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Epidemic models can play a major role in understanding the spread of diseases and their control. These mathematical models have plenty of significance in various scientific domains, including public health, to investigate disease propagation and ecology. This article explains the dynamics of SIR epidemic model of arbitrary order with aid of a precise numerical approach called Genocchi wavelet collocation method. The main purpose of this investigation is to explore and discover the results for system of nonlinear ordinary differential equations arising in the considered mathematical model and to investigate the dynamical aspects of SIR model via Caputo fractional derivative which is non-local in behaviour. The projected method depicts rapid algorithms and is extremely precise, reliable, and uses fewer computational resources. Also, this method is simpler than the other traditional numerical methods as it merges the operational matrix with the collocation method in order to transform fractional-order problem into algebraic equations which enables to obtain satisfactory results. The approximate solution obtained using proposed algorithm exposes the nature of their interactions. Furthermore, the numerical outcomes are represented through graphs for different fractional order and compared the results with Runge–Kutta method and residual power series method. The projected technique is very effective, accurate, free from controlling parameters and consume less time to investigate nonlinear complications arising in diverse fields of epidemical and biological models. Ultimately, the current study help to inspect the wild class of models and their performance which are occurring in real world.

In this paper, we have considered the fractional typhoid disease model and obtained the numerical approximation of the model via the innovative wavelet scheme called the Genocchi wavelet collocation method (GWCM) with the help of Caputo fractional derivative for the fractional order. The approach under consideration is a powerful tool for obtaining numerical solutions to fractional-order nonlinear differential equations. The GWCM approach yields accurate solutions that are very close to exact solutions for highly nonlinear problems by avoiding data rounding and just computing a few terms. The Genocchi wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main benefit of this method lies in its capability to reduce the computational complexity associated with solving systems of ODEs, resulting in accurate and efficient solutions. The results of the developed technique, the RK4 method, and the ND solver have been compared. The numerical outcomes demonstrate that the implemented technique is incredibly effective and precise for solving the Typhoid model of fractional order. This paper contributes to numerical analysis by introducing the Genocchi wavelet method as a robust tool for solving biological models.

The present paper suggests a novel, efficient operational matrix technique on the basis of block-pulse functions and Genocchi wavelets to solve time-fractional telegraph equations considering Dirichlet boundary conditions. First, a brief overview of the Genocchi polynomials, corresponding wavelets, and fundamental characteristics is presented. Then, the same functions and their suitable characteristics are employed to formulate the Genocchi wavelet-like operational matrices of fractional integration. Using the suggested technique, the fractional model is reduced into a system of algebraic equations, which is solvable by employing the classical Newton’s iteration technique. A comparison is made between the estimated solutions of the time-fractional telegraph equation and the present approaches, such as the Legendre wavelet and the Fibonacci wavelet method. According to the numerical results, accurate results are obtained using the Genocchi method, and therefore, it is computationally more effective compared to the present approaches.