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A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies
Mathematical analysis of a hybrid model: Impacts of individual behaviors on the spreading of an epidemic
1. | Laboratoire des Sciences du Numérique, LS2N UMR CNRS 6004, Université de Nantes, France |
2. | Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
3. | UMR IDEES, CNRS, 25 rue Philippe Lebon BP 1123, 76063 Le Havre Cedex, UK |
In this paper, we investigate the well-posedness and dynamics of a class of hybrid models, obtained by coupling a system of ordinary differential equations and an agent-based model. These hybrid models intend to integrate the microscopic dynamics of individual behaviors into the macroscopic evolution of various population dynamics models, and can be applied to a great number of complex problems arising in economics, sociology, geography and epidemiology. Here, in particular, we apply our general framework to the current COVID-19 pandemic. We establish, at a theoretical level, sufficient conditions which lead to particular solutions exhibiting irregular oscillations and interpret those particular solutions as pandemic waves. We perform numerical simulations of a set of relevant scenarios which show how the microscopic processes impact the macroscopic dynamics.
References:
[1] |
M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler and A. Vespignani, Comparing wide-scale computational modeling Approaches to epidemic: Agent-based versus structured MetaPopulation models, BMC Infectious Diseases, 10 (2010), Article number: 190.
doi: 10.1186/1471-2334-10-190. |
[2] |
L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011. |
[3] |
L. J. S. Allen and E. J. Allen,
A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.
doi: 10.1016/S0040-5809(03)00104-7. |
[4] |
A. Banos, N. Corson, B. Gaudou, V. Laperrière and S. R. Coyrehourcq,
The importance of being hybrid for spatial epidemic models: a multi-scale approach, Systems, 3 (2015), 309-329.
doi: 10.3390/systems3040309. |
[5] |
A. Banos, C. Lang and M. Nicolas, Agent-based Spatial Simulation with NetLogo, Elsevier, 2017.
doi: 10.1016/C2015-0-01299-0. |
[6] |
G. Cantin, Nonidentical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp.
doi: 10.1142/S0218127417502133. |
[7] |
G. Cantin and C. J. Silva,
Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Mathematics, 4 (2019), 1145-1169.
doi: 10.3934/math.2019.4.1145. |
[8] |
Centers for Disease Control and Prevention, Coronavirus Disease 2019 (COVID-19), 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/prevention.html. |
[9] |
V. Colizza and A. Vespignani,
Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, Journal of theoretical biology, 251 (2008), 450-467.
doi: 10.1016/j.jtbi.2007.11.028. |
[10] |
E. Delisle, C. Rousseau, B. Broche, I. Leparc-Goffart and ot hers,
Chikungunya outbreak in Montpellier, France, September to October 2014, Eurosurveillance, 20 (2015), 21108.
doi: 10.2807/1560-7917.ES2015.20.17.21108. |
[11] |
Direção Geral da Saúde – COVID-19, Ponto de Situação Atual em Portugal, 2021. Available from: https://covid19.min-saude.pt/ponto-de-situacao-atual-em-portugal/. |
[12] |
M. Dolfinand M. Lachowicz,
Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.
doi: 10.3934/nhm.2015.10.877. |
[13] |
A. Ducrot and P. Magal,
Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.
doi: 10.1017/S0308210507000455. |
[14] |
J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: Mathematical and computational explorations, PLoS One, 3 (2008), e3955.
doi: 10.1371/journal.pone.0003955. |
[15] |
European Centre for Disease Prevention and Control, Guidelines for the Implementation of Non-Pharmaceutical Interventions Against COVID-19, 2020. Available from: https://www.ecdc.europa.eu/en/publications-data/covid-19-guidelines-non-pharmaceutical-interventions. |
[16] |
L. Fahse, C. Wissel and V. Grimm,
Reconciling classical and individual-based approaches in theoretical population ecology: A protocol for extracting population parameters from individual-based models, The American Naturalist, 152 (1998), 832-856.
doi: 10.1086/286212. |
[17] |
S. Galam, Sociophysics: A Physicist's Modeling of Psycho-political Phenomena, Understanding Complex Systems. Springer, New York, 2012.
