# The Viral State Dynamics of the Discrete-Time NIMFA Epidemic Model

## TL;DR

This paper analyzes the discrete-time NIMFA epidemic model on directed networks, demonstrating its stability and ability to accurately reflect real-world viral spread dynamics in digital data collection contexts.

## Contribution

It provides a rigorous analysis of the discrete-time NIMFA model, proving stability, monotonicity, and bounding the viral state, which enhances understanding of epidemic dynamics in digital data settings.

## Key findings

- Viral state is increasing and bounded by the steady-state.
- Steady-state is globally exponentially stable.
- Linear systems can bound viral state evolution.

## Abstract

The majority of research on epidemics relies on models which are formulated in continuous-time. However, real-world epidemic data is gathered and processed in a digital manner, which is more accurately described by discrete-time epidemic models. We analyse the discrete-time NIMFA epidemic model on directed networks with heterogeneous spreading parameters. In particular, we show that the viral state is increasing and does not overshoot the steady-state, the steady-state is globally exponentially stable, and we provide linear systems that bound the viral state evolution. Thus, the discrete-time NIMFA model succeeds to capture the qualitative behaviour of a viral spread and provides a powerful means to study real-world epidemics.