# American Institute of Mathematical Sciences

## Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico

 1 Univ. Bordeaux, IMB, UMR 5251, Talence F-33400, France 2 CNRS, IMB, UMR 5251, Talence F-33400, France 3 Department of Mathematics, Vanderbilt University, Nashville, TN, USA

Received  January 2019 Revised  April 2019 Published  September 2019

A deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The model is focused on outbreaks that arise from a small number of infected individuals in sub-regions of the geographical setting. The goal is to understand how spatial heterogeneity influences the transmission dynamics of susceptible and infected populations. The model consists of a system of partial differential equations with a diffusion term describing the spatial spread of an underlying microbial infectious agent. The model is applied to simulate the spatial spread of the 2016-2017 seasonal influenza epidemic in Puerto Rico. In this simulation, the reported case data from the Puerto Rican Department of Health are used to implement a numerical finite element scheme for the model. The model simulation explains the geographical evolution of this epidemic in Puerto Rico, consistent with the reported case data.

Citation: Pierre Magal, Glenn F. Webb, Yixiang Wu. Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019223
##### References:
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Google Scholar [10] N. F. Britton, An integral for a reaction-diffusion system, Appl. Math. Lett., 4 (1991), 43-47. doi: 10.1016/0893-9659(91)90120-K. Google Scholar [11] V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284. doi: 10.1137/0135022. Google Scholar [12] S. Cauchemez, P. Horby and A. Fox, et al., Influenza infection rates, measurement errors and the interpretation of paired serology, PLOS Pathog., 8 (2012), e1003061. doi: 10.1371/journal.ppat.1003061. Google Scholar [13] S. Charaudeau, P. Khashayar and P.-Y. Boelle, Commuter mobility and the spread of infectious diseases: Application to influenza in France, PLOS One, 9 (2014), e83002. doi: 10.1371/journal.pone.0083002. Google Scholar [14] V. Charu, S. Zeger and J. Gog, et al., Human mobility and the spatial transmission of influenza in the United States, PLOS Comput. Biol., 13 (2017), e1005382. doi: 10.1371/journal.pcbi.1005382. Google Scholar [15] B. J. Coburn, G. W. Bradley and S. Blower, Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1), BMC Med., 7 (2009). doi: 10.1186/1741-7015-7-30. Google Scholar [16] 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). Google Scholar [17] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373. doi: 10.1016/j.jde.2017.03.045. Google Scholar [18] A. Ducrot and T. Giletti, Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552. doi: 10.1007/s00285-013-0713-3. Google Scholar [19] L. 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Glass, Assessing the impact of airline travel on the geographic spread of pandemic influenza, Eur. J. Epidemiol., 18 (2003), 1065-1072. Google Scholar [28] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539. Google Scholar [29] L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, PNAS USA, 101 (2004), 15124-15129. doi: 10.1073/pnas.0308344101. Google Scholar [30] A. Huppert, O. Barnea and G. Katriel, et al., Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel, PLOS One, 7 (2012).Google Scholar [31] A. Kallen, P. Arcuri and and J. D. Murray, A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393. doi: 10.1016/S0022-5193(85)80276-9. Google Scholar [32] I. Z. Kiss, J. C. Miller and P. L. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Interdisciplinary Applied Mathematics, 46, Springer Nature, 2017. doi: 10.1007/978-3-319-50806-1. Google Scholar [33] T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960. doi: 10.1080/00036811.2016.1199796. Google Scholar [34] J. P. LaSalle, Some extensions of Liapunov's second method., IRE Transactions on Circuit Theory, 7 (1960), 520-527. Google Scholar [35] H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044. Google Scholar [36] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. Google Scholar [37] E. T. Lofgren, J. B. Wenger and N. H. Fefferman, et al., Disproportional effects in populations of concern for pandemic influenza: insights from seasonal epidemics in Wisconsin, 1967-2004, Influenza Other Resp., 4 (2010), 205-212. doi: 10.1111/j.1750-2659.2010.00137.x. Google Scholar [38] I. M. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. Cummings and M. E. Halloran, Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. Google Scholar [39] P. Magal, O. Seydi and G. F. Webb, Final size of an epidemic for a two-group SIR model, SIAM J. Appl. Math., 76 (2016), 2042-2059. doi: 10.1137/16M1065392. Google Scholar [40] P. Magal, O. Seydi and G. F. Webb, Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission, Math. Biosci., 301 (2018), 59-67. doi: 10.1016/j.mbs.2018.03.020. Google Scholar [41] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. Google Scholar [42] P. Magal and G. F. Webb, The parameter identification problem for SIR epidemic models: identifying unreported cases, J. Math. Biol., 77 (2018), 1629-1648. doi: 10.1007/s00285-017-1203-9. Google Scholar [43] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. Google Scholar [44] N. Masuda and P. Holme, Temporal Network Epidemiology, Theoretical Biology, Springer Nature Singapore, 2017. doi: 10.1007/978-981-10-5287-3. Google Scholar [45] S. Merler and M. Ajelli, The role of population heterogeneity and human mobility in the spread of pandemic influenza, Proc. R. Soc. B, 277 (2010), 557-565. doi: 10.1098/rspb.2009.1605. Google Scholar [46] M. Moorthy, D. Castronovo and A. 