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arxiv: 2605.19933 · v1 · pith:CJG6EDIHnew · submitted 2026-05-19 · 🧮 math.PR

A Tight Epidemic Threshold for Competing Stochastic Infection Processes with Mutually Exclusive Immunity

Nicolas Klodt , Martin S. Krejca This is my paper

Pith reviewed 2026-05-20 04:33 UTC · model grok-4.3

classification 🧮 math.PR
keywords IRIR processepidemic thresholdcompeting infectionssurvival timesupermartingaleperfectly mixed graphsjumbled graphsSIR models
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The pith

Two competing infections with mutually exclusive immunity exhibit a tight epidemic threshold on perfectly mixed graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The IRIR process consists of two SIR infections that compete on the same graph, with each vertex immune only to the infection type it most recently contracted. The central result is a tight epidemic threshold on perfectly mixed graphs: the survival time until both infections are gone switches from at most quasi-linear in the number of vertices n to at least super-polynomial in n. This threshold applies when the graph has uniform edge density p for every nonempty subset of vertices. The super-polynomial persistence also holds on the broader class of jumbled graphs, which encompasses typical Erdős–Rényi graphs above a certain density.

Core claim

Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size n to at least super-polynomial in n. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value p ∈ (0, 1]. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of k∈ω(ln²(n)). The proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a superma

What carries the argument

A potential function derived from a Lyapunov function of the transition equilibrium that converts IRIR configurations into a supermartingale with drift in a large region.

If this is right

  • Below the threshold the survival time of the IRIR process is at most quasi-linear in n on perfectly mixed graphs.
  • Above the threshold the survival time is at least super-polynomial in n with high probability.
  • The super-polynomial lower bound carries over to jumbled graphs that include Erdős–Rényi graphs of average degree omega(log squared n).
  • A systematic derivation produces the required potential from a Lyapunov function of the equilibrium of the transition rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In networks that closely approximate uniform edge density, competing pathogens with type-specific immunity may persist far longer than single-infection models predict.
  • The supermartingale construction from equilibrium Lyapunov functions may extend to persistence questions in other multi-state Markov chains on graphs.
  • Finite-n simulations should reveal a sharp increase in observed survival times as the control parameter crosses the threshold value.

Load-bearing premise

There exists a potential function that converts the IRIR configurations into a supermartingale with drift in a sufficiently large region of the state space.

What would settle it

Running the IRIR process many times on a large perfectly mixed graph with parameters just above the threshold and finding that survival time stays polynomial in n in a positive fraction of runs would disprove the super-polynomial lower bound.

read the original abstract

Stochastic infection processes are continuous-time Markov chains on graphs that assign each vertex one of multiple states, such as susceptible, infected, or recovered. Depending on the model, vertices change their state based on random transition rates and the states of their neighbors, resulting in a variety of complex dynamics. The body of rigorous literature is rich for processes that consider a single infection. In contrast, the setting with at least two infections, where the same state exists for different types, allows for far more transition combinations, leaving several interesting models entirely unexplored. We address this shortcoming in the literature by defining the IRIR process, in which two SIR processes run on the same graph and each vertex is immune only to its most recent infection. We study the survival time of the IRIR process, that is, the time until no infected vertex remains, with mathematical rigor. Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size $n$ to at least super-polynomial in $n$. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value $p \in (0, 1]$. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of $k\in \omega(\ln^2(n))$. Our proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a supermartingale with drift in a large region, implying the lower bound. We detail how to systematically derive such a potential, based on a Lyapunov function of the transition equilibrium of the process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript defines the IRIR process, a continuous-time Markov chain on a graph in which two SIR-type infections compete and each vertex acquires immunity exclusively to its most recent infection. On perfectly mixed graphs (constant edge density p for every nonempty vertex subset) the authors prove a tight epidemic threshold: below the threshold the survival time is at most quasi-linear in n, while above it the survival time is at least super-polynomial in n with high probability. The super-polynomial lower bound is obtained by constructing a potential V from a Lyapunov function L of the deterministic mean-field equilibrium; the resulting process is shown to be a supermartingale with negative drift on a large region of state space. The argument extends to jumbled graphs and therefore applies to Erdős–Rényi graphs with average degree ω(log² n).

