Sustainability & Social Segmentation in Social Media Contagion: A Mathematical and Computational Study on Dual Effects of Individual Needs & Peer Influence

Dibyajyoti Mallick; Priya Chakraborty; Sayantari Ghosh

REVIEW 4 major objections 5 minor 37 references

Sustainability & Social Segmentation in Social Media Contagion: A Mathematical and Computational Study on Dual Effects of Individual Needs & Peer Influence

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that peer influence, modeled as a nonlinear relapse rate proportional to the product of addicted and temporarily inert populations, creates a backward bifurcation and a bistable regime, allowing social media addiction to…

desk verdict A plausible social-media-addiction model with peer-induced relapse and a backward bifurcation, but the current manuscript overreaches: a sign error in the bistability condition, an unproven global-stability claim, and a non-conservative reaction-diffusion system that undermines the pattern-formation result. read the letter →

arxiv 2411.18709 v1 pith:4TU2K4S2 submitted 2024-11-27 physics.soc-ph

Dibyajyoti Mallick , Priya Chakraborty , Sayantari Ghosh This is my paper · ORCID

classification physics.soc-ph MSC 92D3037G1035K57 PACS 89.65.-s87.23.Ge

keywords socialmediaaddictionepidemiccompartmentalmodelbackwardbifurcationbistabilitypeerinfluencerelapsereaction-diffusionpatternformationcomplexnetworks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper builds an SAI_tI_p epidemic model of social media addiction that splits the population into susceptible, addicted, temporarily inert, and permanently inert individuals. Its central claim is that peer influence—written as a nonlinear relapse term proportional to the product of addicted and temporarily inert populations—turns the addiction-free state into a backward bifurcation and produces a bistable regime. In that regime the endemic addicted state coexists with the addiction-free state even when the basic reproduction number R0 is below 1, so a peer-pressure-driven relapse can keep addiction alive below the classic epidemic threshold. The paper also reports that in a reaction-diffusion version on a 2D lattice, the bistability combined with diffusion generates long-lived spatial clustering, which the authors interpret as social segmentation of addicted and inert populations.

What carries the argument

The carrying object is the SAI_tI_p compartmental model (susceptible S, addicted A, temporary inert I_t, permanent inert I_p), specifically the nonlinear relapse term γ2 A I_t that makes the transition from temporary inert back to addicted proportional to the product of the addicted and inert subpopulations. This mass-action (bilinear) peer-influence term is what generates the backward (saddle-node) bifurcation and the bistable window R_c < R0 < 1, with thresholds γ2_th and R_c derived from the endemic equilibrium's quadratic pA²+qA+r=0. The analysis uses the next-generation matrix to define R0 and a reaction-diffusion extension (with diffusion only for A and I_t) to produce the reported spatiotemporal clustering.

What would settle it

A direct test would be to fit the model to longitudinal social-media-use data with and without a saturating peer-relapse function and check whether the best-fitting interaction is linear in the product A I_t. More concretely, if a frequency-dependent relapse rate γ2 A I_t / (1 + ω A) or γ2 A I_t/N (rather than γ2 A I_t) removes the backward bifurcation in the same parameter regime, the paper's central claim that peer influence per se sustains addiction below R0=1 would be falsified for those more realistic contact laws. Alternatively, an agent-based simulation with per-contact peer influence rather than density-dependent influence would show whether the bistability survives social heterogeneity.

Watch

Extended reading notes

Core claim

The paper's core discovery is that the peer-influenced relapse rate γ2 A I_t, which is absent from the reproduction number R0, controls the qualitative outcome of the model. When γ2 is small, the system shows a forward transcritical bifurcation at R0=1 and the addiction-free equilibrium is the only stable state below threshold. When γ2 is large enough (above a derived threshold γ2_th), a saddle-node bifurcation appears at R0=1, giving a backward bifurcation with a bistable interval R_c < R0 < 1. Consequently, even though R0 measures the average number of new addicts from one addict as below one, the endemic addiction state can persist, and the system exhibits hysteresis: whether the population is addicted or addiction-free depends on history and initial conditions. The authors further demonstrate that with diffusion, this bistability yields pattern formation and long transients of spatial clusters rather than fast convergence to a homogeneous steady state.

