\begin{thebibliography}{10}

\bibitem{aldousStoppingTimesTightness1978}
David Aldous.
\newblock Stopping {{Times}} and {{Tightness}}.
\newblock {\em The Annals of Probability}, 6(2):335--340, April 1978.

\bibitem{anderssonDensityDependentJump2000}
H{\aa}kan Andersson and Tom Britton.
\newblock Density dependent jump {{Markov}} processes.
\newblock In {\em Stochastic {{Epidemic Models}} and {{Their Statistical
  Analysis}}}, Lecture {{Notes}} in {{Statistics}}. {Springer}, {New York, NY},
  2000.

\bibitem{ballReproductionNumbersEpidemic2016}
Frank~G. Ball, Lorenzo Pellis, and Pieter Trapman.
\newblock Reproduction numbers for epidemic models with households and other
  social structures {{II}}: {{Comparisons}} and implications for vaccination.
\newblock {\em Mathematical Biosciences}, 274:108--139, April 2016.

\bibitem{ballEpidemicsRandomIntersection2014}
Frank~G. Ball, David~J. Sirl, and Pieter Trapman.
\newblock Epidemics on random intersection graphs.
\newblock {\em The Annals of Applied Probability}, 24(3):1081--1128, June 2014.

\bibitem{bansayeEpidemiologicalFootprintContact2023}
Vincent Bansaye, Fran{\c c}ois Deslandes, Madeleine Kubasch, and Elisabeta
  Vergu.
\newblock The epidemiological footprint of contact structures in models with
  two levels of mixing.
\newblock Available on arXiv: https://arxiv.org/abs/2303.05287, 2023.

\bibitem{barnardEdgeBasedCompartmentalModelling2018}
Rosanna~C. Barnard, Istvan~Z. Kiss, Luc Berthouze, and Joel~C. Miller.
\newblock Edge-{{Based Compartmental Modelling}} of an {{SIR Epidemic}} on a
  {{Dual-Layer Static}}\textendash{{Dynamic Multiplex Network}} with {{Tunable
  Clustering}}.
\newblock {\em Bulletin of Mathematical Biology}, 80(10):2698--2733, October
  2018.

\bibitem{billingsleyConvergenceProbabilityMeasures1999}
Patrick Billingsley.
\newblock {\em Convergence of Probability Measures}.
\newblock Wiley Series in Probability and Statistics. {{Probability}} and
  Statistics Section. {Wiley}, {New York}, 2nd ed edition, 1999.

\bibitem{brittonChapterGeneralTwoLevel2019}
Tom Britton and Etienne Pardoux.
\newblock Chapter 3 {{A General Two-Level Mixing Model}}.
\newblock In Tom Britton and Etienne Pardoux, editors, {\em Stochastic
  {{Epidemic Models}} with {{Inference}}}, Lecture {{Notes}} in
  {{Mathematics}}, pages 159--213. {Springer International Publishing}, {Cham},
  2019.

\bibitem{decreusefondLargeGraphLimit2012}
Laurent Decreusefond, Jean-St{\'e}phane Dhersin, Pascal Moyal, and Viet~Chi
  Tran.
\newblock Large graph limit for an {{SIR}} process in random network with
  heterogeneous connectivity.
\newblock {\em The Annals of Applied Probability}, 22(2), April 2012.

\bibitem{delvallerafoDiseaseDynamicsMean2021}
Mar{\'i}a {del Valle Rafo}, Juan~Pablo Di~Mauro, and Juan~Pablo Aparicio.
\newblock Disease dynamics and mean field models for clustered networks.
\newblock {\em Journal of Theoretical Biology}, 526:110554, October 2021.

\bibitem{dilauroImpactContactStructure2021}
Francesco Di~Lauro, Luc Berthouze, Matthew~D. Dorey, Joel~C. Miller, and
  Istv{\'a}n~Z. Kiss.
\newblock The {{Impact}} of {{Contact Structure}} and {{Mixing}} on {{Control
  Measures}} and {{Disease-Induced Herd Immunity}} in {{Epidemic Models}}: {{A
  Mean-Field Model Perspective}}.
\newblock {\em Bulletin of Mathematical Biology}, 83(11):117, November 2021.

\bibitem{fengEpidemiologicalModelsNonExponentially2007}
Zhilan Feng, Dashun Xu, and Haiyun Zhao.
\newblock Epidemiological {{Models}} with {{Non-Exponentially Distributed
  Disease Stages}} and {{Applications}} to {{Disease Control}}.
\newblock {\em Bulletin of Mathematical Biology}, 69(5):1511--1536, July 2007.

\bibitem{forienRecentAdvancesEpidemic2022}
Raphael Forien, Guodong Pang, and Etienne Pardoux.
\newblock Recent {{Advances}} in {{Epidemic Modeling}}: {{Non-Markov Stochastic
  Models}} and their {{Scaling Limits}}.
\newblock {\em The Graduate Journal of Mathematics}, 7(2):19--75, September
  2022.

