Roberto Imbuzeiro Oliveira (IMPA)
with Gabriel Leite Baptista da Silva (Groningen)
and Daniel Valesin (Groningen)
Quantitative Methods Seminar @ Krannert
Nov. 30th, 2021
The contact process
A graph with healthy and infected vertices.
The contact process
Infected vertices become healthy at rate 1 (Poisson process).
The contact process
The contact process
No contagion because vertex is already infected.
The contact process
Vertex becomes healthy.
The contact process
Nothing happens because already healthy.
The contact process
Nothing happens because starting point is healthy.
The contact process
Contagion
Definition
Why study the contact process?
Simple model for contagion and epidemics
Susceptible – Infected – Susceptible (SIS) model with spatial component
More generally: a model for “stuff that spreads” but also tends to disappear
Basic facts
Epidemic threshold
Large finite boxes
Random regular graphs
Power-law graphs are epidemic-prone
What happens if the graph changes over time?
Does it make it easier or harder for the epidemic to survive?
Switching
Switching connects all graphs with given degree sequence
Switching dynamics
Main result (modulo **)
Intuition: a branching process?
Each individual
has children
according to a fixed
probability
distribution,
Independently.
Maybe contact is
like this, but with
deaths.
(contagion tree)
A branching process with deaths?
Not quite, because of:
However, switching graph makes collisions less common.
Interpretation
Deriving “practical” insights from toy models of epidemics seems in very bad taste these days.
But maybe one can say this:
a randomly changing graph seems to make stuff spread faster.
Remainder of talk
Background and local models
Previous results for finite and infinite static graphs. “Local models.”
The local model in the dynamical graph setting
Definition, analysis, strict inequality for critical parameter.
From local model to actual dynamics in a finite graph
Only a brief sketch.
Background and local models
Infinite graphs
This is where the literature starts.
Allow for true phase transitions between certain death x survival.
Infinite lattices
Strong survival: finite sets reinfected infinitely many times.
Infinite trees: intermediate phase
Weak survival: goes extinct over finite sets, but not over full tree
From infinite to finite graphs
Large boxes
Large finite boxes (restatement)
Random regular graphs
Random regular graphs (restatement)
Recall main result for today (modulo **)
Main proof steps in our case
Find a local limit and study it
This is the herds process described next.
Prove strict inequality for critical parameter
Edge rewiring is a “branching mechanism”.
Prove something similar for process with truncated trees
Makes the last step easier.
Compare “locally” with dynamics in the finite graph.
The local limit of the process over the dynamical graph
Heuristics
Another tree in the graph.
A large tree in the graph
Switch edges?
Swap subtrees!
The local limit
Herd splits
Herd process = contact + branching
A new herd evolves independently from all other herds.
In particular, it may also split into new herds.
Built-in branching mechanism.
Makes it easier for the epidemic to survive.
If contact creates “many” particles, herds survive forever.
Contact process is dominated
Key result about herds process
Proof sketch (I)
Proof sketch (II)
From the herds process to finite graphs
Proof steps
Find a local limit and study it
In our case this is the herds process, a new process described next.
Prove strict inequality for critical parameter
Edge rewiring serves as a kind of “branching mechanism”.
Prove something similar for process with truncated trees
Makes the last step easier.
Compare “locally” with dynamics in the finite graph.
Truncated herds process
Herd creation is more complicated
Key result about h-herds process
Proof idea: if herds survives forever, there’s a positive chance of creating many particles within subtrees of bounded diameter 2h. Branching guarantees h-herds can survive forever.
Sketch of end of proof
Find a local limit and study it
In our case this is the herds process, a new process described next.
Prove strict inequality for critical parameter
Edge rewiring serves as a kind of “branching mechanism”.
Prove something similar for process with truncated trees
Makes the last step easier.
Compare “locally” with dynamics in the finite graph.
Can show that there is enough contagion between “nice sets of bounded diameter” . Multitype branching process & stuff.
Conclusion
and some open problems
Our main result
Proof goes through novel herd process over infinite graphs.
General idea of finding a local limit for the process,
but nontrivial truncation of the process is required.
Open problems
Thank you!