Load-bearing premise

The central result depends on the choice that peer influence drives relapse through the bilinear term γ2 A I_t — the product of the addicted and temporarily inert populations — and that only the addicted and temporarily inert populations diffuse in space; if peer influence saturates or all compartments diffuse, the sub-threshold endemic state and the spatial clustering may not occur.

Editorial extensions

If this is right

Interventions that only aim to push the reproduction number below 1 will not eradicate addiction if peer-influenced relapse is strong, because the system can remain in the endemic state for R0 below 1.
Because the bistable region widens as γ2 increases, reducing peer pressure (lower γ2) shrinks the parameter window in which addiction can survive, and can restore the addiction-free state as the only outcome.
The model predicts hysteresis: after addiction has become endemic, temporarily reducing risk factors to bring R0 below 1 may not eliminate the addiction unless the system is pushed past the bistable fold point.
In a spatially structured society, diffusion of addictive behavior combined with bistability produces spatial segregation of addicted and inert clusters rather than a homogeneous mix, so local pockets of addiction can persist even as the overall population approaches an endemic steady state.
Network simulations with heterogeneous contacts show residual addictive populations even in parameter regions where the deterministic model predicts an addiction-free steady state, because some individuals never fully recover.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If peer influence is better represented by a saturating (frequency-dependent or Michaelis-Menten type) function instead of the bilinear mass-action term, the backward bifurcation and sub-threshold endemic state may disappear; testing this sensitivity would indicate how robust the central claim is to the form of peer pressure.
The reaction-diffusion model only lets A and I_t diffuse, so local population conservation fails (S and I_p remain fixed in space); permitting all compartments to diffuse could change the pattern formation and the interpretation of social segmentation.
The model's bistability suggests an intervention that reduces γ2 below γ2_th could eliminate the endemic state; this yields a testable prediction that anti-peer-pressure campaigns (e.g., reduced social media exposure) can act as a control parameter in the same way vaccination acts in epidemic models.
The same mechanism might apply to other self-reinforcing social behaviors with relapse, such as smoking, gambling, or misinformation sharing, where peer influence on relapse rather than initial exposure may be what sustains the behavior below the classic threshold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is reasonable: add a peer-influence relapse term to an SIR-type addiction model, and you get a backward bifurcation and hysteresis, so addiction can persist even when R0<1. That is a well-known mechanism in epidemic models, but it is applied here to social media addiction with a clear compartmental setup (susceptible, addictive, temporary inert, permanent inert) and dual relapse from self-craving and peers. The reproduction number is derived cleanly, and the phase diagram in γ2–R space is a helpful way to package the bistability region. This is a genuine new combination of ingredients, and the network simulations and pattern-formation section are a natural extension.

But the manuscript as written has several load-bearing problems. First, the abstract claims the addiction-free equilibrium is globally stable when R0<1, but the paper only proves local stability. There is no Lyapunov function or other global argument anywhere, so that claim should be downgraded. Second, Section 4.1 states that bistability requires q>0 in the quadratic for the endemic equilibrium. For two positive roots, the sum of roots must be positive, which requires q<0 when p>0. Their own parameter set gives q<0, so the stated condition is a sign error. It is fixable, but it is in a central condition. Third, Eq. (9) for β_c is garbled and appears without derivation; I could not verify it, and a referee should ask for a clean derivation or a corrected formula. Fourth, the reaction-diffusion system is not locally conservative: only A and I_t diffuse, so the sum S+A+I_t+I_p deviates from 1 wherever those Laplacians are nonzero. The stress-test note is right. The cluster formation in Figs. 5–7 is therefore an artifact of a non-physical local source/sink, not a robust property of the model. Giving all compartments the same diffusion coefficient would fix conservation but remove the pattern mechanism, so the advertised social-segmentation result is fragile. Finally, the network section says that with no peer effect the deterministic steady state for A is zero, but the parameters used (γ1=0.015, β=0.0542, μ=0.01, α=0.01, σ=0.2) give R0>1, so the disease-free state is not stable. That inconsistency should be resolved.

None of this kills the central qualitative point. A corrected model—fixed sign condition, verified formulas, and a conserving spatial formulation—would likely still show peer-driven bistability. The errors are the kind a careful revision can fix. The manuscript deserves a serious referee: the model is well-motivated, the analysis is mostly transparent, and the topic matters. My recommendation is to send it to peer review, with explicit instructions to address the global-stability overclaim, the q sign error, the derivation of Eq. (9), and the conservation flaw in the PDE system.