Significance. If the supermartingale construction is uniform, the result supplies the first rigorous tight threshold for competing infections with mutually exclusive immunity on dense random graphs. The systematic derivation of the potential from the mean-field Lyapunov function is a methodological contribution that may extend to other multi-type infection models. The explicit applicability to Erdős–Rényi graphs with moderately dense regimes strengthens the practical relevance.

major comments (1)
  1. [Proof of the super-polynomial lower bound (via potential construction from mean-field Lyapunov function)] The central lower-bound argument asserts that the potential V, built from the Lyapunov function L of the transition equilibrium, yields a supermartingale with strictly negative drift on a 'large region' of configurations. Because the super-polynomial survival claim rests on this property holding with high probability, the manuscript must explicitly control the second-order terms arising from the continuous-time Markov chain generator when empirical densities fluctuate by O(1/√n) or when one infection type dominates. Without a uniform bound on these fluctuation terms (or an explicit error analysis showing they remain o(1) relative to the negative drift), the supermartingale property may fail on a set of positive probability, undermining the claimed threshold.
minor comments (2)
  1. [Abstract] The abstract states that the result applies to 'perfectly mixed graphs' with density p ∈ (0,1]; a brief remark clarifying whether p may depend on n (while remaining bounded away from 0 and 1) would help readers interpret the regime.
  2. [Statement of main results] The extension to jumbled graphs is mentioned but not quantified; stating the precise discrepancy tolerance (e.g., o(1) or O(1/log n)) would make the Erdős–Rényi corollary fully self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the supermartingale construction. We address the concern below and will incorporate the requested clarification in the revision.

read point-by-point responses

  1. Referee: The central lower-bound argument asserts that the potential V, built from the Lyapunov function L of the transition equilibrium, yields a supermartingale with strictly negative drift on a 'large region' of configurations. Because the super-polynomial survival claim rests on this property holding with high probability, the manuscript must explicitly control the second-order terms arising from the continuous-time Markov chain generator when empirical densities fluctuate by O(1/√n) or when one infection type dominates. Without a uniform bound on these fluctuation terms (or an explicit error analysis showing they remain o(1) relative to the negative drift), the supermartingale property may fail on a set of positive probability, undermining the claimed threshold.

    Authors: We agree that an explicit uniform bound on the second-order fluctuation terms is needed to complete the argument. In the revised manuscript we will add a dedicated error-analysis subsection. Using standard concentration bounds on the empirical densities together with a case distinction near the axes (where one infection dominates), we will show that the generator error is at most O(1/√n) while the negative drift inherited from the mean-field Lyapunov function remains bounded below by a positive constant on the region of interest. The resulting supermartingale inequality then holds with high probability, preserving the super-polynomial lower bound. revision: yes

Circularity Check

0 steps flagged

No circularity: potential derived systematically from mean-field Lyapunov equilibrium

full rationale

The lower-bound proof constructs a potential V from a Lyapunov function L of the deterministic transition equilibrium (mean-field ODE) and verifies supermartingale drift on a large region for perfectly mixed graphs. This is a standard, non-circular technique that uses external graph-density properties and does not reduce the claimed threshold to a fitted parameter, self-definition, or self-citation chain. The super-polynomial survival follows directly from the supermartingale property without the result being presupposed in the construction. No load-bearing self-citations or ansatz smuggling appear in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the model definition of the IRIR process as two coupled continuous-time Markov chains with mutually exclusive recent-immunity and on standard martingale theory applied to graph-density assumptions for perfectly mixed graphs.

axioms (2)
  • domain assumption The IRIR process is a continuous-time Markov chain whose transitions depend on the states of neighboring vertices and the most recent infection type at each vertex.
    This is the foundational definition of the process under study.
  • domain assumption Perfectly mixed graphs have uniform edge density p between every non-empty subset of vertices.
    This graph property is required for the threshold statement and the supermartingale construction.
invented entities (1)
  • IRIR process no independent evidence
    purpose: Model two competing SIR infections on the same graph with immunity restricted to the most recent infection type.
    Newly introduced model to capture mutually exclusive immunity not covered by prior single-infection or standard multi-infection processes.

pith-pipeline@v0.9.0 · 5868 in / 1716 out tokens · 74205 ms · 2026-05-20T04:33:06.225478+00:00 · methodology

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