# Random genetic drift diffusion model and deterministic and stochastic models of epidemics

Thesis

Spring 5-31-1985

## Degree Name

Master of Science in Applied Mathematics - (M.S.)

## Department

Mathematics

Rose Ann Dios

John Tavantzis

Roman Wolodymyr Voronka

## Abstract

In the Random Genetic Drift Diffusion model two approaches are taken. First we examined a discrete model that represent a relatively idealised version of the phenomena. Hefurther make the assumption that the Population reproduces itself and then dies, thus maintaining a finite population size at all times. If at a given locus there are two possible allels A and B and if X(t) is the number of A type in the genetic pool of size 2N, then 2N-X(t) is the number of B type. We then proceed to obtain a probability density function of X( t) by an Exact method and the Monte Carlo method.

Based on a χ2 for each χ2 generation examined there are no significant difference between the results obtained from either method. However, for large N (N > 20) the Exact method is cumbersome. and as a result the Monte Carlo is more appropriate for such N.

As a second approach, we approximated the Discrete model for large N with a Diffusion model (a singular parabolic partial differential equation) where x and t are assumed continuous. By separation of variables we obtained the Hypergeometric equation which has an infinite series solution. From this we obtained the probability density as a function of gene frequency and compare these results with those of the previous methods (Discrete model). We found that there is favourable comparison between all three methods and in particular between the Diffusion Approximation and the Monte Carlo.

The Monte Carlo method was also utilized in the Stochastic models of Epidemics. The models we examined are the Chain Binomial models of Reed-Frost and Greenwood. We confirmed that for a household of 3 and smaller, both models are indistinguishable, whereas a household of 5 produced different chains based on the inherent assumptions in each model.

Establishing the existence of a threshold population size, we used a continuous model (Deterministic Theory). This approach resulted in a system of nonlinear ordinary differential equations. The solution of which using the Runge-Kutta (order four) established a relative removal rate above which no epidemic seems to occur, as well as demonstrate the existence of a threshold population size.

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