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REVIEW 4 major objections 6 minor 45 references

Effect of Vaccine Dose Intervals: Considering Immunity Levels, Vaccine Efficacy, and Strain Variants for Disease Control Strategy

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The interval between vaccine doses is a tunable control parameter: 3–8 month gaps minimize infection-plus-vaccination cost, while gaps beyond 9 months drive costs up sharply.

desk verdict A useful operational question buried in a model that does not close: the 3–8 month optimal interval is not supported as written. read the letter →

arxiv 2505.21132 v1 pith:CA37TRQ7 submitted 2025-05-27 q-bio.PE cond-mat.softphysics.bio-ph

Samiran Ghosh , Malay Banerjee , Amit K Chattopadhyay This is my paper
classification q-bio.PEcond-mat.softphysics.bio-ph MSC 92D3092C60
keywords vaccinedoseintervalimmuno-epidemicmodelhysteresiseffectcomorbidityefficacyimmunitywaningbasicreproductionnumberoptimalvaccinationschedule
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

This paper argues that the timing between vaccine doses is not a logistics detail but a control parameter that determines how much disease spreads and how much a vaccination program costs. Using an epidemic model in which infectivity, recovery, death, and immunity all depend on time since infection or vaccination, the authors compare campaigns spaced 4, 8, and 12 months apart and then minimize a cost function that balances infections against the number of campaigns. Their central quantitative result is that the cost stays near its minimum when successive doses are 3 to 8 months apart, and rises abruptly when the gap exceeds 9 months. They interpret this as a hysteresis effect: frequent boosters add little immunity once earlier doses have shaped the immune response, so fewer, well-spaced campaigns can achieve the same control at lower cost and with less intrusion.

What carries the argument

The machinery is an integro-differential immuno-epidemic model with time-since-infection distributed infectivity, recovery, and death, plus a vaccination-driven immunity variable. New infections obey $J(t) = \frac{S(t)}{N}\int_0^t \beta(t-\eta)J(\eta)\,d\eta$; immunity is a convolution $m(t) = \frac{1}{N}\sum_i \int_0^t \phi_i(t-\eta)V_i'(\eta)\,d\eta$; and the optimization is carried by the cost function $J(a) = c\int_0^T I(t;a)\,dt + d\,T/a$, where $a$ is the uniform gap between campaigns. The dose-gap $a$ is the control that carries the argument, and the reproduction number is derived as $R_0 = \int_0^{\tau} \beta(x)\,dx$ in the no-immunity limit.

What would settle it

Re-run the model with an explicit equation updating the susceptible population once infections are split into the healthy and higher-risk streams, and see whether the minimum of $J(a)$ still lies at 3–8 months; if it moves or disappears, the recommended window depends on an unstated bookkeeping choice.

Watch

Extended reading notes

Core claim

The paper's central claim is that the gap between successive vaccine doses is a first-order epidemic-control parameter. For a fixed time horizon and a cost that charges both infections and each vaccination campaign, the combined cost $J(a)$ stays near its minimum for gaps between 3 and 8 months and climbs abruptly once the gap exceeds 9 months. The mechanism offered is hysteresis: each dose builds an immune memory that makes later doses, if given too soon, add little extra protection, so frequent vaccination is not only more expensive but largely redundant. The model also predicts that coexisting strains—represented through a higher infectivity multiplier $\kappa$ on a comorbid (higher-risk) class—produce more, smaller epidemic peaks and shrink the range of safe gaps, while higher vaccine efficacy and faster vaccination rates stretch that range.

Load-bearing premise

The recommendation rests on the model's bookkeeping that keeps the susceptible population as a leftover of the other compartments after infections are split into healthy and higher-risk streams, even though no evolution equation for that leftover is stated.

