Geodesic
Non-Markovian modelling highlights the importance of age structure on Covid-19 epidemiological dynamics
Bastien Reyné1 ; Quentin Richard1 ; Christian Selinger1, 2 ; Mircea T. Sofonea1 ; Ramsès Djidjou-Demasse1 ; Samuel Alizon1, 3
1 MIVEGEC, Univ. Montpellier, CNRS, IRDw, Montpellier, France.
2 Swiss Tropical and Public Health Institute (Swiss TPH), 4002 Basel, Switzerland.
3 Center for Interdisciplinary Research in Biology (CIRB), College de France, CNRS, INSERM, Université PSL, Paris, France.
Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 7 Cet article a éte moissonné depuis la source EDP Sciences

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The Covid-19 pandemic outbreak was followed by a huge amount of modelling studies in order to rapidly gain insights to implement the best public health policies. Most of these compartmental models involved ordinary differential equations (ODEs) systems. Such a formalism implicitly assumes that the time spent in each compartment does not depend on the time already spent in it, which is at odds with the clinical data. To overcome this “memoryless” issue, a widely used solution is to increase and chain the number of compartments of a unique reality (e.g. have infected individual move between several compartments). This allows for greater heterogeneity and thus be closer to the observed situation, but also tends to make the whole model more difficult to apprehend and parameterize. We develop a non-Markovian alternative formalism based on partial differential equations (PDEs) instead of ODEs, which, by construction, provides a memory structure for each compartment thereby allowing us to limit the number of compartments. We apply our model to the French 2021 SARS-CoV-2 epidemic and, while accounting for vaccine-induced and natural immunity, we analyse and determine the major components that contributed to the Covid-19 hospital admissions. The results indicate that the observed vaccination rate alone is not enough to control the epidemic, and a global sensitivity analysis highlights a huge uncertainty attributable to the age-structured contact matrix. Our study shows the flexibility and robustness of PDE formalism to capture national COVID-19 dynamics and opens perspectives to study medium or long-term scenarios involving immune waning or virus evolution.
DOI : 10.1051/mmnp/2022008

Bastien Reyné 1 ; Quentin Richard 1 ; Christian Selinger 1, 2 ; Mircea T. Sofonea 1 ; Ramsès Djidjou-Demasse 1 ; Samuel Alizon 1, 3

1 MIVEGEC, Univ. Montpellier, CNRS, IRDw, Montpellier, France.
2 Swiss Tropical and Public Health Institute (Swiss TPH), 4002 Basel, Switzerland.
3 Center for Interdisciplinary Research in Biology (CIRB), College de France, CNRS, INSERM, Université PSL, Paris, France. 
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Bastien Reyné; Quentin Richard; Christian Selinger; Mircea T. Sofonea; Ramsès Djidjou-Demasse; Samuel Alizon. Non-Markovian modelling highlights the importance of age structure on Covid-19 epidemiological dynamics. Mathematical modelling of natural phenomena, Tome 17 (2022), article  no. 7. doi: 10.1051/mmnp/2022008

[1] S. Alizon, S. Haim-Boukobza, V. Foulongne, L. Verdurme, S. Trombert-Paolantoni, E. Lecorche, B. Roquebert, M.T. Sofonea Rapid spread of the SARS-CoV-2 Delta variant in some French regions, June 2021 Eurosurveillance 2021 2100573

[2] S. Alizon, M.T. Sofonea SARS-CoV-2 virulence evolution: Avirulence theory, immunity and trade-offs Journal of Evolutionary Biology 2021 1867 1877

[3] R.M. Anderson and R.M. May, Infectious diseases of humans: dynamics and control. Oxford University Press (1992).

[4] V.C. Barclay, T. Smieszek, J. He, G. Cao, J.J. Rainey, H. Gao, A. Uzicanin, M. Salathé Positive network assortativity of influenza vaccination at a high school: implications for outbreak risk and herd immunity PLOS ONE 2014 e87042

[5] T. Beneteau, B. Elie, M.T. Sofonea, S. Alizon Estimating dates of origin and end of COVID-19 epidemics Peer Community Journal 2021 e70

[6] C. Bidot, M. Lamboni, H. Monod multisensi: Multivariate Sensitivity Analysis 2018 R package version 2.1-1

[7] G. Béraud, S. Kazmercziak, P. Beutels, D. Levy-Bruhl, X. Lenne, N. Mielcarek, Y. Yazdanpanah, P.-Y. Boëlle, N. Hens, B. Dervaux The French connection: the first large population-based contact survey in France relevant for the spread of infectiousdiseases PLOS ONE 2015 e0133203

[8] F. Brauer, C. Castillo-Chavez and Z. Feng, vol. 32 of Mathematical models in epidemiology. Springer (2019).