\bibitem{fournierMicroscopicProbabilisticDescription2004}
Nicolas Fournier and Sylvie M{\'e}l{\'e}ard.
\newblock A microscopic probabilistic description of a locally regulated
  population and macroscopic approximations.
\newblock {\em The Annals of Applied Probability}, 14(4):1880--1919, November
  2004.

\bibitem{hebert-dufresnePropagationDynamicsNetworks2010}
Laurent {H{\'e}bert-Dufresne}, Pierre-Andr{\'e} No{\"e}l, Vincent Marceau,
  Antoine Allard, and Louis~J. Dub{\'e}.
\newblock Propagation dynamics on networks featuring complex topologies.
\newblock {\em Physical Review E}, 82(3):036115, September 2010.

\bibitem{houseDeterministicEpidemicModels2008}
Thomas House and Matt~J. Keeling.
\newblock Deterministic epidemic models with explicit household structure.
\newblock {\em Mathematical Biosciences}, 213(1):29--39, May 2008.

\bibitem{houseInsightsUnifyingModern2011}
Thomas House and Matt~J. Keeling.
\newblock Insights from unifying modern approximations to infections on
  networks.
\newblock {\em Journal of the Royal Society Interface}, 8(54):67--73, January
  2011.

\bibitem{jacobsenLargeGraphLimit2018}
Karly~A. Jacobsen, Mark~G. Burch, Joseph~H. Tien, and Grzegorz~A. Rempa{\l}a.
\newblock The large graph limit of a stochastic epidemic model on a dynamic
  multilayer network.
\newblock {\em Journal of Biological Dynamics}, 12(1):746--788, January 2018.

\bibitem{jansonLawLargeNumbers2014}
Svante Janson, Malwina Luczak, and Peter Windridge.
\newblock Law of large numbers for the {{SIR}} epidemic on a random graph with
  given degrees.
\newblock {\em Random Structures \& Algorithms}, 45(4):726--763, 2014.

\bibitem{joffeWeakConvergenceSequences1986}
Anatole Joffe and Michel Metivier.
\newblock Weak convergence of sequences of semimartingales with applications to
  multitype branching processes.
\newblock {\em Advances in Applied Probability}, 18(1):20--65, March 1986.

\bibitem{jourdainLevyFlightsEvolutionary2012}
Benjamin Jourdain, Sylvie M{\'e}l{\'e}ard, and Wojbor~A. Woyczynski.
\newblock L\'evy flights in evolutionary ecology.
\newblock {\em Journal of Mathematical Biology}, 65(4):677--707, October 2012.

\bibitem{kissNecessarySufficientConditions2022}
Istvan~Z. Kiss, Eben Kenah, and Grzegorz~A. Rempala.
\newblock Necessary and sufficient conditions for exact closures of epidemic
  equations on configuration model networks, August 2022.

\bibitem{kissMathematicsEpidemicsNetworks2017}
Istv{\'a}n~Z. Kiss, Joel~C. Miller, and P{\'e}ter~L. Simon.
\newblock {\em Mathematics of {{Epidemics}} on {{Networks}}: {{From Exact}} to
  {{Approximate Models}}}, volume~46 of {\em Interdisciplinary {{Applied
  Mathematics}}}.
\newblock {Springer International Publishing}, {Cham}, 2017.

\bibitem{kurtzEpidemicModels1981}
Thomas~G. Kurtz.
\newblock Epidemic models.
\newblock In {\em Approximation of Population Processes}, number~36 in
  {{CBMS-NSF}} Regional Conference Series in Applied Mathematics. {Society for
  Industrial and Applied Mathematics}, {Philadelphia, Pa}, 1981.

\bibitem{lloydRealisticDistributionsInfectious2001}
Alun~L. Lloyd.
\newblock Realistic {{Distributions}} of {{Infectious Periods}} in {{Epidemic
  Models}}: {{Changing Patterns}} of {{Persistence}} and {{Dynamics}}.
\newblock {\em Theoretical Population Biology}, 60(1):59--71, August 2001.

\bibitem{lucaImpactRegularSchool2018}
Giancarlo~De Luca, Kim~Van Kerckhove, Pietro Coletti, Chiara Poletto, Nathalie
  Bossuyt, Niel Hens, and Vittoria Colizza.
\newblock The impact of regular school closure on seasonal influenza epidemics:
  A data-driven spatial transmission model for {{Belgium}}.
\newblock {\em BMC Infectious Diseases}, 18(1):29, January 2018.

\bibitem{maEffectiveDegreeHousehold2013}
Junling Ma, Pauline {van den Driessche}, and Frederick~H. Willeboordse.
\newblock Effective degree household network disease model.
\newblock {\em Journal of Mathematical Biology}, 66(1):75--94, January 2013.

