# Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model
## 1 Introduction  
There are various serious diseases produced by viruses, of which influenza is one of them that primarily attacks the upper respiratory portions, bronchi, nose, throat and sometimes disturbs the lungs. The influenza is not a fatal illness, and most people recover within one to two weeks without medical care. This disease is a serious risk to older people or those with serious illnesses such as cancer, diabetes, heart, kidney problems and lung disease. Among these people, infection can lead to serious problems of primary diseases, such as pneumonia causing death. The epidemic rate of influenza is reported as between 5% and 15% per year of the population, which is affected by upper respiratory tract infections. Worldwide, the annual epidemics are witnessed between 3 and 5 million cases of serious illness and the number of deaths is reported to be around 250,000 and 500,000 [1]. Many mathematical epidemiological models are illustrated by the ordinary nonlinear autonomous differential systems, which designate the assumptions of the model that theparameters are time independent. In such systems, variables refer to recovered, infected, transmitted and susceptible disease vectors. [PERSON] et al. [2] suggested a step-by-step differential transformation approach to solving the disease-resistant influenza virus model. [PERSON] et al. [3] presented mathematical investigations of a susceptible-infectious-quarantine-recovered (SIQR) influenza model with imperfect quarantine. [PERSON] and [PERSON] [4] introduced a numerical technique to solve a fractional pandemic influenza model. [PERSON] et al. [5] presented multi-objective optimization models for allocating patients during a pandemic influenza outbreak. [PERSON] et al. [6] provided an analysis of two models of avian influenza outbreaks relating fractal-fractional derivatives with power and memorabilia from Mittag-Leffler. [PERSON] et al. [7] designed and discussed a fractional epidemiological model for simulating influenza A outbreak. [PERSON] et al. [8] researched the impact of media coverage on human influenza transmission dynamics. [PERSON] et al. [9] discussed the dynamics of infection and virus-induced apoptosis in the production of influenza vaccines in cellular culture. [PERSON] et al. [10] presented a network flow system for managing inventory and distributing influenza vaccines in a healthcare supply chain. [PERSON] et al. [11] used genetic algorithms to discuss the optimal vaccination plans for pandemic influenza. [PERSON] et al. [12] introduced the optimum control applications for influenza and pulmonary congestion with antiviral resistance. The influenza disease nonlinear system has four categories, susceptible (\(S(u)\)), infectious (\(I(u)\)), recovered (\(R(u)\)) and cross-immune (\(C(u)\)). The mathematical design of the nonlinear influenza disease system is written as follows [13]:  
\[\left\{ \begin{array}{ll}S^{\prime}(u)=\mu-(\beta I(u)+\mu)S(u)+\gamma C(u),&S (0)=a_{1},\\ I^{\prime}(u)=\beta(S(t)+\sigma C(u))I(u)-(\alpha+\mu)I(u),&I(0)=a_{2},\\ R^{\prime}(u)=\alpha I(u)+\beta(1-\sigma)C(u)I(u)-(\mu+\delta)R(u),&R(0)=a_{3}, \\ C^{\prime}(u)=\delta R(u)-(\gamma+\beta I(u)+\mu)C(u),&C(0)=a_{4},\end{array} \right., \tag{1}\]  
where \(\beta\) shows the transmission rate for the susceptible to the infected individual, and \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(a_{4}\) are the initial conditions. The infected, infectious and cross-immune are signified as \(\gamma^{-1}\), \(\delta^{-1}\) and \(\alpha^{-1}\), respectively. \(\sigma\) shows the exposed cross-immune individuals, who are shifted in a unit time to transmittable subpopulations [14]. The recently reported studies addressing the different aspect of influenza nonlinear modelling can be seen in [15; 16; 17; 18; 19]. The aim of this work is to solve the above nonlinear influenza disease model using the stochastic capabilities of artificial neural networks (ANNs), genetic algorithms (GA) and the active set method (ASM), i.e., ANN-GA-ASM.