Flip bifurcation analysis and mathematical modeling of cholera disease by taking control measures

Abstract

Abstract To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases which are spread in the world wide. The objective of the research study is to assess the early diagnosis and treatment of cholera virus by implementing remedial methods with and without the use of drugs. A mathematical model is built with the hypothesis of strengthening the immune system, and a ABC operator is employed to turn the model into a fractional-order model. A newly developed system SEIBR, which is examined both qualitatively and quantitatively to determine its stable position as well as the verification of flip bifurcation has been made for developed system. The local stability of this model has been explored concerning limited observations, a fundamental aspect of epidemic models. We have derived the reproductive number using next generation method, denoted as “ $$R_{0}$$ R 0 ”, to analyze its impact rate across various sub-compartments, which serves as a critical determinant of its community-wide transmission rate. The sensitivity analysis has been verified according to its each parameters to identify that how much rate of change of parameters are sensitive. Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values which is advanced tool for reliable bounded solution. Also the error analysis has been made for developed scheme. Simulations have been made to see the real behavior and effects of cholera disease with early detection and treatment by implementing remedial methods without the use of drugs in the community. Also identify the real situation the spread of cholera disease after implementing remedial methods with and without the use of drugs. Such type of investigation will be useful to investigate the spread of virus as well as helpful in developing control strategies from our justified outcomes.

Keywords

• Mathematical modeling
• Boundedness
• Non-local kernel
• Uniqueness
• Flip bifurcation
• Atangana–Toufik