\bibitem{meleardStochasticModelsStructured2015}
Sylvie Meleard and Vincent Bansaye.
\newblock {\em Stochastic {{Models}} for {{Structured Populations}}: {{Scaling
  Limits}} and {{Long Time Behavior}}}.
\newblock {Springer International Publishing}, {Cham}, 2015.

\bibitem{mendez-britoSystematicReviewEmpirical2021}
Alba {Mendez-Brito}, Charbel El~Bcheraoui, and Francisco {Pozo-Martin}.
\newblock Systematic review of empirical studies comparing the effectiveness of
  non-pharmaceutical interventions against {{COVID-19}}.
\newblock {\em Journal of Infection}, 83(3):281--293, September 2021.

\bibitem{millerPercolationEpidemicsRandom2009}
Joel~C. Miller.
\newblock Percolation and epidemics in random clustered networks.
\newblock {\em Physical Review E}, 80(2):020901, August 2009.

\bibitem{millerNotePaperErik2011}
Joel~C Miller.
\newblock A note on a paper by {{Erik Volz}}: {{SIR}} dynamics in random
  networks.
\newblock {\em Journal of Mathematical Biology}, 62:349--358, 2011.

\bibitem{newmanRandomGraphsClustering2009}
Mark E.~J. Newman.
\newblock Random graphs with clustering.
\newblock {\em Physical Review Letters}, 103(5):058701, July 2009.

\bibitem{pellisThresholdParametersModel2009}
Lorenzo Pellis, Neil~M. Ferguson, and Christophe Fraser.
\newblock Threshold parameters for a model of epidemic spread among households
  and workplaces.
\newblock {\em Journal of The Royal Society Interface}, 6(40):979--987,
  November 2009.

\bibitem{pellisEpidemicGrowthRate2011}
Lorenzo Pellis, Neil~M. Ferguson, and Christophe Fraser.
\newblock Epidemic growth rate and household reproduction number in communities
  of households, schools and workplaces.
\newblock {\em Journal of Mathematical Biology}, 63(4):691--734, October 2011.

\bibitem{roelly-coppolettaCriterionConvergenceMeasure1986}
Sylvie {Roelly-Coppoletta}.
\newblock A criterion of convergence of measure-valued processes: Application
  to measure branching processes.
\newblock {\em Stochastics: An International Journal of Probability and
  Stochastic Processes}, 17(1-2):43--65, 1986.

\bibitem{sherborneMeanfieldModelsNonMarkovian2018}
Neil Sherborne, Joel~C. Miller, Konstantin~B. Blyuss, and Istvan~Z. Kiss.
\newblock Mean-field models for non-{{Markovian}} epidemics on networks.
\newblock {\em Journal of Mathematical Biology}, 76(3):755--778, February 2018.

\bibitem{simoySociallyStructuredModel2021}
Mario~Ignacio Simoy and Juan~Pablo Aparicio.
\newblock Socially structured model for {{COVID-19}} pandemic: Design and
  evaluation of control measures.
\newblock {\em Computational and Applied Mathematics}, 41(1):14, December 2021.

\bibitem{st-ongeHeterogeneousTransmissionGroups2023}
Guillaume {St-Onge}, Laurent {H{\'e}bert-Dufresne}, and Antoine Allard.
\newblock Heterogeneous transmission in groups induces a superlinear force of
  infection, February 2023.

\bibitem{tranModelesParticulairesStochastiques2006}
Viet~Chi Tran.
\newblock {\em {Mod\`eles particulaires stochastiques pour des probl\`emes
  d'\'evolution adaptative et pour l'approximation de solutions statistiques}}.
\newblock PhD thesis, Universit\'e de Nanterre - Paris X, December 2006.

\bibitem{tranBalladeForetsAleatoires2014}
Viet~Chi Tran.
\newblock {\em {Une ballade en for\^ets al\'eatoires}}.
\newblock {Habilitation a Diriger les Recherches}, Universit\'e Lille 1,
  November 2014.

\bibitem{volzSIRDynamicsRandom2008}
Erik~M. Volz.
\newblock {{SIR}} dynamics in random networks with heterogeneous connectivity.
\newblock {\em Journal of Mathematical Biology}, 56(3):293--310, March 2008.

\bibitem{volzEffectsHeterogeneousClustered2011}
Erik~M. Volz, Joel~C. Miller, Alison Galvani, and Lauren Ancel~Meyers.
\newblock Effects of {{Heterogeneous}} and {{Clustered Contact Patterns}} on
  {{Infectious Disease Dynamics}}.
\newblock {\em PLoS Computational Biology}, 7(6):e1002042, June 2011.

\bibitem{wilkinsonRelationshipsMessagePassing2017}
Robert~R. Wilkinson, Frank~G. Ball, and Kieran~J. Sharkey.
\newblock The relationships between message passing, pairwise,
  {{Kermack}}\textendash{{McKendrick}} and stochastic {{SIR}} epidemic models.
\newblock {\em Journal of Mathematical Biology}, 75(6-7), April 2017.

\end{thebibliography}
