Probabilistic Discrete‐Time Models for Spreading Processes in Complex Networks: A Review

Granell, Clara; Gómez, Sergio; Gómez‐Gardeñes, Jesús; Arenas, Alex

doi:10.1002/andp.202400078

Title: Probabilistic Discrete‐Time Models for Spreading Processes in Complex Networks: A Review

Journal Article · 24 July 2024 · Annalen der Physik (Leipzig)

DOI: https://doi.org/10.1002/andp.202400078 · OSTI ID:2466285

Granell, Clara ^[1]; Gómez, Sergio ^[1]; Gómez‐Gardeñes, Jesús ^[2];

^[1]

Departament d'Enginyeria Informàtica i Matemàtiques Universitat Rovira i Virgili Tarragona 43007 Spain
Department of Condensed Matter Physics University of Zaragoza Zaragoza 50010 Spain, GOTHAM Lab–Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza Zaragoza 50018 Spain

Abstract Research into network dynamics of spreading processes typically employs both discrete and continuous time methodologies. Although each approach offers distinct insights, integrating them can be challenging, particularly when maintaining coherence across different time scales. This review focuses on the Microscopic Markov Chain Approach (MMCA), a probabilistic f ramework originally designed for epidemic modeling. MMCA uses discrete dynamics to compute the probabilities of individuals transitioning between epidemiological states. By treating each time step—usually a day—as a discrete event, the approach captures multiple concurrent changes within this time frame. The approach allows to estimate the likelihood of individuals or populations being in specific states, which correspond to distinct epidemiological compartments. This review synthesizes key findings from the application of this approach, providing a comprehensive overview of its utility in understanding epidemic spread.

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Sponsoring Organization:: USDOE

Grant/Contract Number:: AC05-76RL01830

OSTI ID:: 2466285

Journal Information:: Annalen der Physik (Leipzig), Journal Name: Annalen der Physik (Leipzig) Journal Issue: 10 Vol. 536; ISSN 0003-3804

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: Germany

Language:: English

References (89)

