Global dynamics of an impulsive vector-borne disease model with time delays

Rong Ming; Xiao Yu; Rong Ming; Xiao Yu

doi:10.3934/mbe.2023926

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 12: 20939-20958. doi: 10.3934/mbe.2023926

Previous Article Next Article

Research article

Global dynamics of an impulsive vector-borne disease model with time delays

Rong Ming ,
Xiao Yu ^,

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Academic Editor: Yuan Lou

Received: 14 October 2023 Revised: 12 November 2023 Accepted: 14 November 2023 Published: 21 November 2023

In this paper, we investigate a time-delayed vector-borne disease model with impulsive culling of the vector. The basic reproduction number $\mathcal{R}_0$ of our model is first introduced by the theory recently established in ^[1]. Then the threshold dynamics in terms of $\mathcal{R}_0$ are further developed. In particular, we show that if $\mathcal{R}_0 < 1$ , then the disease will go extinct; if $\mathcal{R}_0 > 1$ , then the disease will persist. The main mathematical approach is based on the uniform persistent theory for discrete-time semiflows on some appropriate Banach space. Finally, we carry out simulations to illustrate the analytic results and test the parametric sensitivity on $\mathcal{R}_0$ .

Keywords:

Citation: Rong Ming, Xiao Yu. Global dynamics of an impulsive vector-borne disease model with time delays[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20939-20958. doi: 10.3934/mbe.2023926

Related Papers:

[1]	Yuhua Ma, Ye Tao, Yuandan Gong, Wenhua Cui, Bo Wang . Driver identification and fatigue detection algorithm based on deep learning. Mathematical Biosciences and Engineering, 2023, 20(5): 8162-8189. doi: 10.3934/mbe.2023355
[2]	Zulqurnain Sabir, Hafiz Abdul Wahab, Juan L.G. Guirao . A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models. Mathematical Biosciences and Engineering, 2022, 19(1): 663-687. doi: 10.3934/mbe.2022030
[3]	Qingwei Wang, Xiaolong Zhang, Xiaofeng Li . Facial feature point recognition method for human motion image using GNN. Mathematical Biosciences and Engineering, 2022, 19(4): 3803-3819. doi: 10.3934/mbe.2022175
[4]	Rami AL-HAJJ, Mohamad M. Fouad, Mustafa Zeki . Evolutionary optimization framework to train multilayer perceptrons for engineering applications. Mathematical Biosciences and Engineering, 2024, 21(2): 2970-2990. doi: 10.3934/mbe.2024132
[5]	Biyun Hong, Yang Zhang . Research on the influence of attention and emotion of tea drinkers based on artificial neural network. Mathematical Biosciences and Engineering, 2021, 18(4): 3423-3434. doi: 10.3934/mbe.2021171
[6]	Ravi Agarwal, Snezhana Hristova, Donal O’Regan, Radoslava Terzieva . Stability properties of neural networks with non-instantaneous impulses. Mathematical Biosciences and Engineering, 2019, 16(3): 1210-1227. doi: 10.3934/mbe.2019058
[7]	Jia-Gang Qiu, Yi Li, Hao-Qi Liu, Shuang Lin, Lei Pang, Gang Sun, Ying-Zhe Song . Research on motion recognition based on multi-dimensional sensing data and deep learning algorithms. Mathematical Biosciences and Engineering, 2023, 20(8): 14578-14595. doi: 10.3934/mbe.2023652
[8]	Chun Li, Ying Chen, Zhijin Zhao . Frequency hopping signal detection based on optimized generalized S transform and ResNet. Mathematical Biosciences and Engineering, 2023, 20(7): 12843-12863. doi: 10.3934/mbe.2023573
[9]	Yongquan Zhou, Yanbiao Niu, Qifang Luo, Ming Jiang . Teaching learning-based whale optimization algorithm for multi-layer perceptron neural network training. Mathematical Biosciences and Engineering, 2020, 17(5): 5987-6025. doi: 10.3934/mbe.2020319
[10]	Jiliang Lv, Chenxi Qu, Shaofeng Du, Xinyu Zhao, Peng Yin, Ning Zhao, Shengguan Qu . Research on obstacle avoidance algorithm for unmanned ground vehicle based on multi-sensor information fusion. Mathematical Biosciences and Engineering, 2021, 18(2): 1022-1039. doi: 10.3934/mbe.2021055

Abstract

1. Introduction

According to the reports of the World Health Organization (WHO), there are many deaths occur using the tobacco epidemic and its prey most of the disabled persons during some past few years. The tobacco epidemic not only disturbs the individuals, but also become a reason for increases the health care cost, delays financial development and decreases the families' budgets ^[1]. The chain smoking is a big reason of death due to oral cavity cancer, bladder, larynx, esophagus, lung, stomach, pancreas, renal pelvis and cervix. The smoking also creates the problems of heart, chronic obstructive, lung weakness, breathing diseases, peripheral vascular and less weight of newly born children. WHO also reports that the reasoning of unproductive pregnancies, peptic ulcer disease and increase the infant mortality rate is due to smoking ^[2]. The termination of smoking is an instantaneous health support and dramatically reduces the danger of many deathly diseases and improve the respiratory system of the younger. It is the obligation of the higher authorities to teach their people and aware the communities about the drawbacks of smoking as well as develop an active policy to control this habit. Castillo et al. ^[3] discussed the mathematical model to avoid the smoking by considering the population into two kings, i.e., smokers (S) and those individuals who left smoking permanently (QP). In addition, Shoromi et al. ^[4] presented a new group temporary smoker (QT) in this mathematical model, which is defined as:

$\left\{\begin{array}{l}{P}^{\text{'}}\left(\mathit{\Omega } \right) = \mu \left(1-P\left(\mathit{\Omega } \right)\right)-\beta P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right), P\left(0\right) = {l}_{1}, \\ {S}^{\text{'}}\left(\mathit{\Omega } \right) = \beta P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)-(\gamma +\mu )S\left(\mathit{\Omega } \right)+\alpha {Q}_{T}\left(\mathit{\Omega } \right), S\left(0\right) = {l}_{2}, \\ {Q}_{T}^{\text{'}}\left(\mathit{\Omega } \right) = \gamma (1-\sigma )S\left(\mathit{\Omega } \right)-(\mu +\alpha ){Q}_{T}\left(\mathit{\Omega } \right), {Q}_{T}\left(0\right) = {l}_{3}, \\ {Q}_{P}^{\text{'}}\left(\mathit{\Omega } \right) = -\mu {Q}_{P}\left(\mathit{\Omega } \right)+\gamma \sigma S\left(\mathit{\Omega } \right), {Q}_{P}\left(0\right) = {l}_{4}, \end{array}\right.$

(1)

where $P\left(\mathit{\Omega } \right)$ , $S\left(\mathit{\Omega } \right)$ , ${Q}_{T}\left(\mathit{\Omega } \right)$ and ${Q}_{P}\left(\mathit{\Omega } \right)$ indicate the Potential smoker (P) group, Smoker (S) group, Temporary smoker (QT) group and Permanent smoker (QS) group at time $\mathit{\Omega }$ . Whereas, $\sigma , \alpha , \gamma , \beta$ and $\mu$ represent the positive values of the constants. Furthermore, ${l}_{1}$ , ${l}_{2}$ , ${l}_{3}$ and ${l}_{4}$ designate the initial conditions (ICs) of the nonlinear smoke model (1).

The aim of this work is to investigate the nonlinear smoke model numerically to exploit a stochastic framework called Gudermannian neural works (GNNs) ^[5,6,7,8] along with the optimization procedures of global/local search terminologies based Genetic algorithm (GA) and interior-point approach (IPA), i.e., GNNs-GA-IPA. The development of the numerical solvers has been reported in various proposals for the solution of the linear/nonlinear differential models with their own applicability, stability and significance ^{[9,10,11,12,13]}, however recently artificial intelligence based numerical computing platform are introduction as a promising alternatives ^{[14,15,16,17,18]}. Whereas, GNNs-GA-IPA is never been applied before to solve the nonlinear smoke system. Some recent proposals of the stochastic solvers are functional singular system ^[19], nonlinear SIR dengue fever model ^[20], mathematical models of environmental economic systems ^[21], prey-predator nonlinear system ^[22], Thomas-Fermi system ^[23], mosquito dispersal model ^[24], transmission of heat in human head ^[25], multi-singular fractional system ^[26] and nonlinear COVID-19 model ^[27]. Some novel prominent features of the current investigations are provided as:

● The GNNs are explored efficaciously using the hybrid optimization paradigm based on GA-IPA for solving the nonlinear smoke model.

● The consistent overlapped outcomes obtained by GNNs-GA-IPA and the Runge-Kutta numerical results validate the correctness and exactness of the proposed scheme.

● The authorization of the performance is accomplished through different statistical valuations to attain the numerical outcomes of the nonlinear smoke system.

