Fractal and Fractional SIS model for syphilis data

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.