Application of a New Generalized Fractional Derivative and Rank of Control Measures on Cholera Transmission Dynamics

Abstract

In this study, the mathematical model of the cholera epidemic is formulated and analyzed to show the impact of Vibrio cholerae in reserved freshwater. Moreover, the results obtained from applying the new fractional derivative method show that, as the order of the fractional derivative increases, cholera-preventing behaviors also increase. Also, the finding of our study shows that the dynamics of Vibrio cholerae can be controlled if continuous treatment is applied in reserved freshwater used for drinking purposes so that the intrinsic growth rate of Vibrio cholerae in water is less than the natural death of Vibrio cholerae. We have applied the stability theory of differential equations and proved that the disease-free equilibrium is asymptotically stable if $R_0<1$ , and the intrinsic growth rate of the Vibrio cholerae bacterium population is less than its natural death rate. The center manifold theory is applied to show the existence of forward bifurcation at the point $R_0=1$ and the local stability of endemic equilibrium if $R_0>1$ . Furthermore, the performed numerical simulation results show that, as the rank of control measures applied increases from no control, weak control, and strong control measures, the recovered individuals are 55.02, 67.47, and 674.7, respectively. Numerical simulations are plotted using MATLAB software package.