A Well-Posed Fractional Order Cholera Model with Saturated Incidence Rate
Isa Abdullahi Baba, Usa Wannasingha Humphries, Fathalla A. Rihan
Abstract
A fractional-order cholera model in the Caputo sense is constructed. The model is an extension of the Susceptible-Infected-Recovered (SIR) epidemic model. The transmission dynamics of the disease are studied by incorporating the saturated incidence rate into the model. This is particularly important since assuming that the increase in incidence for a large number of infected individualsis equivalent to a small number of infected individualsdoes not make much sense. The positivity, boundedness, existence, and uniqueness of the solution of the model are also studied. Equilibrium solutions are computed, and their stability analyses are shown to depend on a threshold quantity, the basic reproduction ratio (R0). It is clearly shown that if R0<1, the disease-free equilibrium is locally asymptotically stable, whereas if R0>1, the endemic equilibrium exists and is locally asymptotically stable. Numerical simulations are carried out to support the analytic results and to show the significance of the fractional order from the biological point of view. Furthermore, the significance of awareness is studied in the numerical section.