Fractal and Fractional SIS model for syphilis data

TL;DR

This paper extends the SIS epidemiological model using fractal and fractional derivatives, providing explicit solutions, numerical analysis, and fitting to Brazilian syphilis data, showing the fractal model's superior data description.

Contribution

Introduces a fractal and fractional SIS model with explicit solutions and demonstrates its effectiveness in fitting real syphilis data, improving correlation.

Findings

Fractal model yields higher correlation with data.

Two fractal orders improve model fit.

Estimated recovery period is 11.6 days.

Abstract

This work studies the SIS model extended by fractional and fractal derivatives. We obtain explicit solutions for the standard and fractal formulations; for the fractional case, we study numerical solutions. As a real data example, we consider the Brazilian syphilis data from 2011 to 2021. We fit the data by considering the three variations of the model. Our fit suggests a recovery period of 11.6 days and a reproduction ratio ($R_0$) equal to 6.5. By calculating the correlation coefficient ($r$) between the real data and the theoretical points, our results suggest that the fractal model presents a higher $r$ compared to the standard or fractional case. The fractal formulation is improved when two different fractal orders with distinguishing weights are considered. This modification in the model provides a better description of the data and improves the correlation coefficient.