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Within-day human movement requires a new family of reproduction numbers that closed-population R(t) estimators systematically miss.

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2026-07-31 04:56 UTC pith:TKPMTUBM

load-bearing objection Solid first-principles PDE-to-NGM taxonomy for mobility-conditioned R(t); operational claims and the abstract’s mobile-phone line outrun the synthetic evidence. the 4 major comments →

arxiv 2607.28514 v1 pith:TKPMTUBM submitted 2026-07-30 q-bio.QM math.DS

Multi-scale measures of time-varying epidemic spread on human mobility networks

Cathal Mills , Benjamin Reddy , William S. Hart , Robin N. Thompson , Kris V. Parag , Moritz U. G. Kraemer , Christl A. Donnelly , Ben Lambert This is my paper
classification q-bio.QM math.DS
keywords instantaneous reproduction numberhuman mobilityrenewal equationnext-generation matrixmulti-scale epidemic controltype reproduction numberwithin-day movementmobility networks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard real-time estimates of the instantaneous reproduction number treat each place as a closed, well-mixed population and therefore cannot say how people generate or acquire infections when they spend fractions of a day in different locations. This paper builds a network model from infection-age partial differential equations whose boundary conditions encode time-varying presence of susceptibles and infecteds, then derives mechanism-led renewal equations and a next-generation matrix from those equations. The spectral radius of that matrix is the network-level instantaneous reproduction number; its entries yield pairwise, inward, outward, meeting-location, and type reproduction numbers, plus elasticities and transient indicators. On simulated dense-urban, sparse-rural, and hub-amplified networks the authors show these quantities correct biases in independent location-specific estimators and rank corridors and locations for targeted control of different strength, type, and duration. A reader who accepts the argument gains multi-scale indicators that match how real outbreaks are shaped by continuous mobility rather than by static patches.

Core claim

From a mobility-informed system of infection-age PDEs with boundary conditions that mix susceptible and infected fractional presence across locations, the authors derive renewal equations and a next-generation matrix whose spectral radius is the network instantaneous reproduction number R(t). They further define inward, outward, pairwise, meeting-location, and type reproduction numbers, kernels, generation-time distributions, elasticities, and transient measures that quantify transmission at network, location, corridor, and venue scales and that closed-population location-specific estimators cannot recover.

What carries the argument

Mobility-informed infection-age PDEs with fractional-presence boundary conditions, which produce a next-generation matrix whose spectral radius is network-level R(t) and whose structure generates the full family of multi-scale reproduction numbers, elasticities, and transient indicators.

Load-bearing premise

The demonstrations and bias comparisons rest on simulated mobility networks and a deterministic large-population model with homogeneous mixing inside each place and movement that does not depend on who is infected, not on fitted inference from real stay-time mobile-phone data.

What would settle it

Apply the framework to real anonymised GPS stay-time data and multi-location incidence for an actual outbreak; if independent closed-population R(t) estimators do not systematically misattribute local versus imported transmission relative to the mobility-derived inward, outward, and network quantities, or if elasticity rankings fail to identify corridors whose control reduces network growth as predicted, the central operational claim does not hold.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Closed-population per-location R(t) estimators systematically misattribute local versus imported transmission when within-day movement is ignored.
  • Network R(t) as the spectral radius of the mobility next-generation matrix, used with a risk-averse average of outward reproduction numbers, can avoid false stability signals from simple averaging.
  • Elasticities of network R with respect to pairwise pathways rank travel corridors by control payoff per unit effort after accounting for network structure.
  • Type reproduction numbers, when defined, identify minimal sets of locations whose control is sufficient to drive network R below one.
  • Meeting-location reproduction numbers can flag high-footfall venues that residence-based metrics miss.

Where Pith is reading between the lines

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

  • Operational use will depend on whether high-resolution fractional-time mobility can be obtained and regularised so contact rates and presence probabilities remain identifiable from incidence alone.
  • The same PDE-to-matrix construction could be embedded in existing real-time R(t) software once multivariate latent states and reporting delays are handled.
  • Transient measures such as reactivity and amplification envelopes will matter most in asymmetric hub cities, where one-generation spikes can occur even when asymptotic network R is below one.
  • Allowing movement to depend on infection status (for example symptomatic stay-home behaviour) would reorder outward and type rankings in ways the current simulations do not capture.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper derives a mobility-informed age-structured PDE system for infection densities by residence location, with boundary conditions that encode within-day presence of susceptibles and infecteds across meeting locations. From the method of characteristics it obtains mechanism-led renewal equations and a family of instantaneous kernels and reproduction numbers: pairwise R_kj(t), outward and inward R, meeting-location R, type reproduction numbers, and a network-level R(t) as the spectral radius of the next-generation matrix, plus elasticities and transience diagnostics (reactivity, mixing ratio, amplification envelopes, risk-averse E(t)). On synthetic dense-urban, sparse-national, and hub-amplified networks it illustrates multi-scale indicators and shows that closed-population and incidence-weighted estimators can bias network-level signals relative to the NGM quantities. The central claim is that these quantities correct unsuitable closed-population location-specific R(t) tools and enable spatially and temporally targeted control.

Significance. If the derivations and operational interpretations hold under realistic mobility and surveillance constraints, the work would give epidemic response a coherent multi-scale R taxonomy grounded in within-day movement rather than closed patches or ad hoc import terms. Strengths include a clear PDE-to-renewal-to-NGM pipeline, standard use of Perron–Frobenius and type-reproduction constructions, explicit tables linking quantities to control uses, numerical solver checks (mass balance, BC residual, GT truncation), and open code. The contribution is primarily theoretical and methodological; significance for real-time policy depends on whether the indicators remain identifiable and stable with imperfect stay-time data—an issue the Discussion acknowledges but the applications do not yet stress-test.

