{"id":"fcd87869-b010-4466-aba9-110a111432fd","arxiv_id":"2505.21540","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":15,"one_line_summary":"A mathematical model coupling fake news dynamics to antimicrobial resistance is proposed, but the stability analysis has major gaps and the claimed resistance thresholds are never derived.","lead":"This preprint adds two equations for antimicrobial resistance to an existing six-compartment model of fake news spreading, and presents a stability analysis of the combined system. The authors aim to show that false beliefs about antibiotics could accelerate drug resistance and to identify targets for public health campaigns.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim unsupported: the appended RAM equations are never analyzed, and at the misinformation-free equilibrium the model has a stable resistance-endemic state (x7=Rmax), so controlling misinformation does not reduce resistance in the model.","rationale":"The reader's weakest_assumption focused on the unverified rank-one perturbation argument for the stability of E2 (Section 3). That is a genuine technical defect, but it concerns only the six-compartment misinformation subsystem; even a fully correct stability proof of E2 would not establish the paper's headline conclusion about antimicrobial resistance, because the RAM variables x7 and x8 are never analyzed. The more fundamental, load-bearing problem is the complete absence of analysis for the extended 8D system: no equilibria involving x7 and x8 are computed, no stability conditions for the RAM subsystem are derived, and no threshold relating misinformation to resistance is obtained. This gap is highlighted by a concrete internal counterexample: in the misinformation-free limit, the RAM subsystem has a locally stable equilibrium at maximal resistance (x7=Rmax), directly contradicting the claim that controlling misinformation reduces resistance. The reader's rationale did mention that 'the resistance model is never analyzed for thresholds,' so there is partial agreement on the broader issue, but the stated weakest_assumption is not the most decisive one. Since the central claim is unsupported and the model's own dynamics point in the opposite direction, the rejection verdict is appropriate; no change to the reader's verdict is needed.","tokens_in":8529,"tokens_out":8983,"duration_ms":75880,"concrete_test":"Set the misinformation transmission and contact rates to zero (β3=β4=η=0), keeping all other parameters from Table 1, and solve the equilibrium equations of the full 8D system (1)–(16). Verify that (x1,…,x6)=(Λ/μ,0,0,0,0,0) together with (x7,x8)=(Rmax, σΛRmax/(μ μA))=(1,4000) is an equilibrium; then compute the eigenvalues of the (x7,x8) subsystem Jacobian (approximately −0.1496 and −0.05) and confirm local stability by numerical integration. If this holds, the model's central claim is internally contradicted; if resistance decays instead, the conclusion may still be salvageable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised conclusion—'controlling misinformation can effectively reduce antimicrobial resistance development'—is not derived from any analysis of the coupled system. Section 3 performs stability analysis only for equilibria E1–E7 of the original six-compartment misinformation model; the RAM equations (15)–(16) are appended in Section 3.1 but never subjected to equilibrium, stability, or threshold analysis. The numerical simulations in Section 4 are also six-dimensional (Figures 1–7 list only x1–x6). Thus the promised 'critical thresholds that determine ... how resistance develops' are never established. More seriously, a direct equilibrium computation contradicts the claim under the model's own parameters. At the misinformation-free state (x3=x4=x5=x6=0, x1=Λ/μ, with β3=β4=η=0 so this state is stable), equations (15)–(16) admit the equilibrium x7=Rmax=1 and x8=σΛRmax/(μ μA)=4000. The Jacobian of the (x7,x8) subsystem at this point has eigenvalues approximately −0.1496 and −0.05, so this resistance-endemic equilibrium is locally stable. Even if the rank-one perturbation issue for E2 were resolved, the central claim would still fail: the model itself predicts maximal resistance under perfect misinformation control.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends a six-compartment, two-strain delay-differential-equation model of fake-news spreading by appending two ordinary differential equations for antimicrobial resistance prevalence (x7) and inappropriate antibiotic consumption (x8). It claims to establish stability of the misinformation-free equilibrium E1 and of an interior equilibrium E2, to identify thresholds for misinformation persistence and resistance development, and to conclude that controlling misinformation reduces antimicrobial resistance. Numerical simulations of the six-compartment system are presented in Figures 1–7.","tokens_in":8844,"tokens_out":7627,"duration_ms":51397,"significance":"If the analysis were correct, the paper would contribute a useful framework for linking behavioral dynamics of misinformation with biological dynamics of antimicrobial resistance, a genuinely important public-health topic. The construction of the coupled system is a reasonable modeling step, and the use of Lyapunov–Krasovskii functionals and characteristic equations is appropriate in direction. However, the central claim is not established: the appended RAM equations are never analyzed, and a direct equilibrium computation shows that the misinformation-free state supports a locally stable maximum-resistance equilibrium. The stability arguments for