[9] R. Carnell lhs: Latin Hypercube Samples 2020 R package version 1.1.1

[10] R. Challen, E. Brooks-Pollock, J.M. Read, L. Dyson, K. Tsaneva-Atanasova, L. Danon Risk of mortality in patients infected with SARS-CoV-2 variant of concern 202012/1: matched cohort study BMJ 2021 n579

[11] N.G. Davies, S. Abbott, R.C. Barnard, C.I. Jarvis, A.J. Kucharski, J.D. Munday, C.A.B. Pearson, T.W. Russell, D.C. Tully, A.D. Washburne, T. Wenseleers, A. Gimma, W. Waites, K.L.M. Wong, K. V. Zandvoort, J.D. Silverman, K. Diaz-Ordaz, R. Keogh, R.M. Eggo, S. Funk, M. Jit, K.E. Atkins, W.J. Edmunds Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England Science 2021 eabg3055

[12] N.G. Davies, C.I. Jarvis, W.J. Edmunds, N.P. Jewell, K. Diaz-Ordaz, R.H. Keogh Increased mortality in community-tested cases of SARS-CoV-2 lineage B.1.1.7. Nature 2021 270 274

[13] N.G. Davies, P. Klepac, Y. Liu, K. Prem, M. Jit, R.M. Eggo Age-dependent effects in the transmission and control of COVID-19 epidemics Nat. Med 2020 1205 1211

[14] O. Diekmann, J.A. Heesterbeek, J.A. Metz On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations J. Math. Biol 1990 365 382

[15] D. Eddelbuettel, R. François Rcpp: Seamless R and C++ integration J. Stat. Softw 2011 1 18

[16] M. Ehrhardt, J. Gašper, S. Kilianová SIR-based mathematical modeling of infectious diseases with vaccination and waning immunity J. Comput. Sci 2019 101027

[17] L. Ferretti, C. Wymant, M. Kendall, L. Zhao, A. Nurtay, L. Abeler-Dörner, M. Parker, D. Bonsall, C. Fraser Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing Science 2020 eabb6936

[18] T.J. Hagenaars, C.A. Donnelly, N.M. Ferguson Spatial heterogeneity and the persistence of infectious diseases J. Theor. Biol 2004 349 359

[19] S. Haim-Boukobza, B. Roquebert, S. Trombert-Paolantoni, E. Lecorche, L. Verdurme, V. Foulongne, C. Selinger, Y. Michalakis, M.T. Sofonea, S. Alizon Detecting rapid spread of sars-CoV-2 variants Emerg. Infectious Dis 2021 eid2705.210397

[20] J. Hellewell, S. Abbott, A. Gimma, N.I. Bosse, C.I. Jarvis, T.W. Russell, J.D. Munday, A.J. Kucharski, W.J. Edmunds, F. Sun, S. Flasche, B.J. Quilty, N. Davies, Y. Liu, S. Clifford, P. Klepac, M. Jit, C. Diamond, H. Gibbs, K. V. Zandvoort, S. Funk, R.M. Eggo Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts Lancet Global Health 2020 e488 e496

[21] H.W. Hethcote The mathematics of infectious diseases SIAM Rev 2000 599 653

[22] F. Hoppenstaedt Mathematical theories of populations: demographics, genetics and epidemics SIAM 1975

[23] N. Hozé, J. Paireau, N. Lapidus, C. Tran Kiem, H. Salje, G. Severi, M. Touvier, M. Zins, X. De Lamballerie, D. Lévy-Bruhl, F. Carrat, S. Cauchemez Monitoring the proportion of the population infected by SARS-CoV-2 using age-stratified hospitalisation and serological data: a modelling study Lancet Public Health 2021 e408 e415

[24] H. Inaba On a new perspective of the basic reproduction number in heterogeneous environments J. Math. Biol 2012 309 348

[25] H. Inaba, Age-structured population dynamics in demography and epidemiology. Springer (2017).

