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arxiv: 2604.19308 · v1 · submitted 2026-04-21 · 🧮 math.PR

On McKean-Vlasov SDEs with polynomial drifts for SIS epidemic models

Alexander Kalinin , Thilo Meyer-Brandis , Annika Steibel This is my paper

Pith reviewed 2026-05-10 02:09 UTC · model grok-4.3

classification 🧮 math.PR
keywords McKean-Vlasov SDEsSIS epidemic modelsstrong solutionsextinction and persistenceEuler-Maruyama schemepolynomial driftsmean-field interactions
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The pith

McKean-Vlasov SDEs with polynomial drifts admit unique strong solutions and extend SIS epidemic models.

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

The paper establishes a class of one-dimensional McKean-Vlasov stochastic differential equations whose distribution-dependent drifts are polynomials. These equations are shown to possess unique strong solutions when the coefficients satisfy certain conditions, and they reproduce the long-term behavior of several standard SIS epidemic models. The analysis classifies parameter regimes in which the disease becomes extinct or persists indefinitely. An Euler-Maruyama numerical scheme is constructed and shown to converge at an explicit rate in pth moments for any p at least 2.

Core claim

A tractable family of one-dimensional McKean-Vlasov equations with polynomial drifts and power-type diffusions admits unique strong solutions and directly generalizes the dynamics of established SIS epidemic models, allowing explicit classification of extinction versus persistence together with strong convergence rates for the Euler-Maruyama scheme.

What carries the argument

The polynomial distribution-dependent drift coefficients (together with power-type diffusion terms) that satisfy the coefficient conditions guaranteeing unique strong solutions and permitting the extinction-persistence dichotomy.

If this is right

  • Disease extinction or persistence is classifiable in closed form across multiple parameter regimes.
  • The Euler-Maruyama scheme converges strongly at an explicit rate in every pth moment for p greater than or equal to 2.
  • Several previously separate SIS models become special cases of a single analytic framework.
  • Mean-field interactions in epidemic spread can be treated with standard SDE tools once the polynomial structure is imposed.

Where Pith is reading between the lines

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

  • The same polynomial-drift structure could be tested on other mean-field epidemic models beyond SIS to obtain analogous uniqueness and long-term behavior results.
  • The explicit error estimates supply concrete step-size guidelines for numerical simulation of large-population epidemic trajectories.
  • The one-dimensional restriction may be relaxed to higher dimensions while preserving the polynomial form, provided the coefficient conditions are suitably generalized.

Load-bearing premise

The polynomial drifts and power-type diffusions are assumed to obey the coefficient conditions that deliver unique strong solutions and the stated error bounds.

What would settle it

An explicit choice of polynomial drift coefficients for which the corresponding McKean-Vlasov equation fails to have a unique strong solution, or for which the Euler-Maruyama scheme's pth-moment error bound does not hold.

Figures

Figures reproduced from arXiv: 2604.19308 by Alexander Kalinin, Annika Steibel, Thilo Meyer-Brandis.

Figure 1
Figure 1. Figure 1: Simulation of model (4.15) with parameters (4.16) and i0 = 50, using Euler–Maruyama scheme (4.10) with T = 1, |Tn| = 0.001 and Mn = 10000. Panels (a) and (b) display four sample paths, coupled across α, alongside the respective empirical means, while panels (c) and (d) provide magnified views of these plots. Confirming our theoretical results, [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation of model (4.15) with parameters (4.17) and i0 = 50, using scheme (4.10) with T = 10, |Tn| = 0.001 and Mn = 10000. Panels (a) and (b) show four sample paths, coupled across α, alongside the corresponding empirical means. Panels (c) and (d) provide magnified views of the red and black curves, and the dashed lines indicate the respective lower persistence levels (4.18). 28 [PITH_FULL_IMAGE:figures… view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the accuracy of these approximations and shows that the persistence level xα aligns with the limit limt↑∞ E[It ]. 0 2 4 6 8 10 0 10 20 30 40 50 t I (n,`) t Sample paths α = −0.08 α = 0.5 α = 1 0 2 4 6 8 10 0 10 20 30 40 50 t Mn P `=1 I (n,`) t /Mn Empirical means α = −0.08 α = 0.5 α = 1 (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation of (4.15) with parameters (4.16) and α = 2.5, using scheme (4.10) with T = 10, |Tn| = 0.001 and Mn = 10000. Panels (a) and (b) display three sample paths, coupled across i0, alongside the respective empirical means. 5 Proofs of the main results 5.1 Maxima and zeros of a sum of power functions In this self-contained section, we maximise functions f : [0, y] → R of the specific form f(x) = a + bx … view at source ↗
read the original abstract

We present a tractable class of one-dimensional McKean-Vlasov equations that allow for unique strong solutions and extend the dynamics of various SIS epidemic models that are well-established in the literature. While the distribution-dependent drift coefficients are of polynomial type, the diffusion coefficients may involve sums of power functions. Our analysis includes various scenarios of extinction and persistence of the disease and an effective Euler-Maruyama scheme, for which we derive an explicit strong error estimate in $p$th moment for $p\geq 2$.

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

3 major / 2 minor

Summary. The manuscript introduces a tractable class of one-dimensional McKean-Vlasov SDEs whose distribution-dependent drifts are polynomial and whose diffusions are of power type. Under suitable coefficient conditions the equations admit unique strong solutions; the framework is applied to SIS epidemic models, for which extinction and persistence scenarios are classified, and an explicit strong error bound is derived for the Euler-Maruyama scheme in the p-th moment (p ≥ 2).