The benefits, merits and noteworthy contributions of the GNNs-GA-IPA are simply implemented to solve the nonlinear smoke model, understanding easiness, operated efficiently and inclusive with reliable applications in diversified fields.

The remaining parts of the current work are studied as: Section 2 defines the procedures of GNNs-GA-IPA along with and statistical measures. Section 3 indicates the results simulations. Section 4 provided the final remarks and future research reports.

2. Designed methodology

In this section, the proposed form of the GNNs-GA-IPA is presented in two steps to solve the nonlinear smoke model as:

● A merit function is designed using the differential system and ICs of the nonlinear smoke system.

● The necessary and essential settings are provided for the optimization procedures of GA-IPA to solve the nonlinear smoke model.

2.1. Structure of GNN-GA-IPA

In this section, the mathematical formulations to solve the nonlinear smoke model-based groups, Potential smoker ( $\widehat{P}$ ), Temporary smoker ( ${\widehat{Q}}_{T}$ ), Permanent smoker ( ${\widehat{Q}}_{S}$ ) and Smoker ( $\widehat{S}$ ) are presented. The proposed results of these groups of the nonlinear smoke model, $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ are represented by $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ together with their derivatives are written as:

$\begin{array}{*{20}{c}} \left[\widehat{P}\right(\mathit{\Omega } ), \widehat{S}(\mathit{\Omega } ), {\widehat{Q}}_{T}(\mathit{\Omega } ), {\widehat{Q}}_{S}(\mathit{\Omega } \left)\right] = \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}{\rm T}({w}_{P, i}\mathit{\Omega } +{h}_{P, i}), {\sum }_{i = 1}^{m}{a}_{S, i}{\rm T}({w}_{S, i}\mathit{\Omega } +{h}_{S, i}), \\ {\sum }_{i = 1}^{m}{a}_{{Q}_{T}, i}{\rm T}({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i}), {\sum }_{i = 1}^{m}{a}_{{Q}_{S}, i}{\rm T}({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})\end{array}\right], \\ \left[{\widehat{P}}^{\text{'}}\right(\mathit{\Omega } ), {\widehat{S}}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{T}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{S}^{\text{'}}(\mathit{\Omega } \left)\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}{{\rm T}}^{\text{'}}({w}_{P, i}\mathit{\Omega } +{h}_{P, i}), {\sum }_{i = 1}^{m}{k}_{S, i}{{\rm T}}^{\text{'}}({w}_{S, i}\mathit{\Omega } +{h}_{S, i}), \\ {\sum }_{i = 1}^{m}{k}_{{Q}_{T}, i}{{\rm T}}^{\text{'}}({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i}), {\sum }_{i = 1}^{m}{k}_{{Q}_{S}, i}{{\rm T}}^{\text{'}}({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})\end{array}\right]. \end{array}$

(2)

where W is the unknown weight vector, written as:

$W = [{W}_{P}, {W}_{S}, {W}_{{Q}_{T}}, {W}_{{Q}_{S}}] , \;{\rm{for}} \; {W}_{P} = [{k}_{P}, {\omega }_{P}, {h}_{P}] , {W}_{S} = [{k}_{S}, {\omega }_{S}, {h}_{S}] , {W}_{{Q}_{T}} = [{k}_{{Q}_{T}}, {\omega }_{{Q}_{T}}, {h}_{{Q}_{T}}]$

and $\;{W}_{{Q}_{S}} = [{k}_{{Q}_{S}}, {\omega }_{{Q}_{S}}, {h}_{{Q}_{S}}]$ , where

$\begin{array}{l} {k}_{P} = [{k}_{P, 1}, {k}_{P, 2}, ..., {k}_{P, m}], {k}_{S} = [{k}_{S, 1}, {k}_{S, 2}, ..., {k}_{S, m}], {k}_{{Q}_{T}} = [{k}_{{Q}_{T}, 1}, {k}_{{Q}_{T}, 2}, ..., {k}_{{Q}_{T}, m}], {k}_{{Q}_{S}} = \\ [{k}_{{Q}_{S}, 1}, {k}_{{Q}_{T}, 2}, ..., {k}_{{Q}_{T}, m}], {w}_{P} = [{w}_{P, 1}, {w}_{P, 2}, ..., {w}_{P, m}], {w}_{S} = [{w}_{S, 1}, {w}_{S, 2}, ..., {w}_{S, m}] {w}_{{Q}_{T}} = \\ [{w}_{{Q}_{T}, 1}, {w}_{{Q}_{T}, 2}, ..., {w}_{{Q}_{T}, m}], {w}_{{Q}_{S}} = [{w}_{{Q}_{S}, 1}, {w}_{{Q}_{T}, 2}, ..., {w}_{{Q}_{T}, m}], {h}_{P} = [{h}_{P, 1}, {h}_{P, 2}, ..., {h}_{P, m}] {h}_{S} =\\ [{h}_{S, 1}, {h}_{S, 2}, ..., {h}_{S, m}], {h}_{{Q}_{T}} = [{h}_{{Q}_{T}, 1}, {h}_{{Q}_{T}, 2}, ..., {h}_{{Q}_{T}, m}], {h}_{{Q}_{S}} = [{h}_{{Q}_{S}, 1}, {h}_{{Q}_{T}, 2}, ..., {h}_{{Q}_{T}, m}] \end{array}$

The Gudermannian function $T\left(\mathit{\Omega } \right) = 2{\mathit{tan}}^{-1}\left[\mathit{exp}(\mathit{\Omega } )\right]-0.5\pi$ is applied in the above model, this GNNs have never been implemented before the solve this model.

$\begin{array}{*{20}{c}} \left[\begin{array}{cc}\widehat{P}\left(\mathit{\Omega } \right), & \widehat{S}\left(\mathit{\Omega } \right), \\ {\widehat{Q}}_{T}\left(\mathit{\Omega } \right), & {\widehat{Q}}_{S}\left(\mathit{\Omega } \right)\end{array}\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{P, i}\mathit{\Omega } +{h}_{P, i})}-\frac{\pi }{2}\right), {\sum }_{i = 1}^{m}{k}_{S, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{S, i}\mathit{\Omega } +{h}_{S, i})}-\frac{\pi }{2}\right), \\ {\sum }_{i = 1}^{m}{k}_{{Q}_{T}, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i})}-\frac{\pi }{2}\right), {\sum }_{i = 1}^{m}{k}_{{Q}_{P}, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})}-\frac{\pi }{2}\right)\end{array}\right], \\ \left[\begin{array}{cc}{\widehat{P}}^{\text{'}}\left(\mathit{\Omega } \right), & {\widehat{S}}^{\text{'}}\left(\mathit{\Omega } \right), \\ {\widehat{Q}}_{T}^{\text{'}}\left(\mathit{\Omega } \right), & {\widehat{Q}}_{S}^{\text{'}}\left(\mathit{\Omega } \right)\end{array}\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}2{k}_{P, i}{w}_{P, i}\left(\frac{{e}^{({w}_{p, i}\mathit{\Omega } +{n}_{p, i})}}{1+{\left({e}^{({w}_{p, i}\mathit{\Omega } +{n}_{p, i})}\right)}^{2}}\right), {\sum }_{i = 1}^{m}2{k}_{S, i}{w}_{S, i}\left(\frac{{e}^{({w}_{S, i}\mathit{\Omega } +{n}_{S, i})}}{1+{\left({e}^{({w}_{S, i}\mathit{\Omega } +{n}_{S, i})}\right)}^{2}}\right), \\ {\sum }_{i = 1}^{m}2{k}_{{Q}_{T}, i}{w}_{{Q}_{T}, i}\left(\frac{{e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{n}_{{Q}_{T}, i})}}{1+{\left({e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{n}_{{Q}_{T}, i})}\right)}^{2}}\right), {\sum }_{i = 1}^{m}2{k}_{{Q}_{S}, i}{w}_{{Q}_{S}, i}\left(\frac{{e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{n}_{{Q}_{S}, i})}}{1+{\left({e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{n}_{{Q}_{S}, i})}\right)}^{2}}\right)\end{array}\right], \end{array}$

(3)

For the process of optimization, a fitness function is given as:

${\mathit{\Xi } }_{Fit} = {\mathit{\Xi } }_{1}+{\mathit{\Xi } }_{2}+{\mathit{\Xi } }_{3}+{\mathit{\Xi } }_{4}+{\mathit{\Xi } }_{5},$

(4)

${\mathit{\Xi } }_{2} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\widehat{S}}_{i}^{\text{'}}-\beta {\widehat{P}}_{i}{\widehat{S}}_{i}+\mu {\widehat{S}}_{i}+\gamma {\widehat{S}}_{i}-\alpha {\left({\widehat{Q}}_{T}\right)}_{i}\right]}^{2},$

(5)

${\mathit{\Xi } }_{1} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\widehat{P}}_{i}^{\text{'}}-\mu +\mu {\widehat{P}}_{i}+\beta {\widehat{P}}_{i}{\widehat{S}}_{i}\right]}^{2},$

