Purpose - The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic. Design/methodology/approach - The differential transform method (DTM) is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations of these problems. Findings - Results obtained using the scheme presented here agree well with the results obtained by the Adomian decomposition and power series methods. Some plots are presented to show the reliability and simplicity of the method. Originality/value - This paper is believed to represent a new application for DTM on solving systems of nonlinear ordinary differential equations. © Emerald Group Publishing Limited.