Local Stability of SIRB Equilibrium Points

Abstract

This paper analyzes the local stability of a cholera transmission model of SIRB type
(Susceptible–Infected–Recovered–Bacteria), incorporating vaccination and treatment.
The disease-free equilibrium (DFE) and endemic equilibrium (EE) were derived,
and their stability was investigated using the Jacobian matrix and Routh–Hurwitz
criteria. Results show that the DFE is locally asymptotically stable when R0 < 1,
ensuring disease elimination, while for R0 > 1 the DFE becomes unstable and the
system converges to a stable EE. A numerical example with biologically realistic parameters confirmed the theoretical findings. The study concludes that reducing R0
below unity through vaccination and improved sanitation is essential for sustainable
cholera control.