# The Linear Stability and Basic Reproduction Numbers for Autonomous FDEs

## TL;DR

This paper establishes the stability equivalence between certain autonomous functional differential equations and their delay-free counterparts, and develops a theory of basic reproduction numbers for autonomous FDEs, with an application to tick population dynamics.

## Contribution

It introduces a new stability equivalence result for autonomous FDEs and formulates a general theory of basic reproduction numbers for these equations.

## Key findings

- Proved stability equivalence between delayed and delay-free systems.
- Developed a theory of basic reproduction number $\\mathcal{R}_0$ for autonomous FDEs.
- Applied the theory to a population model of black-legged ticks.

## Abstract

In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $\mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $\mathcal{R}_0$.