(6)

${\mathit{\Xi } }_{3} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\left({\widehat{Q}}_{T}^{\text{'}}\right)}_{i}+\gamma \sigma {\widehat{S}}_{i}-\gamma {\widehat{S}}_{k}+\alpha {\left({\widehat{Q}}_{T}\right)}_{i}+\mu {\left({\widehat{Q}}_{T}\right)}_{i}\right]}^{2},$

(7)

${\mathit{\Xi } }_{4} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\left({\widehat{Q}}_{P}^{\text{'}}\right)}_{i}-\gamma \sigma {\widehat{S}}_{i}+\mu {\left({\widehat{Q}}_{P}\right)}_{i}\right]}^{2},$

(8)

${\mathit{\Xi } }_{5} = \frac{1}{4}\left[{\left({\widehat{P}}_{0}-{l}_{1}\right)}^{2}+{\left({\widehat{S}}_{0}-{l}_{2}\right)}^{2}+{\left(({\widehat{Q}}_{T}{)}_{0}-{l}_{3}\right)}^{2}++{\left(({\widehat{Q}}_{S}{)}_{0}-{l}_{4}\right)}^{2}\right],$

(9)

where ${\widehat{P}}_{i} = P\left({\mathit{\Omega } }_{i}\right), {\widehat{S}}_{i} = S\left({\mathit{\Omega } }_{i}\right), ({Q}_{T}{)}_{i} = {Q}_{T}\left({\mathit{\Omega } }_{i}\right), ({Q}_{S}{)}_{i} = {Q}_{S}\left({\mathit{\Omega } }_{i}\right), {\rm T}N = 1,$ and ${\mathit{\Omega } }_{i} = ih$ . The error functions ${\mathit{\Xi } }_{1},$ ${\mathit{\Xi } }_{2},$ ${\mathit{\Xi } }_{3}$ and ${\mathit{\Xi } }_{4}$ are related to system (1), while, ${\mathit{\Xi } }_{5}$ is based on the ICs of the nonlinear smoke model (1).

2.2. Optimization procedures: GA-IPA

In this section, the performance of the scheme is observed using the optimization process of GA-IPA to solve the nonlinear smoke system. The designed GNNs-GA-IPA methodology based on the nonlinear smoke system is illustrated in Figure 1.

Figure 1. Designed framework of the GNNs-GA-IPA to solve the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

GA is known as a famous global search optimization method applied to solve the solve the constrained/unconstrained models efficiently. It is commonly implemented to regulate the precise population outcomes for solving the various stiff and complex models using the optimal training process. For the best solutions of the model, GA is implemented through the process of selection, reproduction, mutation and crossover procedures. Recently, GA is applied in many famous applications that can be seen in ^{[28,29,30,31,32]} and references cited therein.

IPA is a local search, rapid and quick optimization method, implemented to solve various reputed complex and non-stiff models efficiently. IPA is implemented in various models like phase-field approach to brittle and ductile fracture ^[33], multistage nonlinear nonconvex programs ^[34], SITR model for dynamics of novel coronavirus ^[35], viscoplastic fluid flows ^[36] and security constrained optimal power flow problems ^[37]. To control the Laziness of the global search method GA, the process of hybridization with the IPA is applied for solving the nonlinear smoke model. The detailed pseudocode based on the GNNs-GA-IPA is provided in Table 1.

Table 1. The procedure of optimization using the GNNs-GA-IPA for the nonlinear smoke model.

GA process starts Inputs: The chromosomes of the same network elements are denoted as: W = [ $k$ , $w$ , $h$ ] Population: Chromosomes set is represented as: $W=[{W}_{P}, {W}_{S}, {W}_{{Q}_{T}}, {W}_{{Q}_{S}}]$ , ${W}_{P}=[{k}_{P}, {\omega }_{P}, {h}_{P}]$ , ${W}_{S}=[{k}_{S}, {\omega }_{S}, {h}_{S}]$ , ${W}_{{Q}_{T}}=\left[{k}_{{Q}_{T}}, {\omega }_{{Q}_{T}}, {h}_{{Q}_{T}}\right]$ and ${W}_{{Q}_{S}}=[{k}_{{Q}_{S}}, {\omega }_{{Q}_{S}}, {h}_{{Q}_{S}}]$ , W is the weight vector Outputs: The best global weight values are signified as: W_GA-Best Initialization: For the chromosome's assortment, adjust the W Assessment of FIT: Adjust " ${\mathit{\Xi } }_{Fit}$ " in the population (P) for each vector values using Eqs (4)–(9). ● Stopping standards: Terminate if any of the criteria is achieved [ ${\mathit{\Xi } }_{Fit}$ = 10^-19], [Generataions = 120] [TolCon = TolFun = 10^-18], [StallLimit = 150], & [Size of population = 270]. Move to [storage] Ranking: Rank precise W in the particular population for ${\mathit{\Xi } }_{Fit}$ . Storage: Save ${\mathit{\Xi } }_{Fit}$ , iterations, W_GA-Best, function counts and time. GA procedure End
Start of IPA Inputs: W_GA-Best is selected as an initial point. Output: W_GA-IPA shows the best weights of GA-IPA. Initialize: W_GA-Best, generations, assignments and other standards. Terminating criteria: Stop if [ ${\mathit{\Xi } }_{Fit}$ = 10-20], [TolFun = 10-18], [Iterations = 700], [TolCon = TolX = 10-22] and [MaxFunEvals = 260000] attained. Evaluation of Fit: Compute the values of W and E for Eqs (4)–(9). Amendments: Normalize 'fmincon' for IPA, compute ${\mathit{\Xi } }_{Fit}$ for Eqs (4)–(9). Accumulate: Transmute W_GA-IPA, time, iterations, function counts and ${\mathit{\Xi } }_{Fit}$ for the IPA trials.
IPA process End

| Show Table

DownLoad: CSV

3. Performance measures

The mathematical presentations using the statistical operators with "variance account for (VAF)", "Theil's inequality coefficient (TIC)", "mean absolute deviation (MAD)" and "semi interquartile (SI) range" together with the global operators G.VAF, G-TIC G-MAD are accessible to solve the nonlinear smoke model, written as:

$\left\{\begin{array}{l}[\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\begin{array}{l}\left(1-\frac{\mathit{var}\left({P}_{m}-{\widehat{P}}_{m}\right)}{\mathit{var}\left({P}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({S}_{m}-{\widehat{S}}_{m}\right)}{\mathit{var}\left({S}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{T}\right)}_{m}}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{S}\right)}_{m}}\right)\times 100\end{array}\right], \\ [\text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\left|\begin{array}{l}100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \\ 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}\end{array}\right|\right].\end{array}\right.$

(10)

$\begin{array}{*{20}{c}} {\left[\text{T.I.}{\text{C}}_{P}, \text{T.I.}{\text{C}}_{S}, \text{T.I.}{\text{C}}_{{Q}_{T}}, \text{T.I.}{\text{C}}_{{Q}_{S}}\right] = }\\ {\left[\begin{array}{l}\frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({P}_{m}-{\widehat{P}}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{P}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\widehat{P}}_{m}^{2}}\right)}, \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({S}_{m}-{\widehat{S}}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{S}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\widehat{S}}_{m}^{2}}\right)}, \\ \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({Q}_{T}\right)}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({\widehat{Q}}_{T}\right)}_{m}^{2}}\right)}, \\ \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({Q}_{S}\right)}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({\widehat{Q}}_{S}\right)}_{m}^{2}}\right)}\end{array}\right],} \end{array}$

(11)

$\begin{array}{l} \;\;\;\;\;[\text{M.A.}{\text{D}}_{P}, \text{M.A.}{\text{D}}_{S}\text{, M.A.}{\text{D}}_{{Q}_{T}}\text{, M.A.}{\text{D}}_{{Q}_{S}}] = \\\left[\begin{array}{l}{\sum }_{m = 1}^{n}\left|{P}_{m}-{\widehat{P}}_{m}\right|, {\sum }_{m = 1}^{n}\left|{S}_{m}-{\widehat{S}}_{m}\right|, \\ {\sum }_{m = 1}^{n}\left|{\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right|, {\sum }_{m = 1}^{n}\left|{\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right|\end{array}\right] \end{array}$

(12)

$\left\{\begin{array}{l}\text{S}\text{.}\text{I}\text{ }\text{R}\text{a}\text{n}\text{g}\text{e} = -0.5\times \left({Q}_{1}-{Q}_{3}\right), \\ {Q}_{1}\text{ = }{\mathrm{ }1}^{st}\text{q}\text{u}\text{a}\text{r}\text{t}\text{i}\text{l}\text{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q}_{3} = {3}^{rd}\text{q}\text{u}\text{a}\text{r}\text{t}\text{i}\text{l}\text{e}\text{.}\end{array}\right.$

(13)

$\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ are the approximate form of the solutions.

