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

The majority of research on epidemics relies on models which are formulated in continuous-time. However, processing real-world epidemic data and simulating epidemics is done digitally and the continuous-time epidemic models are usually approximated by discrete-time models. In general, there is no guarantee that properties of continuous-time epidemic models, such as the stability of equilibria, also hold for the respective discrete-time approximation. 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 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.