# Exact Solution to a Dynamic SIR Model

## TL;DR

This paper derives an exact solution for a generalized SIR epidemic model with time-dependent coefficients, extending classical solutions to continuous and discrete time domains, and analyzes the model's limiting behavior with biological implications.

## Contribution

It introduces an exact solution method for a generalized SIR model with time-dependent coefficients in both continuous and discrete settings.

## Key findings

- Exact solutions for time-dependent SIR models in continuous time.
- Discrete epidemic system solutions matching continuous behavior.
- Analysis of limiting behavior with biological relevance.

## Abstract

We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible-infected-removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.