In this study, the solution of the nonlinear influenza disease system (NIDS) is presented using the Morlet wavelet neural networks (MWNNs) together with the optimisation procedures of the hybrid process of global/local search approaches. The genetic algorithm (GA) and sequential quadratic programming (SQP), that is, GASQP, are executed as the global and local search techniques. The mathematical form of the NIDS depends upon four groups: susceptible
Keywords
 Nonlinear influenza disease system
 Morlet wavelet neural networks
 sequential quadratic programming
 Runge–Kutta
 genetic algorithms
 numerical measures
Influenza is a serious disease caused by viruses that affect the lungs, the upper breathing organs, throat, bronchi and nose. Its recovery rate is very high, but only with proper medical care. A high infection rate is noticed in older adults or in those with severe kidney, heart and lung problems, diabetes or cancer. The epidemic rate of influenza is estimated to be between 5% and 15% per year of the population, caused by upper respiratory tract disease. Worldwide, annual epidemics are seen with 3–5 million illness cases and the number of deaths is estimated at around 250,000–500,000 [1]. In mathematical form, epidemiological systems are demonstrated by the ordinary autonomous nonlinear differential systems with assumptions of the timeindependent parameters. Such biological models involve the variable state to the infected, recovered, susceptible and transmitted vectors.
A number of schemes have been tested to solve the nonlinear influenza disease system (NIDS). A few of them are as follows. Astuti et al. [2] suggested a differential transformation stepbystep scheme to solve the influenza virus diseaseresistant system. Alzahrani et al. [3] proposed a numerical approach for solving a fractional influenza pandemic system. Erdem et al. [4] discussed the influenza SIQR model using the imperfect quarantine system. Sun et al. [5] stated an optimisation based multiobjective system to allocate patients during an influenza epidemic. GonzálezParra et al. [6] designed a fractional epidemiologic system based on simulating an influenza A epidemic. Ghanbari et al. [7] worked towards examining the two systems of avian influenza epidemics related to the derivatives of fractal–fractional with memorabilia and power from Mittag–Leffler. Tchuenche et al. [8] attempted to enhance the media coverage impacts on the dynamics of human influenza communications. SchulzeHorsel et al. [9] worked on the dynamics of infection as well as virusinduced apoptosis using the influenza vaccine production in cellular philosophy. Hovav et al. [10] discussed the system of network flow to manage the inventory and allocating of influenza vaccines in a healthcare supply chain management. Patel et al. [11] applied genetic algorithms (GAs) for optimal vaccination strategies for the pandemic influenza system. Kanyiri et al. [12] worked on the applications of optimal control for influenza along with the antiviral resistance and pulmonary congestion.
NIDS is divided into four groups: (i) susceptible
where,
The purpose of the current investigation is to treat NIDS numerically by using the Morlet wavelet neural networks (MWNNs) together with the optimisation procedures in the hybrid process of global/local search approaches. The GA and sequential quadratic programming (SQP), that is, GASQP, are executed as global and local search techniques. There are various applications in which stochastic computing approaches have been applied, such as COVID19based SITR dynamics [15, 16], singular fractional models [17, 18], preypredator model [19], delay singular functional model [20, 21], dengue fever model [22], higher order nonlinear singular systems [23–25], nonlinear mosquito release system in heterogeneous atmosphere [26] and multisingular differential systems [27, 28]. Keeping in view these recognised submissions, the authors are motivated to solve NIDS using the MWNNS and GASQP. Some motivational factors of MWNNs using the GASQP are briefly as follows:
The proposed MWNNs using GASQP provides impressive numerical solutions of NIDS.
Steady, reliable and stable numerical outcomes of NIDS authenticate the worth of the proposed form of MWNNs using GASQP.
The values of the absolute error (AE) show the best performances, which demonstrate the consistency of the proposed MWNNs using GASQP.
The numerical performance of the scheme is certified using different statistical annotations to solve the NIDS for multiple independent runs.
The proposed MWNNs using GASQP is smoothly executed to solve NIDS with inclusive, easytounderstand and smooth operations.