4. Results and simulations

The current investigations are associated to solve the nonlinear smoke model. The relative performance of the obtained solutions with the Runge-Kutta results is tested to show the exactness of the GNNs-GA-IPA. Moreover, the statistical operator's performances are used to validate the accuracy, reliability and precision of the proposed GNNs-GA-IPA. The updated form of the nonlinear smoke model given in the system (1) along with its ICs using the appropriate parameter values is shown as:

$\left\{\begin{array}{l}{P}^{\text{'}}\left(\mathit{\Omega } \right) = 20-\left(20P\left(\mathit{\Omega } \right)+0.003P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)\right), P\left(0\right) = 0.3, \\ {S}^{\text{'}}\left(\mathit{\Omega } \right) = 0.003P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)-20.3S\left(\mathit{\Omega } \right)+3{Q}_{T}\left(\mathit{\Omega } \right), S\left(0\right) = 0.5, \\ {Q}_{T}^{\text{'}}\left(\mathit{\Omega } \right) = 0.15S\left(\mathit{\Omega } \right)-23{Q}_{T}\left(\mathit{\Omega } \right), {Q}_{T}\left(0\right) = 0.1, \\ {Q}_{P}^{\text{'}}\left(\mathit{\Omega } \right) = 0.15S\left(\mathit{\Omega } \right)-20{Q}_{P}\left(\mathit{\Omega } \right), {Q}_{P}\left(0\right) = 0.2, \end{array}\right.$

(14)

A fitness function for the nonlinear smoke model (14) is written as:

$\begin{array}{*{20}{c}} {{\mathit{\Xi } }_{Fit} = }\\ {\frac{1}{N}{\sum }_{m = 1}^{N}\left(\begin{array}{l}{\left[{\widehat{P}}_{m}^{\text{'}}+20{\widehat{P}}_{m}-20+0.003{\widehat{P}}_{m}{\widehat{S}}_{m}\right]}^{2}+{\left[{\widehat{S}}_{m}^{\text{'}}+20.3{\widehat{S}}_{m}-0.003{\widehat{P}}_{m}{\widehat{S}}_{m}-3({Q}_{T}{)}_{m}\right]}^{2}\\ +{\left[({Q}_{T}^{\text{'}}{)}_{m}+23({Q}_{T}{)}_{m}-0.15{\widehat{S}}_{m}\right]}^{2}+{\left[({Q}_{P}^{\text{'}}{)}_{m}+20({Q}_{P}{)}_{m}-0.15{\widehat{S}}_{m}\right]}^{2}\end{array}\right) }\\ {+\frac{1}{4}\left[{\left({\widehat{P}}_{0}-0.3\right)}^{2}+{\left({\widehat{S}}_{0}-0.5\right)}^{2}+{\left(({Q}_{T}{)}_{0}-0.1\right)}^{2}+{\left(({Q}_{P}{)}_{0}-0.2\right)}^{2}\right]. } \end{array}$

(15)

The performance of the scheme is observed based on the nonlinear smoke system using the GNNs-GA-IPA for 20 independent executions using 30 numbers of variables. The proposed form of the solution based on the nonlinear smoke model is provided in the arrangement of best weights using the below equations for each group of the nonlinear smoke model and the graphical illustrations of these weights are plotted in Figure 2.

$\begin{array}{*{20}{c}} {\widehat{P}\left(\mathit{\Omega } \right) = 7.2965(2{\mathit{tan}}^{-1}{e}^{(-5.343\mathit{\Omega } -18.462)}-0.5\pi )-}\\ {2.7636(2{\mathit{tan}}^{-1}{e}^{1.2606\mathit{\Omega } +0.5134)}-0.5\pi )}\\ {+15.294(2{\mathit{tan}}^{-1}{e}^{(1.1551\mathit{\Omega } +2.0601)}-0.5\pi )+}\\ {8.7324(2{\mathit{tan}}^{-1}{e}^{(19.759\mathit{\Omega } +2.9789)}-0.5\pi ) }\\ {-7.1646(2{\mathit{tan}}^{-1}{e}^{(2.5198\mathit{\Omega } +5.9852)}-0.5\pi )-}\\ {0.0053(2{\mathit{tan}}^{-1}{e}^{(-0.246\mathit{\Omega } -1.7692)}-0.5\pi )}\\ {-2.6104(2{\mathit{tan}}^{-1}{e}^{(0.2323\mathit{\Omega } +1.5497)}-0.5\pi )-}\\ {1.4677(2{\mathit{tan}}^{-1}{e}^{(02.782\mathit{\Omega } +2.0890)}-0.5\pi ) }\\ {+2.4715(2{\mathit{tan}}^{-1}{e}^{(-18516\mathit{\Omega } -3.253)}-0.5\pi )-}\\ {2.9189(2{\mathit{tan}}^{-1}{e}^{(10.272\mathit{\Omega } -7.2465)}-0.5\pi ),} \end{array}$

(16)

$\begin{array}{*{20}{c}} {\widehat{S}\left(\mathit{\Omega } \right) = 7.6303(2{\mathit{tan}}^{-1}{e}^{(-0.143\mathit{\Omega } +0.4605)}-0.5\pi )+}\\ {9.6607(2{\mathit{tan}}^{-1}{e}^{(0.264\mathit{\Omega } +0.8726)}-0.5\pi )}\\ {-1.8710(2{\mathit{tan}}^{-1}{e}^{(0.3252\mathit{\Omega } -1.0967)}-0.5\pi )+}\\ {0.0451(2{\mathit{tan}}^{-1}{e}^{(-1.961\mathit{\Omega } +8.4961)}-0.5\pi )}\\ {+5.4711(2{\mathit{tan}}^{-1}{e}^{(-9.8369\mathit{\Omega } -1.533)}-0.5\pi )-}\\ {1.7275(2{\mathit{tan}}^{-1}{e}^{(-9.2138\mathit{\Omega } -2.037)}-0.5\pi ) }\\ {+0.2908(2{\mathit{tan}}^{-1}{e}^{(-2.4236\mathit{\Omega } -0.378)}-0.5\pi )-}\\ {1.9171(2{\mathit{tan}}^{-1}{e}^{(-9.2376\mathit{\Omega } -0.360)}-0.5\pi )}\\ {+2.7144(2{\mathit{tan}}^{-1}{e}^{(2.8428\mathit{\Omega } -7.8525)}-0.5\pi )-}\\ {5.3000(2{\mathit{tan}}^{-1}{e}^{(-4.6148\mathit{\Omega } +4.164)}-0.5\pi ),} \end{array}$

(17)

$\begin{array}{*{20}{c}} {{\widehat{Q}}_{T}\left(\mathit{\Omega } \right) = -0.563(2{\mathit{tan}}^{-1}{e}^{(-3.245\mathit{\Omega } +2.3616)}-0.5\pi )-}\\ {0.022(2{\mathit{tan}}^{-1}{e}^{(-8.9859+1.2373)}-0.5\pi )}\\ {-0.5360(2{\mathit{tan}}^{-1}{e}^{(3.2982\mathit{\Omega } -2.4088)}-0.5\pi )-}\\ {2.1382(2{\mathit{tan}}^{-1}{e}^{(-3.87\mathit{\Omega } -3.100)}-0.5\pi )}\\ {-4.7812(2{\mathit{tan}}^{-1}{e}^{(-4.1182\mathit{\Omega } -4.994)}-0.5\pi )+}\\ {0.9505(2{\mathit{tan}}^{-1}{e}^{(-0.038\mathit{\Omega } -0.241)}-0.5\pi ) }\\ {+17.841(2{\mathit{tan}}^{-1}{e}^{(-14.7536\mathit{\Omega } -4.557)}-0.5\pi )-}\\ {3.4575(2{\mathit{tan}}^{-1}{e}^{(4.347\mathit{\Omega } +0.7343)}-0.5\pi ) }\\ {+11.8361(2{\mathit{tan}}^{-1}{e}^{(4.5451\mathit{\Omega } +2.013)}-0.5\pi )+}\\ {2.717(2{\mathit{tan}}^{-1}{e}^{(3.6697\mathit{\Omega } +7.33831)}-0.5\pi ), } \end{array}$

(18)

