Analytical and Numerical Boundedness of a Model with Memory Effects for the Spreading of Infectious Diseases: A Theoretical Study of Rabies

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.