ETH Zürich, Combinatorics meeting
Joost Jorritsma,
joint with Tim Hulshof and Júlia Komjáthy
University of Technology Eindhoven
Normally, I'd talk about...
Random Graphs (Probability Theory) with
-
Temporal Structure
Preferential Attachment Graphs -
Spatial Structures
Geometric Inhomogeneous Random Graph -
Stochastic Processes on random graphs
e.g. Random Walk/Epidemic Models - Theoretical focus, not directly applicable
Júlia Komjáthy, Tim Hulshof
https://arxiv.org/abs/2005.06880
...is about...
- Theoretical modeling of epidemics similar to COVID-19
- Qualitative comparisons
- Effect of:
- network topology
- space/geometry
- interventions
- temporary immunity
...is not about...
- Predictions/data oriented modeling for COVID-19
- Quantitative analysis
This talk...
Compartmental model
Two phases:
- Subcritical: R0=β/γ≤1:
epidemic vanishes -
Supercritical: R0>1:
exponential incline, long survival - Stationary solution
Susceptible
Infected
Temporary Immune
βI(t)/N
γ
η
Mean-field model:
- S(t),I(t),T(t)∈R
- well-mixed population
Susceptible
Infected
Infect neighbor w.p. β
w.p. γ
w.p. η
Network approach
Temporary Immune
Choice of network
Geometric Inhomogeneous Random Graph
- Spatially induced clustering,
heavy-tailed degree distribution.
- n nodes uniformly in [0,n]2.
- Power-law weight/fitness for vertices:
- Condition on weights & locations (α>1):
P(x↔y∣Wx,Wy)≍min{1,c(∥x−y∥2WxWy)α}.
\mathbb{P}(x\leftrightarrow y \mid W_x, W_y) \asymp \min\Big\{1, c \left(\frac{W_xW_y}{\|x-y\|^2}\right)^\alpha\Big\}.
P(W≥w)≍w−(τ−1),τ>2.
\mathbb{P}(W\ge w) \asymp w^{-(\tau-1)}, \qquad \tau>2.
- Smaller τ: more variability in weights/degrees.
- Smaller α: more long-range connections.
Temporary immunity:
Two types of supercriticality
Parameters:
β=.225,
γ=.2
mean deg. ≈9
τ>3
τ∈(2,3)
η=0.001
η=0.005
=η
Modeling interventions: remove edges
- Remove edges before start epidemic
- Resulting epidemic still supercritical
- "Rule" per intervention strategy:
- Social dist.: Random
- No travel: Long edges
- No hubs: Removing edges from large-degree nodes
- Match intervention parameters s.t. average degree after intervention is comparable (approx. 60% edges remain)
No interv.
Social dist.
No travel
No hubs
No travel works best for the first peak
Graph with power-law exponent τ>3
- Travel restrictions bring down first peak, but elongate the outbreak
- Absence of long-range edges: Average graph distance increases from logarithmic to polynomial in network size
- First peak insensitive for immunity
- Similar figure for graph G1 (τ∈(2,3))
Critical immune duration decreases under interventions (I)
Graph with power-law exponent τ>3
- Critical immunity length is shortest under limitation of degree and travel restrictions.
- Absence of hubs: larger oscillations
- First peak for travel restrictions is as high as second peak under degree limitation
Critical immune duration decreases under interventions (II)
Graph with power-law exponent τ∈(2,3)
- Without intervention the system immediately equilibriates.
- Critical immunity length is smallest under limitation of degree, travel restrictions outperform social distancing.
- No oscillations for social distancing. Presence of larger hubs brings the system to equilibrium quicker.
Main conclusions
- Epidemics on networks with underlying geometry show richer behaviour than compartmental models.
- Network topology plays a profound role.
- Networks allow for intuitive modeling of interventions.
- Travel restriction works best but they elongate epidemic.
- Limiting node degree might result in high second peak
Thank you for your attention!
https://arxiv.org/abs/2005.06880
Long survival probability under interventions
Higher degree-spread diminishes amplitude
No travel works well for later peaks
Higher degree-spread diminishes amplitude
No travel works well for later peaks
Higher degree-spread diminishes amplitude
Compartmental model
- Additional states could be added
- "Mean-field" model: everything is averaged out:
individuals are equal, everybody is weakly connected, individuals are "real-valued"
- Only way of modeling interventions:
Set β:=β(t),γ:=γ(t),N:=N(t). - Unclear how and why interventions work
- What is the most effective intervention?
Susceptible
Infected
Recovered
βNI(t)S(t)
\beta \frac{I(t)S(t)}{N}
γ
\gamma
Network-based approach
Susceptible
Infected
Recovered
Infect neighbor w.p. β
Heal w.p. γ
- Contact network as a graph
- Infectious nodes infect neighbors
- Models "coincide" for complete graph and scaling β
- Interventions: remove edges
Network-based approach
Susceptible
Infected
Recovered
Heal w.p. γ
- Contact network as a graph
- Infectious nodes infect neighbors
- Models "coincide" for complete graph and scaling β
- Interventions: remove edges
- At random (social dist.)
Infect neighbor w.p. β
Network-based approach
Susceptible
Infected
Recovered
Heal w.p. γ
- Contact network as a graph
- Infectious nodes infect neighbors
- Models "coincide" for complete graph and scaling β
- Interventions: remove edges
- At random (social dist.)
- Euclidean length (no travel)
Infect neighbor w.p. β
Network-based approach
Susceptible
Infected
Recovered
Heal w.p. γ
- Contact network as a graph
- Infectious nodes infect neighbors
- Models "coincide" for complete graph and scaling β
- Interventions: remove edges
- At random (social dist.)
- Euclidean length (no travel)
- Degree (limit # contacts)
Infect neighbor w.p. β
ODE approach
Susceptible
Infected
Recovered
Infect neighbor w.p. β
Heal w.p. γ
Susceptible w.p. η
- Immunity unclear for COVID-19
- Natural model for "second" peaks
Network approach
Network topology
Two graphs with mean degree 8
Network approach
Network topology
Mean degree 8, 160000 nodes
Network approach
Network topology
Two graphs with mean degree 8
Polynomial growth
"Faster" growth
Network approach
Network topology
Desired features
- "close" nodes are
likely to be connected - High variability in degrees
- "Important" nodes are
likely to be connected
Random graph model (α>1):
pij≈c(∥xi−xj∥2WiWj)α
p_{ij} \approx c\Big(\frac{W_iW_j}{\|x_i-x_j\|^2}\Big)^\alpha
Results
- Network vs. Compartmental
- New phases
- Oscillations
-
Effect of social dist., no travel, no hubs
- How to compare?
- First peak
- Later peaks
- Probability of survival
Supercritical epidemic dies out in network for η small
Single peak, extinction
large graphs, 100 runs
Survival
Four networks
Compartmental,
η small and large
Geometry introduces "immunity boundaries", herd immunity
Supercritical epidemic can survive in network for η large
Four networks
Compartmental,
η small and large
Single peak, survival
large graphs, 100 runs
Survival
Supercritical epidemic can survive in network for η large
Oscillations in three networks
Absence of long edges: larger amplitude
Results
-
Network vs. Compartmental
- New phases
- Oscillations
-
Effect of social dist., no travel, no hubs
- How to compare?
- First peak
- Later peaks
- Probability of survival
Not all interventions are equal for the second peak ETH Z ü rich, Combinatorics meeting Joost Jorritsma, joint with Tim Hulshof and Júlia Komjáthy University of Technology Eindhoven
Combinatorics meeting: Corona Interventions on GIRGs
By joostjor
Combinatorics meeting: Corona Interventions on GIRGs
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