$\begin{array}{*{20}{c}} {{\widehat{Q}}_{S}\left(\mathit{\Omega } \right) = 0.0762(2{\mathit{tan}}^{-1}{e}^{(2.9990\mathit{\Omega } +2.8605)}-0.5\pi )+}\\ {2.8390(2{\mathit{tan}}^{-1}{e}^{(8.3177\mathit{\Omega } +0.6804)}-0.5\pi ) }\\ {+7.6702(2{\mathit{tan}}^{-1}{e}^{(-0.0481\mathit{\Omega } +1.9373)}-0.5\pi )-}\\ {0.4142(2{\mathit{tan}}^{-1}{e}^{(-0.309\mathit{\Omega } -0.401)}-0.5\pi )}\\ {-6.2827(2{\mathit{tan}}^{-1}{e}^{(1.2291\mathit{\Omega } +3.5918)}-0.5\pi )+}\\ {0.5077(2{\mathit{tan}}^{-1}{e}^{(1.0574\mathit{\Omega } +1.0166)}-0.5\pi )}\\ {+2.2061(2{\mathit{tan}}^{-1}{e}^{(1.5133\mathit{\Omega } +10.7487)}-0.5\pi )-}\\ {0.2776(2{\mathit{tan}}^{-1}{e}^{(1.7337\mathit{\Omega } +5.1928)}-0.5\pi )}\\ {-5.3623(2{\mathit{tan}}^{-1}{e}^{(8.4564\mathit{\Omega } +1.3100)}-0.5\pi )+}\\ {0.0684(2{\mathit{tan}}^{-1}{e}^{(-4.1583\mathit{\Omega } -1.6646)}-0.5\pi ), } \end{array}$

(19)

Figure 2. Best weight vectors along with the result comparisons of the mean and best results with reference results to solve the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

A fitness function shown in the model (15) is optimized along with the hybridization of GA-IPA for the nonlinear smoke system. The proposed form of the outcomes is found using the above systems (16)–(19) for 30 variables between 0 to 1 input along with step size 0.1. The solutions of the nonlinear smoke model along with the best weight vector values are illustrated in Figures 2(a–d). The comparison of the mean and best outcomes with the reference Runge-Kutta solutions is provided in Figures 2(e–h) to solve the nonlinear smoke system. It is noticed that the mean and best results obtained by the GNNs-GA-IPA are overlapped with the reference results to solve each group of the nonlinear smoke model, which authenticate the exactness of the designed GNNs-GA-IPA. illustrates the values of the absolute error (AE) for each group of the nonlinear smoke model. It is observed that the values of the best AE for the group of potential smokers, smoker; temporary smoker and permanent smoker lie around 10-05-10-07, 10-05-10-06, 10-04-10-07 and 10-04-10-06, respectively. While, the mean AE values for these groups of the nonlinear smoke model found around 10-03-10-04, 10-03-10-05, 10-02-10-04 and 10-03-10-04, respectively. signifies the performance measures based on the operators EVAF, MAD and TIC to solve each group of the nonlinear smoke model. It is specified in the plots that the best values of the EVAF, MAD and TIC performances of each group of the nonlinear smoke model lie around 10-04-10-08, 10-03-10-05 and 10-08-10-09, respectively. The best performances of the EVAF, MAD and TIC for the $\widehat{P}\left(\mathit{\Omega } \right)$ and $\widehat{S}\left(\mathit{\Omega } \right)$ groups lie around 10-08-10-09, 10-05-10-06 and 10-09-10-10, respectively. The best performances of the EVAF, MAD and TIC for the ${\widehat{Q}}_{T}\left(\mathit{\Omega } \right)$ group of the nonlinear smoke model found around 10-05-10-06, 10-04-10-06 and 10-08-10-10 and the best performances of the EVAF, MAD and TIC for the ${\widehat{Q}}_{P}\left(\mathit{\Omega } \right)$ group of the nonlinear smoke model found around 10-07-10-08, 10-05-10-06 and 10-09-10-10. One can accomplish from the indications that the designed GNNs-GA-IPA is precise and accurate.

Figure 3. AE values for each group of the nonlinear smoke system.

DownLoad: Full-Size Img PowerPoint

Figure 4. Performances based on EVAF, MAD and TIC values to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

The graphic illustrations based on the statistical performances are provided in Figures 5– to find the convergence along with the boxplots and the histograms to solve the nonlinear smoke model. shows the performance of TIC for twenty runs to solve each group of the nonlinear smoke model. It is observed that most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-07-10-10. The MAD performances are illustrated in that depicts most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-03-10-05. The EVAF performances are illustrated in that depicts most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-04-10-08. The best trial performances using the GNNs-GA-IPA are calculated suitable for the TIC, EVAF and MAD operators.

Figure 5. Convergence of the TIC values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

Figure 6. Convergence of the MAD values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

Figure 7. Convergence of the EVAF values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

The routines for different statistical operators, Maximum (Max), Median, Minimum (Min), standard deviation (STD) and SIR are provided in – to validate the accurateness and precision to solve the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups of the nonlinear smoke system. The Max operators indicate the worst solutions, whereas the Min operators show the best results using 20 independent runs. For the group $P\left(\mathit{\Omega } \right)$ , $S\left(\mathit{\Omega } \right)$ , ${Q}_{T}\left(\mathit{\Omega } \right)$ and ${Q}_{S}\left(\mathit{\Omega } \right)$ of the nonlinear smoke model, the 'Max' and 'Min' standards lie around 10-03-10-04 and 10-06-10-08, while the Median, SIR, STD and Mean standards lie around 10-04-10-05. These small values designate the worth and values of the GNNs-GA-IPA to solve each group of the nonlinear smoke model. One can observe through these calculate measures, that the designed GNNs-GA-IPA is precise, accurate and stable.

Table 2. Statistical measures for the nonlinear smoke model of group

$P\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	P( $\mathit{\Omega }$ )
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	1.9698E-03	1.5128E-07	1.3179E-05	4.3988E-04	2.8322E-05	1.5255E-04
0.1	2.2927E-03	4.0452E-05	3.7091E-04	7.5870E-04	3.5757E-04	7.1580E-04
0.2	2.8517E-04	2.5715E-06	8.0076E-05	1.0291E-04	8.7298E-05	1.0480E-04
0.3	3.7540E-04	5.0221E-06	4.6631E-05	9.6615E-05	5.7759E-05	8.0291E-05
0.4	3.1095E-04	7.2215E-07	2.5169E-05	7.9915E-05	4.8322E-05	6.4497E-05
0.5	3.7039E-04	1.0187E-07	4.0958E-05	9.9187E-05	4.0049E-05	7.4737E-05
0.6	3.4788E-04	2.7497E-06	3.6783E-05	8.6814E-05	5.3154E-05	7.3405E-05
0.7	7.2524E-04	3.8591E-06	2.6741E-05	1.6887E-04	5.3778E-05	9.7130E-05
0.8	3.1439E-04	2.3193E-06	3.0315E-05	8.9765E-05	3.8282E-05	6.8701E-05
0.9	8.2385E-04	3.6579E-06	3.6581E-05	1.8035E-04	2.2583E-05	8.9806E-05
1	3.9210E-04	8.9543E-07	1.4825E-05	1.1497E-04	3.4368E-05	6.7763E-05

| Show Table

DownLoad: CSV

Table 3. Statistical measures for the nonlinear smoke model of group

$S\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	$S\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	2.2510E-03	8.8331E-07	1.5049E-05	4.3988E-04	6.5800E-05	2.4625E-04
0.1	3.5749E-03	1.2621E-05	4.2711E-04	7.5870E-04	5.9138E-04	9.2693E-04
0.2	5.8967E-04	6.8508E-06	5.0534E-05	1.0291E-04	3.8159E-05	1.0389E-04
0.3	3.9412E-04	4.5897E-07	5.5860E-05	9.6615E-05	4.8082E-05	9.7091E-05
0.4	1.8148E-04	1.4261E-06	3.2627E-05	7.9915E-05	4.6977E-05	6.1366E-05
0.5	3.8406E-04	3.0499E-06	5.0425E-05	9.9187E-05	4.5676E-05	9.6411E-05
0.6	3.3693E-04	1.0786E-06	3.2525E-05	8.6814E-05	2.6474E-05	5.9807E-05
0.7	3.8976E-04	5.9658E-08	3.4605E-05	1.6887E-04	3.0731E-05	7.1943E-05
0.8	4.6451E-04	1.8083E-07	2.9100E-05	8.9765E-05	1.9639E-05	5.1752E-05
0.9	3.7056E-04	5.8454E-07	3.1691E-05	1.8035E-04	4.5808E-05	7.8891E-05
1	3.4425E-04	3.5833E-08	3.7245E-05	1.1497E-04	2.5924E-05	6.3636E-05

| Show Table

DownLoad: CSV

Table 4. Statistical measures for the nonlinear smoke model of group

${Q}_{T}\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	${Q}_{T}\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	6.7379E-02	2.0435E-06	7.4367E-05	4.3988E-04	1.2547E-03	6.6320E-03
0.1	8.8212E-03	1.5933E-04	6.6896E-04	7.5870E-04	6.4736E-04	1.4858E-03
0.2	1.2208E-03	1.0177E-06	1.6681E-04	1.0291E-04	1.3929E-04	2.7755E-04
0.3	1.2039E-03	6.2163E-06	3.5966E-05	9.6615E-05	7.5659E-05	2.0075E-04
0.4	5.2048E-04	1.5004E-06	9.3199E-05	7.9915E-05	1.4812E-04	1.7703E-04
0.5	7.9163E-04	4.1214E-06	7.5760E-05	9.9187E-05	9.4190E-05	1.7072E-04
0.6	7.5113E-04	3.4495E-06	5.6836E-05	8.6814E-05	1.5865E-04	1.8051E-04
0.7	6.9564E-04	2.1601E-06	8.7570E-05	1.6887E-04	1.2891E-04	1.6565E-04
0.8	8.5659E-04	2.3356E-06	5.6424E-05	8.9765E-05	6.5358E-05	1.2238E-04
0.9	8.5832E-04	1.8869E-06	8.6636E-05	1.8035E-04	1.2103E-04	1.8340E-04
1	5.9922E-04	9.8962E-07	3.9188E-05	1.1497E-04	4.8965E-05	1.1550E-04