major comments (4)
  1. [Abstract; §2.3–2.4; §3.2] Abstract and §3.2 claim application “alongside mobile phone data,” but §2.3–2.4 and all results use only simulated f_jk (distance–attractiveness base matrix, DoW scales, lognormal noise). No anonymised GPS stay-time series is analysed. The phrase should be removed or replaced by a clear statement that mobility is synthetic; otherwise the operational claim that the framework corrects estimators and designs interventions in real outbreaks overreaches the evidence.
  2. [§3.2.2; Fig. 4; SI Figs. 10–26] Bias comparisons (Fig. 4; SI 10–26) show that if incidence is generated by the authors’ PDE and NGM, then R_ind, aggregate, population-weighted, and incidence-weighted estimators mis-signal relative to ρ(R(t)) and E(t). That is internal consistency, not external validation. A load-bearing claim that the family “corrects” existing tools needs at least one transfer check: e.g. misspecified or noisy f_jk, infection-dependent mobility, or confounded χ_kl, or a real stay-time case study. Without that, the precision claimed for corridor/location targeting does not follow from the (sound) derivations alone.
  3. [§3.1.7; §3.2.1; Tables 1–2] §3.2.1 and Figs. 3K, SI 28–30 note that R^j_type(t) is often undefined before the incidence peak (ρ(R_JJ)≥1), precisely when targeting is most needed. Group type numbers help only partially. The manuscript should either restrict operational guidance for type R to regimes where it is defined, or supply a default fallback (e.g. elasticities + E(t) + σ) with explicit decision rules, so Tables 1–2 do not oversell type R as a general real-time control target.
  4. [§2.4; eqs. 4–7; SI Fig. 6] Core modelling assumptions—homogeneous mixing within each meeting location, infection-status-independent f_jl, and a fixed infectiousness profile (§2.4; eqs. 4–7)—are stated but not stress-tested against alternatives that would change kernels and rankings (e.g. symptom-reduced mobility, home-vs-away contact structure beyond λ_B=0.3λ_W). Sensitivity in SI 6 varies R0, seeding, and p(a_E) but not these structural choices. At least one such counterfactual is needed if multi-scale intervention rankings are presented as robust.
minor comments (5)
  1. [§3.1.2–3.1.3] Notation occasionally swaps index order (K_kj vs matrix rows as infectees); a single convention stated once near eq. (12) and (22) would help.
  2. [Figure 3 caption] Figure 3 panel lettering in the caption (F reused; I/J vs G/H) does not match the plotted panels cleanly.
  3. [§3.2.2; eq. (57)] Eq. (57) for R_ind uses a 7-day Cori-style smoother in text but the displayed formula is the raw ratio; clarify which is plotted in Fig. 3I–J and SI 26.
  4. [Appendix E] Appendix E (vector-borne extension) is long and unused in results; consider shortening or moving fully to SI with a one-paragraph pointer in the main text.
  5. [Title page; References] Typos: “Kris V . Parag” spacing in author list; “birello” citation casing; occasional double spaces in §1.

Circularity Check

0 steps flagged

No significant circularity: mechanism-led definitions from PDEs/NGM, not fits relabeled as predictions.

full rationale

The paper’s load-bearing chain is a standard first-principles construction: infection-age PDEs with mobility-informed boundary conditions (eqs. 4–6) yield, via characteristics, renewal equations and non-negative kernels K_kj built from f, S, and λ (eqs. 8–13); pairwise R_kj are integrals of those kernels (eq. 14); the network instantaneous reproduction number is defined as the spectral radius of the assembled NGM (eqs. 22–23), with type numbers following the classical Heesterbeek–Roberts construction (eqs. 53–56). These are definitions and derived operators, not independent empirical targets recovered from fitted inputs. Illustrative applications use simulated mobility networks and chosen epidemiological parameters (R0 = 1.5, fixed gamma infectiousness, POLYMOD-scaled contacts) to show dynamics and bias relative to closed-population estimators inside the authors’ own DGP; nothing is fit to a data subset and then “predicted.” Self-citations (prior renewal/R(t)/vector-borne work by overlapping authors; Parag risk-averse mean; classical NGM/type-R literature) supply lineage and tools but do not force the multi-scale taxonomy via an unverified uniqueness theorem. Abstract phrasing about mobile-phone data is an evidence overclaim, not a circular derivation step. Score 0 is appropriate.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

Load-bearing content is mostly standard epidemic math (renewal kernels, NGM spectral radius, type reproduction numbers, Perron–Frobenius) plus domain modelling choices for within-day mobility and frequency-dependent transmission. Free parameters appear in the illustrative simulations (R0, contact rates, infectiousness profile, synthetic commuting fractions), not as fitted constants smuggled into a universal law. No new physical entity is postulated; ‘invented’ items are defined epidemiological indices.

free parameters (4)
  • R0 (initial network transmissibility scale) = 1.5 (main); sensitivity 1.5–3.0
    Set to 1.5 in main simulations; varied in sensitivity. Scales overall transmission and shapes incidence trajectories used to illustrate indicators.
  • Within-/between-location contact rates λW, λB = λW=13.03; λB=0.30×λW
    λW from POLYMOD (13.03/day); λB=0.30×λW by assumption of reduced away intensity. Directly enters λ_kl^E and thus all R quantities in applications.
  • Infectiousness profile p(aE) = Gamma(5.5, 1.8), trunc 25d
    Gamma mean 5.5 d, SD 1.8 d, truncated at 25 d (SARS-CoV-2-like). Separability makes many generation-time distributions collapse to this profile.
  • Synthetic mobility parameters (c_j, distance decay δ, DoW scales, attractiveness) = e.g. c_j∈{0.40,0.35,0.28,0.18}; δ=7km urban
    Commuting fractions by node type, exponential radii, gravity/radiation-like weights, day-of-week multipliers and lognormal noise define f_jk(t) and therefore all spatial R(t) patterns.
axioms (6)
  • domain assumption Infection-age densities obey conservation PDEs ∂E_j/∂t+∂E_j/∂aE=0 with mobility-dependent force-of-infection boundary conditions.
    Standard McKendrick–von Foerster structure extended to multi-location presence; §3.1.1 eqs. 4–6.
  • standard math Network instantaneous reproduction number is the spectral radius of the next-generation matrix of between-location R_kj(t); threshold governs asymptotic per-generation growth of the linearised system.
    Classical Diekmann–Heesterbeek NGM theory plus Perron–Frobenius; eqs. 22–23, Appendix C.
  • domain assumption Frequency-dependent transmission λ_kl^E(t,aE)=χ_kl(t)·p_k(aE)/N_l^eff(t) with effective population from presence-weighted residents.
    One common contact formulation; eq. 7. Alternatives (density-dependent, home-status conditioning) noted but not primary.
  • domain assumption Within each meeting location, contacts are homogeneously mixed (unless further conditioned); movement may be taken independent of infection status in applications.
    Stated modelling choice in §2.4 and §3.1.1; limits realism of contact structure.
  • standard math Type reproduction number R_j^type uses the Heesterbeek–Roberts construction on the mobility NGM and is defined only when the background subnetwork is subcritical.
    Imported from type-reproduction theory; eqs. 53–56, Appendix D.
  • domain assumption Deterministic large-population limit is adequate for the illustrated location sizes; stochastic fade-out and observation error are deferred.
    Discussion explicitly limits suitability when local populations are small.
invented entities (1)
  • Family of mobility-conditioned instantaneous kernels/R measures (R_kj, R_out, R_in, R_meeting, R_type, R_network) and associated elasticities/transience indicators (σ, s, A(n), E(t)) no independent evidence
    purpose: Provide multi-scale control-relevant summaries of transmission on a time-varying mobility network beyond single closed-population R(t).
    Defined mathematically from the NGM/kernels rather than postulated as new biological objects; independent operational meaning requires external mobility+incidence data.

pith-pipeline@v1.2.0-daily-grok45 · 66262 in / 4082 out tokens · 81493 ms · 2026-07-31T04:56:40.054187+00:00 · methodology

