Existence and Uniqueness of Cholera Model with Vaccination
Metet K. Nelson *
Department of Mathematics, Maasai Mara University, Narok, Kenya.
Tireito K. Frankline
Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya.
*Author to whom correspondence should be addressed.
Abstract
This study investigates the existence and uniqueness of solutions for a fractional-order cholera transmission model incorporating vaccination. The model divides the human population into susceptible, vaccinated, infected, and recovered classes, while also considering human-associated vibrios and environmental vibrios as pathogen-related compartments. The classical integer-order model is first formulated using assumptions on recruitment, indirect environmental transmission, vaccination, recovery, loss of immunity, and pathogen dynamics. The model is then extended using the Caputo–Fabrizio fractional derivative to represent memory effects in cholera transmission dynamics. Basic concepts of fractional derivatives, fractional integrals, and fixed-point theory are introduced as the mathematical foundation for the analysis. The Caputo–Fabrizio model is converted into an equivalent Volterra-type integral formulation, and the vector field associated with the model is examined for Lipschitz continuity. Under suitable boundedness and contraction conditions, fixedpoint arguments are used to support the existence and uniqueness of solutions for the proposed fractional initial-value problem. The analysis provides a theoretical basis for studying cholera dynamics with vaccination
Keywords: Cholera transmission, vaccination, Caputo–Fabrizio derivative, fractional-order model, existence, uniqueness, fixed-point theory