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Abstract This study presents a computational investigation of a stochastic Zika virus along with optimal control model using the Legendre spectral collocation method (LSCM). By accumulation of stochasticity into the model through the proposed stochastic differential equations, we appropriating the random fluctuations essential in the progression and disease transmission. The stability, convergence and accuracy properties of the LSCM are conscientiously analyzed and also demonstrating its strength for solving the complex epidemiological models. Moreover, the study evaluates the various control strategies, such as treatment, prevention and treatment pesticide control, and identifies optimal combinations that the intervention costs and also minimize the proposed infection rates. The basic properties of the given model, such as the reproduction number, were determined with and without the presence of the control strategies. For $$R_01$$ R 0 > 1 , then endemic equilibrium is satisfied, in this case the disease spread in the population at higher scale. The fundamental findings acknowledge the significant impact of stochastic phonemes on the robustness and effectiveness of control strategies that accelerating the need for cost-effective and multi-faceted approaches. In last the results provide the valuable insights for public health department to enabling more impressive mitigation of Zika virus outbreaks and management in real-world scenarios.

This study presents a computational investigation of a stochastic Zika virus along with optimal control model using the Legendre spectral collocation method (LSCM). By accumulation of stochasticity into the model through the proposed stochastic differential equations, we appropriating the random fluctuations essential in the progression and disease transmission. The stability, convergence and accuracy properties of the LSCM are conscientiously analyzed and also demonstrating its strength for solving the complex epidemiological models. Moreover, the study evaluates the various control strategies, such as treatment, prevention and treatment pesticide control, and identifies optimal combinations that the intervention costs and also minimize the proposed infection rates. The basic properties of the given model, such as the reproduction number, were determined with and without the presence of the control strategies. For R 0 < 0 , the model satisfies the disease-free equilibrium, in this case the disease die out after some time, while for R 0 > 1 , then endemic equilibrium is satisfied, in this case the disease spread in the population at higher scale. The fundamental findings acknowledge the significant impact of stochastic phonemes on the robustness and effectiveness of control strategies that accelerating the need for cost-effective and multi-faceted approaches. In last the results provide the valuable insights for public health department to enabling more impressive mitigation of Zika virus outbreaks and management in real-world scenarios. [ABSTRACT FROM AUTHOR]

In this paper, a combination of the quasilinearization and the Legendre spectral collocation methods is introduced to approximate the solution of the nonlinear functional Volterra integral equations. Throughout this process, the quasilinearization method converts the nonlinear functional Volterra integral equation to a sequence of linear integral equations. Then, in each iteration, the obtained linear integral equation is solved using the Legendre spectral collocation method. After that, a convergence analysis is discussed in detail. Finally, several numerical examples are included to demonstrate the capability and validity of the proposed method. [ABSTRACT FROM AUTHOR]

Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.

The Zika virus model (ZIKV) is mathematically modeled to create the perfect control strategies. The main characteristics of the model without control strategies, in particular reproduction number, are specified. Based on the basic reproduction number, if R 0 < 0 , then ZIKV satisfies the disease-free equilibrium. If R 0 > 1 , then ZIKV satisfies the endemic equilibrium. We use the maximum principle from Pontryagin's. This describes the critical conditions for optimal control of ZIKV. Notwithstanding, due to the prevention and treatment of mosquito populations without spraying, people infected with the disease have decreased dramatically. Be that as it may, there has been no critical decline in mosquitoes contaminated with the disease. The usage of preventive treatments and insecticide procedures to mitigate the spread of the proposed virus showed a more noticeable centrality in the decrease in contaminated people and mosquitoes. The application of preventive measures including treatment and insecticides has emerged as the most ideal way to reduce the spread of ZIKV. Best of all, to decrease the spread of ZIKV is to use avoidance, treatment and bug spraying simultaneously as control methods. Moreover, for the numerical solution of such stochastic models, we apply the spectral technique. The stochastic or random phenomenons are more realistic and make the model more informative with the additive information. Throughout this paper, the additive term is assumed as additive white noise. The Legendre polynomials and applications are implemented to transform the proposed system into a nonlinear algebraic system. [ABSTRACT FROM AUTHOR]

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R 0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R 0 = 1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally. [ABSTRACT FROM AUTHOR]

In this research work, we present an exciting mathematical analysis of a stochastic model, using a standard incidence function, for infectious disease hepatitis C transmission dynamics. In this model, we divided the infected population into three different classes with two different infection stages known as chronic class and acute class while the third is an isolation class. We also presents briefly the Legendre spectral method for the numerical solution to the proposed model. It is observed that the disease-free equilibrium is asymptotically stable, when basic reproduction number R 0 < 1. It is also shown that the proposed model has a stable endemic equilibrium when the reproduction number R 0 > 1. Also, sensitivity analysis is carried out to study and identify the effect of parameters on R0. Moreover, we have performed numerical simulations to study the influence of disease free equilibrium and endemic equilibrium. Legendre polynomial and Legendre weight function are used to solve the proposed stochastic system numerically. Numerical results are compared against the basic reproduction number. [ABSTRACT FROM AUTHOR]

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre's polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters. [ABSTRACT FROM AUTHOR]

In this article, a novel approach for solving systems of nonlinear partial differential equations over a large time domain is presented. The method is termed a multidomain bivariate spectral quasilinearization method (MD‐BSQLM). This method makes use of the Legendre‐Gauss‐Lobatto grid points, a linearization technique, and the spectral collocation method to approximate functions defined by bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. We demonstrate the use of the MD‐BSQLM technique by solving a system of nonlinear partial differential equations of boundary layer flow. Numerical experiments are conducted to show applicability and accuracy of the methods. Grid independence tests establish the method's accuracy, convergence, and validity. The solution for the limiting case is used to validate the method's accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for solving second-order fractional integral equations. By using of spectral collocation method based on Gauss–Legendre–Lobatto points, a system of algebraic equations is obtained and then, we get a solution based on Legendre polynomials. If L 2 norm is used, this proposed method has a favorable convergence rate, which is shown by some examples. [ABSTRACT FROM AUTHOR]

In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre-Gauss-Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method. [ABSTRACT FROM AUTHOR]

We propose an efficient Legendre-Gauss collocation algorithm for second-order nonlinear ordinary differential equations (ODEs). We also design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partial differential equations (PDEs), which can be implemented in a synchronous parallel fashion. Numerical results indicate the high accuracy and effectiveness of the suggested algorithms. [ABSTRACT FROM AUTHOR]