Analytical and Numerical Boundedness of a Model with Memory Effects for the Spreading of Infectious
1Analytical and Numerical Boundedness of a Model with Memory Effects for the Spreading of Infectious Diseases: A Theoretical Study of Rabies
Layman Abstract : This study focuses on improving a mathematical rabies spread model by converting it into a fractional order epidemic model, which provides a more flexible way to analyze disease dynamics. Instead of using traditional mathematical methods, the study applies Caputo fractional order derivatives to better capture the complexity of rabies transmission.
Rabies remains a serious public health issue, especially in areas with large dog populations and limited vaccination or treatment options. Preventive measures include pre-exposure vaccination and post-exposure prophylaxis (PEP) for people bitten by infected animals.
The study examines the positivity and stability of the new model using Laplace transformations and calculates key factors such as steady states and the basic reproductive number, which helps predict the disease’s spread. Both local and global stability are analyzed to understand how rabies can be controlled over time.
To verify the model, a numerical method is developed to simulate rabies outbreaks. The accuracy and reliability of this method are tested through mathematical proofs and computer simulations, confirming that the model is positive, bounded, and converges to realistic steady states.
In summary, this study provides a more precise and stable mathematical approach for understanding rabies transmission, which could help improve disease control strategies and vaccination policies in affected regions.
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Original Abstract : In this study, an integer order rabies model is converted into the fractional order epidemic model is converted into the fractional order epidemic model. To this end, the Caputo fractional order derivatives are plugged in place of classical derivatives. Rabies is a primary health problem in many populations dense with dogs, specifically in regions where there less or no preventive measures are adopted (vaccination and remedy) for puppies and human beings. Remedy after exposure to the rabies virus is referred to as post-exposure prophylaxis (PEP) and vaccination earlier than exposure to the infection is known as pre-exposure prophylaxis. The positivity and boundedness of the fractional order mathematical model is investigated by applying Laplace transformation and its inversion. To study the qualitative behavior of the non-integer rabies model, two steady states and the basic reproductive number of the underlying model are worked out. The local and global stability is investigated at both the steady states of the fractional order epidemic model. After analytic treatment, a structure preserving numerical template is constructed to solve numerically the fractional order epidemic model. Moreover, the positivity and boundedness of the numerical scheme is examined. Lastly, numerical experiment and simulations are accomplished to substantiate the significant traits of the projected numerical design. Consequences of the study are highlighted in the closing section. The significant features of numerical design are the positivity, boundedness and convergence towards accurate steady states. These traits of the numerical design are identified by establishing some standard results. Moreover, simulations are presented to validate all the key features of the novel numerical design.
View Book: https://doi.org/10.9734/bpi/mcsru/v3/4482
#Caputo_operator #rabies_infection #grunwald_letnikov_approximation #nonstandard_finite_difference #simulations