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read the original abstract

Human movement drives the spatial spread and persistence of many infectious diseases, yet existing theory and real-time operational tools for inferring the instantaneous reproduction number R(t) often assume static and/or homogeneously mixing populations and cannot describe how individuals generate and acquire infections heterogeneously based on their movement patterns within a day. Renewal equations underpin many such popular estimators of R(t), and here, we develop a network-based modelling framework from which we derive new mechanism-led renewal equations and control indicators for outbreaks of infectious diseases. These equations directly integrate within-day human movement to rigorously define a family of instantaneous reproduction numbers; inward, outward, and type R(t) for individual locations, R(t) between locations, R(t) at meeting locations, and R(t) for the entire mobility network. These quantities correct for the unsuitability of existing location-specific R(t) estimators that operate in closed, static populations. Applying our framework to epidemics on diverse types of networks alongside mobile phone data, we demonstrate how our new framework's outputs provide new, multi-scale control indicators at the network, location, and transmission corridor scales, and can be used to design targeted disease control interventions including the strength, type, and length of intervention required across space and time. We capture the biasing effects of different existing ways to measure location-specific and network-level transmission potential without capturing within-day human movements. This generalisable framework redefines reproduction numbers in real-world outbreaks that are shaped by individuals moving across connected locations, enabling more spatially and temporally precise interventions.

Figures

Figures reproduced from arXiv: 2607.28514 by Benjamin Reddy, Ben Lambert, Cathal Mills, Christl A. Donnelly, Kris V. Parag, Moritz U. G. Kraemer, Robin N. Thompson, William S. Hart.

Figure 1
Figure 1. Figure 1: Schematic of key concepts for our framework of modelling epidemics in moving populations A: A snapshot of an epidemic generation on a mobility network where we describe how infections occur by infected individuals (red) infecting susceptible individuals (blue), possibly due to movement of these individuals between locations. B: The renewal equation (left) for directly transmitted and vector-borne diseases … view at source ↗
Figure 2
Figure 2. Figure 2: Overall human mobility and epidemic dynamics across space and time: A) Human mobility network in a dense-urban setting (main Scenario A) with ten nodes/locations (circles) connected by edges denoting the time-averaged probability f jk that residents of location j are in location k, with row sums equal to 1. Edge widths and colour denote f jk and node size and colours denote population N j for each location… view at source ↗
Figure 3
Figure 3. Figure 3: Taxonomy of reproduction numbers, generation time distributions, and related quantities: A) Probability densities for different types of generation time distributions which overlap across space and time due to assumptions of infection age independence for contact rates and movement patterns and location-invariant infectiousness profiles. B) and C) are the outward and inward reproduction numbers respectivel… view at source ↗
Figure 4
Figure 4. Figure 4: Bias in different estimators of network-level epidemic transmissibility: We compare our framework’s network-level reproduction number R(t) and risk-averse reproduction number E(t) alongside an approach that weights independent, per-location estimates R j ind(t) by their relative transmissibility (i.e. R j ind(t) P k Rk ind(t) ). Rows denote scenarios A, B, and C, described in Materials and Methods and Resu… view at source ↗
Figure 5
Figure 5. Figure 5: Transience and importation analyses across space and time: A) Mixing time ratio s(t) and day-of-week (DoW) scaling used for human movement simulation. B) Reactivity σ(t) and network-level R(t) over time. C) Amplification envelope using the ℓ 1 or ℓ 2 norm, A(n) or A1(n) respectively, divided by Rn for n generations at early, peak, and late stages of the outbreak. D) Magnitudes of the top three eigenvalues … view at source ↗

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Reference graph

Works this paper leans on

116 extracted references · 48 canonical work pages

  1. [1]

    Naser Lessani, Zhenlong Li, Fengrui Jing, Shan Qiao, Jiajia Zhang, Bankole Olatosi, and Xiaoming Li

    M. Naser Lessani, Zhenlong Li, Fengrui Jing, Shan Qiao, Jiajia Zhang, Bankole Olatosi, and Xiaoming Li. Human mobility and the infectious disease transmission: a systematic review.Geo-spatial Information Science, 27 (6):1824–1851, November 2024. ISSN 1009-5020, 1993-5153. doi: 10.1080/10095020.2023.2275619. URL https://www.tandfonline.com/doi/full/10.1080...

    arXiv 2024

  2. [2]

    Moritz U. G. Kraemer, Chia-Hung Yang, Bernardo Gutierrez, Chieh-Hsi Wu, Brennan Klein, David M. Pigott, Open COVID-19 Data Working Group, Louis Du Plessis, Nuno R. Faria, Ruoran Li, William P. Hanage, John S. Brownstein, Maylis Layan, Alessandro Vespignani, Huaiyu Tian, Christopher Dye, Oliver G. Pybus, and Samuel V . 28 Scarpino. The effect of human mobi...

    work page doi:10.1126/scien 2020

  3. [3]

    Bicout, and Charin Modchang

    Suttikiat Changruenngam, Dominique J. Bicout, and Charin Modchang. How the individual human mobility spatio- temporally shapes the disease transmission dynamics.Scientific Reports, 10(1):11325, July 2020. ISSN 2045-2322. doi: 10.1038/s41598-020-68230-9. URLhttps://www.nature.com/articles/s41598-020-68230-9

    work page doi:10.1038/s41598-020-68230-9 2020

  4. [4]

    Natalya Kostandova, Ronan Corgel, Shweta Bansal, Sophie Bérubé, Eimear Cleary, Chelsea Hansen, Matt D. T. Hitchings, Bernardo García-Carreras, Lauren Gardner, Moritz U. G. Kraemer, Shengjie Lai, Yao Li, Amanda C. Perofsky, Giulia Pullano, Jonathan M. Read, Gabriel Ribeiro Dos Santos, Henrik Salje, Saki Takahashi, Cécile Viboud, Jasmine Wang, Derek A. T. C...

    2025

  5. [5]

    Pierre Nouvellet, Sangeeta Bhatia, Anne Cori, Kylie E. C. Ainslie, Marc Baguelin, Samir Bhatt, Adhiratha Boonyasiri, Nicholas F. Brazeau, Lorenzo Cattarino, Laura V . Cooper, Helen Coupland, Zulma M. Cucunuba, Gina Cuomo-Dannenburg, Amy Dighe, Bimandra A. Djaafara, Ilaria Dorigatti, Oliver D. Eales, Sabine L. Van Elsland, Fabricia F. Nascimento, Richard G...

    2021

  6. [6]

    Buckee, Satchit Balsari, Jennifer Chan, Mercè Crosas, Francesca Dominici, Urs Gasser, Yonatan H

    Caroline O. Buckee, Satchit Balsari, Jennifer Chan, Mercè Crosas, Francesca Dominici, Urs Gasser, Yonatan H. Grad, Bryan Grenfell, M. Elizabeth Halloran, Moritz U. G. Kraemer, Marc Lipsitch, C. Jessica E. Metcalf, Lauren Ancel Meyers, T. Alex Perkins, Mauricio Santillana, Samuel V . Scarpino, Cecile Viboud, Amy Wesolowski, and Andrew Schroeder. Aggregated...