Editorial extensions

If this is right

  • Campaigns can be spaced 3–8 months apart without sacrificing epidemic control, so the number of vaccine drives—and their direct and indirect costs—can be cut.
  • Waiting 12 months or more between doses is predicted to let immunity wane enough that new epidemic peaks appear, so overly long gaps are not a free saving.
  • Higher vaccine efficacy and faster vaccination rates widen the window of acceptable gaps, while the presence of multiple strains (larger $\kappa$) narrows it.
  • The same model structure can be re-fitted to other diseases by replacing the immunity, viral-load, recovery, and death curves, making the dosing-interval question testable outside COVID-19.
  • If adopted, the hysteresis view implies that 'boost as often as possible' is not just wasteful but may be counterproductive, because extra doses add little once the immune plateau is reached.

Reading between the lines

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

  • A natural extension is to make vaccine efficacy dose-dependent: real-world primary and booster doses differ, and the paper's assumption of equal efficacy across doses may shorten or lengthen the optimal window.
  • The 3–8 month window is derived from a particular fitted immunity-waning curve; re-running the cost minimization with the alternative Gaussian fit reported in the paper would show how sensitive the window is to the choice of $\phi(t)$.
  • The model treats campaigns as evenly spaced; an adaptive schedule that spaces early doses more tightly and later doses more loosely might push the cost minimum even lower than the fixed-gap optimum.
  • A policy-facing test would compare model-predicted infection costs under 4-, 8-, and 12-month booster schedules against real-world data from a country that adopted each schedule.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This manuscript develops an immuno-epidemiological model with time-since-infection distributed transmission, recovery, and death rates, combined with vaccination-induced and infection-acquired immunity functions that are fitted to clinical data. The model is extended by a comorbid class P(t) and a parameter κ that is claimed to capture multiple strains, and the authors formulate a single-parameter cost function J(a) = c∫I dt + dT/a over the gap a between successive vaccination campaigns. The central conclusions are that coexisting strains increase the number of epidemic peaks, that vaccination gaps of 3–8 months are near-optimal with costs rising sharply beyond 9 months (Figure 9, Section 4.3), and that multiple doses produce a 'hysteresis effect' in immunity that makes frequent vaccination unnecessary. The quantitative claims are generated exclusively by numerical simulation of the comorbid model (10)–(15), so the well-posedness of that model is decisive for the paper's conclusions.

Significance. The modeling direction is sensible: time-distributed kernels are more realistic than classical SIR rates, and the authors fit vaccine-induced immunity (Eqs. 21–22), viral load (Eq. 24), and recovery/death kernels (Eq. 25) to published data, reporting SSE, R², RMSE, and confidence intervals for most fits. The R0 derivation in Section 2.3 for the renewal equation is standard and correct. The qualitative premise that dose interval is a relevant control lever is plausible and practically important. However, the load-bearing quantitative results rest on a comorbid model that is not mathematically closed, a cost function with arbitrary unsensitized weights, and a 'hysteresis' interpretation that is not supported by any mechanism in the model. The authors themselves state in Section 5 that the parameter values in Table 1 have no supporting references and that the predictions await future validation. As it stands, the central recommendation of an optimal 3–8 month dosing interval is not established.

