# Application of Fractional Calculus in the Field of Population Genetics

## Disciplines

Applied Mathematics | Diseases

## Abstract (300 words maximum)

Theoretical population genetics bridges mathematics and evolutionary biology. A key innovation in the field was the development of the coalescent theory by the probabilistic JFC Kingman in 1982. The Kingman's coalescent models how alleles from a specific population with no mechanism to introduce genetic variation to further generations may have originated from a common ancestor. In this research, we study the extension of Kingman's coalescent, which is based on fractional calculus. This extension helps us detect potential population heterogeneity and improve our understanding of the evolution of long versus short lived organisms under selection. By utilizing the definition of the fractional coalescent, which is based on the Mittag-Leffler function and its relationship to the exponential function, it was concluded that when a population is synchronized in the same environment, the exponential function is used. Moreover, when the population isn't synchronized, the Mittag-Leffler function would be used. This study can be used to analyze a multitude of population genetics aspects, such as microbes and viruses. The Covid-19 virus can be used as an example to model this study: Individuals who are vaccinated against Covid and those that aren’t vaccinated that live in the same environment. Still, the virus that ends up on a person that is vaccinated will have by chance fewer offspring than the virus inside the individual that was not vaccinated. This fractional coalescent allows for differences in the environment, unlike Kingman's coalescent that assumes all virus particles have the same chance of survival.

Keywords: Kingman's coalescent; Fractional calculus; Fractional coalescent; Exponential function; Mittag-Leffler function; Population heterogeneity

## Academic department under which the project should be listed

CSM - Mathematics

## Primary Investigator (PI) Name

Dr. Somayeh Mashayekhi

## Share

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Application of Fractional Calculus in the Field of Population Genetics

Theoretical population genetics bridges mathematics and evolutionary biology. A key innovation in the field was the development of the coalescent theory by the probabilistic JFC Kingman in 1982. The Kingman's coalescent models how alleles from a specific population with no mechanism to introduce genetic variation to further generations may have originated from a common ancestor. In this research, we study the extension of Kingman's coalescent, which is based on fractional calculus. This extension helps us detect potential population heterogeneity and improve our understanding of the evolution of long versus short lived organisms under selection. By utilizing the definition of the fractional coalescent, which is based on the Mittag-Leffler function and its relationship to the exponential function, it was concluded that when a population is synchronized in the same environment, the exponential function is used. Moreover, when the population isn't synchronized, the Mittag-Leffler function would be used. This study can be used to analyze a multitude of population genetics aspects, such as microbes and viruses. The Covid-19 virus can be used as an example to model this study: Individuals who are vaccinated against Covid and those that aren’t vaccinated that live in the same environment. Still, the virus that ends up on a person that is vaccinated will have by chance fewer offspring than the virus inside the individual that was not vaccinated. This fractional coalescent allows for differences in the environment, unlike Kingman's coalescent that assumes all virus particles have the same chance of survival.

Keywords: Kingman's coalescent; Fractional calculus; Fractional coalescent; Exponential function; Mittag-Leffler function; Population heterogeneity