  7. [7]

    Jorge L. B. Araújo, Rafael Bomfim, Cesar I. N. Sampaio Filho, Luciano P. G. Cavalcanti, Antonio S. Lima Neto, José S. Andrade, and Vasco Furtado. The impact of COVID-19 mobility restrictions on dengue transmission in urban areas.PLOS Neglected Tropical Diseases, 18(11):e0012644, November 2024. ISSN 1935-2727. doi: 10.1371/journal.pntd.0012644. URLhttps://...

    work page doi:10.1371/journal.pntd.0012644 2024

  8. [8]

    You Li, Xin Wang, Harry Campbell, and Harish Nair. The association of community mobility with the time-varying reproduction number (R) of SARS-CoV-2: a modelling study across 330 local UK authorities.The Lancet Digital Health, 3(10):e676–e683, October 2021. ISSN 25897500. doi: 10.1016/S2589-7500(21)00144-8. URL https://linkinghub.elsevier.com/retrieve/pii...

    work page doi:10.1016/s2589-7500(21)00144-8 2021

  9. [9]

    Kraemer, Pierre Nouvellet, and Anne Cori

    Jack Wardle, Sangeeta Bhatia, Moritz U.G. Kraemer, Pierre Nouvellet, and Anne Cori. Gaps in mobility data and implications for modelling epidemic spread: A scoping review and simulation study.Epidemics, 42:100666, March 2023. ISSN 17554365. doi: 10.1016/j.epidem.2023.100666. URL https://linkinghub.elsevier. com/retrieve/pii/S1755436523000026

    arXiv 2023

  10. [10]

    Buckee, Kenth Engø-Monsen, and C

    Amy Wesolowski, Caroline O. Buckee, Kenth Engø-Monsen, and C. J. E. Metcalf. Connecting Mobility to Infectious Diseases: The Promise and Limits of Mobile Phone Data.Journal of Infectious Diseases, 214 (suppl 4):S414–S420, December 2016. ISSN 0022-1899, 1537-6613. doi: 10.1093/infdis/jiw273. URL https: //academic.oup.com/jid/article-lookup/doi/10.1093/infd...

    work page doi:10.1093/infdis/jiw273 2016

  11. [11]

    Ferguson, Christophe Fraser, and Simon Cauchemez

    Anne Cori, Neil M. Ferguson, Christophe Fraser, and Simon Cauchemez. A new framework and software to estimate time-varying reproduction numbers during epidemics.American Journal of Epidemiology, 178(9): 1505–1512, November 2013. ISSN 1476-6256, 0002-9262. doi: 10.1093/aje/kwt133. URL https://academic. oup.com/aje/article-lookup/doi/10.1093/aje/kwt133

    doi:10.1093/aje/kwt133 2013

  12. [12]

    Thompson, Katharine Sherratt, Hamish P

    Sam Abbott, Joel Hellewell, Robin N. Thompson, Katharine Sherratt, Hamish P. Gibbs, Nikos I. Bosse, James D. Munday, Sophie Meakin, Emma L. Doughty, June Young Chun, Yung-Wai Desmond Chan, Flavio Finger, Paul Campbell, Akira Endo, Carl A. B. Pearson, Amy Gimma, Tim Russell, CMMID COVID modelling group, Stefan Flasche, Adam J. Kucharski, Rosalind M. Eggo, ...

  13. [13]

    Parag, Robin N

    Kris V . Parag, Robin N. Thompson, and Christl A. Donnelly. Are epidemic growth rates more informative than reproduction numbers?Journal of the Royal Statistical Society Series A: Statistics in Society, 185:S5–S15, November 2022. ISSN 0964-1998, 1467-985X. doi: 10.1111/rssa.12867. URL https://academic.oup.com/ jrsssa/article/185/Supplement_1/S5/7069470

    work page doi:10.1111/rssa.12867 2022

  14. [14]

    doi: 10.12688/wellcomeopenres.16006.2

    ISSN 2398-502X. doi: 10.12688/wellcomeopenres.16006.2. URL https://wellcomeopenresearch. org/articles/5-112/v2. 29

    doi:10.12688/wellcomeopenres.16006.2

  15. [15]

    A structured epidemic model incorporating geographic mobility among regions

    Lisa Sattenspiel and Klaus Dietz. A structured epidemic model incorporating geographic mobility among regions. Mathematical Biosciences, 128(1-2):71–91, July 1995. ISSN 00255564. doi: 10.1016/0025-5564(94)00068-B. URLhttps://linkinghub.elsevier.com/retrieve/pii/002555649400068B

    arXiv 1995

  16. [16]

    Kris V . Parag. Improved estimation of time-varying reproduction numbers at low case incidence and between epidemic waves.PLOS Computational Biology, 17(9):e1009347, September 2021. ISSN 1553-7358. doi: 10.1371/journal.pcbi.1009347. URLhttps://dx.plos.org/10.1371/journal.pcbi.1009347

    work page doi:10.1371/journal.pcbi.1009347 2021

  17. [17]

    Soriano-Paños, L

    D. Soriano-Paños, L. Lotero, A. Arenas, and J. Gómez-Gardeñes. Spreading Processes in Multiplex Metapopula- tions Containing Different Mobility Networks.Physical Review X, 8(3):031039, August 2018. ISSN 2160-3308. doi: 10.1103/PhysRevX.8.031039. URLhttps://link.aps.org/doi/10.1103/PhysRevX.8.031039

    doi:10.1103/physrevx.8.031039 2018

  18. [18]

    Reaction–diffusion processes and metapopulation models in heterogeneous networks.Nature Physics, 3(4):276–282, April 2007

    Vittoria Colizza, Romualdo Pastor-Satorras, and Alessandro Vespignani. Reaction–diffusion processes and metapopulation models in heterogeneous networks.Nature Physics, 3(4):276–282, April 2007. ISSN 1745-2473, 1745-2481. doi: 10.1038/nphys560. URLhttps://www.nature.com/articles/nphys560

    work page doi:10.1038/nphys560 2007

  19. [19]

    Kiss, Joel C

    István Z. Kiss, Joel C. Miller, and Péter L. Simon.Mathematics of Epidemics on Networks: From Exact to Approximate Models, volume 46 ofInterdisciplinary Applied Mathematics. Springer International Publishing, Cham, 2017. ISBN 978-3-319-50804-7 978-3-319-50806-1. doi: 10.1007/978-3-319-50806-1. URL http: //link.springer.com/10.1007/978-3-319-50806-1

    doi:10.1007/978-3-319-50806-1 2017

  20. [20]