major comments (4)
  1. [§2.2.3, Eqs. (10)–(15)] The comorbid model is not closed and is internally inconsistent. After introducing the comorbid class P(t), the paper replaces the mass-balance equation (1) with the algebraic relation (12), S(t) = N − (I + D + P + m1N), and gives no evolution equation for S(t). Differentiating (12) and substituting (13)–(15) yields dS/dt = −J1 + (1−b)Rn + N d(m2 − m1)/dt. The last term contains vaccination-rate contributions (via V_i′(t)) and acquired-immunity convolutions (via ψ and Rn) that do not cancel; specifically N d(m2−m1)/dt = (1−2α)Σφ_i(0)V_i′(t) + [bε − (1−b)]ψ(t)Rn(t), which for the fitted α=0.8, b=0.2, ε=0.7 has coefficients −0.6 and −0.66 rather than the −α and +(1−b) structure expected from a mass balance on S. The algebraic constraint is therefore incompatible with the flow equations unless additional unstated assumptions hold, and the simulated trajectories depend on an unspecified discretization choice. In addition, Eq. (13) defines P(t) as a difference of cumulative quantities, so P can become negative, rendering J2(t) and the κP(t) term in Eq. (14) nonsensical; and recovered individuals are double-counted, since they appear in R via Eq. (15) and simultaneously in m1N inside Eq. (12). Because the cost curves in Figure 9 and the interval recommendation are generated from this system, the central quantitative claim is not supported.
  2. [§4.3, Eq. (27), Fig. 9] The optimal-gap recommendation depends on the arbitrary cost weights c = 0.01 and d = 5 through their ratio c/d, as J(a) = c∫I dt + dT/a. No sensitivity analysis over c/d is reported, so the flat minimum at 3–8 months and the sharp rise beyond 9 months may shift substantially under other weightings. The symbol c is also used for the transmission proportionality constant (Table 1, Eq. 24), creating ambiguity in the captions of Figures 9–12. Since the 3–8 month window is the paper's central result, the authors should either estimate the cost weights from data or show how the location of the optimum changes as c/d varies.
  3. [§4.2; Abstract and §5] The claimed 'hysteresis effect in immunity levels' is not demonstrated. Immunity in the model is a linear convolution (Eqs. 7–8), and the dynamics (10)–(15) contain no mechanism for rate-independent path dependence; the observation that 4-month and 8-month schedules yield similar epidemic progression (Fig. 8) is at most evidence of diminishing returns from overlapping vaccine responses. The term 'hysteresis' is used in the Abstract and Discussion as if it were an established model output, but no quantitative definition, mechanism, or diagnostic of hysteresis is provided anywhere in the manuscript.
  4. [§3.3, Eq. (23); §4.1, Fig. 7; §5] Key inputs to the optimization are poorly identified, and the stated robustness is not substantiated. The acquired-immunity fit ψ(t) has an essentially unconstrained location parameter (b3 = −206.6 with 95% CI spanning (−704.3, 291)) and no goodness-of-fit statistics are reported for it, unlike the fits in Eqs. (21), (22), and (24). Table 1 lists no sources for the parameters that most directly shape the model (N, V0, L, k, c, α, κ, b, ε), and Section 5 concedes that 'there are no known references to confirm the parameter values predicted.' The sensitivity analysis in Section 4.1 perturbs only c3 over a narrow interval and measures only I(t), never the optimized quantity J(a), so the Discussion's assertion that the model is 'robust against changes in parameter values' is unsupported. These identification gaps propagate directly into the 3–8 month recommendation.
minor comments (6)
  1. [§2.3, Note] The note 'assuming a reproduction substituting of the form S(t) = N − ϵaλt' is garbled; presumably S(t) = N − εa e^{λt} is intended, and the exponent should be written without ambiguity.
  2. [§3.3, p. 14] The sentence 'ϕj ≡ ϕ and ψj ≡ ψ, for j = 1, 2, · · ·, N' uses N, the population size, where K, the number of doses from Eq. (8), is clearly intended.
  3. [§2.3, Eq. (19) vs §3.4, Eq. (24)] R0 is defined by truncating β at an assumed disease duration τ, but the fitted viral-load kernel (24) is a Gaussian with support on all positive times; the truncation is never reconciled with the fitted β, and no computed value of R0 from the estimated parameters is reported.
  4. [§4.3, Eqs. (26)–(27)] The constraint '0 ≤ m(t;n) ≤ 1' is stated in the minimization problem but is never enforced, checked, or discussed in the results; the definition of J(n) as the result of a minimization over n is circular notation, and the passage from (26) to (27) via T = an is stated imprecisely.
  5. [Table 1] The table entry 'r(t), d(t) and death rates and death rates' contains a duplicated phrase, and the survival probability p0 = 0.9975 used in Eq. (25) does not appear in the table.
  6. [§3.3, Fig. 3] No goodness-of-fit statistics (R², SSE, RMSE) are given for the ψ(t) fit, in contrast to the fits in Figures 2 and 4; given the extremely wide confidence intervals reported in the text, this omission is material to assessing the fit.