[26] B. Iooss, S.D. Veiga, A. Janon, G. Pujol, K. Boumhaout, T. Delage, R.E. Amri, J. Fruth, L. Gilquin, J. Guillaume, M.I. Idrissi, L. Le Gratiet, P. Lemaitre, A. Marrel, A. Meynaoui, B.L. Nelson, F. Monari, R. Oomen, O. Rakovec, B. Ramos, O. Roustant, E. Song, J. Staum, R. Sueur, T. Touati, F. Weber sensitivity: Global Sensitivity Analysis of Model Outputs 2021 R package version 1.25.0

[27] M.J. Keeling, E.M. Hill, E.E. Gorsich, B. Penman, G. Guyver-Fletcher, A. Holmes, T. Leng, H. Mckimm, M. Tamborrino, L. Dyson, M.J. Tildesley Predictions of COVID-19 dynamics in the UK: Short-term forecasting and analysis of potential exit strategies PLOS Comput. Biol 2021 e1008619

[28] M.J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals. Princeton University Press (2008).

[29] W.O. Kermack, A.G. Mckendrick A contribution to the mathematical theory of epidemics Proc. Roy. Soc. London A 1927 700 721

[30] C.T. Kiem, C.R. Massonnaud, D. Levy-Bruhl, C. Poletto, V. Colizza, P. Bosetti, A. Fontanet, A. Gabet, V. Olié, L. Zanetti, P.-Y. Boëlle, P. Crépey, S. Cauchemez A modelling study investigating short and medium-term challenges for COVID-19 vaccination: From prioritisation to the relaxation of measures EClinicalMedicine 2021 101001

[31] A.J. Kucharski, T.W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, R.M. Eggo, F. Sun, M. Jit, J.D. Munday, N. Davies, A. Gimma, K. Van Zandvoort, H. Gibbs, J. Hellewell, C.I. Jarvis, S. Clifford, B.J. Quilty, N.I. Bosse, S. Abbott, P. Klepac, S. Flasche Early dynamics of transmission and control of COVID-19: a mathematical modelling study Lancet Infect. Diseases 2020 553 558

[32] N. Lefrancq, J. Paireau, N. Hozé, N. Courtejoie, Y. Yazdanpanah, L. Bouadma, P.-Y. Boëlle, F. Chereau, H. Salje, S. Cauchemez Evolution of outcomes for patients hospitalised during the first 9 months of the SARS-CoV-2 pandemic in France: a retrospective national surveillance data analysis Lancet Regl. Health 2021 100087

[33] A.L. Lloyd Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods Proc. Royal Soc. Lond. Ser B: Biol. Sci 2001 985 993

[34] A.L. Lloyd, R.M. May Spatial heterogeneity in epidemic models J. Theor. Biol 1996 1 11

[35] Y. Ma, S. Pei, J. Shaman, R. Dubrow, K. Chen Role of meteorological factors in the transmission of SARS-CoV-2 in the United States Nat. Commun 2021 3602

[36] P. Magal, Compact attractors for time-periodic age-structured populationmodels (2001).

[37] A. Mazzoni, N.D. Lauria, L. Maggi, L. Salvati, A. Vanni, M. Capone, G. Lamacchia, E. Mantengoli, M. Spinicci, L. Zammarchi, S.T. Kiros, A. Rocca, F. Lagi, M.G. Colao, P. Parronchi, C. Scaletti, L. Turco, F. Liotta, G.M. Rossolini, L. Cosmi, A. Bartoloni, F. Annunziato First-dose mRNA vaccination is sufficient to reactivate immunological memory to SARS-CoV-2 in subjects who have recovered from COVID-19 J. Clin. Invest 2021 e149150

[38] S. Moore, E.M. Hill, M.J. Tildesley, L. Dyson, M.J. Keeling Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study Lancet Infect. Dis 2021 793 802

[39] H. Nishiura Time variations in the generation time of an infectious disease: Implications for sampling to appropriately quantify transmission potential Math. Biosci. Eng 2010 851 869

[40] A. Pazy, Vol. 44 of Semigroups of linear operators and applications to partial differential equations. Springer Science Business Media (2012).

[41] Public Health England, COVID-19 vaccine surveillance report - week 23. Tech. rep. Public Health England (2021).

[42] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2021).

