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Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics

Robin Persoons, Mattia Sensi, Bastian Prasse, and Piet Van Mieghem
Phys. Rev. E 109, 034308 – Published 19 March 2024
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  • INTRODUCTION
  • THE UPPER-TRANSITION TIME
  • BOUNDS FOR THE UPPER-TRANSITION TIME
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times T̲(r) and T¯(r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T¯(r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T¯(r) is almost entirely determined by the basic reproduction number R0 of the network. The value of the upper-transition time T¯(r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

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    • Received 3 November 2023
    • Accepted 15 February 2024
    • Corrected 3 April 2024

    DOI:https://doi.org/10.1103/PhysRevE.109.034308

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    EpidemicSpreading
    1. Physical Systems
    Erdos-Renyi graphEvolving networksSocial networks
    Networks

    Corrections

    3 April 2024

    Correction: The affiliation indicators for the second, third, and last authors were not ascribed properly during the production cycle and have been corrected.

    Authors & Affiliations

    Robin Persoons1,*, Mattia Sensi2,3, Bastian Prasse4, and Piet Van Mieghem1

    • 1Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
    • 2MathNeuro Team, Inria at Université Côte d'Azur, 2004 Rte des Lucioles, 06410 Biot, France
    • 3Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
    • 4European Centre for Disease Prevention and Control (ECDC), Gustav III's Boulevard 40, 169 73 Solna, Sweden

    • *Corresponding author: r.d.l.persoons@tudelft.nl

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    Vol. 109, Iss. 3 — March 2024

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    • Figure 1
      Figure 1

      Timescale separation. The time between two changes in the graph Δt is increasing on the t axis. The transition times T̲(r) and T¯(r) create the borders of the annealed and quenched regimes by including processes that are approximately annealed or quenched to the respective regimes, based on the accuracy tolerance r. The annealed regime is bounded from below by Δt=0. The intermediate regime lies between the transition times T̲(r) and T¯(r). The quenched regime extends from Δt=T¯(r) to infinity

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    • Figure 2
      Figure 2

      NIMFA SIS on a time-varying network for different interupdate times Δt. Parameters: graph size N=50; infection rate β=0.1; curing rate δ=1. The time-varying network is given by a sequence of Erdős-Rényi graphs, where the link density p is chosen uniformly at random from the interval [0.4,0.6]. The solid red line (small interupdate time Δt=0.001) shows the averaging behavior when nearing the annealed regime. The dashed black line (medium interupdate time Δt=1) shows the irregularity of the intermediate regime. The dotted blue line (large interupdate time Δt=10) shows the convergence on each topology from the quenched regime.

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    • Figure 3
      Figure 3

      Comparison of NIMFA SIS and Markovian SIS on time-varying networks for different interupdate times Δt. Left: Δt=10, middle: Δt=1, right: Δt=0.01. The solid blue line corresponds to the NIMFA process and the dashed red line to the Markovian process. Parameters are the same for NIMFA and Markovian. The infection rate β=0.1 and curing rate δ=1. The time-varying network is given by a sequence of Erdős-Rényi graphs, with N=50 and link density p picked uniformly at random from the interval [0.4,0.6]. The graph sequence for NIMFA and Markovian SIS are the same in each panel, but the sequences are different between panels.

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    • Figure 4
      Figure 4

      Ten different NIMFA SIS processes on time-varying networks with long interupdate times (i.e., tk−tk−1 is large). The first graph G1 on the interval [t0,t1), is different for each process: the different graphs are ER graphs with N=50 and link density p={0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8.0.9,1}. The second graph G2 on the interval [t1,t2) is the same for each process and is the complete bipartite graph K25,25. The final graph G3 on the interval [t2,t3) is also the same for each process and is the complete graph on 50 nodes K50. The graph G1 has visibly negligible influence on the interval [t2,t3) when G2 and G3 are fixed due to the Markov-like memorylessness property with respect to previous graphs that defines the quenched regime.

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    • Figure 5
      Figure 5

      The upper-transition time T¯(r) vs the basic reproduction number R0 for different values of the starting prevalence y(0). The chosen parameters are a graph size N=50, accuracy tolerance r=10−4, and infection rate β=0.1. Each graph is an Erdős-Rényi graph Gp(N) with link density p chosen uniformly at random: p∼Unif(0,1). For R0>1, the small values of y(0) correspond to the top curves, i.e., to a higher upper-transition time T¯(r). For R0<1, the small values of y(0) correspond to the bottom curves, i.e., to a lower upper-transition time T¯(r). For y(0)=10−4 the upper-transition time T¯(r) is zero per definition. To determine the upper-transition time T¯(r) we approximate the steady-state prevalence y∞ with y∞≈y(tmax)=y(104). The inset shows the dips below the curves in more detail.

