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.
Highlights
Cholera is a waterborne acute gastrointestinal infection characterized by diarrhea and vomiting, which kills within an hour if not treated [1]
Cholera is caused by drinking water or eating food contaminated by a bacterium called Vibrio cholerae (V. cholerae) [2, 3]
Most of the diseases are controlled by public health organizations including cholera which is characterized with severe vomiting and diarrhea [4]. e incubation period of cholera is less than five days [5]
Summary
Cholera is a waterborne acute gastrointestinal infection characterized by diarrhea and vomiting, which kills within an hour if not treated [1]. Codeco developed the first basic mathematical model for cholera infection. A new fractional derivative was developed by Hattaf [9] and applied to analyze the memory effect on the HIV-infected population. The new fractional derivative has not been applied to study the cases in the cholera infection. We modified the model developed in [5], we focus on the logistic growth model of Vibrio cholerae concentration in reserved freshwater used for drinking purposes, and the contribution of infected individuals to the environment is assumed to be properly managed. We developed an optimal control mathematical model that takes into consideration the loss of immunity after cholera infection recovery. We have applied a new fractional derivative and analyzed the relationship between the order of the derivative and the extinction status of cholera infection
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