Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis

Abstract

Psychological diseases and their treatment are problems related to public health. According to data from the World Health Organization, about a billion people have either mental illness or substance use disorder problems in 2017. Mental, neurological diseases and substance use disorders account for 30 percent of the global non-fatal disease burden and 10 percent of the global disease burden. It is noted that in the world Dec 2005 and 2015, the incidence of mental health diseases increased by about 16 percent. In this study, we have created a fractional-order mathematical modeling for the population of individuals suering from psychological diseases in a society. In this model, the total population was divided into three compartments: individuals who did not receive psychological treatment (S), individuals who received psychological support (P) and individuals who recovered after completing psychological treatment (R). As a fractional derivative, we used the Caputo derivative definitions. Numerical solutions were obtained with the help of Euler method by performing stability analysis related to the fractional SPR model created for the mathematical model of psychological patients. Thus, it was interpreted by creating dynamics for the number of individuals with psychological problems in a population.

Keywords

• fractional psychological disease model
• mathematical modeling
• euler method
• caputo derivative
• stability analysis.