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

Abstract

The majority of research on epidemics relies on models which are formulated in continuoustime. 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.

Time-dependent solution of the NIMFA equations around the epidemic threshold

“…Thus, a linear convergence of the error term ξ(t) to zero, i.e., ξ(t) 2 = O (R 0 − 1), would not be sufficient to show that the viral state v(t) converges to c(t)v ∞ when R 0 ↓ 1. 7 In Prasse and Van Mieghem (2019), an analogous statement has been proved for the discrete-time version of the NIMFA equations (2).…”

“…3). Three results of Prasse and Van Mieghem (2019) are worth mentioning, since we believe that they could also apply to NIMFA (1) in continuous time. First, the steady-state v ∞ is exponentially stable.…”

“…The basic reproduction number R 0 only provides a coarse description of the dynamics of NIMFA (1). Recently (Prasse and Van Mieghem 2019), we analysed the viral state dynamics for the discrete-time version of NIMFA (1), provided that the initial viral state v(0) is small (see also Assumption 2 in Sect. 3).…”

“…Furthermore, NIMFA (1) has been generalised to time-varying parameters. Paré et al (2017) consider that the infection rates 3 β i j (t) depend continuously on time t. Rami et al (2013) consider a switched model in which both the infection rates β i j (t) and the curing rates δ i (t) change with time t. NIMFA (1) in discrete time has been analysed in Ahn and Hassibi (2013), Paré et al (2018), Prasse and Van Mieghem (2019) and Liu et al (2020).…”

“…Under Assumption 2, the steady-state v ∞ is exponentially stable for NIMFA in discrete time (Prasse and Van Mieghem 2019). If the steady-state v ∞ is exponentially stable, then the error vector ξ(t) goes to zero exponentially fast, since ξ(t) is orthogonal to v ∞ .…”