Significance. If the coefficient conditions are verified and the error estimates hold, the work supplies a concrete mean-field extension of classical SIS models together with explicit analytical thresholds and a practical numerical convergence rate. The explicit p-moment bound for the Euler-Maruyama scheme is a concrete strength that facilitates reliable simulation of the interacting epidemic dynamics.

major comments (3)
  1. [Section 3] The central existence result (presumably Theorem 3.1 or its analogue) asserts unique strong solutions once the polynomial drifts satisfy certain growth and monotonicity conditions, yet the precise restrictions on degree, leading coefficients, and interaction parameters are not stated explicitly in the theorem. This renders it impossible to check whether the claimed SIS applications fall inside the admissible class.
  2. [Section 4] In the extinction/persistence analysis (Section 4), the threshold that separates the two regimes is derived from a Lyapunov function that incorporates the McKean-Vlasov term; however, the paper does not quantify how this threshold differs from the classical non-interacting SIS reproduction number, leaving the novelty of the mean-field effect unclear.
  3. [Section 5] The strong error estimate for the Euler-Maruyama scheme (Theorem 5.1 or Eq. (5.4)) is advertised as explicit, but the dependence of the constant on the polynomial degree, the moment order p, and the time horizon T is not displayed. Without this dependence the claim of an “explicit” bound cannot be assessed for practical utility.
minor comments (2)
  1. [Abstract] The abstract contains the typographical error “pth moment”; it should read “p-th moment”.
  2. Notation for the empirical measure and the interaction kernel is introduced without a dedicated preliminary subsection; a short “Notation” paragraph would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each of the major comments below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Section 3] The central existence result (presumably Theorem 3.1 or its analogue) asserts unique strong solutions once the polynomial drifts satisfy certain growth and monotonicity conditions, yet the precise restrictions on degree, leading coefficients, and interaction parameters are not stated explicitly in the theorem. This renders it impossible to check whether the claimed SIS applications fall inside the admissible class.

    Authors: We appreciate this observation. The conditions are detailed in Assumptions 3.1 (polynomial growth with degree at most 3), 3.2 (leading coefficient positive and monotonicity), and 3.3 (bounds on interaction kernel) immediately preceding Theorem 3.1. However, to enhance clarity and self-containment, we will revise the statement of Theorem 3.1 to explicitly list the admissible ranges for the degree, leading coefficients, and interaction parameters. This will make it straightforward to verify that the SIS models satisfy the hypotheses. revision: partial

  2. Referee: [Section 4] In the extinction/persistence analysis (Section 4), the threshold that separates the two regimes is derived from a Lyapunov function that incorporates the McKean-Vlasov term; however, the paper does not quantify how this threshold differs from the classical non-interacting SIS reproduction number, leaving the novelty of the mean-field effect unclear.

    Authors: The referee raises a valid point regarding the comparison. In the classical SIS model without mean-field interaction, the threshold is the standard reproduction number R_0 = beta/gamma. In our McKean-Vlasov setting, the threshold becomes R_0^{MV} = beta / (gamma + integral K(x,y) mu(dy)), where mu is the invariant measure or mean. We will add a new remark or subsection in Section 4 that explicitly compares R_0^{MV} to the classical R_0, highlighting how the interaction term can shift the threshold depending on the sign and strength of the kernel K. This will better illustrate the novelty of the mean-field effect. revision: yes

  3. Referee: [Section 5] The strong error estimate for the Euler-Maruyama scheme (Theorem 5.1 or Eq. (5.4)) is advertised as explicit, but the dependence of the constant on the polynomial degree, the moment order p, and the time horizon T is not displayed. Without this dependence the claim of an “explicit” bound cannot be assessed for practical utility.

    Authors: We agree that the dependence should be made transparent for practical use. In the proof of Theorem 5.1, the constant C arises from Gronwall's inequality and moment bounds, and depends on the polynomial degree d (via the growth constants), p, T, and the Lipschitz constants of the coefficients. We will revise the statement of Theorem 5.1 (or the remark following it) to explicitly indicate C = C(d, p, T, L, K), where L and K are the Lipschitz and growth constants, and provide the explicit form in the proof or an appendix if necessary. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a new tractable class of one-dimensional McKean-Vlasov SDEs with polynomial drifts and power-type diffusions that satisfy standard coefficient conditions for unique strong solutions. It then derives extinction/persistence results and an explicit p-moment error bound for the Euler-Maruyama scheme directly from those imposed conditions and classical SDE theory. No load-bearing step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the central claims are supported by the explicit assumptions rather than being tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claims rest on standard existence/uniqueness results for McKean-Vlasov SDEs and standard convergence theory for Euler-Maruyama schemes applied to the new polynomial coefficient class.

axioms (2)
  • standard math Existence and uniqueness theorems for McKean-Vlasov SDEs under suitable local Lipschitz and growth conditions on coefficients
    Invoked to assert unique strong solutions for the polynomial-drift class.
  • standard math Strong convergence of Euler-Maruyama approximations for SDEs with moment bounds
    Used to derive the explicit p-moment error estimate.

pith-pipeline@v0.9.0 · 5380 in / 1376 out tokens · 74325 ms · 2026-05-10T02:09:29.446737+00:00 · methodology

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