Referee Report

4 major / 5 minor

Summary. The paper proposes a four-compartment SAItIp ODE model for social media addiction, with relapse driven both by individual craving (γ1 I_t) and by peer influence (γ2 A I_t). The authors derive the basic reproduction number via the next-generation matrix, analyze equilibria and bifurcations, perform network simulations, and study a reaction-diffusion version on a 2D lattice. The main advertised results are (i) that the addiction-free equilibrium is globally stable when the reproduction number is below unity in the absence of peer influence, (ii) that peer influence creates a backward bifurcation and a bistable window in which addiction persists below the epidemic threshold, and (iii) that diffusion produces long-transient social clustering and segmentation.

Significance. If established, the paper would provide a concrete mechanism by which peer-induced relapse sustains social media addiction below the classic epidemic threshold and produces spatial segregation of addicted and inert populations. The model formulation and the next-generation matrix computation are standard and clearly presented, and the connected-component analysis of cluster size is a useful quantitative probe. However, the manuscript contains several load-bearing technical errors: the global stability claim is not proved, the stated bistability condition in Section 4.1 is inconsistent with the polynomial as written, the network simulations are run in a parameter regime where the claimed bistability is absent, and the reaction-diffusion system in Section 6 does not conserve the local population. These problems undermine the paper's central claims.

major comments (4)

[Abstract and Section 3] The abstract states that the addiction-free equilibrium is 'globally stable' whenever the addictive-generation number is less than unity, but Section 3 proves only local asymptotic stability via the next-generation matrix. No Lyapunov function or other global argument is supplied, and the text itself at the end of Section 3 refers only to the basin of attraction of E0. The global claim must either be proved or removed from the abstract.
[Section 4.1, Eq. (7) and Eq. (8)] The condition stated for bistability is incorrect. With p = αβσγ2 + μβγ2 > 0 as defined in Eq. (7), two positive roots of p A^2 + q A + r = 0 require q < 0 and r > 0 (together with q^2 - 4 p r > 0), not q > 0 as claimed. Using the values in Fig. 3 (μ=0.01, β=0.0542, σ=0.2, α=0.01, γ1=0.015, γ2=0.85) gives p>0 and q≈−3.4×10^-4, i.e. the opposite sign. The subsequent derivation of γ2_th from q=0 is thus presented with an internally inconsistent sign convention, and the bistability region in Fig. 2(c) needs to be re-derived.
[Section 5, Figs. 3 and 4] The network simulations are performed in a parameter regime that is not bistable. For μ=0.01, β=0.0542, σ=0.2, α=0.01, γ1=0.015, the reproduction number from Eq. (5) is R ≈ 3.57, far above 1, so E0 is unstable and the system has a single endemic equilibrium. The statement that 'even though in deterministic analysis when there is no peer effect, the steady state value related to A population was zero' is contradicted by the authors' own model: for this β, R>1, so a positive endemic equilibrium exists even with γ2=0. The network results therefore do not demonstrate bistability or sub-threshold persistence.
[Section 6.1, Eq. (11)] The reaction-diffusion system does not conserve the local population. Summing the four equations in Eq. (11) gives ∂t(S+A+I_t+I_p) = μ(1−N) + D_A ∇²A + D_I ∇²I_t, where N = S+A+I_t+I_p. With the initial condition in Eq. (12) and spatially non-uniform A, the Laplacian terms are generically nonzero, so N immediately evolves away from 1. The compartments are therefore no longer normalized fractions of a fixed population, and the cluster patterns in Figs. 5–7 are produced by a non-conservative PDE. This directly affects the advertised 'social segmentation' result and requires a corrected formulation.

minor comments (5)

[Section 6.1, Eq. (12) and Fig. 6] The initial conditions are inconsistent: Eq. (12) sets I_t=0 and I_p=1−S−A−I_t, while the Fig. 6 caption sets I_t=1−S−A−I_p and I_p=0. Please harmonize the notation.
[Section 4, Fig. 2(c)] The description of the phase diagram says the green region is enclosed by Eq. (8), Eq. (10), and R=1, but the text describing the yellow and green regions is confusing. Please clarify which region corresponds to which combination of conditions.
[Eq. (9) and Eq. (10)] Eq. (9) has unbalanced parentheses, and Eq. (10) is missing a closing parenthesis. Please correct the typography.
[Section 4] The sentence 'the system poses three solutions, out of which two are physically achievable' is imprecise about the number of equilibria. Please state explicitly how many endemic equilibria coexist with E0 in the bistable window.
[General] The paper does not discuss the sensitivity of the main conclusions to the mass-action form γ2 A I_t. Since a saturating or frequency-dependent peer-influence term could eliminate the backward bifurcation, a brief discussion of this modeling choice would strengthen the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bistability and clustering results follow algebraically from the explicitly stated bilinear relapse assumption, and self-citations are contextual rather than load-bearing.