The remainder of the paper is categorised as follows: Section 2 depicts the designed MWNNs using the GASQP methodology along with statistical procedures. Section 3 provides the results and simulation. Section 4 deals with the final comments and future research directions.
The structure of the MWNNs–GASQP is presented in this section based on two phases for solving NIDS:
A merit function is proposed based on MWNNs using the GASQP to solve NIDS.
Some major settings are provided to improve the merit function using the methodology of GASQP.
The mathematical design to solve NIDS is divided into four groups: susceptible (
The updated form of NIDS using the Morlet function
A merit function is given as:
where
This section provides a detailed procedure of the designed MWNNs together with GASQP for solving NIDS. The designed MWNNs structure using GASQP for solving NIDS is shown in Figure 1.
GA was first applied by professor John Holland in 1975 [21] to present a simple representation of natural selection. GA grows with the population of applicant results. A geneticbased search initiates with a random (initial) population, then operators, as crossover, selection and mutation. It is applied one after another to get a new chromosome generation in which the projected excellence over all the chromosomes is improved over that of the preceding generation. This procedure is repeated till the termination standard is encountered, and the best values of the chromosomes of the final generation are described as the terminal solution. The evolutionary algorithms based on GA are broadly applied by researchers due to their capability of controlling the effectiveness, robustness, divergencefree, not to become fixed in local minima, consistent and efficient as compared with other mathematical heuristic solvers. Recently, GA is being applied in the network anomaly detection system [32], wellhead back pressure control system [33], optimising bank lending decisions [34], green vehicle routing systems [35], adaptive anomalybased intrusion detection system [36], population initialisation with dispatching rules [37], heat conduction system [38], path planning in a dynamic field [39], Thomas–Fermi system [40] and nonlinear HIV infection system [41].
SQP is one of the efficient, local search, speedy and rapid optimisation scheme generally applied to solve constrained/unconstrained systems. SQP is executed in numerous optimisation models of numerous complexes as well as nonstiff systems. Presently, it is used to investigate the guidewire deformation in blood vessels [42], in the power system stabiliser design [43], optimal control of rapid cooperative rendezvous [44], 3D deformable prostate model pose estimation in minimally invasive surgery [45], deterministic constrained production optimisation of hydrocarbon reservoirs [46], prediction differential system [47] and in the optimisation of an auxetic jounce bumper [48]. To switch the sluggishness of GA, hybridisation of the GASQP process is implemented along with the necessary steps, as provided in Table 1.
Optimisation through MWNNs–GASQP to solve NIDS.
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GA, genetic algorithm; Max, maximum; MWNNs, Morlet wavelet neural networks; NIDS, nonlinear influenza disease system; SQP, sequential quadratic programming.
The mathematical notations using the statisticalbased operators, including ‘mean absolute deviation (MAD)’, ‘variance account for (VAF)’, ‘Theil’s inequality coefficient (TIC)’ and ‘semi interquartile (S.I.R)’, along with their global performances accessible to solve the biological based NIDS, are written as:
where,
The comparative presentations of the obtained numerical results and the Runge–Kutta solutions are specialised to check the accuracy of the MWNNs–GASQP. Furthermore, the statistical representations are specified to check the precision, accuracy and reliability of the proposed scheme. The efficient form of NIDS using the suitable parameters is accessible as:
A merit function based on NIDS (14) is written as:
The demonstration of the performance is presented to solve NIDS using the designed MWNNs–GASQP for multiple trials along with 30 variations. The obtained solutions of NIDS are stated in the form of best weight vector values, which are presented in Eqs 16–19. The graphical representations of these best weight vector values are illustrated in Figure 2.
