Sains Malaysiana 46(10)(2017): 2007–2017

Differential Transformation Method (DTM) for Solving SIS and SI Epidemic Models

(Kaedah Transformasi Pembezaan (DTM) untuk Menyelesaikan Model Epidemik SIS dan SI)

M.Z. AHMAD1*, D. ALSARAYREH1, A. ALSARAYREH1 & I. QARALLEH2

1Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis Indera Kayangan, Malaysia

2Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan

Received: 23 May 2016/Accepted: 13 February 2017

ABSTRACT

In this paper, the differential transformation method (DTM) is employed to find the semi-analytical solutions of SIS and SI epidemic models for constant population. Firstly, the theoretical background of DTM is studied and followed by constructing the solutions of SIS and SI epidemic models. Furthermore, the convergence analysis of DTM is proven by proposing two theorems. Finally, numerical computations are made and compared with the exact solutions. From the numerical results, the solutions produced by DTM approach the exact solutions which agreed with the proposed theorems. It can be seen that the DTM is an alternative technique to be considered in solving many practical problems involving differential equations.

Keywords: Differential transformation method (DTM); exact solution; semi-analytical solution; SIS model; SI model

ABSTRAK

Dalam kajian ini, kaedah transformasi pembezaan (DTM) telah digunakan untuk mencari penyelesaian separuh-analisis bagi model epidemik SIS dan SI untuk populasi malar. Pertama, latar belakang teori DTM telah dikaji dan diikuti dengan membina penyelesaian bagi model epidemik SIS dan SI. Selain itu, analisis konvergensi DTM telah dibuktikan melalui cadangan dua teorem. Akhir sekali, pengiraan berangka dibuat dan dibandingkan dengan penyelesaian tepat. Daripada keputusan berangka, penyelesaian yang diperoleh melalui DTM adalah hampir sama dengan penyelesaian tepat yang bersetuju dengan teorem yang dicadangkan. Boleh dikatakan bahawa DTM adalah satu teknik alternatif yang boleh dipertimbangkan untuk menyelesaikan banyak masalah praktikal yang melibatkan persamaan pembezaan.

Kata kunci: Kaedah transformasi pembezaan (DTM); model SIS; model SI; penyelesaian tepat; penyelesaian separuh-analitik

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*Corresponding author; email: mzaini.imk@gmail.com

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