full rationale

The derivation chain is self-contained. Section 3 computes R0 via standard next-generation matrices (Eqs. 3-5) and uses it only for local stability of the addiction-free equilibrium; this does not presuppose the bistability result. Section 4 derives the endemic-equilibrium polynomial pA^2+qA+r=0 (Eq. 7) and then obtains gamma2_th from q=0 and Rc from q^2-4pr=0 (Eqs. 8-10); these are algebraic consequences of the explicitly nonlinear relapse term gamma2*A*I_t in Eq. (2), not fitted parameters renamed as predictions. The network and reaction-diffusion simulations in Sections 5-6 are numerical explorations of the same model, not independent empirical validations. Cited works by the same authors (e.g., refs. [20] and [21]) are used as background and do not carry the load of the central claim; no uniqueness theorem or fitted parameter is imported from them. Two non-circular weaknesses should be flagged separately: the abstract claims global stability of E0 for R0<1, while Section 3 proves only local stability; and the reaction-diffusion system in Eq. (11) does not conserve total population because only A and I_t diffuse, so N=S+A+I_t+I_p evolves under D_A*laplacian(A)+D_I*laplacian(I_t), and the normalized compartments can leave [0,1]. These are correctness/completeness concerns, not reductions of outputs to inputs.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

All model parameters are chosen by hand; none are fitted to empirical data. The key nonlinear peer-relapse term is introduced ad hoc to produce the bistability phenomenon. The reaction-diffusion assumption introduces a conservation inconsistency.

free parameters (10)

mu (birth/death rate) = 0.01
Chosen by hand for all simulations.
beta (addiction rate) = 0.0542
Chosen to put R0 above 1 in the baseline simulations; not fitted to data.
alpha (rate leaving SMA) = 0.01
Chosen by hand.
gamma1 (self-craving relapse rate) = 0.015
Chosen by hand.
gamma2 (peer influence relapse rate) = 0.85 (high peer effect), 0.015 (low)
Varied to demonstrate bistability; not data-derived.
sigma (fraction permanent recovery) = 0.2
Chosen by hand.
D_A (addict diffusion coefficient) = 0.05 (some runs 0.02)
Chosen for the reaction-diffusion simulations.
D_I (temporary inert diffusion coefficient) = 0.0015
Chosen by hand, smaller than D_A.
k1, k2 (initial condition scaling) = 0.9, 0.1
Set so initial population sums to 1.
A_th (cluster threshold) = (A_max + A_min)/2
Arbitrary threshold for connected component analysis.

assumptions (7)

domain assumption Closed, homogeneously mixed population for the ODE model
Section 2 states N individuals in a closed homogeneously mixed population.
ad hoc to paper Peer influence on relapse is bilinear (mass-action): rate gamma2*A*I_t
This functional form is introduced to capture peer effects and is the source of the backward bifurcation.
standard math Exponentially distributed compartment residence times (standard ODE epidemic modeling)
Implicit in the compartmental ODE formulation.
domain assumption Normalization S+A+I_t+I_p=1 for the ODE
Eq. (1) and used to close the system.
domain assumption Random network with N=10000 and average degree 5 approximates a heterogeneous society
Section 5 uses this network without sensitivity analysis.
ad hoc to paper Reaction-diffusion with only A and I_t diffusing under no-flux boundary conditions is a valid spatial extension
Section 6; this breaks local total population conservation because S and I_p do not diffuse.
standard math Next-generation matrix method determines invasion threshold
Section 3 uses the method from van den Driessche and Watmough.