    Watts, Roby Muhamad, Daniel C

    Duncan J. Watts, Roby Muhamad, Daniel C. Medina, and Peter S. Dodds. Multiscale, resurgent epidemics in a hierarchical metapopulation model.Proceedings of the National Academy of Sciences, 102(32):11157– 11162, August 2005. doi: 10.1073/pnas.0501226102. URL https://www.pnas.org/doi/10.1073/pnas. 0501226102. Publisher: Proceedings of the National Academy o...

    work page doi:10.1073/pnas.0501226102 2005

  21. [21]

    Controllability of complex networks.Nature, 473(7346):167–173, May 2011

    Yang-Yu Liu, Jean-Jacques Slotine, and Albert-László Barabási. Controllability of complex networks.Nature, 473(7346):167–173, May 2011. ISSN 0028-0836, 1476-4687. doi: 10.1038/nature10011. URL https://www. nature.com/articles/nature10011

    work page doi:10.1038/nature10011 2011

  22. [22]

    Networks and epidemic models.Journal of the Royal Society Interface, 2(4): 295–307, September 2005

    Matt J Keeling and Ken T.D Eames. Networks and epidemic models.Journal of the Royal Society Interface, 2(4): 295–307, September 2005. ISSN 1742-5689. doi: 10.1098/rsif.2005.0051. URL https://www.ncbi.nlm.nih. gov/pmc/articles/PMC1578276/

    arXiv 2005

  23. [23]

    Estimates of the reproduction ratio from epidemic surveillance may be biased in spatially structured populations.Nature Physics, 20(7):1204–1210, July 2024

    Piero Birello, Michele Re Fiorentin, Boxuan Wang, Vittoria Colizza, and Eugenio Valdano. Estimates of the reproduction ratio from epidemic surveillance may be biased in spatially structured populations.Nature Physics, 20(7):1204–1210, July 2024. ISSN 1745-2473, 1745-2481. doi: 10.1038/s41567-024-02471-7. URL https://www.nature.com/articles/s41567-024-02471-7

    work page doi:10.1038/s41567-024-02471-7 2024

  24. [24]

    Rtglm: Unifying estimation of the time-varying reproduction number, R, under the Generalised Linear and Additive Models.Epidemics, 53:100857, December 2025

    Pierre Nouvellet. Rtglm: Unifying estimation of the time-varying reproduction number, R, under the Generalised Linear and Additive Models.Epidemics, 53:100857, December 2025. ISSN 17554365. doi: 10.1016/j.epidem. 2025.100857. URLhttps://linkinghub.elsevier.com/retrieve/pii/S1755436525000453

    arXiv 2025

  25. [25]

    Clapham, and Swapnil Mishra

    Mousumi Roy, Hannah E. Clapham, and Swapnil Mishra. Incorporating human mobility to enhance epidemic response and estimate real-time reproduction numbers.PLOS Computational Biology, 21(11):e1013642, November

  26. [26]

    Spatially explicit effective reproduction numbers from incidence and mobility data

    Cristiano Trevisin, Enrico Bertuzzo, Damiano Pasetto, Lorenzo Mari, Stefano Miccoli, Renato Casagrandi, Marino Gatto, and Andrea Rinaldo. Spatially explicit effective reproduction numbers from incidence and mobility data. Proceedings of the National Academy of Sciences, 120(20):e2219816120, May 2023. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.2219816120...

    work page doi:10.1073/pnas.2219816120 2023

  27. [27]

    Pflug, and Simone Pigolotti

    Samuel Cure, Florian G. Pflug, and Simone Pigolotti. Exponential rate of epidemic spreading on complex networks. Physical Review E, 111(4):044311, April 2025. ISSN 2470-0045, 2470-0053. doi: 10.1103/PhysRevE.111.044311. URLhttps://link.aps.org/doi/10.1103/PhysRevE.111.044311

    work page doi:10.1103/physreve.111.044311 2025

  28. [28]

    M. G. Roberts and J. A. P. Heesterbeek. A new method for estimating the effort required to control an infectious disease.Proceedings of the Royal Society of London. Series B: Biological Sciences, 270(1522):1359–1364, July 30

  29. [29]

    A mobility- adjusted framework for regional Rt estimation: enhancing spatial interpretation of transmission dynamics.BMC Infectious Diseases, 25(1):1473, November 2025

    Young Kim, Junwoo Jo, Byul Nim Kim, Arsen Abdulali, Gerardo Chowell, and Sunmi Lee. A mobility- adjusted framework for regional Rt estimation: enhancing spatial interpretation of transmission dynamics.BMC Infectious Diseases, 25(1):1473, November 2025. ISSN 1471-2334. doi: 10.1186/s12879-025-11889-0. URL https://bmcinfectdis.biomedcentral.com/articles/10....

    work page doi:10.1186/s12879-025-11889-0 2025

  30. [30]

    D. C. P. Jorge, J. F. Oliveira, J. G. V . Miranda, R. F. S. Andrade, and S. T. R. Pinho. Estimating the effective reproduction number for heterogeneous models using incidence data.Royal Society Open Science, 9(9):220005, September 2022. ISSN 2054-5703. doi: 10.1098/rsos.220005. URL https://royalsocietypublishing.org/ doi/10.1098/rsos.220005

    work page doi:10.1098/rsos.220005 2022

  31. [31]

    Kris V . Parag. How to Measure the Controllability of an Infectious Disease?Physical Review X, 14(3):031041, September 2024. ISSN 2160-3308. doi: 10.1103/PhysRevX.14.031041. URL https://link.aps.org/doi/10. 1103/PhysRevX.14.031041

    work page doi:10.1103/physrevx.14.031041 2024

  32. [32]

    Early Estimation of the Reproduction Number in the Presence of Imported Cases: Pandemic Influenza H1N1-2009 in New Zealand.PLoS ONE, 6(5):e17835, May 2011

    Michael George Roberts and Hiroshi Nishiura. Early Estimation of the Reproduction Number in the Presence of Imported Cases: Pandemic Influenza H1N1-2009 in New Zealand.PLoS ONE, 6(5):e17835, May 2011. ISSN 1932-

    2009

  33. [33]

    Heesterbeek and M.G

    J.A.P. Heesterbeek and M.G. Roberts. The type-reproduction number T in models for infectious disease control. Mathematical Biosciences, 206(1):3–10, March 2007. ISSN 00255564. doi: 10.1016/j.mbs.2004.10.013. URL https://linkinghub.elsevier.com/retrieve/pii/S0025556405001549

    work page doi:10.1016/j.mbs.2004.10.013 2007

  34. [34]

    Thompson, J.E

    R.N. Thompson, J.E. Stockwin, R.D. Van Gaalen, J.A. Polonsky, Z.N. Kamvar, P.A. Demarsh, E. Dahlqwist, S. Li, E. Miguel, T. Jombart, J. Lessler, S. Cauchemez, and A. Cori. Improved inference of time-varying reproduction numbers during infectious disease outbreaks.Epidemics, 29:100356, December 2019. ISSN 17554365. doi: 10.1016/j.epidem.2019.100356. URL ht...

    arXiv 2019

  35. [35]

    Fraiberger, Till Koebe, Sune Lehmann, Juan Murillo, Alex Pentland, Phuong N Pham, Frédéric Pivetta, Jari Saramäki, Samuel V

    Nuria Oliver, Bruno Lepri, Harald Sterly, Renaud Lambiotte, Sébastien Deletaille, Marco De Nadai, Emmanuel Letouzé, Albert Ali Salah, Richard Benjamins, Ciro Cattuto, Vittoria Colizza, Nicolas De Cordes, Samuel P. Fraiberger, Till Koebe, Sune Lehmann, Juan Murillo, Alex Pentland, Phuong N Pham, Frédéric Pivetta, Jari Saramäki, Samuel V . Scarpino, Michele...