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    • Figure 6
      Figure 6

      On the left y axis the prevalence of a NIMFA process on a time-varying network (solid blue) is compared with the prevalence of the “quenched prediction” of each interval [tm−1,tm) starting from y∞(Gm−1) (dashed red) for different interupdate times Δt. The top figure has Δt=1, the middle has Δt=5, and the bottom one has Δt=10. On the right y axis the absolute error (dotted black) between the prediction and the process is shown. Each graph Gm is an ER graph with link density p∈[0.3,0.8] and the graphs Gm are the same between subfigures.

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    • Figure 7
      Figure 7

      On the left y axis the prevalence of a NIMFA process on a time-varying network (solid blue) is compared with the prevalence of the “quenched prediction” of each interval [tm−1,tm) starting from y∞(Gm−1) (dashed red) with interupdate times Δt=10. When y∞(Gm−1)=0, we set y(tm)=ru instead. On the right y axis the absolute error (dotted black) between the prediction and the process is shown. Graphs are specifically chosen for the epidemic to be both above and below the epidemic threshold on different intervals. The issue illustrated in Lemma t1 is visible: if the prevalence decreases much below y=ru, the error in the prediction is large.

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    • Figure 8
      Figure 8

      Error between the prevalence y(t) and the steady-state prevalence y∞ [approximated by y(tmax)] vs time. The dashed red line is the function y(t)=11+t. Each of the blue curves corresponds to a decay process starting in y(0)=u on different ER(N,p) graphs where p∼Unif(0,1). The effective infection rate τ is varied for the different figures.

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    • Figure 9
      Figure 9

      Comparison of the analytically derived bounds and numerically determined upper-transition time T¯(r), with r=10−4. The upper-transition time T¯(r) is shown for y(0)=1 in black with T¯D for decay and for y(0)=r in red with T¯G for growth. The blue lines are the bounds on decay (black) and the green lines are the bounds on growth (red). The lower bounds are indicated with the symbol Lx and the upper bounds are indicated with Ux, where x is D for decay or G for growth. The upper bound UD is given by Eq. (13) for R0<R and (11) for R0≥R. Here the intersection point R is given by (18). The lower bound LD is given by (28), and the lower bound LG is given by (26). The upper bound UG is given by (20).

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    • Figure 10
      Figure 10

      On the left y axis the prevalence of a NIMFA process on a time-varying network (solid blue) is compared with the prevalence of the “quenched prediction” of each interval [tm−1,tm) starting from y∞(Gm−1) (dashed red). The interupdate times Δt equal the (modified) upper bound (23) in the top figure and the lower bound (28) in the bottom figure. On the right y axis the absolute error (dotted black) between the prediction and the process is shown. Each graph Gm is an ER graph with link density p∈[0.3,0.8] and the graphs Gm are the same between subfigures.

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    • Figure 11
      Figure 11

      The upper-transition time T¯(r) vs basic reproduction number R0 for three different random graph types. Shown is the decay process starting in the all-infected initial state vector V(0)=u, for N=50 and β=0.1. In blue, 3000 Erdős-Rényi (ER) graphs with uniformly distributed parameters [Gp(N) with p∼ Unif(0,1)]. In black, 3000 Barabási-Albert (BA) graphs with uniformly distributed parameters (m0∼ Unif{1,N} and m∼ Unif{1,m0}). In red, 3000 Watts-Strogatz (WS) graphs with uniformly distributed parameters [K∼ Unif{1,⌊N−12⌋} and βWS∼ Unif(0,1)]. In the inset, the same figure with log-linear axes is shown. In log scale, for R0>2, there is a significant difference from BA graphs to ER and WS graphs, which are on top of each other and below the BA graphs.

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    • Figure 12
      Figure 12

      The derivative convergence time t* vs the basic reproduction number R0 for different values of the starting prevalence y(0). The chosen parameters are a graph size N=50, accuracy tolerance r*=10−7, infection rate β=0.1, and step size h=0.01. Each graph is a ER(N,p) graph with p∼Unif(0,1). For R0>1, the small values of y(0) are on top. For R0<1, the small values of y(0) are at the bottom. The inset shows the dips below the curves in more detail.

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    • Figure 13
      Figure 13

      Comparison of the upper-transition time T¯(r) [thick (dark) blue lines] and the derivative convergence time t*(r*) [thin (light) red lines] for different values of r and r*. The initial infection probability vector V(0)=u and time-step parameter h=0.01 are constant throughout all simulations. The values of the accuracy tolerance r are 10−1,10−2,10−3,10−4 and the values of the accuracy tolerance r* are 10−1,10−2,⋯,10−7,10−8.

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