The obtained outputs are calculated using Eqs 16–19 within the range of 0–1, to indicate the numerical outcomes for each group of NIDS. Figure 2(a–d) illustrates the weight vector plots based on the best solutions to solve NIDS. A comparison of the best and mean solutions using MWNNs–GASQP with the reference Runge–Kutta results are presented in Figure 2(e–h) for solving NIDS. It is indicated in these plots that the obtained results through MWNNs–GASQP overlapped with the reference solutions for each group of NIDS. This overlapping of the results indicates the excellence and precision of the designed MWNNs–GASQP. The AE plots for each group of NIDS are provided in Figure 3. One can find that the best values of the AE for the groups susceptible
The graphical plots based on the statistical measures to authorise the performance of convergence are given in Figure 5 to solve NIDS. The performance of the TIC operator using multiple executions to solve NIDS. It is observed the maximum (Max) number of trials based on the
The statistical representations are provided in Tables 2–5, based on the operators Minimum (Min), standard deviation (STD), Mean, Max, Median (Med) and S.I.R to authenticate the precision and accurateness for solving each group of NIDS. The Max values for the
Statistical performances for NIDSbased
Max  Min  Med  Mean  S.I.R  STD  

0  9.97407E−02  6.23657E−06  9.67647E−02  3.15861E−03  6.56937E−04  1.11176E−03 
0.1  3.82797E−03  2.46073E−03  2.63760E−03  3.28652E−02  3.74939E−04  1.50172E−01 
0.2  1.59218E−03  2.35321E−03  2.86086E−04  3.36978E−02  4.41197E−03  1.41380E−01 
0.3  7.52762E−04  2.33924E−03  1.43482E−04  3.35508E−02  7.04131E−03  1.27429E−01 
0.4  1.17681E−03  1.74469E−03  1.49107E−04  3.20838E−02  8.72803E−03  1.14515E−01 
0.5  8.45225E−04  1.43288E−03  8.54966E−05  3.07057E−02  9.01342E−03  1.02999E−01 
0.6  9.64376E−04  6.38340E−04  6.69333E−05  2.97428E−02  9.36168E−03  9.27501E−02 
0.7  1.03468E−03  1.08089E−03  8.11832E−05  2.88456E−02  9.76687E−03  8.35818E−02 
0.8  1.04732E−03  9.68113E−04  6.50947E−05  2.78143E−02  1.02683E−02  7.53055E−02 
0.9  9.93449E−04  6.87790E−04  6.98467E−05  2.65912E−02  1.08763E−02  6.77636E−02 
1  8.66289E−04  3.87491E−04  6.32949E−05  2.51101E−02  1.15130E−02  6.08310E−02 
Max, maximum; Med, median; Min, minimum; NIDS, nonlinear influenza disease system; S.I.R, semi interquartile.
Statistical performances for NIDSbased
Min  Max  Med  Mean  S.I.R  SD  

0  1.07524E−02  1.13718E−05  6.43473E−04  1.84408E−03  1.81226E−02  1.12267E−03 
0.1  1.02042E−01  1.55703E−04  9.97639E−02  2.50909E−03  2.90364E−04  7.08815E−03 
0.2  1.03305E−01  2.36329E−03  1.01710E−01  7.67239E−03  9.27342E−05  2.05399E−02 
0.3  1.01679E−01  2.59357E−04  1.00308E−01  1.30883E−02  1.08303E−04  3.23014E−02 
0.4  1.00068E−01  1.83367E−04  9.69585E−02  1.72975E−02  6.27781E−05  4.25479E−02 
0.5  9.73448E−02  1.67571E−03  9.31885E−02  2.05823E−02  5.26592E−05  5.12053E−02 
0.6  9.38826E−02  1.71384E−03  9.00952E−02  2.32492E−02  5.88914E−05  5.83514E−02 
0.7  8.99105E−02  1.69484E−03  8.65303E−02  2.55060E−02  5.75666E−05  6.41060E−02 
0.8  8.55744E−02  1.52268E−03  8.22485E−02  2.74916E−02  5.39135E−05  6.85933E−02 
0.9  8.09763E−02  1.14571E−03  7.80244E−02  2.93020E−02  4.83546E−05  7.19385E−02 
1  7.61938E−02  5.69917E−04  7.35171E−02  3.10092E−02  3.62539E−05  7.42639E−02 
Max, maximum; Med, median; Min, minimum; NIDS, nonlinear influenza disease system; S.I.R, semi interquartile.
