# 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:
 [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), 1283-1309. doi: 10.1137/060672522. [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), 1434-1442. doi: 10.1093/aje/kwm012. [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, 1991. [4] J. Arino, Diseases in metapopulations. In Z. Ma, Y. Zhou and J. Wu (Eds.), Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 11 (2009), 64-122. [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), 129-142. [6] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Studies, 10 (2003), 175-193. doi: 10.1080/08898480306720. [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), p45 (12pp). doi: 10.1186/1741-7015-7-45. [8] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9. [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), 473-481. doi: 10.1016/S1473-3099(09)70176-8. [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), e41069 (10pp). doi: 10.1371/journal.pone.0041069. [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), e1000436 (13pp). doi: 10.1371/journal.pmed.1000436. [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), p298 (12pp). doi: 10.1186/1471-2334-12-298. [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), e21287 (10pp). doi: 10.1371/journal.pone.0021287. [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), 1857-1866. doi: 10.1098/rspb.2009.1897. [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), e13 (16pp). doi: 10.1371/journal.pmed.0040013. [16] O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2013. [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), 365-382. doi: 10.1007/BF00178324. [18] P. van den Driessche, Deterministic compartmental models: Extensions of basic models, In F. Brauer, P. van den Driessche and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 1945 (2008), 147-157. doi: 10.1007/978-3-540-78911-6_5. [19] P. van den Driessche, Spatial structure: Patch models}. In F. Brauer, P. van den Driessche and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 1945 (2008), 179-189. doi: 10.1007/978-3-540-78911-6_7. [20] P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics, Math. Biosci., 228 (2010), 71-77. doi: 10.1016/j.mbs.2010.08.008. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 164-175. doi: 10.1016/j.mbs.2013.08.014. [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), e1003635 (11pp). doi: 10.1371/journal.pcbi.1003635. [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), 21-48. doi: 10.3934/mbe.2011.8.21. [25] Instituto Nacional de Estadísticas (INE)., Estadísticas Demográficas y Vitales, 2009., Available from: , (). [26] C. Jackson, E. Vynnycky, J. Hawker, B. Olowokure and P. Mangtani, School closures and influenza: Systematic review of epidemiological studies, BMJ open, 3 (2) (2013), e002149 (10pp). doi: 10.1136/bmjopen-2012-002149. [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., April 10, 2014 (560pp). doi: 10.1002/14651858.CD008965.pub3. [28] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721. [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), 212-214. doi: 10.1056/NEJMc0904559. [30] T. Kuniya, Global stability of a multi-group SVIR epidemic model, Nonlin. Anal. Real World Appl., 14 (2013), 1135-1143. doi: 10.1016/j.nonrwa.2012.09.004. [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), 1375-1393. doi: 10.3934/mbe.2014.11.1375. [32] M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canad. Appl. Math. Quart., 17 (2009), 175-187. [33] A. Lowen, S. Mubareka, J. Steel and P. Palese, Influenza virus transmission is dependent on relative