E1 and E2 contain mathematical errors and unverified assumptions, and the numerics contradict the paper's own conditions. The potential significance is therefore not realized in the submitted manuscript.","major_comments":[{"comment":"The RAM subsystem (15)–(16) is appended but never subjected to equilibrium, stability, or threshold analysis; the simulations in Section 4 and Figures 1–7 show only x1–x6. Consequently, the abstract's promise of 'critical thresholds that determine ... how resistance develops' is not fulfilled. More seriously, the model contradicts the paper's own conclusion: at the misinformation-free equilibrium E1 (x3=x4=x5=x6=0, x1=Λ/μ), equations (15)–(16) admit the resistance-endemic equilibrium x7=Rmax=1, x8=σΛRmax/(μ μA)=4000 with Table 1 parameters, and the 2x2 Jacobian there has eigenvalues approximately −0.1496 and −0.05, so this maximal-resistance state is locally stable. The model thus predicts maximum resistance when misinformation is perfectly controlled, directly contradicting the Conclusion's claim that controlling misinformation can effectively reduce resistance development.","section":"3.1 (Eqs. (15)–(16))"},{"comment":"The stability proof for E2 replaces the characteristic determinant (8) by (9), setting a31=a41=0, and invokes a rank-one perturbation theorem from reference [2] on the grounds that these entries are 'small.' The paper never computes a31 and a41 at E2, never states the precise conditions of the theorem, and never verifies them for the parameter values in Table 1. Since a31 and a41 are nonzero partial derivatives of the x3 and x4 equations with respect to x1, the smallness assumption is not automatic; without verification, the factorization into (10)–(11) is not a valid proxy for (8) and the stability conclusion for E2 collapses.","section":"3, Eqs. (8)–(9)"},{"comment":"The Lyapunov–Krasovskii calculation for E1 contains algebraic errors that invalidate the negativity claim. The first term should be (1−x1*/x1)(Λ−μx1) = −μ(x1−x1*)²/x1, but the paper writes −μ(x1*−x1)/x1. The β3 and β4 terms do not combine as claimed: (1−x1/x1*)β3x1x3/(1+α3x1+ε3x3) plus β3x1x3/(...) gives β3x1²x3/(x1*(...)), not β3x1*x3/(...). The η terms yield ηx1*x2 + η(x1−x1*)x2τ1 − μx2, and the middle term is dropped although it can be positive when x1>x1*. Thus the inequality (7) is not derived, and the global asymptotic stability conclusion for E1 is unsupported.","section":"2, Lyapunov derivative"},{"comment":"The numerical results are internally inconsistent and contradict the analytical conditions. Table 1 gives Λ/μ=2000, but Figure 1 lists E1=(1666.67,0,...). For E2, Figure 2 reports x1*=0.5 whereas the paper's own formula x1*=μ/η=0.25 under Table 1. The caption of Figure 2 asserts stability for all τ≥0, but the paper's Routh–Hurwitz condition for the no-delay case, 2μ²>x2*, is violated by orders of magnitude (0.005>1590.74 is false), and no alternative stability proof for these parameter values is supplied. These inconsistencies remove the numerical support for the theoretical claims.","section":"4, Table 1 and Figures 1–2"}],"minor_comments":[{"comment":"The abbreviation RAM for antimicrobial resistance is nonstandard; AMR is the usual acronym, and the body frequently spells out 'antimicrobial resistance' instead of using the abbreviation.","section":"Abstract"},{"comment":"The section begins with a stray 'I' and contains typographical errors, including 'wherex∗1 = Λ µ without any relation' and missing punctuation after the definition of E1.","section":"Section 2"},{"comment":"After deriving equation (12), the text says 'separating the real and imaginary parts in (11)', but the trigonometric system that follows is obtained from equation (10)/(12), not from the determinant (11).","section":"Section 3"},{"comment":"References [17] and [31] are placeholders ('Title, Journal Name, year') and must be completed before submission.","section":"References"},{"comment":"The captions of Figures 5 and 6 are identical and list the same equilibrium vector for E5 and E6, which appears to be a copy-paste error.","section":"Figures 5 and 6"}],"recommendation":"reject","confidential_remarks":"The manuscript is explicitly labeled preliminary, contains placeholder references, and its central policy conclusion is contradicted by a straightforward equilibrium computation of the model's own RAM subsystem. Even a major revision would require reworking the RAM subsystem analysis, verifying the perturbation hypotheses of [2], correcting the Lyapunov calculation, and reconciling the numerical inconsistencies. The paper does not currently meet the standard for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result — that controlling misinformation reduces antimicrobial resistance — is not supported by the model as written. In fact, the model contradicts it: at the misinformation-free equilibrium, the appended resistance equations admit a locally stable resistance-endemic state (x7 = Rmax = 1, x8 = 4000), so perfect misinformation control can produce maximal resistance. That alone sinks the paper's central claim.\n\nThe modeling idea is legitimate. Coupling a six-compartment two-strain fake-news model with delays to two resistance variables is new, and the one-way information-to-resistance structure is a reasonable first step. The authors correctly identify a real gap: most misinformation models and most RAM models are separate. I credit them for the attempt.