.63815E-06	2.31111E-06	6.39825E-07	1.01675E-07	4.31817E-07
1	1.60816E-06	7.03658E-06	2.24844E-06	6.89440E-07	3.11178E-08	6.28636E-07

Table 2.

Statistical operator performances for.$R\left(\theta \right)$

$\theta$	$R\left(\theta \right)$
	Mean	Max	STD	MD	Min	SIR
0	6.99460E-07	4.08335E-06	1.18329E-06	1.28407E-07	4.52558E-11	4.59012E-07
0.05	1.02726E-06	4.77161E-06	1.19940E-06	6.79014E-07	6.14489E-08	4.07815E-07
0.1	1.10895E-06	4.73802E-06	1.41447E-06	4.51034E-07	2.76715E-08	7.40714E-07
0.15	1.43775E-06	5.26152E-06	1.58219E-06	5.56007E-07	1.35583E-07	1.31568E-06
0.2	1.69645E-06	5.42398E-06	1.62995E-06	1.06686E-06	5.07068E-08	1.32827E-06
0.25	1.77104E-06	6.39889E-06	1.64773E-06	1.51742E-06	3.35034E-08	8.96608E-07
0.3	1.65634E-06	6.34479E-06	1.67096E-06	1.48981E-06	1.95450E-08	7.47756E-07
0.35	1.58544E-06	5.54289E-06	1.41214E-06	9.79522E-07	2.67251E-07	8.58269E-07
0.4	1.37958E-06	4.31668E-06	1.08515E-06	9.64483E-07	2.94522E-07	7.07770E-07
0.45	1.09051E-06	2.93435E-06	8.50722E-07	9.42918E-07	6.43934E-08	5.79809E-07
0.5	9.11132E-07	2.70075E-06	8.70778E-07	5.39011E-07	4.37776E-08	7.04615E-07
0.55	1.01544E-06	3.27748E-06	1.07882E-06	5.01748E-07	5.62508E-09	8.23989E-07
0.6	1.37673E-06	4.22142E-06	1.17568E-06	1.05988E-06	1.34298E-09	7.02332E-07
0.65	1.56921E-06	4.66951E-06	1.30563E-06	1.27681E-06	7.53043E-08	8.68021E-07
0.7	1.51554E-06	5.42980E-06	1.40851E-06	1.23761E-06	9.53281E-08	8.49305E-07
0.75	1.31311E-06	5.90948E-06	1.37913E-06	8.22917E-07	3.31786E-08	6.87327E-07
0.8	1.09372E-06	6.05496E-06	1.39529E-06	4.67729E-07	1.63426E-08	6.48860E-07
0.85	1.32931E-06	5.86506E-06	1.46699E-06	8.17396E-07	6.71198E-09	9.54506E-07
0.9	1.68167E-06	5.93889E-06	1.69677E-06	9.63803E-07	3.03388E-07	9.65987E-07
0.95	1.74649E-06	6.86096E-06	1.77784E-06	1.03184E-06	1.67538E-07	9.49239E-07
1	1.28010E-06	4.67303E-06	1.33233E-06	9.98536E-07	5.68242E-08	6.78837E-07

Fig. 2 presents the optimal weights and result comparison for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS. The optimal weight vectors are illustrated plotted in Fig. 2(a and b) for the MRS, while the results are performed in Fig. 2(c and d) of the MRS. These overlapping of the mean, best and worst solutions is performed to check the correctness of the AI based MWNNs-PSOASP. Fig. 2(e) provides the AE performances for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS. For the $I\left(\theta \right)$ and $R\left(\theta \right)$ categories of the MRS, the AE measures are performed around 10⁻⁰⁶ to 10⁻⁰⁸ and 10⁻⁰⁷ to 10⁻⁰⁸. These calculated optimal performances based on the AE represent the exactness of the AI based MWNN-PSOASP.

Fig. 2.

Optimal weights, comparison performances and AE for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS.

Fig. 3 presents the statistical computing measures for the $I\left(\theta \right)$ and $R\left(\theta \right)$ classes of the MRS. The statistical TIC and MSE operators have been used to present the numerical solutions of the MRS. One can observe that the TIC values for $I\left(\theta \right)$ and $R\left(\theta \right)$ classes of the MRS are measured as 10⁻¹⁰-10⁻¹¹ and 10⁻¹¹-10⁻¹². The MSE operator values for both the classes $I\left(\theta \right)$ and $R\left(\theta \right)$ lie as 10⁻¹³-10⁻¹⁴. These performances indicate the correctness of the AI based MWNN-PSOASP for the MRS.

To check the consistency of the AI based MWNN-PSOASP for the MRS, the statistical TIC and MSE interpretations have been illustrated in Fig. 4 -6. Twenty trials have been implemented in input domain [0,1] by taking 10 number of neurons. The TIC convergence values have been plotted as 10⁻⁰⁹-10⁻¹⁰ and 10⁻¹⁰-10⁻¹¹ for $I\left(\theta \right)$ and $R\left(\theta \right)$. Similarly, MSE convergence performances have been derived as 10⁻¹¹-10⁻¹³ and 10⁻¹²-10⁻¹³ for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS. These optimal performances using the AI based MWNN-PSOASP authenticate that the proposed scheme performs well to solve the MRS (see Fig. 5).

Fig. 5.

MSE operator convergence measures performances for $I\left(\theta \right)$ and $R\left(\theta \right)$ of the MRS.

4. Concluding remarks

The purpose of these investigations is to present the numerical solutions of the mathematical robot system to examine the positive coronavirus cases. The design of the artificial intelligence-based Morlet wavelet neural network has been presented first time to solve the mathematical robot system. The mathematical robot system has been categorized into two dynamics, infected $I\left(\theta \right)$ and Robots $R\left(\theta \right)$. Few concluding remarks of this study are presented as:

• •The design of the AI based MWNN along with the optimization efficiency of PSOASM is presented first time to solve the mathematical robot system.
• •The proposed AI based MWNN-PSOAPS is effectively applied to solve the mathematical robot system.
• •For the exactness of the AI based MWNN-PSOIPS, the comparison of the results has been presented by using the proposed and reference solutions.
• •The reliability of the MWNN-PSOASP has been authenticated by extending the data into 20 trials to check the performance of the scheme through the statistical operators with 10 hidden numbers of neurons to solve the MRS.

Future research directions

The proposed AI based MWNN-PSOASP can be implemented to solve various nonlinear, fractional and fluid dynamic systems [[48], [49], [50], [51], [52], [53], [54]].

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.