| Show Table

DownLoad: CSV

Table 5. Statistical measures for the nonlinear smoke model of group

${Q}_{S}\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	${Q}_{S}\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	1.4249E-02	4.6038E-07	1.0193E-05	3.1752E-03	5.3007E-05	7.6325E-04
0.1	7.1456E-03	9.4140E-06	2.0718E-04	1.6018E-03	3.9882E-04	8.1152E-04
0.2	1.4449E-03	2.1615E-07	2.8536E-05	3.6835E-04	3.0447E-05	1.7480E-04
0.3	3.7091E-04	3.2754E-07	2.1791E-05	8.7324E-05	2.6186E-05	5.6609E-05
0.4	1.5545E-03	1.8846E-07	2.2389E-05	3.7612E-04	6.1143E-05	1.5810E-04
0.5	3.7972E-04	3.6651E-06	1.7656E-05	9.8654E-05	3.1268E-05	6.2521E-05
0.6	1.2224E-03	2.4032E-07	2.1050E-05	2.9478E-04	3.3053E-05	1.2732E-04
0.7	7.4101E-04	1.7539E-07	2.4276E-05	1.6625E-04	2.5398E-05	7.7130E-05
0.8	1.0726E-03	2.5213E-06	1.4100E-05	2.3771E-04	3.0541E-05	9.1329E-05
0.9	7.4407E-04	2.5770E-06	3.5282E-05	1.6258E-04	2.8865E-05	7.9482E-05
1	9.7616E-04	7.0693E-07	8.9328E-06	2.1858E-04	1.0034E-05	8.0351E-05

| Show Table

DownLoad: CSV

The global performances of the G.EVAF, G.MAD and G.TIC operators for twenty runs to solve the designed GNNs-GA-IPA are provided in Table 6 to solve each group of the nonlinear smoke model. These Min global G.MAD, G.TIC and G.EVAF performances found around 10-04-10-05, 10-08-10-09 and 10-05-10-07, whereas the SIR global values lie in the ranges of 10-04-10-05, 10-08-10-09 and 10-04-10-07 for all groups of the nonlinear smoke model. These close optimal global measures values demonstrate the correctness, accurateness and precision of the proposed GNNs-GA-IPA.

Table 6. Global measures based on the MAD, TIC and EVAF values to solve each group of the nonlinear smoke model.

Class	(G.MAD)		(G.TIC)		(G.EVAF)
Class	Min	SIR	Min	SIR	Min	SIR
$P(\mathit{\Omega } )$	1.2499E-04	9.6058E-05	7.2597E-09	7.4459E-09	2.6155E-07	7.5331E-07
$S(\mathit{\Omega } )$	9.7570E-05	1.3353E-04	8.3068E-08	1.0766E-08	1.3879E-06	5.2293E-06
${Q}_{T}\left(\mathit{\Omega } \right)$	1.4111E-04	2.5334E-04	9.1566E-09	1.8187E-08	3.9553E-05	5.1487E-04
${Q}_{S}\left(\mathit{\Omega } \right)$	6.0979E-05	4.1189E-05	3.6367E-09	4.8515E-09	1.1015E-06	9.3296E-06

| Show Table

DownLoad: CSV

5. Reference style, citation, and cross-reference

The current investigations are related to solve the nonlinear smoke model by exploiting the Gudermannian neural networks using the global and local search methodologies, i.e., GNNs-GA-IPA. The smoke model is a system of nonlinear equations contain four groups temporary smokers, potential smokers, permanent smokers and smokers. For the numerical outcomes, a fitness function is established using all groups of the nonlinear smoke model and its corresponding ICs. The optimization of the fitness function using the hybrid computing framework of GNNs-GA-IPA for solving each group of the nonlinear smoke model. The Gudermannian function is designed as a merit function along with 30 numbers of variables. The overlapping of the proposed mean and best outcomes is performed with the Runge-Kutta reference results for each group of the nonlinear smoke model. These matching and reliable results to solve the nonlinear smoke model indicate the exactness of the designed GNNs-GA-IPA. In order to show the precision and accuracy of the proposed GNNs-GA-IPA, the statistical performances based on the TIC, MAD and EVAF operators have been accessible for twenty trials using 10 numbers of neurons. To check the performance analysis, most of the runs based on the statistical TIC, MAD and EVAF performances show a higher level of accuracy to solve each group of the nonlinear smoke model. The valuations using the statistical gages of Max, Min, Mean, STD, Med and SIR further validate the value of the proposed GNNs-GA-IPA. Furthermore, global presentations through SIR and Min have been applied for the nonlinear smoke model.

In future, the designed GNNs-GA-IPA is accomplished to solve the biological nonlinear systems ^[38], singular higher order model ^[39], fluid dynamics nonlinear models ^[40] and fractional differential model ^[41].

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-12.

Conflict of interest

All authors of the manuscript declare that there have no potential conflicts of interest.

