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Society must understand, model, and forecast infectious disease transmission patterns in order to prevent pandemics. Mathematical models and computer technology may help us better understand the pandemic and create more systematic and effective infection management strategies. This study offers a novel perspective through a compartmental model that incorporates fractional calculus. The first scenario is based on proportional fractional definitions, considering compartmental individuals of susceptible, moving susceptible, exposed, infected, hospitalized, and recovered. Through an extension of this derivative, they decimated the model to integer order. We extended the deterministic model to a stochastic extension to capture the uncertainty or variance in disease transmission. It can develop an appropriate Lyapunov function to detect the presence and uniqueness of positive global solutions. Next, we discuss how the epidemic model might have become extinct. In our theoretical study, we demonstrated that a sufficiently outrageous amount of noise can cause a disease to become extinct. A modest level of noise, on the other hand, promotes the persistence of diseases and their stationary distribution. The Khasminskii method was used to determine the stationary distribution and ergodicity of the model.

In the view of significant exposure of multifarious interval-valued fuzzy numbers in neoteric studies, different measures of interval-valued generalized hexagonal fuzzy numbers (IVGHFN) associated with assorted membership functions (MF) are explored in this article. Considering the symmetricity and asymmetricity of the hexagonal fuzzy structures, the idea of MF is generalized a bit more, to nonlinear membership functions. The construction of level sets, accordingly for each case of linear and nonlinear MF are also carried out. In addition, the concepts of generalized Hukuhara (gH) differentiability for the interval-valued generalized hexagonal fuzzy functions (IVGHFF) are also the main features of this framework. Illustratively, the developed intellects are implemented on a logistic population growth problem, by taking ecological functions as IVGHFFs. For the further numerical demonstrations of the model, artificial neural network with simulated annealing (ANNSA) algorithm is utilized.

This work proposes a new trend for determining the numerical solutions of complex real life models by the profitable implementation of a nature inspired optimization technique, which is enthused by the pollination strategy of the flowering plants. The design methodology optimizes the determined residual fitness function achieved by the utilization of the generalized Taylor series for the deliberated fractional order differential model. Results of numerical experimentation achieved by the proposed pollination enthused residual optimization (PERO) technique are compared with some former numerical and metaheuristic techniques. Moreover, a detailed performance analysis is performed via statistical inference based on hundred independent runs.

This endeavor aims to present a new comprehensive technique to compute the approximate solutions of Painleve type differential equations (P-type-DE) with arbitrary order. The purpose is achieved by the idea of combining rational approximation with the cuckoo search algorithm (RACS). The cuckoo search algorithm (CSA) is mainly a meta-heuristic optimization technique inspired by the brood parasitism of cuckoo birds. It utilizes Levy flights for the searching procedure with few controlling variables, which makes its global search capacity extremely robust to optimize the objective function. Here, the key role of the objective function is played by the fitness function of P-type-DE, which is structured by a trial solution having unknown weights. Furthermore, a comparative study is also conducted among the calculated results, the exact and other reference solutions, which demonstrates the rightness and computational intelligence of RACS for the non-linear differential equations of arbitrary order.

In this trying time for the world battling different variants of the COVID'19 pandemic, different intervention strategies are being taken by government, to limit the spread of infection. Closing educational institutes, stay at home orders, campaigns for emphasis on vaccination, usage of medical mask and frequently sanitizing hands, etc. are the endeavors made by the authorities to decrease the number of cases in the country. In this regard, the contribution aims to help the decision-makers to identify a potential prevention strategy, based on public acceptance and intervention effectiveness. To achieve this objective, feasible judgments of professionals from three different sectors are brought together through meetings. Opinions, based on ten criteria, are recorded in linguistic form for prioritizing six alternatives. The linguistic terms are then evaluated and manipulated by entailing triangular fuzzy numbers and a group multi-criteria decision making (GMCDM) approach. After using the fuzzy analytical hierarchy process (F-AHP) for the complex decisions, the fuzzy VIšekriterijumsko KOmpromisno Rangiranje method (F-VIKOR) is utilized to attain the closest ideal stratagem. Consequently, through the ranking orders of defuzzified scores, intuitive preference of compromise solutions is suggested. The tactic gaining more priority with respect to the group utility to the majority and F-VIKOR index is complete lockdown for the short term. Furthermore, a comparison analysis is also added in the discussion to verify the attained prioritized outcomes. This comparative study is carried out through the technique for order of preference by similarity to ideal solution (TOPSIS), which evidently produces the same preference of alternatives. In addition, this strategy can be apparently discovered to be an effective strategy adopted by different countries in successfully decreasing the number of cases.

Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of change and constant solutions of each compartment are studied with incorporation of a special case of proportional fractional derivative (PFD). In addition, existence and uniqueness of the system are also comprehensively elaborated. Moreover, novel dynamic assessment of the system is carried out in context with the fractional order index. Succinctly, the manuscript accomplishes cyclic epidemiological behavior of the infectious disease due to the delay in treatment of the infected individuals.

The key objective of this paper is to study the imprecise biological complexities in the interaction of two species pertaining to harvesting threshold. It is explained by taking the prey–predator model with imprecise biological parameters and fractional order generalized Hukuhara (fgH) differentiability. In this vain, different possible systems of the model are constructed, according to the increasing and decreasing behavior of population growth. Feasibility and stability analyses of equilibrium points of the stated models are also discussed by means of variational matrix with Routh–Hurwitz conditions. In addition, the numerical elaborations are carried out by taking parametric expansion of fuzzy fractional Laplace transform (FFLT). This significantly helps the researchers in using a novel approach to analyze the constant solutions of the dynamical systems in the presence of fractional index. This would allow the avoidance of any intricacy that occurs while solving fractional order derivatives. Furthermore, this attempt also provides numerical and pictorial results, obtained through some well-known methods, namely fifth-forth Runge–Kutta method (FFRK), Grunwald–Letnikov’s definition (GL) and Adams–Bashforth method (ABM) that are deemed appropriate to scrutinize the dynamics of the system of equations.

This paper presents a bio-evolutionary metaheuristic approach to study the harmonically oscillating behavior of the Duffing equation. The proposed methodology is an amalgamation of the artificial neural network with the firefly algorithm. A novelty in the activation of neurons of artificial neural network is described using the cosine function with the angular frequency. Chronologically, artificial neural network approximates discretizes the nonlinear functions of the governing problem, which then undergoes an optimization process by the firefly algorithm that then later generates the effective values of the unknown parameters. Generally, the algorithm and implementation of the scheme are assimilated by considering an application of Duffing-harmonic oscillator. Some error measurements, in order to discuss the convergence and accuracy of the scheme, are also visualized through tables and graphs. An effective optimized relationship between the angular frequency and amplitude is derived and its results are depicted in a tabular form. The comparison of the proposed methodology is also deliberated by homotopy perturbation method. Moreover, the geometrical illustration of the trajectories of the dynamic system is also added in the phase plane for different values of amplitude and angular frequency.

In present time, the whole world is in the phase of war against the deadly pandemic COVID'19 and working on different interventions in this regard. Variety of strategies are taken into account from ground level to the state to reduce the transmission rate. For this purpose, the epidemiologists are also augmenting their contribution in structuring such models that could depict a scheme to diminish the basic reproduction number. These tactics also include the awareness campaigns initiated by the stakeholders through digital, print media and etc. Analyzing the cost and profit effectiveness of these tactics, we design an optimal control dynamical model to study the proficiency of each strategy in reducing the virulence of COVID'19. The aim is to illustrate the memory effect on the dynamics of COVID'19 with and without prevention measures through fractional calculus. Therefore, the structure of the model is in line with generalized proportional fractional derivative to assess the effects at each chronological change. Awareness about using medical mask, social distancing, frequent use of sanitizer or cleaning hand and supportive care during treatment are the strategies followed worldwide in this fight. Taking these into consideration, the optimal objective function proposed for the surveillance mitigation of COVID'19, is contemplated as the cost function. The effect analysis is supported through graphs and tabulated values. In addition, sensitivity inspection of basic reproduction number is also carried out with respect to different values of fractional index and cost function. Ultimately, social distancing and supportive care of infected are found to be significant in decreasing the basic reproduction number more rapidly.

