Abstract
Cholera is an infectious intestinal disease which occurs as a result of poor sanitation and lack of basic education in its transmission. It is characterized by profuse vomiting and severe diarrhea when an individual eats food or drinks water contaminated with the Vibrio cholerae. A dynamic mathematical model that explicitly simulates the transmission mechanism of cholera by taking into account the role of control measures and the environment in the transmission of the disease is developed. The model comprises two populations: the human population and bacteria population. The next-generation method is used to compute the basic reproduction number, R0. Both the disease-free and endemic equilibria are shown to be locally and globally stable for R0 values less than unity and unstable otherwise. Necessary conditions of the optimal control problem were analyzed using Pontryagin’s maximum principle with control measures such as educational campaign and treatment of water bodies used to optimize the objective function. Numerical values of model parameters were estimated using the nonlinear least square method. The model simulations confirm the significant role played by control measures (education and treatment of water bodies) and the bacteria in the environment in the transmission dynamics as well as reducing the spread of cholera.
Highlights
Cholera is an infectious intestinal disease characterized by profuse vomiting and acute watery diarrhea which is caused by eating food or drinking water contaminated with a bacterium called Vibrio cholerae [1, 2]
If educating individuals c1 on reducing the contact between infected person and susceptible individuals, as well as infected individuals understanding the mode of transmission and prevention of cholera, is the only control measure being applied, that is, c1 ≠ 0 and c2 0, the basic reproduction number is given by
Strategy I: Control with Educational Campaign and Treatment of Water Bodies. e strategy applied is to obtain the optimal control simulations that describes the effectiveness of the two control measures, that is, c1 ≠ 0 and c2 ≠ 0, when applied on the infectious class. e basic reproduction number obtained by applying this strategy is given by Rc0 0.4470
Summary
Cholera is an infectious intestinal disease characterized by profuse vomiting and acute watery diarrhea which is caused by eating food or drinking water contaminated with a bacterium called Vibrio cholerae [1, 2]. They used the Zimbabwean data to derive estimates of the basic reproductive number (R0) of cholera on a regional basis In another development, Tuite et al [18] built an SIR compartmental transmission model that characterized the population as susceptible to infection, infected, and infectious to others. They computed the basic reproduction number and the global dynamics of the system They considered control parameters which were immunization coverage rate and environmental management especially drinking water, they failed to apply optimal control analysis. Ey incorporated two control strategies which were human educational campaign and treatment of water They did not analytically calculate their model to determine the basic reproduction number which serves as a threshold value for the dynamics of the system and the disease. Some control measures were applied to ascertain the transmission dynamics of cholera
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