humidity and temperature, PLoS Pathog., 3 (2007), 1470-1476. doi: 10.1371/journal.ppat.0030151. [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), 5650-5652. [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), 858-865. doi: 10.1086/597073. [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), e1002225 (10pp). doi: 10.1371/journal.ppat.1002225. [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), 19456 (9pp). [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. Biosci., 246 (2013), 47-54. doi: 10.1016/j.mbs.2013.08.009. [39] L. Sattenspiel, The Geographic Spread of Infectious Diseases: Models and Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2009. doi: 10.1515/9781400831708. [40] L. Sattenspiel and K. Dietz, A structured epidemic model incorporating geographic mobility among regions, Math. Biosci., 128 (1995), 71-91. doi: 10.1016/0025-5564(94)00068-B. [41] D. L. Schanzer, J. M. Langley, T. Dummer and S. Aziz, The geographic synchrony of seasonal influenza: A waves across Canada and the United States, PLoS One, 6 (2011), e21471 (8pp). doi: 10.1371/journal.pone.0021471. [42] C. Schuck-Paim, C. Viboud, L. Simonsen, M. A. Miller, F. E. Moura, R. M. Fernandes, M. L. Carvalho and W. J. Alonso, Were equatorial regions less affected by the 2009 influenza pandemic? The Brazilian experience, PLoS One, 7 (2012), e41918 (10pp). doi: 10.1371/journal.pone.0041918. [43] J. Shaman and M. Kohn, Absolute humidity modulates influenza survival, transmission, and seasonality, Proc. Natl. Acad. Sci. USA, 106 (2009), 3243-3248. doi: 10.1073/pnas.0806852106. [44] J. Shaman, V. Pitzer, C. Viboud, B. Grenfell and M. Lipsitch, Absolute humidity and the seasonal onset of influenza in the continental United States, PLoS Biol., 8 (2010), e1000316 (13pp). [45] L. Simonsen, P. Spreeuwenberg, R. Lustig, R. J. Taylor, D. M. Fleming, M. Kroneman, M. D. Van Kerkhove, A. W. Mounts, W. J. Paget and GLaMOR Collaborating Teams, Global mortality estimates for the 2009 Influenza Pandemic from the GLaMOR project: a modeling study, PLoS Med., 10 (2013), e1001558 (17pp). doi: 10.1371/journal.pmed.1001558. [46] J. Steel, P. Palese and A. Lowen, Transmission of a 2009 pandemic influenza virus shows a sensitivity to temperature and humidity similar to that of an H3N2 seasonal strain, J. Virol., 85 (2011), 1400-1402. doi: 10.1128/JVI.02186-10. [47] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Applic., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [48] J. Tamerius, M. I. Nelson, S. Z. Zhou, C. Viboud, M. A. Miller and W. J. Alonso, Global influenza seasonality: Reconciling patterns across temperate and tropical regions, Environ. Health Perspect., 119 (2011), 439-445. doi: 10.1289/ehp.1002383. [49] C. Viboud, O. N. Bjornstad, D. L. Smith, L. Simonsen, M. A. Miller and B. T. Grenfell, Synchrony, waves, and spatial hierarchies in the spread of influenza, Science, 312 (2006), 447-451. doi: 10.1126/science.1125237. [50] E. Vynnycky and R. E. White, An Introduction to Infectious Disease Modelling, Oxford University Press, 2010. [51] J. B. Wenger and E. N. Naumova, Seasonal synchronization of influenza in the United States older adult population, PLoS One, 5 (2010), e10187 (11pp). doi: 10.1371/journal.pone.0010187. [52] H. Yu, S. Cauchemez, C. A. Donnelly, L. Zhou, L. Feng, N. Xiang, J. Zheng, M. Ye, Y. Huai, Q. Liao, Z. Peng, Y. Feng, H. Jiang, W. Yang, Y. Wang, N. M. Ferguson and Z. Feng, Transmission dynamics, border entry screening, and school holidays during the 2009 influenza A (H1N1) pandemic, China, Emerg. Infect. Dis., 18 (2012), 758-766. doi: 10.3201/eid1805.110356.

