arxiv: 2604.10742 · v1 · submitted 2026-04-12 · ⚛️ physics.soc-ph · math.DS· nlin.AO

Recognition: unknown

Epidemic Transmission Modelling on the Birth-death Evolving Network with Indirect Contacts

Minyu Feng , Yuhan Li , J\"urgen Kurths

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:34 UTC · model grok-4.3

classification ⚛️ physics.soc-ph math.DSnlin.AO

keywords epidemic transmissionbirth-death evolving networkindirect contactsSIRS epidemicpopulation migrationMarkovian queueing networknetwork co-evolution

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The pith

Newly-created indirect contacts on birth-death evolving networks facilitate epidemic transmission in SIRS models.

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

The paper studies the impact of a network that adds and removes nodes over time on the spread of an SIRS epidemic, where individuals cycle through susceptible, infected, recovered, and susceptible states again. This birth-death process models population migration and the formation of indirect contacts between people who do not share a direct link. The authors build a Markovian queueing network to track changes in the sizes of the different epidemic compartments as the underlying network evolves. Simulations reveal that the new indirect contacts created by the network dynamics increase epidemic transmission compared to static networks.

Core claim

On the birth-death evolving network with heritable node deletion, the SIRS epidemic process experiences facilitated transmission because newly-created indirect contacts increase opportunities for infection, as shown by simulations tracking population sizes in each epidemic state via a Markovian queueing network model.

What carries the argument

The birth-death evolving network with heritable node deletion and indirect contacts, which models population migration and enables Markovian queueing analysis of epidemic state populations.

If this is right

Population migration modeled through node birth and heritable deletion changes the sizes of susceptible, infected, and recovered groups over time.
Newly-created indirect contacts increase the rate at which the epidemic spreads compared to networks without such dynamics.
The co-evolution of the network and the epidemic process must be modeled jointly rather than assuming a fixed network structure.
Analyses of the Markovian queueing network yield explicit expressions for the variation of population sizes across epidemic states.

Where Pith is reading between the lines

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

Real-world epidemic forecasts may underestimate spread if they ignore how migration creates extra indirect contacts.
The same network mechanism could be applied to study other cyclic epidemic models or diseases with longer infectious periods.
Public health interventions targeting migration patterns might reduce the formation of these facilitating contacts.

Load-bearing premise

The birth-death evolving network with heritable node deletion and indirect contacts sufficiently represents real individual behaviors such as population migration to support conclusions about epidemic facilitation.

What would settle it

A simulation in which the indirect contact mechanism is disabled while keeping all other network birth-death rules fixed shows no increase in epidemic transmission rates or infected population sizes.

Figures

Figures reproduced from arXiv: 2604.10742 by J\"urgen Kurths, Minyu Feng, Yuhan Li.

**Figure 1.** Figure 1: Illustration of the network evolution and epidemic spreading on the evolving network. Sub-figure (a) presents the process of a leaving individual transferring their connections and creating indirect contacts with other individuals. The individual colored green is a newly-coming one who connects to an existing individual a’. The individual colored purple (labeled as a) is the one who are leaving, and dashed… view at source ↗

**Figure 2.** Figure 2: Illustration of a queueing network of epidemic states. There are three service centers representing the three epidemic states 𝑆, 𝐼, and 𝑅 marked by square boxes. Arrows between two service centers indicate transitions from one epidemic state to another, and the Greek alphabets (𝛽, 𝛾, and 𝛼) above arrows indicate transition rates between two epidemic states. For each service center, there is an arrow toward… view at source ↗

**Figure 3.** Figure 3: The population size varying with time in state 𝑆, 𝐼, and 𝑅 with different values of the inflow rate 𝜆 and outflow rate 𝜇. In sub-figure (a), we set three pairs different values of the inflow 𝜆 and outflow rate 𝜇, (3, 0.05), (1.5, 0.025), (0.75, 0.0125). In these settings, the population size of each state fluctuates horizontally. (b) and (c) present a rising tendency in the population size of state 𝐼 and 𝑅… view at source ↗

**Figure 4.** Figure 4: The number of individuals in the three epidemic compartments 𝑆, 𝐼, and 𝑅 varying with time with different values of the infected rate 𝛽, the recovered rate 𝛾, and the reviving rate 𝛼. 𝜆 and 𝜇 are set as 1.5 and 0.025 in each sub-figure. The initial setting in (a) is 𝛽 = 0.005, 𝛾 = 0.7, 𝛼 = 0.4. 𝛾 is halved in (b) and doubled in (c), and 𝛼 is halved in (d) and doubled in (e). 𝛽 is heightened to 0.01 and 0.0… view at source ↗