Modeling Communicable Diseases, Human Mobility, and Epidemics: A Review Soriano‐Paños, David; Cota, Wesley; Ferreira, Silvio C. Annalen der Physik, Vol. 534, Issue 6 https://doi.org/10.1002/andp.202100482	journal	March 2022
Spatial Heterogeneity in Epidemic Models Lloyd, Alun L.; May, Robert M. Journal of Theoretical Biology, Vol. 179, Issue 1 https://doi.org/10.1006/jtbi.1996.0042	journal	March 1996
Exact epidemic models on graphs using graph-automorphism driven lumping Simon, Péter L.; Taylor, Michael; Kiss, Istvan Z. Journal of Mathematical Biology, Vol. 62, Issue 4 https://doi.org/10.1007/s00285-010-0344-x	journal	April 2010
Impacts of information propagation on epidemic spread over different migration routes Wang, Bing; Gou, Min; Han, Yuexing Nonlinear Dynamics, Vol. 105, Issue 4 https://doi.org/10.1007/s11071-021-06791-8	journal	August 2021
Qualitative analyses of communicable disease models Hethcote, Herbert W. Mathematical Biosciences, Vol. 28, Issue 3-4 https://doi.org/10.1016/0025-5564(76)90132-2	journal	January 1976
Some discrete-time SI, SIR, and SIS epidemic models Allen, Linda J. S. Mathematical Biosciences, Vol. 124, Issue 1 https://doi.org/10.1016/0025-5564(94)90025-6	journal	November 1994
Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study Wu, Joseph T.; Leung, Kathy; Leung, Gabriel M. The Lancet, Vol. 395, Issue 10225 https://doi.org/10.1016/S0140-6736(20)30260-9	journal	February 2020
Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study Zhou, Fei; Yu, Ting; Du, Ronghui The Lancet, Vol. 395, Issue 10229 https://doi.org/10.1016/S0140-6736(20)30566-3	journal	March 2020
Epidemiology and transmission of COVID-19 in 391 cases and 1286 of their close contacts in Shenzhen, China: a retrospective cohort study Bi, Qifang; Wu, Yongsheng; Mei, Shujiang The Lancet Infectious Diseases, Vol. 20, Issue 8 https://doi.org/10.1016/S1473-3099(20)30287-5	journal	August 2020
Allocating resources for epidemic spreading on metapopulation networks Zhu, Xuzhen; Liu, Yuxin; Wang, Shengfeng Applied Mathematics and Computation, Vol. 411 https://doi.org/10.1016/j.amc.2021.126531	journal	December 2021
The interconnection between independent reactive control policies drives the stringency of local containment Reyna-Lara, Adriana; Soriano-Paños, David; Arenas, Alex Chaos, Solitons & Fractals, Vol. 158 https://doi.org/10.1016/j.chaos.2022.112012	journal	May 2022
Bifurcation analysis of the Microscopic Markov Chain Approach to contact-based epidemic spreading in networks Arenas, Alex; Garijo, Antonio; Gómez, Sergio Chaos, Solitons & Fractals, Vol. 166 https://doi.org/10.1016/j.chaos.2022.112921	journal	January 2023
Seven challenges for metapopulation models of epidemics, including households models Ball, Frank; Britton, Tom; House, Thomas Epidemics, Vol. 10 https://doi.org/10.1016/j.epidem.2014.08.001	journal	March 2015
Nine challenges in incorporating the dynamics of behaviour in infectious diseases models Funk, Sebastian; Bansal, Shweta; Bauch, Chris T. Epidemics, Vol. 10 https://doi.org/10.1016/j.epidem.2014.09.005	journal	March 2015
Asymmetrically interacting dynamics with mutual confirmation from multi-source on multiplex networks Chen, Jiaxing; Liu, Ying; Tang, Ming Information Sciences, Vol. 619 https://doi.org/10.1016/j.ins.2022.11.033	journal	January 2023
Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations Colizza, Vittoria; Vespignani, Alessandro Journal of Theoretical Biology, Vol. 251, Issue 3 https://doi.org/10.1016/j.jtbi.2007.11.028	journal	April 2008
Invasion threshold in structured populations with recurrent mobility patterns Balcan, Duygu; Vespignani, Alessandro Journal of Theoretical Biology, Vol. 293 https://doi.org/10.1016/j.jtbi.2011.10.010	journal	January 2012