Statistical performances for NIDSbased
Max  Min  Med  Mean  S.I.R  STD  

0  6.92243E−03  5.64028E−05  2.11655E−04  3.15861E−03  9.26468E−04  7.81009E−02 
0.1  1.51990E−02  8.69332E−03  7.09041E−03  3.28652E−02  1.78168E−02  2.56713E−03 
0.2  2.92093E−02  5.01355E−03  2.44201E−02  3.36978E−02  1.51010E−02  3.86235E−04 
0.3  4.14911E−02  5.07207E−03  3.94198E−02  3.35508E−02  1.34338E−02  1.77153E−04 
0.4  5.40406E−02  4.99083E−03  5.19292E−02  3.20838E−02  1.28378E−02  2.18649E−04 
0.5  6.43792E−02  4.44331E−03  6.24911E−02  3.07057E−02  1.23623E−02  1.43347E−04 
0.6  7.31617E−02  4.24610E−03  7.10555E−02  2.97428E−02  1.20395E−02  1.42089E−04 
0.7  8.03111E−02  4.63968E−03  7.80977E−02  2.88456E−02  1.17625E−02  1.55918E−04 
0.8  8.60489E−02  5.44280E−03  8.35771E−02  2.78143E−02  1.16541E−02  1.16323E−04 
0.9  9.05767E−02  6.39784E−03  8.76171E−02  2.65912E−02  1.14688E−02  1.19646E−04 
1  9.40773E−02  7.34102E−03  9.11587E−02  2.51101E−02  1.11275E−02  1.06769E−04 
Max, maximum; Med, median; Min, minimum; NIDS, nonlinear influenza disease system; S.I.R, semi interquartile.
Statistical performances for NIDSbased
Max  Min  Med  Mean  S.I.R  STD  

0  1.48512E−02  2.50654E−03  1.85049E−04  1.84408E−03  3.85021E−04  2.18319E−03 
0.1  1.90381E−01  2.03580E−04  1.86341E−01  2.50909E−03  3.27146E−02  8.05712E−02 
0.2  1.77459E−01  1.05623E−04  1.74888E−01  7.67239E−03  2.46695E−02  8.27583E−02 
0.3  1.60388E−01  7.96272E−06  1.58166E−01  1.30883E−02  2.07562E−02  8.11572E−02 
0.4  1.46077E−01  5.88240E−06  1.41058E−01  1.72975E−02  1.89140E−02  7.90228E−02 
0.5  1.32592E−01  5.20223E−07  1.26230E−01  2.05823E−02  1.73278E−02  7.64063E−02 
0.6  1.19829E−01  3.96760E−07  1.14065E−01  2.32492E−02  1.59848E−02  7.32956E−02 
0.7  1.08056E−01  1.44197E−06  1.03949E−01  2.55060E−02  1.49896E−02  6.99375E−02 
0.8  9.73313E−02  1.37818E−06  9.40407E−02  2.74916E−02  1.37403E−02  6.64160E−02 
0.9  8.76205E−02  1.07879E−06  8.53022E−02  2.93020E−02  1.25955E−02  6.27128E−02 
1  7.88533E−02  3.73973E−07  7.71953E−02  3.10092E−02  1.09303E−02  5.88248E−02 
Max, maximum; Med, median; Min, minimum; NIDS, nonlinear influenza disease system; S.I.R, semi interquartile.
The global operators of [GTIC], [GMAD] and [GEVAF] for multiple independent trials to solve NIDS using the proposed MWNNs–GASQP are tabulated in Table 6. These global performances based on Mean lie around 10^{−02}–10^{−03}, 10^{−06}–10^{−07} and 10^{−01}–10^{−02}, whereas the global S.I.R performances are found around 10^{−02} to 10^{−03}, 10^{−07}–10^{−08} and 10^{−02}–10^{−03} for each group of NIDS. These ideal close performances attained through global measures show the precision, accuracy and correctness of the designed MWNNs–GASQP for solving all the groups of NIDS.
Global operators based on TIC, MAD and EVAF values to solve NIDS.