Circularity Check

2 steps flagged · score 6.0 of 10

Optimal-interval 'prediction' is a readout of the fitted immunity-waning curve; the 'hysteresis' conclusion is built into the convolution definition.

  1. fitted input called prediction [Section 3.3 (Eq. 21) and Section 4.3 (Eq. 27, Fig. 9)]
    "ϕ(t) = a1e^{−((t−b1)/c1)^2}, where, a1 = 0.9411 with 95% CI (0.8886, 0.9937), b1 = 117.8 with 95% CI (113.5, 122), and c1 = 92.44 with 95% CI (86.06, 98.82) (Figure 2a). ... Figure (9) shows that the cost function more or less remains at the minimum when the gap between two successive vaccinations falls within the range of 3 to 8 months. However, the plot takes an interesting turn when the gap between consecutive vaccinations exceeds 9 months."

    The model's immunity m(t) is defined in Eq. (8) as a convolution of the vaccination schedule V'(η) with this same fitted efficacy function ϕ. Since Eq. (12) sets S(t) = N − (I + D + P + m1(t)N), the infection cost inside J(a) (Eq. 27) inherits the timescale of the fitted ϕ. The fitted Gaussian peaks at b1 = 117.8 days ≈ 3.9 months and has width c1 = 92.44 days ≈ 3 months, which is the scale of the reported 3-8 month plateau and the rise beyond 9 months. The optimal interval is thus a direct readout of the fitted immunity-waning curve rather than an independent prediction; refitting ϕ would move the claimed optimum.

  2. self definitional [Section 2.2.1 (Eq. 8) and Section 4.2 (Scenario-1/2 discussion)]
    "m(t) = 1/N ( Σ_{i=1}^K ∫_0^t ϕ_i(t − η)V'_i(η)dη ). ... From Figure 8, we observe that Scenario-1 and Scenario-2 depict almost the same epidemic progression ... both scenarios eventually accord the same level of immunity. This is a key observation that can help us to avoid unnecessary vaccinations."

    Eq. (8) defines population immunity as a linear convolution of dose timing with the per-dose efficacy ϕ. If two doses are separated by less than the width of ϕ, the remaining immunity from the first dose automatically overlaps the second dose's contribution, so the marginal gain of the extra dose is small by construction. The paper names this overlap 'hysteresis' and concludes that frequent vaccination is unnecessary, but no interaction term, threshold, or state-dependent memory appears in Eqs. (8)-(11). The observation that 4-month and 8-month schedules give nearly the same epidemic progression is therefore a restatement of the convolution assumption and the fitted shape of ϕ, not an emergent property of the epidemic dynamics.

full rationale

The R0 derivation (Eqs. 16-19) is a standard dispersion-relation argument and is not circular. The paper's self-citations ([33]-[35]) supply the distributed recovery/death kernels and the immuno-epidemic framework; because those kernels are fit to clinical and experimental data in the cited works, they constitute external evidence rather than an unverified uniqueness claim, so I do not score them as circular per se. However, the central quantitative result—the cost-minimizing dose interval of 3-8 months (Fig. 9)—is a numerical output whose month-scale structure is imposed by the fitted vaccine-efficacy curve ϕ(t) (Eq. 21, b1 = 117.8 days, c1 = 92.44 days). The model defines immunity as a convolution of vaccination pulses with this same ϕ (Eq. 8), and the susceptible pool is then reduced by that immunity (Eq. 12); hence the interval optimization is effectively performed against the fitted waning timescale. The 'hysteresis' language likewise relabels the linear superposition of overlapping waning curves rather than a derived mechanism. Separately, the comorbid system (Eqs. 10-15) is not closed: no evolution equation is given for S(t), Eq. (12) is an algebraic replacement for mass balance, and P(t) in Eq. (13) can become negative; these are correctness risks that make the simulated cost curves less trustworthy, but they are not themselves instances of circularity. I therefore score 6: one or more central 'predictions' reduce by construction to fitted or constitutive inputs, even though the R0 analysis and some parameter sensitivity explorations retain independent content.