    2020

  36. [36]

    Tatem, Qianwen Duan, Elliot N

    Natalya Kostandova, Christine Prosperi, Simon Mutembo, Chola Nakazwe, Harriet Namukoko, Bertha Nachinga, Shengjie Lai, Andrew J. Tatem, Qianwen Duan, Elliot N. Kabalo, Kabondo Makungo, Gershom Chongwe, Innocent Chilumba, Gloria Musukwa, Kalumbu H. Matakala, Mutinta Hamahuwa, Webster Mufwambi, Japhet Matoba, Irene Mutale, Kenny Situtu, Edgar Simulundu, Phi...

    2025

  37. [37]

    Jones, P

    Takahiro Yabe, Nicholas K.W. Jones, P. Suresh C. Rao, Marta C. Gonzalez, and Satish V . Ukkusuri. Mobile phone location data for disasters: A review from natural hazards and epidemics.Computers, Environment and Urban Systems, 94:101777, June 2022. ISSN 01989715. doi: 10.1016/j.compenvurbsys.2022.101777. URL https://linkinghub.elsevier.com/retrieve/pii/S01...

    arXiv 2022

  38. [38]

    Parag and Patrick Hoscheit

    Kris V . Parag and Patrick Hoscheit. Why do border controls fail to stabilise epidemics?, May 2025. URL https://www.medrxiv.org/content/10.1101/2025.05.12.25327440v1. Pages: 2025.05.12.25327440

    work page doi:10.1101/2025.05.12.25327440v1 2025

  39. [39]

    Andrew F. Brouwer. Why the Spectral Radius? An intuition-building introduction to the basic reproduction number.Bulletin of Mathematical Biology, 84(9):96, September 2022. ISSN 0092-8240, 1522-9602. doi: 10.1007/s11538-022-01057-9. URLhttps://link.springer.com/10.1007/s11538-022-01057-9

    work page doi:10.1007/s11538-022-01057-9 2022

  40. [40]

    Keeling, Matthew James Keeling, and Pejman Rohani.Modeling infectious diseases in humans and animals

    Matt J. Keeling, Matthew James Keeling, and Pejman Rohani.Modeling infectious diseases in humans and animals. Princeton Univ. Press, Princeton, NJ, 2008. ISBN 978-0-691-11617-4. 31

    2008

  41. [41]

    Odo Diekmann and J. A. P. Heesterbeek.Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley series in mathematical and computational biology. Wiley, Chichester Weinheim,

  42. [42]

    Anderson and Robert M

    Roy M. Anderson and Robert M. May.Infectious diseases of humans: dynamics and control. Oxford Univ. Press, Oxford, reprinted edition, 2010. ISBN 978-0-19-854040-3

    2010

  43. [43]

    Commentary on the use of the reproduction numberRduring the COVID-19 pandemic.Statistical Methods in Medical Research, 31(9):1675–1685, September 2022

    Carolin Vegvari, Sam Abbott, Frank Ball, Ellen Brooks-Pollock, Robert Challen, Benjamin S Collyer, Ciara Dangerfield, Julia R Gog, Katelyn M Gostic, Jane M Heffernan, T Déirdre Hollingsworth, Valerie Isham, Eben Kenah, Denis Mollison, Jasmina Panovska-Griffiths, Lorenzo Pellis, Michael G Roberts, Gianpaolo Scalia Tomba, Robin N Thompson, and Pieter Trapma...

    2022

  44. [44]

    Inference of epidemic dynamics in the COVID-19 era and beyond.Epidemics, 48:100784, September 2024

    Anne Cori and Adam Kucharski. Inference of epidemic dynamics in the COVID-19 era and beyond.Epidemics, 48:100784, September 2024. ISSN 17554365. doi: 10.1016/j.epidem.2024.100784. URL https://linkinghub. elsevier.com/retrieve/pii/S1755436524000458

    arXiv 2024

  45. [45]

    Bolker, Sebastian Funk, C

    Sang Woo Park, Benjamin M. Bolker, Sebastian Funk, C. Jessica E. Metcalf, Joshua S. Weitz, Bryan T. Grenfell, and Jonathan Dushoff. The importance of the generation interval in investigating dynamics and control of new SARS-CoV-2 variants.Journal of The Royal Society Interface, 19(191):20220173, June 2022. ISSN 1742-5662. doi: 10.1098/rsif.2022.0173. URLh...

    arXiv 2022

  46. [46]

    Diekmann, J

    O. Diekmann, J. A. Heesterbeek, and J. A. Metz. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations.Journal of Mathematical Biology, 28(4): 365–382, 1990. ISSN 0303-6812. doi: 10.1007/BF00178324

    doi:10.1007/bf00178324 1990

  47. [47]

    Van Den Driessche

    Julien Arino and P. Van Den Driessche. A multi-city epidemic model.Mathematical Population Studies, 10(3):175–193, January 2003. ISSN 0889-8480, 1547-724X. doi: 10.1080/08898480306720. URL http: //www.tandfonline.com/doi/abs/10.1080/08898480306720

    work page doi:10.1080/08898480306720 2003

  48. [48]

    Diekmann, J

    O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts. The construction of next-generation matrices for compartmental epidemic models.Journal of The Royal Society Interface, 7(47):873–885, June 2010. ISSN 1742-5689, 1742-5662. doi: 10.1098/rsif.2009.0386. URL https://royalsocietypublishing.org/doi/10. 1098/rsif.2009.0386

    arXiv 2010

  49. [49]

    Delamater, Erica J

    Paul L. Delamater, Erica J. Street, Timothy F. Leslie, Y . Tony Yang, and Kathryn H. Jacobsen. Complexity of the Basic Reproduction Number (R0).Emerging Infectious Diseases, 25(1):1–4, January 2019. ISSN 1080-6059. doi: 10.3201/eid2501.171901

    Pith/arXiv arXiv 2019

  50. [50]

    Reproduction numbers of infectious disease models.Infectious Disease Modelling, 2(3): 288–303, August 2017

    Pauline van den Driessche. Reproduction numbers of infectious disease models.Infectious Disease Modelling, 2(3): 288–303, August 2017. ISSN 2468-0427. doi: 10.1016/j.idm.2017.06.002. URL https://www.sciencedirect. com/science/article/pii/S2468042717300209

    work page doi:10.1016/j.idm.2017.06.002 2017

  51. [51]

