Applications of the Differential Transformation Method and Multi-Step Differential Transformation Method to Solve a Rotavirus Epidemic Model

Abstract

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S, V, I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods can be extended to other compartmental models.

Metode Transformasi Diferensial untuk Menentukan Solusi Persamaan Diferensial Linier Nonhomogen

“…Pada [6], metode ini digunakan untuk menyelesaikan model epidemi SEIR. Beberapa penulis lain yang telah menggunakan MTD dalam menyelesaikan model epidemiologi antara lain [7], [8], [9], dan [10]. MTD memiliki keuntungan karena relatif mudah untuk dipahami dan diimplementasikan.…”

Application of Non-Standard Finite Difference Method on Covid-19 Mathematical Model With Fear of Infection

“…The use of differential equations to model the transmission dynamics of infectious disease can be traced back to 1970 when Daniel Bernoulli justified the use of inoculation to curb the spread of smallpox (Dietz and Heesterbeek, 2002;Foppa, 2017). These models are usually nonlinear (Peter et al, 2020;Gu et al, 2023;Akinyemi et al, 2023;Kambali et al, 2023;Ochi et al, 2023) and are difficult to obtain their exact solution (Onwubuoya et al, 2018b;Riyapan et al, 2021;ur Rehman et al, 2023). Thus, numerical methods are used to obtain approximate solutions.…”

An Approximate Solution to Predator-prey Models Using The Differential Transform Method and Multi-step Differential Transform Method, in Comparison with Results of The Classical Runge-kutta Method

“…The DTM procedure is one of the many numerical methods for systems of ODEs [13], [14], which uses polynomial for approximation to obtain the exact solution representation that is differentiable. An extension of DTM is that the MSDTM procedure [15], [16], [17] is subdivided into intervals of equal step size, and while Runge-Kutta obtains numerical computation for comparative analysis.…”