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Cite this review

Pith. "Pith review of Sustainability & Social Segmentation in Social Media Contagion: A Mathematical and Computational Study on Dual Effects of Individual Needs & Peer Influence." pith.science (2026). https://pith.science/paper/4TU2K4S2

@misc{pith2026241118709,
  author       = {Pith},
  title        = {Pith review of: Sustainability & Social Segmentation in Social Media Contagion: A Mathematical and Computational Study on Dual Effects of Individual Needs & Peer Influence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TU2K4S2}},
  note         = {Machine review of arXiv:2411.18709}
}

read the original abstract

Addiction to internet-based social media has increasingly emerged as a critical social problem, especially among young adults and teenagers. Based on multiple research studies, excessive usage of social media may have detrimental psychological and physical impacts. In this study, we are going to explore mathematically the dynamics of social media addiction behaviour and explore the determinants of compulsive use of social media from the dual perspectives of individual needs or cravings and peer-related factors or peer pressure. The theoretical analysis of the model without the peer pressure effect reveals that the associated addiction-free equilibrium is globally stable whenever a certain threshold, known as the addictive-generation number, is less than unity and unstable when the threshold is greater than unity. We observed how introduction of peer influence adds a sustainability to the dynamics, and causes a multistability, through which addiction-contagion can proliferate, even below the designated critical threshold. Using simulations over model networks, we demonstrate our finding, even in the presence of social heterogeneity. Finally, we use the reaction-diffusion approach to investigate spatio-temporal dynamics in a synthetic society, in the form of a 2D lattice. Instead of a fast convergence to the steady states, we observe a long transient of social clustering and segmentation, represented by spatio-temporal pattern formation. Our model illustrates how the peer influence factor plays a crucial role and concludes that it is required to consider the peer factors while formulating specific strategies that could be more effective against this addiction and its potential adverse outcomes.

Figures

Figures reproduced from arXiv: 2411.18709 by the authors.

**Figure 1.** Block diagram of the proposed model of social media addiction. The notation for compartments (denoted by solid boxes) and transition rates (denoted by arrows with parameters) have been elaborated in the text. So 𝛾1 𝐼𝑡 and 𝛾2𝐴𝐼𝑡 amount must be added here and deducted from 𝐼𝑡 . Here the negative peer influence on relapse has been considered. • People in the Temporary Inert group got over their SMA temporarily and have… view at source ↗

**Figure 2.** Variation in steady state fraction of Addictive population with reproduction number 𝑅 for (a) 𝛾2 = 0.015, when only a single epidemic state persists beyond 𝑅 = 1 and for (b) 𝛾2 = 0.85, when bistability can be observed in the range 𝑅𝑐 to 1. In this figure, the continuous lines indicate stable solutions, and the dashed lines indicate unstable solutions. For these parameter values, we calculated 𝑅𝑐 = 0.85 from equ 10. … view at source ↗

**Figure 3.** Time evolution of all the subpopulations for the proposed 𝑆𝐴𝐼𝑡 𝐼𝑝 model. (a) For 𝛾2 = 0.015 (less peer effect) (b) 𝛾2 = 0.85 (Strong peer effect). The rest of the parameter values for both curves are 𝜇 = 0.01, 𝛽 = 0.0542, 𝜎 = 0.2 and 𝛼 = 0.01 population to get back the addiction to social media. We got a backward (saddle-node) bifurcation at 𝑅 = 1 for a comparatively higher nonlinear relapse as shown in [PITH_FULL_… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Population of different compartments with time for Individual craving vs. peer effects towards SMA (a) Addictive Population (b) Temporary Inert Population (c) Permanent Inert Population. Here for self-seeking 𝛾1 = 0.015 and 𝛾2 = 0 and for peer effects 𝛾1 = 0 and 𝛾2 = 0…

**Figure 5.** Figure 5: The effect of diffusion in the spatiotemporal behavior of the addictive population under peer effect. Set I: The system shows cluster formation in the presence of diffusion of the addicted population to reach the endemic state under peer effect. Yellow islands represen…

**Figure 6.** Figure 6: Under the peer effect, with a diffusion of the addictive behavior, the system shows cluster formation in order to reach the endemic state. Spatiotemporal evolution of permanent inert population (𝐼𝑝 ) dynamics. The initial condition is given by 𝑆(𝑥, 𝑦, 0) = 𝑘1 ; 𝐴(𝑥, 𝑦,…

**Figure 7.** Figure 7: Measure of cluster formation in the presence of peer influence, averaged over 50 realizations. With time we count the number of yellow islands (a), and the lattice points in yellow islands (b), for addiction diffusion coefficient 𝐷𝐴 = 0.05. A visual of the clustered po…

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