show all references

##### 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), 1283-1309. doi: 10.1137/060672522. [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), 1434-1442. doi: 10.1093/aje/kwm012. [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, 1991. [4] J. Arino, Diseases in metapopulations. In Z. Ma, Y. Zhou and J. Wu (Eds.), Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 11 (2009), 64-122. [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), 129-142. [6] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Studies, 10 (2003), 175-193. doi: 10.1080/08898480306720. [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), p45 (12pp). doi: 10.1186/1741-7015-7-45. [8] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9. [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), 473-481. doi: 10.1016/S1473-3099(09)70176-8. [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), e41069 (10pp). doi: 10.1371/journal.pone.0041069. [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), e1000436 (13pp). doi: 10.1371/journal.pmed.1000436. [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), p298 (12pp). doi: 10.1186/1471-2334-12-298. [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), e21287 (10pp). doi: 10.1371/journal.pone.0021287. [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), 1857-1866. doi: 10.1098/rspb.2009.1897. [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), e13 (16pp). doi: 10.1371/journal.pmed.0040013. [16] O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2013. [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), 365-382. doi: 10.1007/BF00178324. [18] P. van den Driessche, Deterministic compartmental models: Extensions of basic models, In F. Brauer, P. van den Driessche and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 1945 (2008), 147-157. doi: 10.1007/978-3-540-78911-6_5. [19] P. van den Driessche, Spatial structure: Patch models}. In F. Brauer, P. van den Driessche and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 1945 (2008), 179-189. doi: 10.1007/978-3-540-78911-6_7. [20] P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics, Math. Biosci., 228 (2010), 71-77. doi: 10.1016/j.mbs.2010.08.008. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 164-175. doi: 10.1016/j.mbs.2013.08.014. [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), e1003635 (11pp). doi: 10.1371/journal.pcbi.1003635. [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), 21-48. doi: 10.3934/mbe.2011.8.21. [25] Instituto Nacional de Estadísticas (INE)., Estadísticas Demográficas y Vitales, 2009., Available from: , (). [26] C. Jackson, E. Vynnycky, J. Hawker, B. Olowokure and P. Mangtani, School closures and influenza: Systematic review of epidemiological studies, BMJ open, 3 (2) (2013), e002149 (10pp). doi: 10.1136/bmjopen-2012-002149. [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., April 10, 2014 (560pp). doi: 10.1002/14651858.CD008965.pub3. [28] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721. [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), 212-214. doi: 10.1056/NEJMc0904559. [30] T. Kuniya, Global stability of a multi-group SVIR epidemic model, Nonlin. Anal. Real World Appl., 14 (2013), 1135-1143. doi: 10.1016/j.nonrwa.2012.09.004. [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), 1375-1393. doi: 10.3934/mbe.2014.11.1375. [32] M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canad. Appl. Math. Quart., 17 (2009), 175-187. [33] A. Lowen, S. Mubareka, J. Steel and P. Palese, Influenza virus transmission is dependent on relative humidity and temperature, PLoS Pathog., 3 (2007), 1470-1476. doi: 10.1371/journal.ppat.0030151. [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), 5650-5652. [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), 858-865. doi: 10.1086/597073. [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), e1002225 (10pp). doi: 10.1371/journal.ppat.1002225. [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), 19456 (9pp). [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. Biosci., 246 (2013), 47-54. doi: 10.1016/j.mbs.2013.08.009. [39] L. Sattenspiel, The Geographic Spread of Infectious Diseases: Models and Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2009. doi: 10.1515/9781400831708. [40] L. Sattenspiel and K. Dietz, A structured epidemic model incorporating geographic mobility among regions, Math. Biosci., 128 (1995), 71-91. doi: 10.1016/0025-5564(94)00068-B. [41] D. L. Schanzer, J. M. Langley, T. Dummer and S. Aziz, The geographic synchrony of seasonal influenza: A waves across Canada and the United States, PLoS One, 6 (2011), e21471 (8pp). doi: 10.1371/journal.pone.0021471. [42] C. Schuck-Paim, C. Viboud, L. Simonsen, M. A. Miller, F. E. Moura, R. M. Fernandes, M. L. Carvalho and W. J. Alonso, Were equatorial regions less affected by the 2009 influenza pandemic? The Brazilian experience, PLoS One, 7 (2012), e41918 (10pp). doi: 10.1371/journal.pone.0041918. [43] J. Shaman