The structure and dynamics of multilayer networks Boccaletti, S.; Bianconi, G.; Criado, R. Physics Reports, Vol. 544, Issue 1 https://doi.org/10.1016/j.physrep.2014.07.001	journal	November 2014
Statistical physics of vaccination Wang, Zhen; Bauch, Chris T.; Bhattacharyya, Samit Physics Reports, Vol. 664 https://doi.org/10.1016/j.physrep.2016.10.006	journal	December 2016
Human mobility: Models and applications Barbosa, Hugo; Barthelemy, Marc; Ghoshal, Gourab Physics Reports, Vol. 734 https://doi.org/10.1016/j.physrep.2018.01.001	journal	March 2018
Coupled disease–behavior dynamics on complex networks: A review Wang, Zhen; Andrews, Michael A.; Wu, Zhi-Xi Physics of Life Reviews, Vol. 15 https://doi.org/10.1016/j.plrev.2015.07.006	journal	December 2015
Epidemics and Rumours Daley, D. J.; Kendall, D. G. Nature, Vol. 204, Issue 4963 https://doi.org/10.1038/2041118a0	journal	December 1964
Generalization of Epidemic Theory: An Application to the Transmission of Ideas Goffman, William; Newill, Vaun A. Nature, Vol. 204, Issue 4955 https://doi.org/10.1038/204225a0	journal	October 1964
Understanding individual human mobility patterns González, Marta C.; Hidalgo, César A.; Barabási, Albert-László Nature, Vol. 453, Issue 7196 https://doi.org/10.1038/nature06958	journal	June 2008
Phase transitions in contagion processes mediated by recurrent mobility patterns Balcan, Duygu; Vespignani, Alessandro Nature Physics, Vol. 7, Issue 7 https://doi.org/10.1038/nphys1944	journal	March 2011
Reaction–diffusion processes and metapopulation models in heterogeneous networks Colizza, Vittoria; Pastor-Satorras, Romualdo; Vespignani, Alessandro Nature Physics, Vol. 3, Issue 4 https://doi.org/10.1038/nphys560	journal	March 2007
Spatial immunization to abate disease spreading in transportation hubs Mazzoli, Mattia; Gallotti, Riccardo; Privitera, Filippo Nature Communications, Vol. 14, Issue 1 https://doi.org/10.1038/s41467-023-36985-0	journal	March 2023
Critical regimes driven by recurrent mobility patterns of reaction–diffusion processes in networks Gómez-Gardeñes, J.; Soriano-Paños, D.; Arenas, A. Nature Physics, Vol. 14, Issue 4 https://doi.org/10.1038/s41567-017-0022-7	journal	December 2017
Don’t close the gates Scarpino, Samuel V. Nature Physics, Vol. 14, Issue 4 https://doi.org/10.1038/s41567-017-0028-1	journal	January 2018
Network clique cover approximation to analyze complex contagions through group interactions Burgio, Giulio; Arenas, Alex; Gómez, Sergio Communications Physics, Vol. 4, Issue 1 https://doi.org/10.1038/s42005-021-00618-z	journal	May 2021
Critical behavior in interdependent spatial spreading processes with distinct characteristic time scales Castioni, Piergiorgio; Gallotti, Riccardo; De Domenico, Manlio Communications Physics, Vol. 4, Issue 1 https://doi.org/10.1038/s42005-021-00631-2	journal	June 2021
Interplay between population density and mobility in determining the spread of epidemics in cities Hazarie, Surendra; Soriano-Paños, David; Arenas, Alex Communications Physics, Vol. 4, Issue 1 https://doi.org/10.1038/s42005-021-00679-0	journal	August 2021
Assembly and Validation of Versatile Transcription Activator-Like Effector Libraries Li, Yi; Ehrhardt, Kristina; Zhang, Michael Q. Scientific Reports, Vol. 4, Issue 1 https://doi.org/10.1038/srep04857	journal	May 2014
Contagion–diffusion processes with recurrent mobility patterns of distinguishable agents Valgañón, P.; Soriano-Paños, D.; Arenas, A. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, Issue 4 https://doi.org/10.1063/5.0085532	journal	April 2022
A metapopulation approach to identify targets for Wolbachia-based dengue control Reyna-Lara, A.; Soriano-Paños, D.; Arias-Castro, J. H. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, Issue 4 https://doi.org/10.1063/5.0087435	journal	April 2022
