# On new systems of rabies virus using vaccination variable with modification

## TL;DR

This paper introduces new mathematical models for rabies transmission that incorporate vaccination using fractal-fractional calculus to better understand disease dynamics.

## Contribution

The novel contribution is the development of fractal-fractional dynamical systems for rabies modeling with vaccination as a variable.

## Key findings

- A five-dimensional ordinary system with vaccination (RVT-Vaccination) was formulated.
- Fractal-fractional differential operators were used to generalize the RVT-Vaccination system.
- The solvability of the systems was analyzed using modified fractal-fractional integral operators.

## Abstract

Usually, a compartmental epidemiological modeling such as the SIR (Susceptible-Infectious-Recovered) or SEIR (Susceptible-Exposed-Infectious-Recovered) model must be modified in order to include a vaccine element in a mathematical representation of rabies. This element may influence the dynamics of disease transmission by accounting for vaccinated persons. In this effort, we suggest new two different dynamical systems of rabies virus utilizing the vaccination as a variable with modification in fractal-fractional calculus. Analysis of data is investigated with several examples. Moreover, the solvability of fractal-fractional dynamic system is given using the two dimensional and three dimensional fractal-fractional differential operators. The solution is suggested by utilizing the modified fractal-fractional integral operator. Finally, we estimate the greatest number of infectious individuals and the time at which this peak occurs in order to examine the peak infection in every scenario. The outcomes for the Baseline case and the altered beginning conditions will be computed and shown. The methodology of this paper is as follows:•Include a vaccine element, as a variable to get five dimensional ordinary system with vaccination (RVT-Vaccination);•Generalize the above system RVT-Vaccination by using the most recent fractal-fractional differential operators, with two and three fractal-fractional powers.•Analyze the three systems and establish the solvability.

Include a vaccine element, as a variable to get five dimensional ordinary system with vaccination (RVT-Vaccination);

Generalize the above system RVT-Vaccination by using the most recent fractal-fractional differential operators, with two and three fractal-fractional powers.

Analyze the three systems and establish the solvability.

Image, graphical abstract