References

[1]	Z. Bai, X. Q. Zhao, Basic reproduction ratios for periodic and time-delayed compartmental models with impulses, J. Math. Biol., 80 (2020), 1095–1117. https://doi.org/10.1007/s00285-019-01452-2 doi: 10.1007/s00285-019-01452-2
[2]	S. Ruan, D. Xiao, J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098–1114. https://doi.org/10.1007/s11538-007-9292-z doi: 10.1007/s11538-007-9292-z
[3]	D. Nash, F. Mostashari, A. Fine, J. Miller, D. O'Leary, K. Murray, et al., The outbreak of West Nile virus infection in the New York City area in 1999, N. Engl. J. Med., 344 (2001), 1807–1814. https://doi.org/10.1056/NEJM200106143442401 doi: 10.1056/NEJM200106143442401
[4]	S. Tang, L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44 (2002), 185–199. https://doi.org/10.1007/s002850100121 doi: 10.1007/s002850100121
[5]	Z. Bai, Y. Lou, X. Q. Zhao, A delayed succession model with diffusion for the impact of diapause on population growth, SIAM J. Appl. Math., 80 (2020), 1493–1519. https://doi.org/10.1137/19M1236448 doi: 10.1137/19M1236448
[6]	S. Gao, L. Chen, Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol., 69 (2007), 731–745. https://doi.org/10.1007/s11538-006-9149-x doi: 10.1007/s11538-006-9149-x
[7]	X. Meng, L. Chen, H. Cheng, Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. Math. Comput., 186 (2007), 516–529. https://doi.org/10.1016/j.amc.2006.07.124 doi: 10.1016/j.amc.2006.07.124
[8]	Y. P. Yang, Y. Xiao, Threshold dynamics for compartmental epidemic models with impulses, Nonlinear Anal. Real World Appl., 13 (2012), 224–234. https://doi.org/10.1016/j.nonrwa.2011.07.028 doi: 10.1016/j.nonrwa.2011.07.028
[9]	X. Xu, Y. Xiao, R. A. Cheke, Models of impulsive culling of vector to interrupt transmission of West Nile virus to host, Appl. Math. Model., 39 (2015), 3549–3568. https://doi.org/10.1016/j.apm.2014.10.072 doi: 10.1016/j.apm.2014.10.072
[10]	Z. Yang, C. Huang, X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, J. Math. Biol., 76 (2018), 1387–1419. https://doi.org/10.1007/s00285-017-1172-z doi: 10.1007/s00285-017-1172-z
[11]	S. A. Gourley, R. Liu, J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Math. Biol., 54 (2007), 309–335. https://doi.org/10.1007/s00285-006-0050-x doi: 10.1007/s00285-006-0050-x
[12]	A. J. Terry, Impulsive culling of a structured population on two patches. J. Math. Biol., 61 (2010), 843–875. https://doi.org/10.1007/s00285-009-0325-0 doi: 10.1007/s00285-009-0325-0
[13]	S. P. Rajasekar, M. Pitchaimani, Q. Zhu, Probing a stochastic epidemic hepatitis C virus model with a chronically infected treated population, Acta Math. Sci., 42 (2022), 2087–2112. https://doi.org/10.1007/s10473-022-0521-1 doi: 10.1007/s10473-022-0521-1
[14]	Y. Zhao, L. Wang, Practical exponential stability of impulsive stochastic food chain system with time-varying delays, Mathematics, 11 (2023), 147. https://doi.org/10.3390/math11010147 doi: 10.3390/math11010147
[15]	Y. Han, Z. Bai, Threshold dynamics of a West Nile virus model with impulsive culling and incubation period., Discrete Contin. Dyn. Syst. B, 27 (2022), 4515–4529. https://doi.org/10.3934/dcdsb.2021239 doi: 10.3934/dcdsb.2021239
[16]	Y. Lou, X. Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573–603. https://doi.org/10.1007/s00332-016-9344-3 doi: 10.1007/s00332-016-9344-3
[17]	X. Wang, X. Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77 (2017), 181–201. https://doi.org/10.1137/15M1046277 doi: 10.1137/15M1046277
[18]	F. B. Wang, R. Wu, X. Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498–1535. https://doi.org/10.1137/18M123616 doi: 10.1137/18M123616
[19]	F. B. Wang, R. Wu, X. Yu, Threshold dynamics of a bat-borne rabies model with periodic incubation periods, Nonlinear Anal. Real World Appl., 61 (2021), 103340. https://doi.org/10.1016/j.nonrwa.2021.103340 doi: 10.1016/j.nonrwa.2021.103340
[20]	X. Liu, G. Ballinger, Continuous dependence on initial values for impulsive delay differential equations, Appl. Math. Lett., 17 (2004), 483–490. https://doi.org/10.1016/S0893-9659(04)90094-8 doi: 10.1016/S0893-9659(04)90094-8
[21]	X. Q. Zhao, Dynamical Systems in Population Biology, 2nd edition, CMS Books in Mathematics, Springer, 2017. https://doi.org/10.1007/978-3-319-56433-3
[22]	G. Ballinger, X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71–93. https://doi.org/10.1080/00036810008840804 doi: 10.1080/00036810008840804
[23]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. https://doi.org/10.1090/surv/041
[24]	D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Harlow (1993), New York. https://doi.org/10.1201/9780203751206
[25]	X. Liang, X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1–40. https://doi.org/10.1002/cpa.20154 doi: 10.1002/cpa.20154
[26]	L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49–57. https://doi.org/10.1007/BF01190081 doi: 10.1007/BF01190081
[27]	T. Kato, Perturbation Theory for Linear Operators, Springer, 1976. https://doi.org/10.1007/978-3-642-66282-9
[28]	X. Liu, X. Shen, Y. Zhang, A comparison principle and stability for large-scale impulsive delay differential systems, ANZIAM J., 47 (2005), 203–235. https://doi.org/10.1017/S1446181100009998 doi: 10.1017/S1446181100009998
[29]	W. Hu, Q. Zhu, H. R. Karimi, Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems, IEEE Trans. Autom. Control, 64 (2019), 5207–5213. https://doi.org/10.1109/TAC.2019.2911182 doi: 10.1109/TAC.2019.2911182
[30]	W. Hu, Q. Zhu, Stability criteria for impulsive stochastic functional differential systems with distributed delay dependent impulsive effects, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 2027–2032. https://doi.org/10.1109/TSMC.2019.2905007 doi: 10.1109/TSMC.2019.2905007
[31]	M. Xia, L. Liu, J. Fang, Y. Zhang, Stability analysis for a class of stochastic differential equations with impulses, Mathematics, 11 (2023), 1541. https://doi.org/10.3390/math11061541 doi: 10.3390/math11061541

This article has been cited by:

1.	Wajaree Weera, Thongchai Botmart, Samina Zuhra, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Salem Ben Said, A Neural Study of the Fractional Heroin Epidemic Model, 2023, 74, 1546-2226, 4453, 10.32604/cmc.2023.033232
2.	Nabeela Anwar, Iftikhar Ahmad, Adiqa Kausar Kiani, Shafaq Naz, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Intelligent solution predictive control strategy for nonlinear hepatitis B epidemic model with delay, 2023, 1745-5030, 1, 10.1080/17455030.2023.2178827
3.	Yue Li, Jianyou Zhao, Zenghua Chen, Gang Xiong, Sheng Liu, A Robot Path Planning Method Based on Improved Genetic Algorithm and Improved Dynamic Window Approach, 2023, 15, 2071-1050, 4656, 10.3390/su15054656
4.	Muneerah AL Nuwairan, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Anwar Aldhafeeri, An advance artificial neural network scheme to examine the waste plastic management in the ocean, 2022, 12, 2158-3226, 045211, 10.1063/5.0085737
5.	Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Nadia Mumtaz, Irwan Fathurrochman, R. Sadat, Mohamed R. Ali, An Investigation Through Stochastic Procedures for Solving the Fractional Order Computer Virus Propagation Mathematical Model with Kill Signals, 2022, 1370-4621, 10.1007/s11063-022-10963-x
6.	Wajaree Weera, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Salem Ben Said, Maria Emilia Camargo, Chantapish Zamart, Thongchai Botmart, An Artificial Approach for the Fractional Order Rape and Its Control Model, 2023, 74, 1546-2226, 3421, 10.32604/cmc.2023.030996
7.	Zulqurnain Sabir, Dumitru Baleanu, Muhammad Asif Zahoor Raja, Evren Hincal, A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model, 2022, 97, 0031-8949, 085002, 10.1088/1402-4896/ac7a6a
8.	Thongchai Botmart, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Wajaree weera, Mohamed R. Ali, R. Sadat, Ayman A. Aly, Ali Saad, A hybrid swarming computing approach to solve the biological nonlinear Leptospirosis system, 2022, 77, 17468094, 103789, 10.1016/j.bspc.2022.103789
9.	J. F. Gómez-Aguilar, Zulqurnain Sabir, Manal Alqhtani, Muhammad Umar, Khaled M. Saad, Neuro-Evolutionary Computing Paradigm for the SIR Model Based on Infection Spread and Treatment, 2022, 1370-4621, 10.1007/s11063-022-11045-8
10.	Watcharaporn Cholamjiak, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Manuel Sánchez-Chero, Dulio Oseda Gago, José Antonio Sánchez-Chero, María-Verónica Seminario-Morales, Marco Antonio Oseda Gago, Cesar Augusto Agurto Cherre, Gilder Cieza Altamirano, Mohamed R. Ali, Artificial intelligent investigations for the dynamics of the bone transformation mathematical model, 2022, 34, 23529148, 101105, 10.1016/j.imu.2022.101105
11.	Zulqurnain Sabir, Salem Ben Said, Qasem Al-Mdallal, Mohamed R. Ali, A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics, 2022, 12, 2045-2322, 10.1038/s41598-022-26566-4
12.	Tianqi Yu, Lei Liu, Yan-Jun Liu, Observer-based adaptive fuzzy output feedback control for functional constraint systems with dead-zone input, 2022, 20, 1551-0018, 2628, 10.3934/mbe.2023123
13.	Prem Junsawang, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Soheil Salahshour, Thongchai Botmart, Wajaree Weera, Novel Computing for the Delay Differential Two-Prey and One-Predator System, 2022, 73, 1546-2226, 249, 10.32604/cmc.2022.028513
14.	Chenyang Hu, Yuelin Gao, Fuping Tian, Suxia Ma, A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem, 2022, 10, 2227-7390, 270, 10.3390/math10020270
15.	Rahman Ullah, Qasem Al Mdallal, Tahir Khan, Roman Ullah, Basem Al Alwan, Faizullah Faiz, Quanxin Zhu, The dynamics of novel corona virus disease via stochastic epidemiological model with vaccination, 2023, 13, 2045-2322, 10.1038/s41598-023-30647-3
16.	Thanasak Mouktonglang, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Saira Bhatti, Thongchai Botmart, Wajaree Weera, Chantapish Zamart, Designing Meyer wavelet neural networks for the three-species food chain model, 2023, 8, 2473-6988, 61, 10.3934/math.2023003
17.	Lihe Hu, Yi Zhang, Yang Wang, Qin Jiang, Gengyu Ge, Wei Wang, A simple information fusion method provides the obstacle with saliency labeling as a landmark in robotic mapping, 2022, 61, 11100168, 12061, 10.1016/j.aej.2022.06.002
18.	Sakda Noinang, Zulqurnain Sabir, Shumaila Javeed, Muhammad Asif Zahoor Raja, Dostdar Ali, Wajaree Weera, Thongchai Botmart, A Novel Stochastic Framework for the MHD Generator in Ocean, 2022, 73, 1546-2226, 3383, 10.32604/cmc.2022.029166
19.	Suthep Suantai, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Watcharaporn Cholamjiak, Numerical Computation of SEIR Model for the Zika Virus Spreading, 2023, 75, 1546-2226, 2155, 10.32604/cmc.2023.034699
20.	Zulqurnain Sabir, Juan L. G. Guirao, A Soft Computing Scaled Conjugate Gradient Procedure for the Fractional Order Majnun and Layla Romantic Story, 2023, 11, 2227-7390, 835, 10.3390/math11040835
21.	Kanit Mukdasai, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Peerapongpat Singkibud, R. Sadat, Mohamed R. Ali, A computational supervised neural network procedure for the fractional SIQ mathematical model, 2023, 1951-6355, 10.1140/epjs/s11734-022-00738-9
22.	Jingzhi Tu, Gang Mei, Francesco Piccialli, An improved Nyström spectral graph clustering using k-core decomposition as a sampling strategy for large networks, 2022, 34, 13191578, 3673, 10.1016/j.jksuci.2022.04.009
23.	Thongchai Botmart, Qusain Hiader, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Wajaree Weera, Stochastic Investigations for the Fractional Vector-Host Diseased Based Saturated Function of Treatment Model, 2023, 74, 1546-2226, 559, 10.32604/cmc.2023.031871
24.	Zulqurnain Sabir, Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion, 2022, 137, 2190-5444, 10.1140/epjp/s13360-022-02869-3
25.	Muhammad Umar, Fazli Amin, Soheil Salahshour, Thongchai Botmart, Wajaree Weera, Prem Junswang, Zulqurnain Sabir, Swarming Computational Approach for the Heartbeat Van Der Pol Nonlinear System, 2022, 72, 1546-2226, 6185, 10.32604/cmc.2022.027970
26.	Thongchai Botmart, Zulqurnain Sabir, Afaf S. Alwabli, Salem Ben Said, Qasem Al-Mdallal, Maria Emilia Camargo, Wajaree Weera, Computational Stochastic Investigations for the Socio-Ecological Dynamics with Reef Ecosystems, 2022, 73, 1546-2226, 5589, 10.32604/cmc.2022.032087
27.	Qusain Haider, Ali Hassan, Sayed M. Eldin, Artificial neural network scheme to solve the hepatitis B virus model, 2023, 9, 2297-4687, 10.3389/fams.2023.1072447
28.	Farhad Muhammad Riaz, Shafiq Ahmad, Juniad Ali Khan, Saud Altaf, Zia Ur Rehman, Shuaib Karim Memon, Numerical Treatment of Non-Linear System for Latently Infected CD4+T Cells: A Swarm- Optimized Neural Network Approach, 2024, 12, 2169-3536, 103119, 10.1109/ACCESS.2024.3429279
29.	Zulqurnain Sabir, Shahid Ahmad Bhat, Muhammad Asif Zahoor Raja, Sharifah E. Alhazmi, A swarming neural network computing approach to solve the Zika virus model, 2023, 126, 09521976, 106924, 10.1016/j.engappai.2023.106924
30.	Chenxu Wang, Mengqin Wang, Xiaoguang Wang, Luyue Zhang, Yi Long, Multi-Relational Graph Representation Learning for Financial Statement Fraud Detection, 2024, 7, 2096-0654, 920, 10.26599/BDMA.2024.9020013
31.	ZULQURNAIN SABIR, DUMITRU BALEANU, MUHAMMAD ASIF ZAHOOR RAJA, ALI S. ALSHOMRANI, EVREN HINCAL, COMPUTATIONAL PERFORMANCES OF MORLET WAVELET NEURAL NETWORK FOR SOLVING A NONLINEAR DYNAMIC BASED ON THE MATHEMATICAL MODEL OF THE AFFECTION OF LAYLA AND MAJNUN, 2023, 31, 0218-348X, 10.1142/S0218348X23400169
32.	Zulqurnain Sabir, Mohamed R. Ali, R. Sadat, Simulation of fractional order mathematical model of robots for detection of coronavirus using Levenberg–Marquardt backpropagation neural network, 2024, 0941-0643, 10.1007/s00521-024-10361-5
33.	MUHAMMAD SHOAIB, RAFIA TABASSUM, KOTTAKKARAN SOOPPY NISAR, MUHAMMAD ASIF ZAHOOR RAJA, FAROOQ AHMED SHAH, MOHAMMED S. ALQAHTANI, C. AHAMED SALEEL, H. M. ALMOHIY, DESIGN OF BIO-INSPIRED HEURISTIC TECHNIQUE INTEGRATED WITH SEQUENTIAL QUADRATIC PROGRAMMING FOR NONLINEAR MODEL OF PINE WILT DISEASE, 2023, 31, 0218-348X, 10.1142/S0218348X23401485
34.	Manal Alqhtani, J.F. Gómez-Aguilar, Khaled M. Saad, Zulqurnain Sabir, Eduardo Pérez-Careta, A scale conjugate neural network learning process for the nonlinear malaria disease model, 2023, 8, 2473-6988, 21106, 10.3934/math.20231075
35.	Soumen Pal, Manojit Bhattacharya, Snehasish Dash, Sang-Soo Lee, Chiranjib Chakraborty, A next-generation dynamic programming language Julia: Its features and applications in biological science, 2024, 64, 20901232, 143, 10.1016/j.jare.2023.11.015
36.	Syed T. R. Rizvi, Aly R. Seadawy, Samia Ahmed, Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice, 2023, 12, 2192-8029, 10.1515/nleng-2022-0337
37.	Juan Luis García Guirao, On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks, 2023, 1, 2956-7068, 177, 10.2478/ijmce-2023-0014
38.	Honghua Liu, Chang She, Zhiliang Huang, Lei Wei, Qian Li, Han Peng, Mailan Liu, Uncertainty analysis and optimization of laser thermal pain treatment, 2023, 13, 2045-2322, 10.1038/s41598-023-38672-y
39.	Zulqurnain Sabir, Salem Ben Said, Qasem Al-Mdallal, An artificial neural network approach for the language learning model, 2023, 13, 2045-2322, 10.1038/s41598-023-50219-9
40.	Nabeela Anwar, Iftikhar Ahmad, Adiqa Kausar Kiani, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Intelligent solution predictive networks for non-linear tumor-immune delayed model, 2024, 27, 1025-5842, 1091, 10.1080/10255842.2023.2227751
41.	Manoj Gupta, Achyuth Sarkar, Stochastic intelligent computing solvers for the SIR dynamical prototype epidemic model using the impacts of the hospital bed, 2024, 1025-5842, 1, 10.1080/10255842.2023.2300684
42.	Zulqurnain Sabir, Atef Hashem, Adnène Arbi, Mohamed Abdelkawy, Designing a Bayesian Regularization Approach to Solve the Fractional Layla and Majnun System, 2023, 11, 2227-7390, 3792, 10.3390/math11173792
43.	Juan Luis García Guirao, On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks, 2023, 0, 2956-7068, 10.2478/ijmce-2023-00014
44.	A. A. Alderremy, J. F. Gómez-Aguilar, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Shaban Aly, Numerical investigations of the fractional order derivative-based accelerating universe in the modified gravity, 2024, 39, 0217-7323, 10.1142/S0217732323501808
45.	A. A. Alderremy, J. F. Gómez-Aguilar, Zulqurnain Sabir, Shaban Aly, J. E. Lavín-Delgado, José R. Razo-Hernández, Numerical performances based artificial neural networks to deal with the computer viruses spread on the complex networks, 2024, 101, 0020-7160, 314, 10.1080/00207160.2024.2326926
46.	Abdulgafor Alfares, Yusuf Abubakar Sha'aban, Ahmed Alhumoud, Machine learning -driven predictions of lattice constants in ABX3 Perovskite Materials, 2025, 141, 09521976, 109747, 10.1016/j.engappai.2024.109747
47.	Kamel Guedri, Rahat Zarin, Mowffaq Oreijah, Samaher Khalaf Alharbi, Hamiden Abd El-Wahed Khalifa, Artificial neural network-driven modeling of Ebola transmission dynamics with delays and disability outcomes, 2025, 115, 14769271, 108350, 10.1016/j.compbiolchem.2025.108350
48.	Farhad Muhammad Riaz, Junaid Ali Khan, A Swarm Optimized ANN-based Numerical Treatment of Nonlinear SEIR System based on Zika Virus, 2025, 2147-9429, 1, 10.2339/politeknik.1543179
49.	Kamel Guedri, Rahat Zarin, Mowffaq Oreijah, Samaher Khalaf Alharbi, Hamiden Abd El-Wahed Khalifa, Modeling syphilis progression and disability risk with neural networks, 2025, 314, 09507051, 113220, 10.1016/j.knosys.2025.113220
50.	Zulqurnain Sabir, Basma Souayeh, Zahraa Zaiour, Alyn Nazal, Mir Waqas Alam, Huda Alfannakh, A machine learning neural network architecture for the accelerating universe based modified gravity, 2025, 29, 11108665, 100635, 10.1016/j.eij.2025.100635
51.	Zhou Yang, Zhi Li, Xikun Jin, Yimin Zhang, Sensitivity analysis of subjective and objective reliability of disc brakes based on the parallel efficient global optimization algorithm, 2025, 0305-215X, 1, 10.1080/0305215X.2025.2475000

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)