An efficient computational numerical approach for nonlinear mathematical influenza disease modelling

: In this study, an advanced computational numerical scheme based on the Levenberg-Marquardt backpropagation (LMB) neural network (NN) process, i.e., LMB-NN is presented for solving the nonlinear mathematical influenza disease model. The nonlinear mathematical influenza disease model depends on four categories named susceptible S( t ), infected I( t ), recovered R( t ) and cross-immune individuals proportion C( t ). Six different cases of the nonlinear mathematical influenza disease model have been numerically treated using the LMB-NN process and the comparison of the results has been presented by using the reference databased solutions designed based on the Adams results. The numerically obtained results of the nonlinear mathematical influenza disease model using the verification, testing, and training procedures are calculated using the LMB-NN process to reduce the functions of mean square error (MSE). For the correctness, competence, effectiveness, and efficiency of the LMB-NN process, the proportional and analysis methods are performed using the analysis of correlation, MSE results, error histograms and regression.


Introduction
There are many serious diseases caused by viruses, among them influenza is one of them that mainly attacks the upper respiratory parts, nose, bronchi, throat and sometimes affects the lungs. Influenza is not a common deathly disease and most people recover in 1-2 weeks without taking any medical cure. Influenza is a serious risk for those who are older age or those who are suffering serious medical illnesses like as diabetes, lung diseases, cancer, heart and kidney problems. In such people, the infection might lead to severe problems of primary illnesses, pneumonia as well as death. The influenza epidemic rate is reported 5-15% per year of the population that is affected by the upper breathing tract infections. Around the world, yearly epidemics are reported between 3-5 million cases of severe illness and the number of deaths is reported around 250,000 and 500,000 [1]. Many mathematical epidemiology problems are demonstrated by the autonomous nonlinear ordinary differential system of equations, which indicates the assumptions of the model parameters are independent of time. In these systems, the variables signify subpopulations of recovered, infected, transmitted disease vectors, and susceptible.
Astuti et al [2] proposed a multi-step differential transform method to solve the influenza virus model with disease resistance. Erdem et al [3] give a mathematical analysis of a SIQR influenza model with imperfect quarantine. Alzahrani and Khan [4] introduced a numerical technique to solve a fractional epidemic influenza model. Sun et al [5] presented a multi-objective optimization model for patient allocation during a pandemic influenza outbreak. Ghanbari et al [6] give an analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories. González-Parra et al [7] presented a fractional-order epidemic model for the simulation of outbreaks of influenza A. Schulze-Horsel et al [8] studied the infection dynamics and virus-induced apoptosis in cell culture-based influenza vaccine production. Tchuenche et al [9] discussed the impact of media coverage on the transmission dynamics of human influenza. Hovav et al [10] discussed a network flow model for inventory management and distribution of influenza vaccines through a healthcare supply chain. Patel et al [11] examined the optimal vaccination strategies for pandemic influenza using genetic algorithms. Kanyiri et al [12] presented the applications of optimal control to influenza pneumonia coinfection with antiviral resistance.
The human population influenza disease model is based on the four classes, susceptible class S(t), infectious class I(t), recovered class R(t) and cross-immune class C(t). The mathematical form of the influenza disease nonlinear model is given as [13]: where the transmission/contact rate for susceptible to infected is β. 1 r , 2 r , 3 r and 4 r represent the initial conditions. The periods based on infected, cross-immune and infectious are denoted as 1   , 1   and 1   , respectively. The exposed cross-immune based individuals are denoted by σ, who are drafted in a unit time into the infectious subpopulation [14].
The aim of the present study is to solve the nonlinear mathematical influenza disease model using the Levenberg-Marquardt backpropagation (LMB) neural network (NN) process i.e., LMB-NN. This method has never been reported to solve the nonlinear mathematical influenza disease model. Six different cases representing the nonlinear mathematical influenza disease model have been numerically treated using the LMB-NN scheme. The comparison is performed based on the reference dataset of Adams results using the LMB-NN scheme. Artificial neural networks have been extensively applied in most advance and recent applications of biological system model [15][16][17][18][19], doubly singular multi-fractional order Lane-Emden system [20], singular Thomas-Fermi model [21], model of heat conduction in human head [22], functional differential models [23][24][25] and higher order nonlinear singular models [26][27][28][29]. The novelty of the presented work is itemized as:  A novel integrated design is presented using the intelligent computing scheme through the designed LMB-NN scheme to find the solutions of the nonlinear mathematical influenza disease model.

