The Volterra delay integral equations have numerous applications in various branches of science, including biology, ecology, physics and modeling of engineering and natural sciences. In many cases, it is difficult to obtain analytical solutions of these equations. So, numerical methods as an efficient approximation method for solving Volterra delay integral equations are of interest to many researchers. In this paper, a numerical method is developed for solving the Hammerstein–Volterra delay integral equation by least squares (LS) approximation method, which is based on Shifted Bernstein polynomials (BPs). This equation is a mathematical model for the spread of certain infectious diseases with a constant rate that varies seasonally. Least squares method is a mathematical model for data fiting which minimizes the sum of squared the difference between an observed value and the value provided by a model. In this paper, the shifted Bernstein polynomials are introduced and then approximation of an arbitrary function by using these polynomials is presented . Also, the Hammerstein–Volterra delay integral equation is introduced and the details of least squares method for solving a mathematical model is presented. Finally, we show the efficiency of the proposed method by solving two numerical examples and comparing the results with other methods.
