Developing a continuous SIR epidemic model and its discrete version using Euler method: Analyzing dynamics with analytical and numerical methods

Abstract

In this paper, we develop a continuous SIR epidemic model and its discrete version using the Euler method. The continuous SIR epidemic model is a widely used mathematical tool for understanding the dynamics of infectious diseases. We derive the system of differential equations that describe the rates of change in the number of individuals in each compartment and discuss the key parameters that influence the dynamics of the model. Next, we use the Euler method to discretize the continuous SIR epidemic model and obtain a discrete‐time version. We analyze the dynamics of the discrete SIR epidemic model using both analytical and numerical methods. We derive the positively invariant set of the model and discuss the initial conditions of the model and how they relate to the total population. We compare the behavior of the continuous and discrete SIR epidemic models and show that the discretized model captures the essential features of the continuous model. We also investigate the impact of key parameters such as the transmission rate, recovery rate, and death rate on the dynamics of the model. Our findings suggest that the discrete SIR epidemic model is a powerful mathematical tool for understanding the dynamics of infectious diseases and developing effective strategies for their control. By studying the behavior of the model under different scenarios, researchers can gain insights into the transmission and prevalence of infectious diseases and develop interventions that reduce their impact on public health.