Assumptions & free parameters 17 free parameters · 8 assumptions · 0 invented entities

The model introduces no new physical or biological entities, but it relies on many handpicked parameters, ad hoc constitutive relations, and imported gamma distributions from the authors' earlier work. The cost function weights are arbitrary and directly determine the reported optimal interval. The claimed hysteresis effect is a label applied to a linear convolution, not a new mechanistic entity.

free parameters (17)
  • Total population N = 1e7
    Chosen without data or justification in Table 1.
  • Initial vaccinated V0 = 500
    Chosen arbitrarily in Table 1.
  • Proportion expected vaccinated L = 0.75
    Assumed in Table 1, no source given.
  • Vaccination rate k = 0.002, 0.003, 0.005
    Selected values in Table 1; no fitting or source.
  • Transmission proportionality constant c = 0.44e-5
    Handpicked in Table 1, no source.
  • Vaccination allocation proportion alpha = 0.8
    Assumed in Table 1, no source.
  • Comorbid susceptibility amplification kappa = 1.1
    Handpicked in Table 1; used to represent multiple strains.
  • Comorbidity proportion b = 0.2
    Assumed in Table 1, no source.
  • Comorbid immunity discount epsilon = 0.7
    Handpicked in Table 1, no source.
  • Survival probability p0 = 0.9975
    Assumed in Section 3.5, no source.
  • Cost weight for infections c (Eq. 26) = 0.01
    Arbitrary scale chosen for Figure 9; central to the 3-8 month optimum.
  • Cost per vaccination campaign d = 5
    Arbitrary scale chosen for Figure 9; central to the 3-8 month optimum.
  • Vaccine immunity Gaussian fit, Eq. 21 = a1=0.9411, b1=117.8, c1=92.44
    Least-squares fit to clinical data from [35,37], used as an input function.
  • Vaccine immunity power-law fit, Eq. 22 = a2=0.01152, b2=1.023, c2=5.01e-6, d2=2.412
    Alternative fit to the same immunity data; used for sensitivity.
  • Acquired immunity fit, Eq. 23 = a3=1.035, b3=-206.6, c3=1133
    Fit to data from [38]; b3 is negative and no goodness of fit is reported.
  • Viral load fit, Eq. 24 = a4=1.829e5, b4=3.136, c4=1.294
    Fit to ex vivo viral load data from [39], treated as proportional to transmission rate.
  • Bimodal gamma recovery/death parameters, Eq. 25 = a1=32.17, b1=0.2206, c1=65.41, d1=0.210; a2=36.03, b2=0.575, c2=140.11, d2=0.276
    Taken directly from the authors' earlier paper [35], not re-fit here.
assumptions (8)
  • standard math Incidence follows the renewal equation J(t) = (S(t)/N) * integral of beta(t-s) J(s) ds.
    This is the central modeling assumption of the paper, introduced in Eq. (3).
  • domain assumption Homogeneous mixing and constant total population size.
    Assumed in Eqs. (1) and (12), and acknowledged in the Discussion.
  • ad hoc to paper Vaccination-generated immunity obeys the constitutive relation m(t) = (1/N) * integral of phi(t-s) V'(s) ds.
    This relation, Eq. (7), is postulated without derivation from an underlying immune process.
  • ad hoc to paper All vaccine doses have uniform efficacy.
    Stated in Section 2.2.1 and listed as a limitation in the Discussion.
  • ad hoc to paper Multiple strains can be represented by a single constant kappa multiplying comorbid infectivity.
    Introduced in Section 2.2.3; no explicit strain dynamics or cross-immunity are modeled.
  • domain assumption Recovery and death rates follow bimodal gamma distributions taken from [35].
    Parameter values are imported from the authors' prior work, Eq. (25).
  • standard math The basic reproduction number is derived in a disease-free setting with no vaccination or immunity.
    Section 2.3 sets I=P=D=m1=0 to derive R0.
  • ad hoc to paper Cost is a linear combination of total infections and number of campaigns.
    The cost function in Eq. (26) is chosen for convenience, not derived from economic data.