doi: 10.1007/978-1-4614-2032-3. |
[18] |
S. Grauwin, E. Bertin, R. Lemoy and P. Jensen,
Competition between collective and individual dynamics, Proceedings of the National Academy of Sciences, 106 (2009), 20622-20626.
doi: 10.1073/pnas.0906263106. |
[19] |
J. K. Hale, Ordinary Differential Equations, Krieger Publishing Company (second edition), 1980. |
[20] |
D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Springer-Verlag Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-11546-2. |
[21] |
A. J. Heppenstall, A. T. Crooks, L. M. See and M. Batty, Agent-based Models of Geographical Systems, Springer Science & Business Media, 2011.
doi: 10.1007/978-90-481-8927-4. |
[22] |
H. W. Hethcote, Three basic epidemiological models, in Applied Mathematical Ecology, Springer, 18 (1989), 119–144.
doi: 10.1007/978-3-642-61317-3_5. |
[23] |
H. W. Hethcote and P. Van den Driessche,
Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.
doi: 10.1007/BF00160539. |
[24] |
D. A. Jones, H. L. Smith and H. R. Thieme,
Spread of viral infection of immobilized bacteria, Networks & Heterogeneous Media, 8 (2013), 327-342.
doi: 10.3934/nhm.2013.8.327. |
[25] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics–I. 1927, Bulletin of mathematical biology, 53 (1991), 33-55.
|
[26] |
K. Klemm, M. Serrano, V. M. Eguíluz and M. San Miguel, A measure of individual role in collective dynamics, Scientific Reports, 2 (2012), Article number: 292, 8pp.
doi: 10.1038/srep00292. |
[27] |
E. Logak and I. Passat,
An epidemic model with nonlocal diffusion on networks, Networks & Heterogeneous Media, 11 (2016), 693-719.
doi: 10.3934/nhm.2016014. |
[28] |
N. Marilleau, C. Lang and P. Giraudoux,
Coupling agent-based with equation-based models to study spatially explicit megapopulation dynamics, Ecological Modelling, 384 (2018), 34-42.
doi: 10.1016/j.ecolmodel.2018.06.011. |
[29] |
C. McPhail and R. T. Wohlstein,
Individual and collective behaviors within gatherings, demonstrations, and riots, Annual Review of Sociology, 9 (1983), 579-600.
doi: 10.1146/annurev.so.09.080183.003051. |
[30] |
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. |
[31] |
M. E. J. Newman and D. J. Watts,
Scaling and percolation in the small-world network model, Physical Review E, 60 (1999), 7332.
doi: 10.1515/9781400841356.310. |
[32] |
N. D. Nguyen, Coupling Equation-based and Individual-based Models in the Study of Complex Systems. A Case Study in Theoretical Population Ecology, Ph.D thesis, Pierre and Marie Curie University, 2010. |
[33] |
F. Schweitzer, Self-organization of Complex Structures: From Individual to Collective Dynamics, Gordon and Breach Science Publishers, Amsterdam, 1997. |
[34] |
C. J. Silva, G. Cantin, C. Cruz, R. Fonseca-Pinto, R. P. Fonseca, E. S. Santos and D. F. M. Torres, Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves, Journal of Mathematical Analysis and Applications, (2021), 125171.
doi: 10.1016/j.jmaa.2021.125171. |
[35] |
C. J. Silva, C. Cruz, D. F. M. Torres et al., Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), Art. 3451, 15 pp.
doi: 10.1038/s41598-021-83075-6. |
[36] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soc., Providence, RI, 2011.
doi: 10.1090/gsm/118. |
[37] |
R. H. Turner, L. M. Killian and others, Collective Behavior, Prentice-Hall Englewood Cliffs, NJ, 1957. |
[38] |
S. Wright, Crowds and Riots: A Study in Social Organization, Sage Publications Beverly Hills, CA, 1978. |
[39] |
P. Yan and G. Chowell, Beyond the initial phase: Compartment models for disease transmission, in Quantitative Methods for Investigating Infectious Disease Outbreaks (Texts in Applied Mathematics), Springer, Cham, 70 (2019), 135–182. |
show all references
References:
[1] |
M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler and A. Vespignani, Comparing wide-scale computational modeling Approaches to epidemic: Agent-based versus structured MetaPopulation models, BMC Infectious Diseases, 10 (2010), Article number: 190.
