# American Institute of Mathematical Sciences

2016, 13(1): 43-65. doi: 10.3934/mbe.2016.13.43

## Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile

 1 CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción 2 School of Public Health, Georgia State University, Atlanta, Georgia, United States 3 Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain 4 GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

Received  July 2014 Revised  July 2015 Published  October 2015

A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal range, is developed to characterize the spatial variation in peak timing of that pandemic as a function of local transmission rates, spatial connectivity assumptions for Chilean regions, and the putative location of introduction of the novel virus into the country. Specifically, a metapopulation SEIR (susceptible-exposed-infected-removed) compartmental model that tracks the transmission dynamics of influenza in 15 Chilean regions is calibrated. The model incorporates population mobility among neighboring regions and indirect mobility to and from other regions via the metropolitan central region (hub region''). The stability of the disease-free equilibrium of this model is analyzed and compared with the corresponding stability in each region, concluding that stability may occur even with some regions having basic reproduction numbers above 1. The transmission model is used along with epidemiological data to explore potential factors that could have driven the spatial-temporal progression of the pandemic. Simulations and sensitivity analyses indicate that this relatively simple model is sufficient to characterize the south-north gradient in peak timing observed during the pandemic, and suggest that south Chile observed the initial spread of the pandemic virus, which is in line with a retrospective epidemiological study. The hub region'' in our model significantly enhanced population mixing in a short time scale.
Citation: Raimund Bürger, Gerardo Chowell, Pep Mulet, Luis M. Villada. Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile. Mathematical Biosciences & Engineering, 2016, 13 (1) : 43-65. doi: 10.3934/mbe.2016.13.43
##### References:
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##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic of the steady states for an SIS epidemic patch model,, SIAM J. Appl. Math., 67 (2007), 1283. doi: 10.1137/060672522. Google Scholar [2] W. J. Alonso, C. Viboud, L. Simonsen, E. W. Hirano, L. Z. Daufenbach and M. A. Miller, Seasonality of influenza in Brazil: A traveling wave from the Amazon to the subtropics,, Amer. J. Epidemiol., 165 (2007), 1434. doi: 10.1093/aje/kwm012. Google Scholar [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford Science Publications, (1991). Google Scholar [4] J. Arino, Diseases in metapopulations. In Z. Ma, Y. Zhou and J. Wu (Eds.),, Modeling and Dynamics of Infectious Diseases, 11 (2009), 64. Google Scholar [5] J. Arino, J. R. Davis, D. Hartley, R. Jordan, J. M. Miller and P. van den Driessche, A multi-species epidemic model with spatial dynamics,, Mathematical Medicine and Biology, 22 (2005), 129. Google Scholar [6] J. Arino and P. van den Driessche, A multi-city epidemic model,, Math. Popul. Studies, 10 (2003), 175. doi: 10.1080/08898480306720. Google Scholar [7] D. Balcan, H. Hu, B. Goncalves, P. Bajardi, C. Poletto, J. J. Ramasco, D. Paolotti, N. Perra, M. Tizzoni, W. Van den Broeck, V. Colizza and A. Vespignani, Seasonal transmission potential and activity peaks of the new influenza A(H1N1): A Monte Carlo likelihood analysis based on human mobility,, BMC Med., 7 (2009). doi: 10.1186/1741-7015-7-45. Google Scholar [8] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Second Ed., (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar [9] S. Cauchemez, N. Ferguson, C. Wachtel, A. Tegnell, G. Saour, B. Duncan and A. Nicoll, Closure of schools during an influenza pandemic},, Lancet Infect. Dis., 9 (2009), 473. doi: 10.1016/S1473-3099(09)70176-8. Google Scholar [10] G. Chowell, S. Echevarría-Zuno, C. Viboud, L. Simonsen, M. A. Miller, I. Fernández-Gárate, C. González-Bonilla and V. H. Borja-Aburto, Epidemiological characteristics and underlying risk factors for mortality during the autumn 2009 pandemic wave in Mexico,, PLoS One, 7 (2012). doi: 10.1371/journal.pone.0041069. Google Scholar [11] G. Chowell, S. Echevarría-Zuno, C. Viboud, L. Simonsen, J. Tamerius, M. A. Miller and V. H. Borja-Aburto, Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico,, PloS Med., 8 (2011). doi: 10.1371/journal.pmed.1000436. Google Scholar [12] G. Chowell, S. Towers, C. Viboud, R. Fuentes, V. Sotomayor, L. Simonsen, M. Miller, M. Lima, C. Villarroel and M. Chiu, The influence of climatic conditions on the transmission dynamics of the 2009 A/H1N1 influenza pandemic in Chile,, BMC Infect. Dis., 12 (2012). doi: 10.1186/1471-2334-12-298. Google Scholar [13] G. Chowell, C. Viboud, C. V. Munayco, J. Gomez, L. Simonsen, M. A. Miller, J. Tamerius, V. Fiestas, E. S. Halsey and V. A. Laguna-Torres, Spatial and temporal characteristics