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Cite this review

Pith. "Pith review of Effect of Vaccine Dose Intervals: Considering Immunity Levels, Vaccine Efficacy, and Strain Variants for Disease Control Strategy." pith.science (2026). https://pith.science/paper/CA37TRQ7

@misc{pith2026250521132,
  author       = {Pith},
  title        = {Pith review of: Effect of Vaccine Dose Intervals: Considering Immunity Levels, Vaccine Efficacy, and Strain Variants for Disease Control Strategy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CA37TRQ7}},
  note         = {Machine review of arXiv:2505.21132}
}
read the original abstract

In this study, we present an immuno-epidemic model to understand mitigation options during an epidemic break. The model incorporates comorbidity and multiple-vaccine doses through a system of coupled integro-differential equations to analyze the epidemic rate and intensity from a knowledge of the basic reproduction number and time-distributed rate functions. Our modeling results show that the interval between vaccine doses is a key control parameter that can be tuned to significantly influence disease spread. We show that multiple doses induce a hysteresis effect in immunity levels that offers a better mitigation alternative compared to frequent vaccination which is less cost-effective while being more intrusive. Optimal dosing intervals, emphasizing the cost-effectiveness of each vaccination effort, and determined by various factors such as the level of immunity and efficacy of vaccines against different strains, appear to be crucial in disease management. The model is sufficiently generic that can be extended to accommodate specific disease forms.

Figures

Figures reproduced from arXiv: 2505.21132 by the authors.

Figure 1
Figure 1. Schematic diagram for the system (10)-(15). 2.3 Basic reproduction number In the beginning of epidemic, assume that I = P = D = m1 = 0. Then, using (2), we can write equation (5) in the following form: S ′ (t) = S N Z t 0 β(t − η)S ′ (η)dη. (16) Suppose, S(t) = N − ϵeλt. Then from the above equation we get, dS(t) dt = −ϵλeλt = N − ϵeλt N Z t 0 β(t − η)(−ϵλ)e ληdη. (17) Now equating the terms with the first power of … view at source ↗
Figure 2
Figure 2. The effectiveness of vaccine-induced immunity [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. The effectiveness of infection-acquired immunity [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Viral load as a function of the days post infection. The blue dots [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Time-distributed rate functions of (a) recovery and (b) death as [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Plot of I(t) for different choice of κ. The associated parameter values are chosen as estimated before and as in Table. 1. (which is almost 5.5 years), where we’re trying to control an epidemic. To explain the effect we consider three scenarios as follows: Scenario-1: …
Figure 7
Figure 7. Figure 7: Plot of I(t) for different choice of κ. (a) κ = 1; (b) κ = 2; (c) κ = 3. The bold colored curves correspond to c3 = 1133 in formula (23). The light colored curves correspond to 20 randomly chosen values of c3 in the interval [1033, 1233]. The other associated parameter…
Figure 8
Figure 8. Figure 8: (a), (b) correspond to Scenario-1; (c), (d) correspond to Scenario [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Plot of cost function J (a) for c = 0.01, d = 5, and all other parameter values are chosen as estimated before and as in Table. 1. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Plot of cost function J (a) for different vaccine efficacy functions. Green: corresponds to formula 21 and Red: corresponds to formula 21 mul￾tiplied by 0.7. The parameter values: c=0.01, d=5, and all other parameter values are chosen as estimated before and as in Tab…
Figure 11
Figure 11. Figure 11: Plot of cost function J (a) for different vaccination rates. The left panel corresponds to formula 20 with the rate of vaccination k = 0.001 (green) and k = 0.003 (red). The right panel corresponds to the plot of the cost function with corresponding colors. The parame…
Figure 12
Figure 12. Figure 12: Plot of cost function J (a) for different values of κ. The green and red curves correspond to κ = 2 and κ = 1 respectively. The parameter values: c=0.01, d=5, and all other parameter values are chosen as estimated before and as in Table. 1. tively control the epidemic…

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