    Estimating individual and household reproduction numbers in an emerging epidemic.PLOS ONE, 2(8):e758, August 2007

    Christophe Fraser. Estimating individual and household reproduction numbers in an emerging epidemic.PLOS ONE, 2(8):e758, August 2007. ISSN 1932-6203. doi: 10.1371/journal.pone.0000758. URL https://dx.plos. org/10.1371/journal.pone.0000758

    doi:10.1371/journal.pone.0000758 2007

  52. [52]

    Thompson, David J

    Ioana Bouros, Robin N. Thompson, David J. Gavaghan, and Ben Lambert. The time-dependent reproduction number for epidemics in heterogeneous populations.Journal of The Royal Society Interface, 22(228):20250095, July 2025. ISSN 1742-5662. doi: 10.1098/rsif.2025.0095. URL https://royalsocietypublishing.org/ doi/10.1098/rsif.2025.0095

    arXiv 2025

  53. [53]

    Green, Neil M

    William D. Green, Neil M. Ferguson, and Anne Cori. Inferring the reproduction number using the renewal equation in heterogeneous epidemics.Journal of The Royal Society Interface, 19(188):20210429, March 2022. ISSN 1742-5662. doi: 10.1098/rsif.2021.0429. URL https://royalsocietypublishing.org/doi/10.1098/ rsif.2021.0429

    arXiv 2022

  54. [54]

    Hart, Moritz U

    Cathal Mills, Tarek Alrefae, William S. Hart, Moritz U. G. Kraemer, Kris V . Parag, Robin N. Thompson, Christl A. Donnelly, and Ben Lambert. Renewal equations for mosquito-borne diseases.Methods in Ecology and Evolution, pages 2041–210X.70110, September 2025. ISSN 2041-210X, 2041-210X. doi: 10.1111/2041-210X.70110. URL https://besjournals.onlinelibrary.wi...

    work page doi:10.1111/2041-210x.70110 2041

  55. [55]

    Van Den Driessche and James Watmough

    P. Van Den Driessche and James Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.Mathematical Biosciences, 180(1-2):29–48, November 2002. ISSN 00255564. doi: 10.1016/S0025-5564(02)00108-6. URL https://linkinghub.elsevier.com/retrieve/ pii/S0025556402001086

    doi:10.1016/s0025-5564(02)00108-6 2002

  56. [56]

    Parag, Robin N

    Nicholas Steyn, Kris V . Parag, Robin N. Thompson, and Christl A. Donnelly. A primer on inference and prediction with epidemic renewal models and sequential Monte Carlo, March 2025. URL http://arxiv.org/abs/2503. 18875. arXiv:2503.18875 [stat]. 32

    arXiv 2025

  57. [57]

    Tatem, David L

    Amy Wesolowski, Nathan Eagle, Andrew J. Tatem, David L. Smith, Abdisalan M. Noor, Robert W. Snow, and Caroline O. Buckee. Quantifying the Impact of Human Mobility on Malaria.Science, 338(6104):267–270, October

  58. [58]

    Mobile phone data highlights the role of mass gatherings in the spreading of cholera outbreaks.Proceedings of the National Academy of Sciences, 113(23):6421–6426, June 2016

    Flavio Finger, Tina Genolet, Lorenzo Mari, Guillaume Constantin De Magny, Noël Magloire Manga, Andrea Rinaldo, and Enrico Bertuzzo. Mobile phone data highlights the role of mass gatherings in the spreading of cholera outbreaks.Proceedings of the National Academy of Sciences, 113(23):6421–6426, June 2016. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.152230...

    doi:10.1073/pnas.1522305113 2016

  59. [59]

    Inference of the SARS-CoV-2 generation time using UK household data

    William S Hart, Sam Abbott, Akira Endo, Joel Hellewell, Elizabeth Miller, Nick Andrews, Philip K Maini, Sebastian Funk, and Robin N Thompson. Inference of the SARS-CoV-2 generation time using UK household data. eLife, 11:e70767, February 2022. ISSN 2050-084X. doi: 10.7554/eLife.70767. URL https://elifesciences. org/articles/70767

    work page doi:10.7554/elife.70767 2022

  60. [60]

    John Edmunds

    Joël Mossong, Niel Hens, Mark Jit, Philippe Beutels, Kari Auranen, Rafael Mikolajczyk, Marco Massari, Stefania Salmaso, Gianpaolo Scalia Tomba, Jacco Wallinga, Janneke Heijne, Malgorzata Sadkowska-Todys, Magdalena Rosinska, and W. John Edmunds. Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases.PLoS Medicine, 5(3):e74, March...

    doi:10.1371/journal.pmed.0050074 2008

  61. [61]

    Nicholas Steyn and Kris V . Parag. Robust uncertainty quantification in popular estimators of the instanta- neous reproduction number, January 2025. URL https://www.medrxiv.org/content/10.1101/2024.10. 22.24315918v2. Pages: 2024.10.22.24315918

    work page doi:10.1101/2024.10 2025

  62. [62]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson.Matrix analysis. Cambridge University Press, New York, NY , second edition, corrected reprint edition, 2017. ISBN 9780521548236 9780521839402

    2017

  63. [63]

    Sinauer, Sunderland, Mass, 2

    Hal Caswell.Matrix population models: construction, analysis, and interpretation. Sinauer, Sunderland, Mass, 2. ed., 2. printing edition, 2002. ISBN 9780878930968

    2002

  64. [64]

    Neubert and Hal Caswell

    Michael G. Neubert and Hal Caswell. Alternatives to Resilience for Measuring the Responses of Eco- logical Systems to Perturbations.Ecology, 78(3):653–665, April 1997. ISSN 0012-9658. doi: 10.1890/ 0012-9658(1997)078[0653:ATRFMT]2.0.CO;2. URL http://doi.wiley.com/10.1890/0012-9658(1997) 078[0653:ATRFMT]2.0.CO;2

    work page doi:10.1890/0012-9658(1997 1997

  65. [65]

    Hal Caswell and Michael G. Neubert. Reactivity and transient dynamics of discrete-time ecological sys- tems.Journal of Difference Equations and Applications, 11(4-5):295–310, April 2005. ISSN 1023-6198, 1563-5120. doi: 10.1080/10236190412331335382. URL http://www.tandfonline.com/doi/abs/10.1080/ 10236190412331335382

    work page doi:10.1080/10236190412331335382 2005

  66. [66]

    Trefethen and Mark Embree.Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

    Lloyd N. Trefethen and Mark Embree.Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ, 2020. ISBN 9780691213101

    2020

  67. [67]

    Sufficient reproduction numbers to prevent recurrent epidemics.Methods in Ecology and Evolution, 16(6):1186–1200, June 2025