doi: 10.1186/1471-2334-10-190. |
[2] |
L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011. |
[3] |
L. J. S. Allen and E. J. Allen,
A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.
doi: 10.1016/S0040-5809(03)00104-7. |
[4] |
A. Banos, N. Corson, B. Gaudou, V. Laperrière and S. R. Coyrehourcq,
The importance of being hybrid for spatial epidemic models: a multi-scale approach, Systems, 3 (2015), 309-329.
doi: 10.3390/systems3040309. |
[5] |
A. Banos, C. Lang and M. Nicolas, Agent-based Spatial Simulation with NetLogo, Elsevier, 2017.
doi: 10.1016/C2015-0-01299-0. |
[6] |
G. Cantin, Nonidentical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp.
doi: 10.1142/S0218127417502133. |
[7] |
G. Cantin and C. J. Silva,
Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Mathematics, 4 (2019), 1145-1169.
doi: 10.3934/math.2019.4.1145. |
[8] |
Centers for Disease Control and Prevention, Coronavirus Disease 2019 (COVID-19), 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/prevention.html. |
[9] |
V. Colizza and A. Vespignani,
Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, Journal of theoretical biology, 251 (2008), 450-467.
doi: 10.1016/j.jtbi.2007.11.028. |
[10] |
E. Delisle, C. Rousseau, B. Broche, I. Leparc-Goffart and ot hers,
Chikungunya outbreak in Montpellier, France, September to October 2014, Eurosurveillance, 20 (2015), 21108.
doi: 10.2807/1560-7917.ES2015.20.17.21108. |
[11] |
Direção Geral da Saúde – COVID-19, Ponto de Situação Atual em Portugal, 2021. Available from: https://covid19.min-saude.pt/ponto-de-situacao-atual-em-portugal/. |
[12] |
M. Dolfinand M. Lachowicz,
Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.
doi: 10.3934/nhm.2015.10.877. |
[13] |
A. Ducrot and P. Magal,
Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.
doi: 10.1017/S0308210507000455. |
[14] |
J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: Mathematical and computational explorations, PLoS One, 3 (2008), e3955.
doi: 10.1371/journal.pone.0003955. |
[15] |
European Centre for Disease Prevention and Control, Guidelines for the Implementation of Non-Pharmaceutical Interventions Against COVID-19, 2020. Available from: https://www.ecdc.europa.eu/en/publications-data/covid-19-guidelines-non-pharmaceutical-interventions. |
[16] |
L. Fahse, C. Wissel and V. Grimm,
Reconciling classical and individual-based approaches in theoretical population ecology: A protocol for extracting population parameters from individual-based models, The American Naturalist, 152 (1998), 832-856.
doi: 10.1086/286212. |
[17] |
S. Galam, Sociophysics: A Physicist's Modeling of Psycho-political Phenomena, Understanding Complex Systems. Springer, New York, 2012.
doi: 10.1007/978-1-4614-2032-3. |
[18] |
S. Grauwin, E. Bertin, R. Lemoy and P. Jensen,
Competition between collective and individual dynamics, Proceedings of the National Academy of Sciences, 106 (2009), 20622-20626.
doi: 10.1073/pnas.0906263106. |
[19] |
J. K. Hale, Ordinary Differential Equations, Krieger Publishing Company (second edition), 1980. |
[20] |
D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Springer-Verlag Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-11546-2. |
[21] |
A. J. Heppenstall, A. T. Crooks, L. M. See and M. Batty, Agent-based Models of Geographical Systems, Springer Science & Business Media, 2011.
doi: 10.1007/978-90-481-8927-4. |
[22] |
H. W. Hethcote, Three basic epidemiological models, in Applied Mathematical Ecology, Springer, 18 (1989), 119–144.
doi: 10.1007/978-3-642-61317-3_5. |
[23] |
H. W. Hethcote and P. Van den Driessche,
Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.
doi: 10.1007/BF00160539. |
[24] |
D. A. Jones, H. L. Smith and H. R. Thieme,
Spread of viral infection of immobilized bacteria, Networks & Heterogeneous Media, 8 (2013), 327-342.