In this paper, a process is devised systematically to scrutinize the scrolling chaotic behaviour of fractional-order Chua's system. The process is composed of fractional Laplace transformation, artificial neural network with Mexican hat wavelet as an activation function and simulated annealing. Sequentially, the parametric expansion of fractional Laplace transform is employed to convert the governing fractional system into an ordinary differential system. Next, artificial neural network and simulated annealing approximate and optimize the attained system and produce accurate solutions. The predictability and elaboration of double scrolling chaotic structures of fractional-order Chua's system are also studied using the Lyapunov exponent and fifth–fourth Runge–Kutta method. Moreover, the mean absolute error and root mean square error are measured for the convergence analysis of the proposed scheme. On the whole, the accurate approximate solutions, the phase plots of Lyapunov exponent spectrum and bifurcation maps of the dynamical evolution of fractional Chua's system are a triumph of this endeavour.

Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different fields of science and engineering, in this paper, nonlinear nth order FDEs are approximated, heuristically. The analysis is carried out on using Chebyshev neural network (ChNN), which is a type of single layer functional link artificial neural network (FLANN). Besides, explication of generalized Hukuhara differentiability (gH-differentiability) is also added for the nth order differentiability of fuzzy-valued functions. Moreover, general formulation of the structure of ChNN for the governing problem is described and assessed on some examples of nonlinear FDEs. In addition, comparison analysis of the proposed method with Runge-Kutta method is added and also portrayed the error bars that clarify the feasibility of attained solutions and validity of the method.

The purpose of this paper is to investigate the pricing European call option valuation problems under the exercise price, maturity, risk-free interest rate, and the volatility function. An advance methodology, Chebyshev simulated annealing neural network (ChSANN), is enforced for the Black-Scholes (B-S) model with boundary conditions. Our scheme is stable and easy to implement on B-S equation, for arbitrary volatility and arbitrary interest rate values. Also, the comparative results demonstrate that the attained approximate solutions are converging towards the exact solution. The graphical results show that the increasing flow of the European call option as the exponential increase takes place in assets. The presented algorithm can be further applied to other financial models with certain boundary conditions. The algorithm of the method shows that the approach can also be easily employed on time-fractional B-S equation.

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

In this paper, analytical investigations of linear fractio nal order fuzzy differential equations are obtained using a newfound operator method. Fuzzy fractional differential equations (FFDEs) subjected to initial conditions are dissected under the assumptions of generalized Hukuhara differentiability in conjunction with Caputo-type fuzzy fractional derivative. Consequently, all the prospects of fractional differentials of fuzzy-valued functions are de duced and discussed in detail under the notion of Caputo-type fuzzy fractional differentiability (CFH-differentiability). Moreover, the novel method is illust rated on constructed systems of FFDEs and convex combination of r -level solutions for each system is measured, explicitly.

In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

In this paper, Sumudu transform is advanced and designed for the solution of linear differential models with uncertainty. For this purpose, Sumudu transform is coupled with fuzzy theory and all its fundamental properties are formalized in fuzzy sense. At last, to demonstrate the accuracy of this approach, fuzzy Sumudu transform is employed to some examples of fuzzy linear differential equations considered under generalized H-differentiability and analytical solutions are obtained efficiently.

In this paper, numerical simulation of linear and nonlinear fuzzy initial value problems of fractional order has been done. A modificaton in the improved Euler’s method for fuzzy fractional order is proposed and utilized. Also, the two component homotopy perturbation method also known as a modified homotopy perturbation method (MHPM) is employed to solve the considered problems. The results obtained from the considered techniques are compared. These methods are illustrated by solving several examples. Effciency and exactness of results worked out from MHPM is examined from the tables and comparatively it is found that MHPM is more accurate and closer to the exact solutions than the existing methods.