of the 2009 A/H1N1 influenza pandemic in Peru,, PLoS One, 6 (2011). doi: 10.1371/journal.pone.0021287. Google Scholar [14] G. Chowell, C. Viboud, L. Simonsen, M. Miller and W. J. Alonso, The reproduction number of seasonal influenza epidemics in Brazil, 1996-2006,, Proc. Biol. Sci., 277 (2010), 1857. doi: 10.1098/rspb.2009.1897. Google Scholar [15] V. Colizza, A. Barrat, M. Barthelemy, A. J. Valleron and A. Vespignani, Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions,, PLoS Med., 4 (2007). doi: 10.1371/journal.pmed.0040013. Google Scholar [16] O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton Series in Theoretical and Computational Biology, (2013). Google Scholar [17] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [18] P. van den Driessche, Deterministic compartmental models: Extensions of basic models,, In F. Brauer, 1945 (2008), 147. doi: 10.1007/978-3-540-78911-6_5. Google Scholar [19] P. van den Driessche, Spatial structure: Patch models}. In F. Brauer, P. van den Driessche and J. Wu (Eds.),, Mathematical Epidemiology, 1945 (2008), 179. doi: 10.1007/978-3-540-78911-6_7. Google Scholar [20] P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics,, Math. Biosci., 228 (2010), 71. doi: 10.1016/j.mbs.2010.08.008. Google Scholar [21] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [22] X. Fei, C. Connell McCluskey and R. Cressman, Spatial spread of an epidemic through public transportation systems with a hub,, Math. Biosci., 246 (2013), 164. doi: 10.1016/j.mbs.2013.08.014. Google Scholar [23] J. R. Gog, S. Ballesteros, C. Viboud, L. Simonsen, O. N. Bjornstad, J. Shaman, D. L. Chao, F. Khan and B. T. Grenfell, Spatial transmission of 2009 pandemic influenza in the US,, PLoS Comput. Biol., 10 (2014). doi: 10.1371/journal.pcbi.1003635. Google Scholar [24] M. Herrera-Valdez, M. Cruz-Aponte and C. Castillo-Chavez, Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different waves of A-H1N1pdm cases observed in México during 2009,, Math. Biosci. Eng., 8 (2011), 21. doi: 10.3934/mbe.2011.8.21. Google Scholar [25] Instituto Nacional de Estadísticas (INE)., Estadísticas Demográficas y Vitales, 2009., Available from: , (). Google Scholar [26] C. Jackson, E. Vynnycky, J. Hawker, B. Olowokure and P. Mangtani, School closures and influenza: Systematic review of epidemiological studies,, BMJ open, 3 (2013). doi: 10.1136/bmjopen-2012-002149. Google Scholar [27] T. Jefferson, M. A. Jones, P. Doshi, C. B. Del Mar, R. Hama, M. J. Thompson, E. A. Spencer, I. Onakpoya, K. R. Mahtani, D. Nunan, J. Howick and C. Heneghan, Neuraminidase inhibitors for preventing and treating influenza in healthy adults and children,, Cochrane Database Syst. Rev., (2014). doi: 10.1002/14651858.CD008965.pub3. Google Scholar [28] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Roy. Soc. A, 115 (1927), 700. Google Scholar [29] K. Khan, J. Arino, W. Hu, P. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Liauw, A. Chan and M. Gardam, Spread of a novel influenza A (H1N1) virus via global airline transportation,, New Engl. J. Med., 361 (2009), 212. doi: 10.1056/NEJMc0904559. Google Scholar [30] T. Kuniya, Global stability of a multi-group SVIR epidemic model,, Nonlin. Anal. Real World Appl., 14 (2013), 1135. doi: 10.1016/j.nonrwa.2012.09.004. Google Scholar [31] T. Kuniya, Y. Muroya and Y. Enatsu, Threshold dynamics of an SIR epidemic model with hybrid and multigroup of patch structures,, Math. Biosci. Eng., 11 (2014), 1375. doi: 10.3934/mbe.2014.11.1375. Google Scholar [32] M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment,, Canad. Appl. Math. Quart., 17 (2009), 175. Google Scholar [33] A. Lowen, S. Mubareka, J. Steel and P. Palese, Influenza virus transmission is dependent on relative humidity and temperature,, PLoS Pathog., 3 (2007), 1470. doi: 10.1371/journal.ppat.0030151. Google Scholar [34] A. Lowen, J. Steel, S. Mubareka and P. Palese, High temperature (30 degrees C) blocks aerosol but not contact transmission of influenza virus,, J. Virol.., 82 (2008), 5650. Google Scholar [35] S. Mubareka, A. Lowen, J. Steel, A. Coates, A. Garcia-Sastre and P. Palese, Transmission of influenza virus via aerosols and fomites in the guinea pig model,, J. Infect. Dis., 199 (2009), 858. doi: 10.1086/597073. Google Scholar [36] L. Opatowski, C. Fraser, J. Griffin, E. de Silva, M. D. Van Kerkhove, E. J. Lyons, S. Cauchemez and N. M. Ferguson, Transmission characteristics of the 2009 H1N1 influenza pandemic: Comparison of 8 Southern hemisphere countries,, PLoS Pathog. 7 (2011), 7 (2011). doi: 10.1371/journal.ppat.1002225. Google Scholar [37] E. Pedroni, M. Garcia, V. Espinola, A. Guerrero, C. Gonzalez, A. Olea, M. Calvo, B. Martorell, M. Winkler and M. Carrasco, Outbreak of 2009 pandemic influenza A/H1N1, Los Lagos, Chile, April-June 2009,, Eurosurveillance 15 (2010), 15 (2010). Google Scholar [38] M. M. Saito, S. Imoto, R. Yamaguchi, H. Sato, H. Nakada, M. Kami, S. Miyano and T. Higuchi, Extension and verification of the SEIR model on the 2009 influenza A (H1N1) pandemic in Japan,, Math. 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