doi: 10.3934/nhm.2013.8.327. |
[25] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics–I. 1927, Bulletin of mathematical biology, 53 (1991), 33-55.
|
[26] |
K. Klemm, M. Serrano, V. M. Eguíluz and M. San Miguel, A measure of individual role in collective dynamics, Scientific Reports, 2 (2012), Article number: 292, 8pp.
doi: 10.1038/srep00292. |
[27] |
E. Logak and I. Passat,
An epidemic model with nonlocal diffusion on networks, Networks & Heterogeneous Media, 11 (2016), 693-719.
doi: 10.3934/nhm.2016014. |
[28] |
N. Marilleau, C. Lang and P. Giraudoux,
Coupling agent-based with equation-based models to study spatially explicit megapopulation dynamics, Ecological Modelling, 384 (2018), 34-42.
doi: 10.1016/j.ecolmodel.2018.06.011. |
[29] |
C. McPhail and R. T. Wohlstein,
Individual and collective behaviors within gatherings, demonstrations, and riots, Annual Review of Sociology, 9 (1983), 579-600.
doi: 10.1146/annurev.so.09.080183.003051. |
[30] |
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. |
[31] |
M. E. J. Newman and D. J. Watts,
Scaling and percolation in the small-world network model, Physical Review E, 60 (1999), 7332.
doi: 10.1515/9781400841356.310. |
[32] |
N. D. Nguyen, Coupling Equation-based and Individual-based Models in the Study of Complex Systems. A Case Study in Theoretical Population Ecology, Ph.D thesis, Pierre and Marie Curie University, 2010. |
[33] |
F. Schweitzer, Self-organization of Complex Structures: From Individual to Collective Dynamics, Gordon and Breach Science Publishers, Amsterdam, 1997. |
[34] |
C. J. Silva, G. Cantin, C. Cruz, R. Fonseca-Pinto, R. P. Fonseca, E. S. Santos and D. F. M. Torres, Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves, Journal of Mathematical Analysis and Applications, (2021), 125171.
doi: 10.1016/j.jmaa.2021.125171. |
[35] |
C. J. Silva, C. Cruz, D. F. M. Torres et al., Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), Art. 3451, 15 pp.
doi: 10.1038/s41598-021-83075-6. |
[36] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soc., Providence, RI, 2011.
doi: 10.1090/gsm/118. |
[37] |
R. H. Turner, L. M. Killian and others, Collective Behavior, Prentice-Hall Englewood Cliffs, NJ, 1957. |
[38] |
S. Wright, Crowds and Riots: A Study in Social Organization, Sage Publications Beverly Hills, CA, 1978. |
[39] |
P. Yan and G. Chowell, Beyond the initial phase: Compartment models for disease transmission, in Quantitative Methods for Investigating Infectious Disease Outbreaks (Texts in Applied Mathematics), Springer, Cham, 70 (2019), 135–182. |





Time sub-interval | |||
(transmission rate) | (transfer from |
(transfer from |
|
Time sub-interval | |||
(transmission rate) | (transfer from |
(transfer from |
|
Parameter | Description | Value |
Recruitment rate | ||
Natural death rate | ||
Modification parameter | ||
Transfer rate from |
||
Fraction of |
||
Transfer rate from |
||
Transfer rate from |
||
Transfer rate from |
||
Class of individuals | Initial condition value | |
Susceptible | ||
Asymptomatic | ||
Active infected | ||
Removed | ||
Protected |
Parameter | Description | Value |
Recruitment rate | ||
Natural death rate | ||
Modification parameter | ||
Transfer rate from |
||
Fraction of |
||
Transfer rate from |
||
Transfer rate from |
||
Transfer rate from |
||
Class of individuals | Initial condition value | |
Susceptible | ||
Asymptomatic | ||
Active infected | ||
Removed | ||
Protected |
Parameter | Region 1 | Region 2 | Region 3 | Region 4 | Region 5 |
Initial condition | Region 1 | Region 2 | Region 3 | Region 4 | Region 5 |
Parameter | Region 1 | Region 2 | Region 3 | Region 4 | Region 5 |
Initial condition | Region 1 | Region 2 | Region 3 | Region 4 | Region 5 |
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