**Figure 5.** Figure 5: The population size of the three epidemic states 𝑆, 𝐼, and 𝑅 varying with time with different values of the infected rate 𝛽. 𝛽 is respectively set as 2 × 10−4 , 3 × 10−4 , 4 × 10−4 , 6 × 10−4 , 7 × 10−4 , 8 × 10−4 in (a)-(f). The termination time is set long enough as 𝑡=1.2× 105 in (a)-(e) and 𝑡=2.4× 105 in (f). Within the observation time, in (a)-(e), the infected population all experience an observable g… view at source ↗

**Figure 6.** Figure 6: Stationary distributions of the population size of three epidemic state 𝑆, 𝐼, and 𝑅, represented by blue, red crosses, and green scatters. The distributions in (a)-(g) subsequently correspond to the population size sequence under the stationary situation in [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: The comparison of the epidemic spreading with and without indirect contacts given different settings of parameters. Dark marked lines and light non-marked lines respectively represent the population sizes with indirect contacts and without indirect contacts. The initial parameters are set in (a), where population sizes with indirect contacts reach stationary, while the population sizes without indirect con… view at source ↗

**Figure 8.** Figure 8: The average number of infected individuals of 10 simulations given different values of the infected rate 𝛽. The dashed vertical line indicates the theoretical threshold of 𝛽 (𝛽𝐶 = 0.000088) given a set of parameters. On the x-axis, the values of 𝛽 is respectively set as 0.2𝛽𝐶 , 0.5𝛽𝐶 , 0.8𝛽𝐶 , 𝛽𝐶 , 1.2𝛽𝐶 , 1.5𝛽𝐶 , 2𝛽𝐶 , 2.5𝛽𝐶 , and 3𝛽𝐶 . Below the threshold, the number of infected individuals equals to 0, … view at source ↗

read the original abstract

Epidemic modelling on complex networks has been studied intensively all the time. The majority of relative research assumes that the time scale of the underlying network evolution is much larger compared to the propagation dynamics on it, while the co-evolution of epidemics and networks needs exploring further. In this paper, we investigate how our recently proposed birth-death evolving network impacts the Susceptible-Infected-Recovered-Susceptible (SIRS) epidemic process. Our evolving network considers the increase and the heritable deletion of nodes, which enables to depicting individual behaviors during an epidemic, e.g., population migration and indirect contacts. To model the above processes, we construct a Markovian queueing network and perform analyses for the variation of population size of different epidemic states. In simulations, we reveal how the population migration and indirect contacts caused by our network dynamic properties influence the population sizes of each epidemic state, and find that newly-created indirect contacts facilitate epidemic transmission.

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.

Desk Editor's Note private letter to a colleague

The paper layers SIRS epidemics onto the authors' prior birth-death network, uses a Markov queueing model for state tracking, and reports simulations where new indirect contacts appear to speed transmission, but the design does not separate that from simultaneous shifts in population size and degree.

read the letter

The core contribution is straightforward: the authors apply their birth-death evolving network (node addition, heritable deletion) to an SIRS process, build a Markovian queueing network to follow the sizes of each epidemic compartment, and run simulations that link the network dynamics to changes in infected fraction. They highlight how migration-like node turnover and the resulting indirect contacts affect spread, which moves beyond the usual static-network assumption in epidemic modeling. That part is cleanly executed and directly extends their earlier framework without obvious internal contradictions.

Referee Report

2 major / 2 minor

Summary. The paper constructs a birth-death evolving network model incorporating node births, heritable deletions (to represent migration), and indirect contacts. It analyzes SIRS epidemic dynamics via a Markovian queueing network that tracks population sizes of susceptible, infected, and recovered states, and uses simulations to conclude that newly created indirect contacts facilitate epidemic transmission.