\n\nThe problems are load-bearing. Section 3 analyzes only the six-compartment system; equations (15)–(16) are appended but never analyzed for equilibria, stability, or thresholds. The abstract promises 'critical thresholds that determine ... how resistance develops' — none are derived. The Lyapunov argument for E1 has a sign/typo issue (the negative term appears without the squared numerator), and the claimed negativity condition is not derived correctly. For E2, the rank-one perturbation from reference [2] is invoked without verifying that a31 and a41 are small at the simulated equilibrium; for the given parameters they are not obviously small, so equation (9) is not a valid proxy for (8). Figure 2 also contradicts the paper's own Routh-Hurwitz condition: with μ=0.05, the condition 2μ^2 > x2^* requires x2^* < 0.005, yet the reported x2^* ≈ 1590. The paper says stable; the condition says unstable.\n\nThe manuscript is clearly unfinished: placeholder references ([17], [31]) and the preliminary-note disclaimer are fine for a preprint but not for a claim of rigorous analysis. There are also formatting errors in the characteristic equation.\n\nWho is this for? Possibly a modeler interested in the formulation, but no reader should rely on the stability results. I would not send this to peer review as is. It needs a complete re-analysis of the full eight-dimensional system, verification of the perturbation condition, and a corrected numerical section. If that is done, the modeling idea could be worth a paper. As it stands, reject.","headline":"Coupling a fake-news compartment model to two resistance equations is new, but the analysis is broken and the model's own equilibrium contradicts the advertised conclusion.","tokens_in":9331,"tokens_out":2383,"would_cite":false,"duration_ms":18209,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34D05","34D20","34D23","92D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that coupling a two-strain fake-news model to antimicrobial resistance variables produces thresholds that decide when misinformation persists and resistance grows, so controlling misinformation should reduce resistance.","keywords":["fake news spreading","antimicrobial resistance","delay differential equations","stability analysis","numerical simulations","misinformation","compartmental model","antibiotic resistance"],"falsifier":"At the simulated equilibrium E2 = (0.5, 1590.742559, 49.864465, 16.465254, 5.205975, 3.888413) with the table's parameters, compute the Jacobian entries a31 and a41 and compare them with the diagonal entries of the block in equation (8); if either coupling is not small in the sense required by [2], then equation (9) is not a valid proxy for equation (8) and E2's stability is not established by the paper's argument. Re-running the five-delay simulation while tracking these entries would settle the claim.","tokens_in":8314,"feed_emoji":"🦠","tokens_out":9853,"duration_ms":73956,"temperature":0.7,"pith_summary":"The paper sets out to connect two dynamics that are usually modeled separately: how false beliefs about antibiotics circulate through a population, and how antimicrobial resistance develops. It extends a two-strain fake-news model with time delays by adding two variables, one for inappropriate antibiotic consumption and one for resistance prevalence, producing an eight-compartment system of delay differential equations. Its central claim is that this system has thresholds that decide whether misinformation spreads and persists, and because resistance is driven by believers of misinformation and suppressed by skeptics, the same thresholds decide whether resistance develops. On that basis the paper concludes that controlling misinformation can effectively reduce antimicrobial resistance. The intended use is to give public health campaigns a mathematical argument for targeting false antibiotic beliefs as part of resistance control.","feed_headline":"Controlling misinformation can curb antibiotic resistance, model says","feed_subtitle":"A two-strain fake-news model with delays sets thresholds for when resistance takes hold.","key_machinery":"The central object is the eight-compartment delayed system (1), (15), (16). Six compartments describe exposure to and belief in two strains of antibiotic misinformation—susceptible, active, believer in strain 1, believer in strain 2, skeptic of strain 1, skeptic of strain 2—using saturating (Holling Type II) incidence and five time delays; two additional variables track inappropriate antibiotic consumption x8 and resistance prevalence x7, with x7 growing from x8 through a saturating, logistic-type term and falling with skeptic numbers. The argument is carried by stability analysis of these equilibria: a Lyapunov-Krasovskii functional establishes global stability of E1 under the three product conditions, and a rank-one perturbation argument from [2] is used to factor the characteristic equation at E2 into lower-order delay equations whose zeros determine stability. This factorization is the mechanism that turns the 6x6 and 8x8 linearizations into tractable threshold conditions.","core_discovery":"The paper argues that the dynamics of antibiotic misinformation and antimicrobial resistance can be captured in one eight-dimensional delay differential system, and that the system's equilibria carry the public-health message. For the misinformation-free equilibrium E1, it claims global asymptotic stability under the parameter conditions Λβ3<μ², Λβ4<μ², and Λη<μ², shown through a Lyapunov-Krasovskii functional. For the endemic equilibrium E2 where both misinformation strains persist, it claims stability for all delays, obtained by a rank-one perturbation argument from [2] that lets the characteristic equation factor into independent blocks. The appended