Methodology
The proposed LMB-NN scheme is provided in two steps for the numerical results of the nonlinear mathematical influenza disease model.
 Necessary explanations are provided to make the data based on LMB-NN scheme.  Execution procedures approve the LMB-NN scheme for solving the nonlinear mathematical influenza disease model.
The proposed LMB-NN scheme is a combination of the NN multi-layer structure together with the optimization LMB process is given in Fig 1. Suggested methodology for the single neuronbased system is provided in Fig 2. The proposed LMB-NN scheme is applied using the 'nftool' in the 'Matlab' software for the proper parts of testing data, hidden neurons, learning schemes and validation data.  with the training data is 80%, testing and validation data is 10%.. The proposed LMB-NN approach is provided in Fig. 3, whereas, the designed LMB based SNN scheme is performed to solve each case of the nonlinear mathematical influenza disease model.  3.465 × 10 -09 , 3.935 × 10 -09 , 3.537 × 10 -09 , 2.444 × 10 -08 and 4.823 × 10 -09 , respectively. Fig. 5 shows the gradient values with the step size, i.e., Mu of the LMB-NN approach for solving the nonlinear mathematical influenza disease model are [9.430 × 10 -08 , 9.959 × 10 -08 , 9.883 × 10 -08 , 3.695 × 10 -07 , 2.498 × 10 -07 and 9.996 × 10 -08 ]. These plotted values show the exactness and the convergence of the LMB-NN approach for solving all the cases of the nonlinear mathematical influenza disease model. Figs 6-11 shows the fitting curves for solving the nonlinear mathematical influenza disease model that shows the obtained results comparability through the designed LMB-NN approach using the reference-based data set of the Adams results. The maximum error is plotted using the validation, testing, and training inputs through LMB-NN approach lie around 10 -07 to 10 -08 for solving the nonlinear mathematical influenza disease model. The EHs are plotted in Fig. 12, whereas the regression plots are drawn in Figs 13 to 18 for each case of the nonlinear mathematical influenza disease model. The co-relation values are implemented to authenticate the regression analysis. It is seen that the correlation values (R) for the nonlinear mathematical influenza disease model lie around 1 that indicates the perfect model form. The validation, training, testing plots signify the correctness of the designed LMB-NN approach for solving the nonlinear mathematical influenza disease model.

Conclusions
The presented work is associated to design a biological nonlinear mathematical influenza disease model using the numerical computing integrated Levenberg-Marquardt backpropagation (LMB) approach and the supervised neural networks (NN). The hidden neurons are used 10 and the training data is used 80 %, while the validation/testing data is used 10% data to adjust the proposed LMB-NN approach to solve each case of the nonlinear mathematical influenza disease model. To check the excellence, brilliance, perfection and exactness of the LMB-NN approach, the overlapping of the obtained results from the designed LMB-NN scheme with the reference databased Adams results are implemented. The values of the performance using the mean square error and convergence for training, best curve, authentication and testing are provided for all cases of the nonlinear mathematical influenza disease model. The performance through co-relation is accomplished for the authentication of the regression process. The gradient values along with the step size are performed for each case of the nonlinear mathematical influenza disease model. Moreover, the correctness and precision are calculated using the graphical as well as numerical conformations of the convergence plots, error-histograms, MSE indexes and regression dynamics, respectively.

Compliance with ethical standards Conflict of interest
The authors also declare that there is no conflict of interest.