The spread of awareness and its impact on epidemic outbreaks Funk, S.; Gilad, E.; Watkins, C. Proceedings of the National Academy of Sciences, Vol. 106, Issue 16 https://doi.org/10.1073/pnas.0810762106	journal	March 2009
A simple model of global cascades on random networks Watts, D. J. Proceedings of the National Academy of Sciences, Vol. 99, Issue 9 https://doi.org/10.1073/pnas.082090499	journal	April 2002
Multiscale mobility networks and the spatial spreading of infectious diseases Balcan, D.; Colizza, V.; Goncalves, B. Proceedings of the National Academy of Sciences, Vol. 106, Issue 51 https://doi.org/10.1073/pnas.0906910106	journal	December 2009
Traffic-driven epidemic spreading in finite-size scale-free networks Meloni, S.; Arenas, A.; Moreno, Y. Proceedings of the National Academy of Sciences, Vol. 106, Issue 40 https://doi.org/10.1073/pnas.0907121106	journal	September 2009
Heterogeneity in social and epidemiological factors determines the risk of measles outbreaks Bosetti, Paolo; Poletti, Piero; Stella, Massimo Proceedings of the National Academy of Sciences, Vol. 117, Issue 48 https://doi.org/10.1073/pnas.1920986117	journal	November 2020
Threshold Models of Collective Behavior Granovetter, Mark American Journal of Sociology, Vol. 83, Issue 6 https://doi.org/10.1086/226707	journal	May 1978
Complex Contagions and the Weakness of Long Ties Centola, Damon; Macy, Michael American Journal of Sociology, Vol. 113, Issue 3 https://doi.org/10.1086/521848	journal	November 2007
Infectious disease dynamics in metapopulations with heterogeneous transmission and recurrent mobility Cota, Wesley; Soriano-Paños, David; Arenas, A. New Journal of Physics, Vol. 23, Issue 7 https://doi.org/10.1088/1367-2630/ac0c99	journal	July 2021
Impact of temporal scales and recurrent mobility patterns on the unfolding of epidemics Soriano-Paños, David; Ghoshal, Gourab; Arenas, Alex Journal of Statistical Mechanics: Theory and Experiment, Vol. 2020, Issue 2 https://doi.org/10.1088/1742-5468/ab6a04	journal	February 2020
Networks Newman, Mark https://doi.org/10.1093/acprof:oso/9780199206650.001.0001	book	March 2010
Multilayer networks Kivela, M.; Arenas, A.; Barthelemy, M. Journal of Complex Networks, Vol. 2, Issue 3 https://doi.org/10.1093/comnet/cnu016	journal	July 2014
A general Markov chain approach for disease and rumour spreading in complex networks Ferraz de Arruda, Guilherme; Aparecido Rodrigues, Francisco; Martín Rodríguez, Pablo Journal of Complex Networks, Vol. 6, Issue 2 https://doi.org/10.1093/comnet/cnx024	journal	August 2017
Networks Newman, Mark Oxford Scholarship Online https://doi.org/10.1093/oso/9780198805090.001.0001	book	July 2018
Networks and epidemic models Keeling, Matt J.; Eames, Ken T. D. Journal of The Royal Society Interface, Vol. 2, Issue 4 https://doi.org/10.1098/rsif.2005.0051	journal	May 2005
When individual behaviour matters: homogeneous and network models in epidemiology Bansal, Shweta; Grenfell, Bryan T.; Meyers, Lauren Ancel Journal of The Royal Society Interface, Vol. 4, Issue 16 https://doi.org/10.1098/rsif.2007.1100	journal	April 2007
Markovian approach to tackle the interaction of simultaneous diseases Soriano-Paños, D.; Ghanbarnejad, F.; Meloni, S. Physical Review E, Vol. 100, Issue 6 https://doi.org/10.1103/PhysRevE.100.062308	journal	December 2019
Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks Silva, Diogo H.; Rodrigues, Francisco A.; Ferreira, Silvio C. Physical Review E, Vol. 110, Issue 1 https://doi.org/10.1103/PhysRevE.110.014302	journal	July 2024
Epidemic dynamics and endemic states in complex networks Pastor-Satorras, Romualdo; Vespignani, Alessandro Physical Review E, Vol. 63, Issue 6 https://doi.org/10.1103/PhysRevE.63.066117	journal	May 2001
Spread of epidemic disease on networks Newman, M. E. J. Physical Review E, Vol. 66, Issue 1 https://doi.org/10.1103/PhysRevE.66.016128	journal	July 2002
