Modeling Cholera Epidemiology Using Stochastic Differential Equations

Abstract

In this study, we extend Codeço’s classical SI-B epidemic and endemic model from a deterministic framework into a stochastic framework. Then, we formulated it as a stochastic differential equation for the number of infectious individuals $I\left(t\right)$ under the role of the aquatic environment. We also proved that this stochastic differential equation (SDE) exists and is unique. The reproduction number, $R_0$ , was derived for the deterministic model, and qualitative features such as the positivity and invariant region of the solution, the two equilibrium points (disease-free and endemic equilibrium), and stabilities were studied to ensure the biological meaningfulness of the model. Numerical simulations were also carried out for the stochastic differential equation (SDE) model by utilizing the Euler-Maruyama numerical method. The method was used to simulate the sample path of the SI-B stochastic differential equation for the number of infectious individuals $I\left(t\right)$ , and the findings showed that the sample path or trajectory of the stochastic differential equation for the number of infectious individuals $I\left(t\right)$ is continuous but not differentiable and that the SI-B stochastic differential equation model for the number of infectious individuals $I\left(t\right)$ fluctuates inside the solution of the SI-B ordinary differential equation model. Another significant feature of our proposed SDE model is its simplicity.