Index  (GMAD)  (GTIC)  (GEVAF)  

Mean  S.I.R  Mean  S.I.R  Mean  S.I.R  
6.66623E−02  1.60694E−03  2.89392E−06  3.44759E−08  7.20734E−02  2.11643E−02  
7.47583E−03  1.14516E−02  9.71475E−07  7.52206E−07  7.09781E−01  2.08050E−03  
9.25308E−02  7.73955E−03  4.16727E−06  2.24806E−07  6.77949E−02  2.90708E−03  
4.47326E−02  1.63317E−02  2.11608E−06  5.07713E−07  8.00158E−01  2.75884E−02 
MAD, mean absolute deviation; NIDS, nonlinear influenza disease system; S.I.R, semi interquartile.
This current work is related to solve NIDS by exploiting MWNNs using the optimisation procedures of the hybrid process of global/local search approaches. The GA as a global approach and SQP as a local search approach have been implemented as an optimisation procedure to solve the nonlinear biological model. The influenza model is based on four groups: susceptible
In future, the proposed MWNNs–GASQP can be used to solve the systems of higher order models, fluids problems and nonlinear biological systems [49–58].
Statistical performances for NIDSbased I(y).
Min  Max  Med  Mean  S.I.R  SD  

0  1.07524E−02  1.13718E−05  6.43473E−04  1.84408E−03  1.81226E−02  1.12267E−03 
0.1  1.02042E−01  1.55703E−04  9.97639E−02  2.50909E−03  2.90364E−04  7.08815E−03 
0.2  1.03305E−01  2.36329E−03  1.01710E−01  7.67239E−03  9.27342E−05  2.05399E−02 
0.3  1.01679E−01  2.59357E−04  1.00308E−01  1.30883E−02  1.08303E−04  3.23014E−02 
0.4  1.00068E−01  1.83367E−04  9.69585E−02  1.72975E−02  6.27781E−05  4.25479E−02 
0.5  9.73448E−02  1.67571E−03  9.31885E−02  2.05823E−02  5.26592E−05  5.12053E−02 
0.6  9.38826E−02  1.71384E−03  9.00952E−02  2.32492E−02  5.88914E−05  5.83514E−02 
0.7  8.99105E−02  1.69484E−03  8.65303E−02  2.55060E−02  5.75666E−05  6.41060E−02 
0.8  8.55744E−02  1.52268E−03  8.22485E−02  2.74916E−02  5.39135E−05  6.85933E−02 
0.9  8.09763E−02  1.14571E−03  7.80244E−02  2.93020E−02  4.83546E−05  7.19385E−02 
1  7.61938E−02  5.69917E−04  7.35171E−02  3.10092E−02  3.62539E−05  7.42639E−02 
Statistical performances for NIDSbased C(y).
Max  Min  Med  Mean  S.I.R  STD  

0  1.48512E−02  2.50654E−03  1.85049E−04  1.84408E−03  3.85021E−04  2.18319E−03 
0.1  1.90381E−01  2.03580E−04  1.86341E−01  2.50909E−03  3.27146E−02  8.05712E−02 
0.2  1.77459E−01  1.05623E−04  1.74888E−01  7.67239E−03  2.46695E−02  8.27583E−02 
0.3  1.60388E−01  7.96272E−06  1.58166E−01  1.30883E−02  2.07562E−02  8.11572E−02 
0.4  1.46077E−01  5.88240E−06  1.41058E−01  1.72975E−02  1.89140E−02  7.90228E−02 
0.5  1.32592E−01  5.20223E−07  1.26230E−01  2.05823E−02  1.73278E−02  7.64063E−02 
0.6  1.19829E−01  3.96760E−07  1.14065E−01  2.32492E−02  1.59848E−02  7.32956E−02 
0.7  1.08056E−01  1.44197E−06  1.03949E−01  2.55060E−02  1.49896E−02  6.99375E−02 
0.8  9.73313E−02  1.37818E−06  9.40407E−02  2.74916E−02  1.37403E−02  6.64160E−02 
0.9  8.76205E−02  1.07879E−06  8.53022E−02  2.93020E−02  1.25955E−02  6.27128E−02 
1  7.88533E−02  3.73973E−07  7.71953E−02  3.10092E−02  1.09303E−02  5.88248E−02 
Statistical performances for NIDSbased S(y).
