A delay mathematical model for the spread and control of water borne diseases.

A non-linear SIRS mathematical model to explore the dynamics of water borne diseases like cholera is proposed and analyzed by incorporating delay in using disinfectants to control the disease. It is assumed that the only way for the spread of infection is ingestion of contaminated water by susceptibles. As the pathogens discharged by infectives reach to the aquatic environment, it is assumed that the growth rate of pathogens is proportional to the number of infectives. Further, it is assumed that disinfectants are introduced to kill pathogens with a rate proportional to the density of pathogens in the aquatic environment. The model is analyzed by using stability theory of delay differential equations. It is found that the model exhibits two equilibria, the disease free equilibrium and the endemic equilibrium. The analysis shows that under certain conditions, the cholera disease may be controlled by using disinfectants but a longer delay in their use may destabilize the system. Numerical simulation is also carried out to confirm the analytical results.