Langevin approach for the dynamics of the contact process on annealed scale-free networks Boguñá, Marián; Castellano, Claudio; Pastor-Satorras, Romualdo Physical Review E, Vol. 79, Issue 3 https://doi.org/10.1103/PhysRevE.79.036110	journal	March 2009
Interacting epidemics on overlay networks Funk, Sebastian; Jansen, Vincent A. A. Physical Review E, Vol. 81, Issue 3 https://doi.org/10.1103/PhysRevE.81.036118	journal	March 2010
Nonperturbative heterogeneous mean-field approach to epidemic spreading in complex networks Gómez, Sergio; Gómez-Gardeñes, Jesús; Moreno, Yamir Physical Review E, Vol. 84, Issue 3 https://doi.org/10.1103/PhysRevE.84.036105	journal	September 2011
Epidemics on interconnected networks Dickison, Mark; Havlin, S.; Stanley, H. E. Physical Review E, Vol. 85, Issue 6 https://doi.org/10.1103/PhysRevE.85.066109	journal	June 2012
Epidemics in networks with nodal self-infection and the epidemic threshold Van Mieghem, Piet; Cator, Eric Physical Review E, Vol. 86, Issue 1 https://doi.org/10.1103/PhysRevE.86.016116	journal	July 2012
Epidemic spreading on interconnected networks Saumell-Mendiola, Anna; Serrano, M. Ángeles; Boguñá, Marián Physical Review E, Vol. 86, Issue 2 https://doi.org/10.1103/PhysRevE.86.026106	journal	August 2012
Contact-based social contagion in multiplex networks Cozzo, Emanuele; Baños, Raquel A.; Meloni, Sandro Physical Review E, Vol. 88, Issue 5 https://doi.org/10.1103/PhysRevE.88.050801	journal	November 2013
Competing spreading processes on multiplex networks: Awareness and epidemics Granell, Clara; Gómez, Sergio; Arenas, Alex Physical Review E, Vol. 90, Issue 1 https://doi.org/10.1103/PhysRevE.90.012808	journal	July 2014
Epidemic spreading in localized environments with recurrent mobility patterns Granell, Clara; Mucha, Peter J. Physical Review E, Vol. 97, Issue 5 https://doi.org/10.1103/PhysRevE.97.052302	journal	May 2018
Dynamical Interplay between Awareness and Epidemic Spreading in Multiplex Networks Granell, Clara; Gómez, Sergio; Arenas, Alex Physical Review Letters, Vol. 111, Issue 12 https://doi.org/10.1103/PhysRevLett.111.128701	journal	September 2013
Triadic Approximation Reveals the Role of Interaction Overlap on the Spread of Complex Contagions on Higher-Order Networks Burgio, Giulio; Gómez, Sergio; Arenas, Alex Physical Review Letters, Vol. 132, Issue 7 https://doi.org/10.1103/PhysRevLett.132.077401	journal	February 2024
Epidemic Spreading in Scale-Free Networks Pastor-Satorras, Romualdo; Vespignani, Alessandro Physical Review Letters, Vol. 86, Issue 14 https://doi.org/10.1103/PhysRevLett.86.3200	journal	April 2001
Absence of Epidemic Threshold in Scale-Free Networks with Degree Correlations Boguñá, Marián; Pastor-Satorras, Romualdo; Vespignani, Alessandro Physical Review Letters, Vol. 90, Issue 2 https://doi.org/10.1103/PhysRevLett.90.028701	journal	January 2003
Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks Barthélemy, Marc; Barrat, Alain; Pastor-Satorras, Romualdo Physical Review Letters, Vol. 92, Issue 17 https://doi.org/10.1103/PhysRevLett.92.178701	journal	April 2004
Universal Behavior in a Generalized Model of Contagion Dodds, Peter Sheridan; Watts, Duncan J. Physical Review Letters, Vol. 92, Issue 21 https://doi.org/10.1103/PhysRevLett.92.218701	journal	May 2004
Epidemic Dynamics on an Adaptive Network Gross, Thilo; D’Lima, Carlos J. Dommar; Blasius, Bernd Physical Review Letters, Vol. 96, Issue 20 https://doi.org/10.1103/PhysRevLett.96.208701	journal	May 2006
Invasion Threshold in Heterogeneous Metapopulation Networks Colizza, Vittoria; Vespignani, Alessandro Physical Review Letters, Vol. 99, Issue 14 https://doi.org/10.1103/PhysRevLett.99.148701	journal	October 2007
Abrupt phase transition of epidemic spreading in simplicial complexes Matamalas, Joan T.; Gómez, Sergio; Arenas, Alex Physical Review Research, Vol. 2, Issue 1 https://doi.org/10.1103/PhysRevResearch.2.012049	journal	February 2020
