#### Event Title

An Application of Compartmental Disease Modelling

#### Advisor(s)

Dr. Patrick Davis, Illinois Mathematics and Science Academy

#### Discipline

Mathematics

#### Start Date

21-4-2021 11:55 AM

#### End Date

21-4-2021 12:20 PM

#### Abstract

As the coronavirus continues to plague the world, researchers are constantly working to model the spread of the disease and determine optimal means of containing the pandemic. Using a compartmental framework, we created a transition model that, by using available resources and ongoing research, was tailored to the COVID-19 pathogen. We classified the US population into different age groups and level of risk, along with how individuals transfer between different states of infection. By utilizing differential equations, we modelled the flow at which the members of the population exit their respective compartments and transition into the next, with the compartments being S (susceptible), I (infectious), Y (symptomatic), R (removed), and V (vaccinated). To create these differential equations, we included several parameters which account for various behaviors of the population members and characteristics of the disease. To analyze the model, we constructed the next generation matrix to systematically determine the value of the basic reproduction number R0_{0}. R0_{0} is defined as the average number of new cases of an infection caused by one typical infected individual in a population consisting of only susceptibles; and it can be calculated by evaluating the dominant eigenvalue of the next-generation matrix of the linearized infected subsystem. If R0_{0} > 1, an epidemic will occur; and if R0_{0} < 1, then it will not. In addition to calculating R0_{0}, we proved the existence, uniqueness and non-negativity of a solution to our model given appropriate initial conditions. Finally, we simulated several scenarios with varied parameter values in order to observe changes in the rate that individuals moved from each compartment.

An Application of Compartmental Disease Modelling

As the coronavirus continues to plague the world, researchers are constantly working to model the spread of the disease and determine optimal means of containing the pandemic. Using a compartmental framework, we created a transition model that, by using available resources and ongoing research, was tailored to the COVID-19 pathogen. We classified the US population into different age groups and level of risk, along with how individuals transfer between different states of infection. By utilizing differential equations, we modelled the flow at which the members of the population exit their respective compartments and transition into the next, with the compartments being S (susceptible), I (infectious), Y (symptomatic), R (removed), and V (vaccinated). To create these differential equations, we included several parameters which account for various behaviors of the population members and characteristics of the disease. To analyze the model, we constructed the next generation matrix to systematically determine the value of the basic reproduction number R0_{0}. R0_{0} is defined as the average number of new cases of an infection caused by one typical infected individual in a population consisting of only susceptibles; and it can be calculated by evaluating the dominant eigenvalue of the next-generation matrix of the linearized infected subsystem. If R0_{0} > 1, an epidemic will occur; and if R0_{0} < 1, then it will not. In addition to calculating R0_{0}, we proved the existence, uniqueness and non-negativity of a solution to our model given appropriate initial conditions. Finally, we simulated several scenarios with varied parameter values in order to observe changes in the rate that individuals moved from each compartment.