Significance. The Markovian queueing network provides a structured framework for analyzing co-evolving epidemics and networks with migration, which is a methodological strength if the derivations are complete. However, the central simulation claim that indirect contacts specifically facilitate transmission would have moderate significance only if the effect can be isolated from concurrent changes in population size N(t) and average degree; without such controls the result risks being driven by overall contact volume rather than the indirect-contact mechanism.

major comments (2)

[Simulation analysis] Simulation results (as described in the abstract and implied analysis): the reported facilitation of epidemic transmission by newly-created indirect contacts is not supported by a controlled comparison that disables the indirect-contact creation rule while holding birth rate, death rate, and resulting N(t) fixed. The birth-death process simultaneously alters total population, average degree, and contact structure, so the observed rise in infected fraction cannot be unambiguously attributed to the indirect character of the new edges.
[Markovian queueing network] Markovian queueing network construction: while the model tracks state populations, no closed-form comparison or sensitivity analysis is provided that holds network statistics (e.g., mean degree or total contacts) constant while toggling only the indirect-contact mechanism. This leaves the analytical support for the facilitation claim incomplete.

minor comments (2)

The abstract states that the network 'enables to depicting individual behaviors' but does not specify how the heritable deletion rule is parameterized or validated against empirical migration data.
Notation for the queueing network states and transition rates should be defined more explicitly in the main text to allow readers to reproduce the population-size equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the analysis and support for our claims.

read point-by-point responses

Referee: [Simulation analysis] Simulation results (as described in the abstract and implied analysis): the reported facilitation of epidemic transmission by newly-created indirect contacts is not supported by a controlled comparison that disables the indirect-contact creation rule while holding birth rate, death rate, and resulting N(t) fixed. The birth-death process simultaneously alters total population, average degree, and contact structure, so the observed rise in infected fraction cannot be unambiguously attributed to the indirect character of the new edges.

Authors: We acknowledge that the current simulations do not include an explicit controlled comparison that disables only the indirect-contact creation rule while holding birth rate, death rate, and N(t) fixed. This is a valid observation, as the birth-death process does couple changes in population size and network structure. In the revised manuscript, we will add new simulation scenarios that turn off indirect contact creation (by setting the corresponding rate parameter to zero) while retaining the same birth and death rates, and we will present side-by-side comparisons of N(t), average degree, and epidemic state populations to isolate the contribution of the indirect-contact mechanism. revision: yes
Referee: [Markovian queueing network] Markovian queueing network construction: while the model tracks state populations, no closed-form comparison or sensitivity analysis is provided that holds network statistics (e.g., mean degree or total contacts) constant while toggling only the indirect-contact mechanism. This leaves the analytical support for the facilitation claim incomplete.

Authors: The Markovian queueing network is formulated to capture the exact transition rates for the co-evolving system, including the indirect contacts generated by the birth-death rules. While a closed-form solution that holds mean degree exactly constant is difficult to obtain due to the inherent coupling, we agree that further analysis is needed. In the revision, we will add a numerical sensitivity study that varies the indirect-contact formation rate and reports the resulting steady-state populations of S, I, and R states together with the associated changes in network statistics such as mean degree and total contacts. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of prior network model; epidemic analysis and facilitation claim are independent

full rationale

The paper references its own earlier work to define the birth-death evolving network with heritable deletion and indirect contacts. However, the Markovian queueing analysis of state populations, the simulation protocol, and the reported finding that newly-created indirect contacts facilitate SIRS transmission are developed from scratch on top of that model. No equation or derivation reduces the facilitation result to a fitted parameter, a self-defined quantity, or a chain of unverified self-citations. The central claim therefore retains independent empirical content from the simulations and queueing analysis.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim rests on parameters controlling network birth and death rates plus the assumption that the system can be treated as a Markovian queueing network; indirect contacts are introduced as a modeled feature without external falsifiable evidence provided in the abstract.

free parameters (2)

node birth rate
Controls addition of new nodes to represent population increase or migration; value not specified in abstract.
node death rate
Controls heritable deletion of nodes to represent population decrease; value not specified in abstract.

axioms (1)

domain assumption The co-evolution of network structure and epidemic states can be captured by a Markovian queueing network.
Invoked to perform analyses for the variation of population sizes of different epidemic states.

invented entities (1)

Indirect contacts via network evolution no independent evidence
purpose: To represent additional transmission opportunities created by node additions and deletions.
Postulated as part of the birth-death network dynamics to explain facilitation of epidemic spread.

pith-pipeline@v0.9.0 · 5469 in / 1380 out tokens · 77799 ms · 2026-05-10T15:34:09.122233+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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