Vector-borne epidemics driven by human mobility Soriano-Paños, David; Arias-Castro, Juddy Heliana; Reyna-Lara, Adriana Physical Review Research, Vol. 2, Issue 1 https://doi.org/10.1103/PhysRevResearch.2.013312	journal	March 2020
Outbreak diversity in epidemic waves propagating through distinct geographical scales Costa, Guilherme S.; Cota, Wesley; Ferreira, Silvio C. Physical Review Research, Vol. 2, Issue 4 https://doi.org/10.1103/PhysRevResearch.2.043306	journal	December 2020
Mathematical Formulation of Multilayer Networks De Domenico, Manlio; Solé-Ribalta, Albert; Cozzo, Emanuele Physical Review X, Vol. 3, Issue 4 https://doi.org/10.1103/PhysRevX.3.041022	journal	December 2013
Dynamics of Interacting Diseases Sanz, Joaquín; Xia, Cheng-Yi; Meloni, Sandro Physical Review X, Vol. 4, Issue 4 https://doi.org/10.1103/PhysRevX.4.041005	journal	October 2014
Analytical Computation of the Epidemic Threshold on Temporal Networks Valdano, Eugenio; Ferreri, Luca; Poletto, Chiara Physical Review X, Vol. 5, Issue 2 https://doi.org/10.1103/PhysRevX.5.021005	journal	April 2015
Epidemic processes in complex networks Pastor-Satorras, Romualdo; Castellano, Claudio; Van Mieghem, Piet Reviews of Modern Physics, Vol. 87, Issue 3 https://doi.org/10.1103/RevModPhys.87.925	journal	August 2015
Virus Spread in Networks Van Mieghem, P.; Omic, J.; Kooij, R. IEEE/ACM Transactions on Networking, Vol. 17, Issue 1 https://doi.org/10.1109/TNET.2008.925623	journal	February 2009
Spreading Processes in Multilayer Networks Salehi, Mostafa; Sharma, Rajesh; Marzolla, Moreno IEEE Transactions on Network Science and Engineering, Vol. 2, Issue 2 https://doi.org/10.1109/TNSE.2015.2425961	journal	April 2015
Effective approach to epidemic containment using link equations in complex networks Matamalas, Joan T.; Arenas, Alex; Gómez, Sergio Science Advances, Vol. 4, Issue 12 https://doi.org/10.1126/sciadv.aau4212	journal	December 2018
EPIDEMIOLOGY: How Viruses Spread Among Computers and People Lloyd, A. L. Science, Vol. 292, Issue 5520 https://doi.org/10.1126/science.1061076	journal	May 2001
The Mathematics of Infectious Diseases Hethcote, Herbert W. SIAM Review, Vol. 42, Issue 4 https://doi.org/10.1137/S0036144500371907	journal	January 2000
Recurrent host mobility in spatial epidemics: beyond reaction-diffusion Belik, V.; Geisel, T.; Brockmann, D. The European Physical Journal B, Vol. 84, Issue 4 https://doi.org/10.1140/epjb/e2011-20485-2	journal	December 2011
Epidemic thresholds in real networks Chakrabarti, Deepayan; Wang, Yang; Wang, Chenxi ACM Transactions on Information and System Security, Vol. 10, Issue 4 https://doi.org/10.1145/1284680.1284681	journal	January 2008
Discrete-time Markov chain approach to contact-based disease spreading in complex networks Gómez, S.; Arenas, A.; Borge-Holthoefer, J. EPL (Europhysics Letters), Vol. 89, Issue 3 https://doi.org/10.1209/0295-5075/89/38009	journal	February 2010
Infectious Disease Modeling of Social Contagion in Networks Hill, Alison L.; Rand, David G.; Nowak, Martin A. PLoS Computational Biology, Vol. 6, Issue 11 https://doi.org/10.1371/journal.pcbi.1000968	journal	November 2010
Effects of human mobility on the spread of Dengue in the region of Caldas, Colombia Ospina-Aguirre, Carolina; Soriano-Paños, David; Olivar-Tost, Gerard PLOS Neglected Tropical Diseases, Vol. 17, Issue 11 https://doi.org/10.1371/journal.pntd.0011087	journal	November 2023
Commuter Mobility and the Spread of Infectious Diseases: Application to Influenza in France Charaudeau, Segolene; Pakdaman, Khashayar; Boëlle, Pierre-Yves PLoS ONE, Vol. 9, Issue 1 https://doi.org/10.1371/journal.pone.0083002	journal	January 2014

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