and M. Kohn, Absolute humidity modulates influenza survival, transmission, and seasonality, Proc. Natl. Acad. Sci. USA, 106 (2009), 3243-3248. doi: 10.1073/pnas.0806852106. [44] J. Shaman, V. Pitzer, C. Viboud, B. Grenfell and M. Lipsitch, Absolute humidity and the seasonal onset of influenza in the continental United States, PLoS Biol., 8 (2010), e1000316 (13pp). [45] L. Simonsen, P. Spreeuwenberg, R. Lustig, R. J. Taylor, D. M. Fleming, M. Kroneman, M. D. Van Kerkhove, A. W. Mounts, W. J. Paget and GLaMOR Collaborating Teams, Global mortality estimates for the 2009 Influenza Pandemic from the GLaMOR project: a modeling study, PLoS Med., 10 (2013), e1001558 (17pp). doi: 10.1371/journal.pmed.1001558. [46] J. Steel, P. Palese and A. Lowen, Transmission of a 2009 pandemic influenza virus shows a sensitivity to temperature and humidity similar to that of an H3N2 seasonal strain, J. Virol., 85 (2011), 1400-1402. doi: 10.1128/JVI.02186-10. [47] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Applic., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [48] J. Tamerius, M. I. Nelson, S. Z. Zhou, C. Viboud, M. A. Miller and W. J. Alonso, Global influenza seasonality: Reconciling patterns across temperate and tropical regions, Environ. Health Perspect., 119 (2011), 439-445. doi: 10.1289/ehp.1002383. [49] C. Viboud, O. N. Bjornstad, D. L. Smith, L. Simonsen, M. A. Miller and B. T. Grenfell, Synchrony, waves, and spatial hierarchies in the spread of influenza, Science, 312 (2006), 447-451. doi: 10.1126/science.1125237. [50] E. Vynnycky and R. E. White, An Introduction to Infectious Disease Modelling, Oxford University Press, 2010. [51] J. B. Wenger and E. N. Naumova, Seasonal synchronization of influenza in the United States older adult population, PLoS One, 5 (2010), e10187 (11pp). doi: 10.1371/journal.pone.0010187. [52] H. Yu, S. Cauchemez, C. A. Donnelly, L. Zhou, L. Feng, N. Xiang, J. Zheng, M. Ye, Y. Huai, Q. Liao, Z. Peng, Y. Feng, H. Jiang, W. Yang, Y. Wang, N. M. Ferguson and Z. Feng, Transmission dynamics, border entry screening, and school holidays during the 2009 influenza A (H1N1) pandemic, China, Emerg. Infect. Dis., 18 (2012), 758-766. doi: 10.3201/eid1805.110356.
 [1] Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501 [2] Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham, George Tita. Latent self-exciting point process model for spatial-temporal networks. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1335-1354. doi: 10.3934/dcdsb.2014.19.1335 [3] Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems and Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036 [4] Dashun Xu, Z. Feng. A metapopulation model with local competitions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 495-510. doi: 10.3934/dcdsb.2009.12.495 [5] Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers, Peter D. Lee. 4D-CT reconstruction with unified spatial-temporal patch-based regularization. Inverse Problems and Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447 [6] Zhun Gou, Nan-jing Huang, Ming-hui Wang, Yao-jia Zhang. A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator. Mathematical Control and Related Fields, 2021, 11 (2) : 291-312. doi: 10.3934/mcrf.2020037 [7] Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1565-1583. doi: 10.3934/mbe.2017081 [8] Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022062 [9] C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603 [10] Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200 [11] Yantao Luo, Zhidong Teng, Xiao-Qiang Zhao. Transmission dynamics of a general temporal-spatial vector-host epidemic model with an application to the dengue fever in Guangdong, China. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022069 [12] Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 [13] Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219 [14] Alan J. Terry. Pulse vaccination strategies in a metapopulation SIR model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 455-477. doi: 10.3934/mbe.2010.7.455 [15] Zhilan Feng, Robert Swihart, Yingfei Yi, Huaiping Zhu. Coexistence in a metapopulation model with explicit local dynamics. Mathematical Biosciences & Engineering, 2004, 1 (1) : 131-145. doi: 10.3934/mbe.2004.1.131 [16] Luca Bolzoni, Rossella Della Marca, Maria Groppi, Alessandra Gragnani. Dynamics of a metapopulation epidemic model with localized culling. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2307-2330. doi: 10.3934/dcdsb.2020036 [17] Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567 [18] C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008 [19] Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173 [20] Suman Ganguli, David Gammack, Denise E. Kirschner. A Metapopulation Model Of Granuloma Formation In The Lung During Infection With Mycobacterium Tuberculosis. Mathematical Biosciences & Engineering, 2005, 2 (3) : 535-560. doi: 10.3934/mbe.2005.2.535

2018 Impact Factor: 1.313