resistance equations make x7 rise with inappropriate use x8 and fall with the number of skeptics x5+x6, so at a stable misinformation equilibrium the resistance level is tied to the believers-versus-skeptics balance. The paper's stated conclusion is that controlling misinformation can effectively reduce antimicrobial resistance development.","pith_inferences":["The stability of E2 is conditional on a small-coupling assumption that the paper does not verify; if the Jacobian entries a31 and a41 at the simulated E2 are not small, the factored equation (9) may not represent the true characteristic equation, so the 'stable for all delays' claim should be treated as conditional.","The title page states that this is a preliminary version and that an extended version with numerical results is forthcoming; until then, the simulation figures are illustrative and the quantitative claims should be read with that caveat.","A natural next step is to convert the threshold inequalities into intervention targets, estimating how much η or β3,β4 must fall for a given population to cross into the misinformation-free region, which the paper does not do.","The resistance subsystem feeds back into misinformation only through the σ term in x8; allowing current resistance levels to change belief-adoption rates would be a testable modification that could alter the thresholds."],"forward_implications":["If the three conditions Λβ3<μ², Λβ4<μ², and Λη<μ² hold, the misinformation-free equilibrium is globally stable, so both misinformation strains die out and the believers that drive inappropriate antibiotic use disappear.","If the endemic equilibrium E2 is stable, both misinformation strains persist at positive levels and inappropriate antibiotic use stays positive, so resistance can develop and persist even while skeptics are present.","Because resistance prevalence increases with believers and decreases with skeptics, the model predicts that growing the skeptic/fact-checking population is a direct lever for lowering resistance.","The simulated instability of the single-strain equilibria E3 and E4 implies that suppressing only one misinformation strain is not enough; both strains must be addressed together.","The thresholds identify measurable campaign targets: driving the transmission rates β3,β4 and contact rate η below μ²/Λ should move the system toward the misinformation-free state."],"supporting_citations":[{"why":"Provides the rank-one perturbation argument used to replace the characteristic equation at E2 by a factorable equation.","marker":"[2]"},{"why":"Supplies Theorem 5.3.1, cited to conclude global asymptotic stability of E1 from the Lyapunov-Krasovskii functional.","marker":"[14]"},{"why":"Supplies the form of the Lyapunov-Krasovskii functional used in the stability proof for E1.","marker":"[26]"},{"why":"Supplies the method used to study the delay equations (13) and (14) that come from the factored characteristic equation.","marker":"[34]"}],"fun_headline_variants":["Curb fake news to curb antibiotic resistance, model says","Misinformation dynamics drive antimicrobial resistance, model shows","Model links controlling misinformation to lower drug resistance","Tackle misinformation to reduce antibiotic resistance, says model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that at the equilibrium where both misinformation strains persist, two coupling strengths in the linearized system are small enough for the rank-one perturbation argument of reference [2] to apply; the paper never verifies this smallness for the equilibrium or for the simulated parameter values.","fun_headline_variants_meta":{"raw":{"variants":["Curb fake news to curb antibiotic resistance, model says","Misinformation dynamics drive antimicrobial resistance, model shows","Model links controlling misinformation to lower drug resistance","Tackle misinformation to reduce antibiotic resistance, says model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000455,"raw_usage":{"total_tokens":2269,"prompt_tokens":910,"completion_tokens":1359,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1297}},"tokens_in":526,"tokens_out":1359,"duration_ms":10687,"temperature":1.0,"reasoning_tokens":1297,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:26:12.077569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At the simulated equilibrium E2 = (0.5, 1590.742559, 49.864465, 16.465254, 5.205975, 3.888413) with the table's parameters, compute the Jacobian entries a31 and a41 and compare them with the diagonal entries of the block in equation (8); if either coupling is not small in the sense required by [2], then equation (9) is not a valid proxy for equation (8) and E2's stability is not established by the paper's argument. Re-running the five-delay simulation while tracking these entries would settle the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the rank-one perturbation argument used to replace the characteristic equation at E2 by a factorable equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 5.3.1, cited to conclude global asymptotic stability of E1 from the Lyapunov-Krasovskii functional."},{"cited_title":"and Zou, X., Global dynamics of a two-strain disease model with latency and saturating incidence rate,Canadian Applied Mathematics Quarterly, 20(1), 2012","cited_arxiv_id":null,"evidence_quote":"Supplies the form of the Lyapunov-Krasovskii functional used in the stability proof for E1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the method used to study the delay equations (13) and (14) that come from the factored characteristic equation."}],"review_version":1}