    Lorenzo Mari, Cristiano Trevisin, Andrea Rinaldo, and Marino Gatto. Sufficient reproduction numbers to prevent recurrent epidemics.Methods in Ecology and Evolution, 16(6):1186–1200, June 2025. ISSN 2041-210X, 2041- 210X. doi: 10.1111/2041-210X.70042. URL https://besjournals.onlinelibrary.wiley.com/doi/10. 1111/2041-210X.70042

    work page doi:10.1111/2041-210x.70042 2025

  68. [68]

    Points of view: Color blindness.Nature Methods, 8(6):441–441, June 2011

    Bang Wong. Points of view: Color blindness.Nature Methods, 8(6):441–441, June 2011. ISSN 1548-7091, 1548-7105. doi: 10.1038/nmeth.1618. URLhttps://www.nature.com/articles/nmeth.1618

    doi:10.1038/nmeth.1618 2011

  69. [69]

    Evaluating mobility restrictions through spatiotemporal effective reproduction number analysis in a multi-patch model with complex mobility data

    Byul Nim Kim, Minchan Choi, Hyosun Lee, and Sunmi Lee. Evaluating mobility restrictions through spatiotemporal effective reproduction number analysis in a multi-patch model with complex mobility data. Epidemics, 54:100884, March 2026. ISSN 17554365. doi: 10.1016/j.epidem.2025.100884. URL https: //linkinghub.elsevier.com/retrieve/pii/S1755436525000726

    arXiv 2026

  70. [70]

    Jewell and Joseph A

    Nicholas P. Jewell and Joseph A. Lewnard. On the use of the Reproduction Number for SARS-COV-2: Estimation, Misinterpretations and Relationships with other Ecological Measures.Journal of the Royal Statistical Society Series A: Statistics in Society, 185(Supplement_1):S16–S27, November 2022. ISSN 0964-1998, 1467-985X. doi: 10. 1111/rssa.12860. URLhttps://a...

    2022

  71. [71]

    Cosner, J.C

    C. Cosner, J.C. Beier, R.S. Cantrell, D. Impoinvil, L. Kapitanski, M.D. Potts, A. Troyo, and S. Ruan. The effects of human movement on the persistence of vector-borne diseases.Journal of Theoretical Biology, 258(4):550–560, June 2009. ISSN 00225193. doi: 10.1016/j.jtbi.2009.02.016. URL https://linkinghub.elsevier.com/ retrieve/pii/S0022519309000757

    work page doi:10.1016/j.jtbi.2009.02.016 2009

  72. [72]

    Dengue transmission dynamics prediction by combining metapopulation 33 networks and Kalman filter algorithm.PLOS Neglected Tropical Diseases, 17(6):e0011418, June 2023

    Qinghui Zeng, Xiaolin Yu, Haobo Ni, Lina Xiao, Ting Xu, Haisheng Wu, Yuliang Chen, Hui Deng, Yingtao Zhang, Sen Pei, Jianpeng Xiao, and Pi Guo. Dengue transmission dynamics prediction by combining metapopulation 33 networks and Kalman filter algorithm.PLOS Neglected Tropical Diseases, 17(6):e0011418, June 2023. ISSN 1935-

    2023

  73. [73]

    Hisashi Inaba and Hiroshi Nishiura. The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model.Mathematical Biosciences, 216(1):77–89, November 2008. ISSN 00255564. doi: 10.1016/j.mbs.2008.08.005. URL https://linkinghub.elsevier. com/retrieve/pii/S0025556408001296

    work page doi:10.1016/j.mbs.2008.08.005 2008

  74. [74]

    Birrell, Joshua Blake, Christopher E

    Lorenzo Pellis, Paul J. Birrell, Joshua Blake, Christopher E. Overton, Francesca Scarabel, Helena B. Stage, Ellen Brooks-Pollock, Leon Danon, Ian Hall, Thomas A. House, Matt J. Keeling, Jonathan M. Read, JUNIPER Consortium, and Daniela De Angelis. Estimation of Reproduction Numbers in Real Time: Conceptual and Statistical Challenges.Journal of the Royal S...

    doi:10.1111/rssa.12955 2022

  75. [75]

    The R = 1 threshold can misclassify epidemic stability, November 2024

    Kris V Parag, Mauricio Santillana, Anne Cori, and Uri Obolski. The R = 1 threshold can misclassify epidemic stability, November 2024. URLhttp://medrxiv.org/lookup/doi/10.1101/2024.10.28.24316306

    work page doi:10.1101/2024.10.28.24316306 2024

  76. [76]

    Mahecha, Jordi Muñoz-Marí, Egbert H

    Jakob Runge, Sebastian Bathiany, Erik Bollt, Gustau Camps-Valls, Dim Coumou, Ethan Deyle, Clark Glymour, Marlene Kretschmer, Miguel D. Mahecha, Jordi Muñoz-Marí, Egbert H. Van Nes, Jonas Peters, Rick Quax, Markus Reichstein, Marten Scheffer, Bernhard Schölkopf, Peter Spirtes, George Sugihara, Jie Sun, Kun Zhang, and Jakob Zscheischler. Inferring causation...

    doi:10.1038/s41467-019-10105-3 2019

  77. [77]

    Ray, Tilmann Gneiting, and Nicholas G

    Johannes Bracher, Evan L. Ray, Tilmann Gneiting, and Nicholas G. Reich. Evaluating epidemic forecasts in an interval format.PLOS Computational Biology, 17(2):e1008618, February 2021. ISSN 1553-7358. doi: 10.1371/journal.pcbi.1008618. URLhttps://dx.plos.org/10.1371/journal.pcbi.1008618

    doi:10.1371/journal.pcbi.1008618 2021

  78. [78]

    Juliette T

    Seth Flaxman, Swapnil Mishra, Axel Gandy, H. Juliette T. Unwin, Thomas A. Mellan, Helen Coupland, Charles Whittaker, Harrison Zhu, Tresnia Berah, Jeffrey W. Eaton, Mélodie Monod, Imperial College COVID-19 Response Team, Pablo N. Perez-Guzman, Nora Schmit, Lucia Cilloni, Kylie E. C. Ainslie, Marc Baguelin, Adhiratha Boonyasiri, Olivia Boyd, Lorenzo Cattari...

    2020

  79. [79]

    How does policy modelling work in practice? A global analysis on the use of epidemiological modelling in health crises, August 2024

    Liza Hadley, Caylyn Rich, Alex Tasker, Olivier Restif, and Sebastian Funk. How does policy modelling work in practice? A global analysis on the use of epidemiological modelling in health crises, August 2024. URL http://medrxiv.org/lookup/doi/10.1101/2024.08.12.24311899

    work page doi:10.1101/2024.08.12.24311899 2024

  80. [80]

    Anderson, Christl A

    Roy M. Anderson, Christl A. Donnelly, T. Deirdre Hollingsworth, Matt J. Keeling, Carolin Vegvari, Rebecca Baggaley, and Rosie Maddren. Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation. Technical report, The Royal Society, A...

    2020

Showing first 80 references.