[43] Q. Richard, S. Alizon, M. Choisy, M.T. Sofonea, R. Djidjou-Demasse Age-structured non-pharmaceutical interventions for optimal control of COVID-19 epidemic PLOS Comput. Biol 2021 e1008776

[44] Q. Richard, M. Choisy, T. Lefèvre, R. Djidjou-Demasse Human-vector malaria transmission model structured by age, time since infection and waning immunity Nonlinear Anal.: Real World Appl 2022 103393

[45] Philosophical Transactions of the Royal Society B 2021

[46] H. Salje, C. Tran Kiem, N. Lefrancq, N. Courtejoie, P. Bosetti, J. Paireau, A. Andronico, N. Hozé, J. Richet, C.-L. Dubost, Y. Le Strat, J. Lessler, D. Levy-Bruhl, A. Fontanet, L. Opatowski, P.-Y. Boelle, S. Cauchemez Estimating the burden of SARS-CoV-2 in France Science 2020 208 211

[47] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Global sensitivity analysis: the primer. Wiley (2008).

[48] A. Sheikh, J. Mcmenamin, B. Taylor, C. Robertson SARS-CoV-2 Delta VOC in Scotland: demographics, risk of hospital admission, and vaccine effectiveness The Lancet 2021 2461 2462

[49] M.T. Sofonea, B. Reyné, B. Elie, R. Djidjou-Demasse, C. Selinger, Y. Michalakis, S. Alizon Memory is key in capturing COVID-19 epidemiological dynamics Epidemics 2021 100459

[50] S.Y.D. Valle, J.M. Hyman, N. Chitnis Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases Math. Biosci. Eng 2013 1475

[51] R. Verity, L.C. Okell, I. Dorigatti, P. Winskill, C. Whittaker, N. Imai, G. Cuomo-Dannenburg, H. Thompson, P.G.T. Walker, H. Fu, A. Dighe, J.T. Griffin, M. Baguelin, S. Bhatia, A. Boonyasiri, A. Cori, Z. Cucunubá, R. Fitzjohn, K. Gaythorpe, W. Green, A. Hamlet, W. Hinsley, D. Laydon, G. Nedjati-Gilani, S. Riley, S. V. Elsland, E. Volz, H. Wang, Y. Wang, X. Xi, C.A. Donnelly, A.C. Ghani, N.M. Ferguson Estimates of the severity of coronavirus disease 2019: a model-based analysis Lancet Infect. Dis. 2020 669 677

[52] A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli, T.J.R. Hughes, A. Patton, A. Reali, T.E. Yankeelov, A. Veneziani Simulating the spread of COVID-19 via a spatially-resolved susceptible–exposed–infected–recovered–deceased (SEIRD) model with heterogeneous diffusion Appl. Math. Lett 2021 106617

[53] J. Wallinga, M. Lipsitch How generation intervals shape the relationship between growth rates and reproductive numbers Proc. Roy. Soc. B: Biol. Sci 2007 599 604

[54] J. Wallinga, P. Teunis, M. Kretzschmar Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents Am. J. Epidemiol 2006 936 944

[55] H. Wang, N. Yamamoto, H. Wang, N. Yamamoto Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona Math. Biosci. Eng. 2020 4891 4904

[56] J.T. Wu, S. Mei, S. Luo, K. Leung, D. Liu, Q. Lv, J. Liu, Y. Li, K. Prem, M. Jit, J. Weng, T. Feng, X. Zheng, G.M. Leung A global assessment of the impact of school closure in reducing COVID-19 spread Philos. Trans. Royal Soc. A 2022 20210124

[57] R.M. Zellweger, T.A. Wartel, F. Marks, M. Song, J.H. Kim Vaccination against SARS-CoV-2 and disease enhancement – knowns and unknowns Exp. Rev. Vaccines 2020 691 698

[58] M. Zhang, J. Xiao, A. Deng, Y. Zhang, Y. Zhuang, T. Hu, J. Li, H. Tu, B. Li, Y. Zhou, J. Yuan, L. Luo, Z. Liang, Y. Huang, G. Ye, M. Cai, G. Li, B. Yang, B. Xu, X. Huang, Y. Cui, D. Ren, Y. Zhang, M. Kang, Y. Li Transmission dynamics of an outbreak of the COVID-19 Delta Variant B.1.617.2 – Guangdong Province, China, May–June 2021 China CDC Weekly 2021 584 586

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