MATHEMATICAL MODEL OF CHOLERA USING STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS

Abstract

This study has attempted to develop and analyze a mathematical model for the spread of cholera disease the case of Ethiopia. To achieve this objective, secondary data sources from Minister of Health and some parameter values are from related published articles. The required data for the study were analyzed by using a MATLAB computer program or Maple Software and the results are presented in the form of tables and graphs. The ndings of the study revealed that we formulated and analyzed a deterministic SIBTRS mathematical model by considering indirect contact transmission and extended to a stochastic SITRS mathematical model by incorporating direct transmission path way for cholera disease dynam- ics. The total population of this model consists: susceptible, infected, treated, bacteria concentration and recovered individuals. First we develop and analyze a deterministic mathematical model of the disease dynamics. Since, the deterministic model does not consider the randomness process of the disease or environmental factors; we have extended the deterministic model to stochastic model by considering sto- chastic environmental factors. Next, the qualitative behavior of this model is analyzed in both approaches by comparing deterministic to stochastic approach. The invariant region, existence of equilibrium points (disease free and endemic equilibrium), basic reproduction number (deterministic and stochastic) and their stabilities (local as well as global stability) of both models are studied. On the other hand, Simula- tion results are done using parameter values to attempt the impact of transmission parameters on cholera disease dynamics. Our numerical simulation results indicated that reducing contact rate, improvement in treatment and the environmental sanitation is vital to eradicate cholera disease.