Max  Min  Med  Mean  S.I.R  STD  

0  9.97407E−02  6.23657E−06  9.67647E−02  3.15861E−03  6.56937E−04  1.11176E−03 
0.1  3.82797E−03  2.46073E−03  2.63760E−03  3.28652E−02  3.74939E−04  1.50172E−01 
0.2  1.59218E−03  2.35321E−03  2.86086E−04  3.36978E−02  4.41197E−03  1.41380E−01 
0.3  7.52762E−04  2.33924E−03  1.43482E−04  3.35508E−02  7.04131E−03  1.27429E−01 
0.4  1.17681E−03  1.74469E−03  1.49107E−04  3.20838E−02  8.72803E−03  1.14515E−01 
0.5  8.45225E−04  1.43288E−03  8.54966E−05  3.07057E−02  9.01342E−03  1.02999E−01 
0.6  9.64376E−04  6.38340E−04  6.69333E−05  2.97428E−02  9.36168E−03  9.27501E−02 
0.7  1.03468E−03  1.08089E−03  8.11832E−05  2.88456E−02  9.76687E−03  8.35818E−02 
0.8  1.04732E−03  9.68113E−04  6.50947E−05  2.78143E−02  1.02683E−02  7.53055E−02 
0.9  9.93449E−04  6.87790E−04  6.98467E−05  2.65912E−02  1.08763E−02  6.77636E−02 
1  8.66289E−04  3.87491E−04  6.32949E−05  2.51101E−02  1.15130E−02  6.08310E−02 
Statistical performances for NIDSbased R(y).
Max  Min  Med  Mean  S.I.R  STD  

0  6.92243E−03  5.64028E−05  2.11655E−04  3.15861E−03  9.26468E−04  7.81009E−02 
0.1  1.51990E−02  8.69332E−03  7.09041E−03  3.28652E−02  1.78168E−02  2.56713E−03 
0.2  2.92093E−02  5.01355E−03  2.44201E−02  3.36978E−02  1.51010E−02  3.86235E−04 
0.3  4.14911E−02  5.07207E−03  3.94198E−02  3.35508E−02  1.34338E−02  1.77153E−04 
0.4  5.40406E−02  4.99083E−03  5.19292E−02  3.20838E−02  1.28378E−02  2.18649E−04 
0.5  6.43792E−02  4.44331E−03  6.24911E−02  3.07057E−02  1.23623E−02  1.43347E−04 
0.6  7.31617E−02  4.24610E−03  7.10555E−02  2.97428E−02  1.20395E−02  1.42089E−04 
0.7  8.03111E−02  4.63968E−03  7.80977E−02  2.88456E−02  1.17625E−02  1.55918E−04 
0.8  8.60489E−02  5.44280E−03  8.35771E−02  2.78143E−02  1.16541E−02  1.16323E−04 
0.9  9.05767E−02  6.39784E−03  8.76171E−02  2.65912E−02  1.14688E−02  1.19646E−04 
1  9.40773E−02  7.34102E−03  9.11587E−02  2.51101E−02  1.11275E−02  1.06769E−04 
Global operators based on TIC, MAD and EVAF values to solve NIDS.
Index  (GMAD)  (GTIC)  (GEVAF)  

Mean  S.I.R  Mean  S.I.R  Mean  S.I.R  
6.66623E−02  1.60694E−03  2.89392E−06  3.44759E−08  7.20734E−02  2.11643E−02  
7.47583E−03  1.14516E−02  9.71475E−07  7.52206E−07  7.09781E−01  2.08050E−03  
9.25308E−02  7.73955E−03  4.16727E−06  2.24806E−07  6.77949E−02  2.90708E−03  
4.47326E−02  1.63317E−02  2.11608E−06  5.07713E−07  8.00158E−01  2.75884E−02 
Optimisation through MWNNs–GASQP to solve NIDS.
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