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

...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: $R_0=\beta/\gamma\le1$:
  epidemic vanishes
• Supercritical: $R_0>1$:
  exponential incline, long survival
• Stationary solution

Mean-field model:

• $S(t), I(t), T(t)\in\mathbb{R}$
• well-mixed population

Network approach

Temporary Immune

Choice of network

Geometric Inhomogeneous Random Graph

• Spatially induced clustering,
  heavy-tailed degree distribution.
   
• $n$ nodes uniformly in $[0,\sqrt{n}]^{2}$.
• Power-law weight/fitness for vertices:
   
   
• Condition on weights & locations ($\alpha>1$):

• Smaller $\tau$: more variability in weights/degrees.
• Smaller $\alpha$: more long-range connections.

Temporary immunity:

Two types of supercriticality

Parameters:

$\beta=.225$,

$\gamma=.2$

mean deg. $\approx 9$

$\eta=0.001$

$\eta=0.005$

$=\eta$

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 $\tau>3$

• 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 ($\tau\in(2,3)$)

Critical immune duration decreases under interventions (I)

Graph with power-law exponent $\tau>3$

• 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 $\tau\in(2,3)$

• 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

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 $\beta:=\beta(t), \gamma:=\gamma(t), N:=N(t)$.
• Unclear how and why interventions work
• What is the most effective intervention?

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

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