## The Key Determinant of Epidemic System Dynamics

This value of the reproduction number can predict the dynamics of a disease within a population. There are various forms of this reproduction number, which each represent a different scenario. These are explored in the accordion below. With a basic reproduction number less that one, we see the number of infected individuals decrease over time as the disease is contained. This can be seen in the first image below. Meanwhile, with a basic reproduction number greater than one, we see the number of cases of the disease increase over time and an epidemic occur. This can be seen in the second image below.

### Basic Reproduction Number

The basic reproduction number represents the number of secondary infections that will be produced by a **single initially infected case** in a **population of susceptible individuals**.

### Control Reproduction Number

The control reproduction number represents the number of secondary infections that will be produced by a **single infected individual** in a **population that includes treatment methods** – both PrEP and ART.

### Effective Control Reproduction Number

The effective control reproduction number represents the number of secondary infections that will be produced by a **single infected individual** in a **population that** **includes people who are not susceptible.** This is based off the premise that both treatment methods are effective, essentially eliminating the susceptible population.

## Derivation of Our Reproduction Number

To derive our reproduction numbers, we found the spectral radius of the next generation matrix of our system. This spectral radius has been shown to be the control reproduction number. In order to find our effective control reproduction number, we set the values of epsilon equal to zero. Further we found the basic reproduction number by setting the uptake rates into PrEP and ART equal to zero. Our derived reproduction numbers can be seen below.