This project discussed the numerical solutions of first-order nonlinear ordinary differential equations (ODEs) that established the epidemic model. The application under consideration is a SEIR model of the influenza outbreak in Australia in 1919. Two proposed methods, the Runge-Kutta fourth order (RK4) and the Euler method, were used, and they proved to be good approximations of the behaviour of each SEIR variable and practically well-suited for solving this problem. The analytical solution was hard to obtain, so this study used ode45 in MATLAB to solve the simulation of the SEIR model. The outcomes of the RK4 and Euler methods were acquired through MATLAB software implementation. Additionally, we intended to evaluate two different ways to determine a good approximation to the behaviour of each of the SEIR variables. To verify good approximation to the behaviour of each SEIR variable, approximation methods implementing different step sizes, which were h = 0.01, 0.1 and 0.25, were compared to the ode45 in MATLAB provided by the SEIR model. The step size had to be smaller for the solution to approximate each SEIR variable’s behaviour better. The RK4 order and Euler methods were compared numerically and illustrated graphically. To examine errors, we concluded by analyzing and computing the errors of the two proposed methods for various step sizes. The RK4 order method was observed to be a good approximation to the behaviour of four systems SEIR in solving the SEIR model using ode45 in MATLAB.