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##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. Google Scholar [2] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Communications in Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar [3] J. Arino, Spatio-temporal spread of infectious pathogens of humans, Infect. Disease Model, 2 (2017), 218-228. doi: 10.1016/j.idm.2017.05.001. Google Scholar [4] J. Arino and K. Khan, Using mathematical modelling to integrate disease surveillance and global air transportation data, in Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases (eds. D. Chen, B. Moulin, and J. Wu), John Wiley & Sons, 2014.Google Scholar [5] J. Arino and S. Portet, Epidemiological implications of mobility between a large urban centre and smaller satellite cities, J. Math. Biol., 71 (2015), 1243-1265. doi: 10.1007/s00285-014-0854-z. Google Scholar [6] D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco and A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, PNAS USA, 106 (2009), 21484-21489. doi: 10.1073/pnas.0906910106. Google Scholar [7] D. Bandaranayake, M. Jacobs and M. Baker, et al., The second wave of 2009 pandemic influenza A (H1N1) in New Zealand, January-October 2010, Eurosurveillance, 16 (2011), 1978.Google Scholar [8] M. Biggerstaff and L. Balluz, Self-reported influenza-like illness during the 2009 H1N1 influenza pandemic, US Morbid. Mortal. Weekly Rep., September 2009 March 2010, 60 (2011), 37.Google Scholar [9] E. Bonabeau, L. Toubiana and A. Flahault, The geographical spread of influenza, Proc. Roy. Soc. Lond. B, 265 (1998), 2421-2425. doi: 10.1098/rspb.1998.0593. Google Scholar [10] N. F. Britton, An integral for a reaction-diffusion system, Appl. Math. Lett., 4 (1991), 43-47. doi: 10.1016/0893-9659(91)90120-K. Google Scholar [11] V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284. doi: 10.1137/0135022. Google Scholar [12] S. Cauchemez, P. Horby and A. Fox, et al., Influenza infection rates, measurement errors and the interpretation of paired serology, PLOS Pathog., 8 (2012), e1003061. doi: 10.1371/journal.ppat.1003061. Google Scholar [13] S. Charaudeau, P. Khashayar and P.-Y. Boelle, Commuter mobility and the spread of infectious diseases: Application to influenza in France, PLOS One, 9 (2014), e83002. doi: 10.1371/journal.pone.0083002. Google Scholar [14] V. Charu, S. Zeger and J. Gog, et al., Human mobility and the spatial transmission of influenza in the United States, PLOS Comput. Biol., 13 (2017), e1005382. doi: 10.1371/journal.pcbi.1005382. Google Scholar [15] B. J. Coburn, G. W. Bradley and S. Blower, Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1), BMC Med., 7 (2009). doi: 10.1186/1741-7015-7-30. Google Scholar [16] 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). Google Scholar [17] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373. doi: 10.1016/j.jde.2017.03.045. Google Scholar [18] A. Ducrot and T. Giletti, Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552. doi: 10.1007/s00285-013-0713-3. Google Scholar [19] L. Dung, Global $L^\infty$ estimates for a class of reaction diffusion systems, Journal of Mathematical Analysis and Applications, 217 (1998), 72-94. doi: 10.1006/jmaa.1997.5702. Google Scholar [20] S. Eubank, H. Guclu, V. S. A. Kumar and M. V. Marathe, Modelling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. doi: 10.1038/nature02541. Google Scholar [21] N. M. Ferguson, D. A. Cummings, S. Cauchemez and C. Fraser, Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437 (2005), 209-214. doi: 10.1038/nature04017. Google Scholar [22] W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, in Structured Population Models in Biology and Epidemiology, 115-164, Lecture Notes in Math., 1936, Math. Biosci. Subser., Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5_3. Google Scholar [23] W. E. Fitzgibbon, J. J. Morgan and G. F. Webb, An outbreak vector-host epidemic model with spatial structure: The 2015-2016 zika outbreak in Rio de Janeiro, Theor. Biol. Med. Model., 14 (2017), 7. doi: 10.1186/s12976-017-0051-z. Google Scholar [24] W. E. Fitzgibbon, J. J. Morgan, G. F. Webb and Y. Wu, A vector-host epidemic model with spatial structure and age of infection, Nonlinear Analysis: Real World Applications, 41 (2018), 692-705. doi: 10.1016/j.nonrwa.2017.11.005. Google Scholar [25] T. C. Germann, K. Kadau, I. M. Longini and C. A. Macken, Mitigation strategies for pandemic influenza in the United States, PNAS USA, 103 (2006), 5935-5940. doi: 10.1073/pnas.0601266103. Google Scholar [26] J. R. Gog, S. Ballesteros, C. Viboud and L. Simonsen, et al., Spatial transmission of 2009 pandemic influenza in the US, PLOS Comput. Biol., 10 (2014), e1003635. doi: 10.1371/journal.pcbi.1003635. Google Scholar [27] R. F. Grais, J. H. Ellis and G. E. Glass, Assessing the impact of airline travel on the geographic spread of pandemic influenza, Eur. J. Epidemiol., 18 (2003), 1065-1072. Google Scholar [28] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539. Google Scholar [29] L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, PNAS USA, 101 (2004), 15124-15129. doi: 10.1073/pnas.0308344101. Google Scholar [30] A. Huppert, O. Barnea and G. Katriel, et al., Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel, PLOS One, 7 (2012).Google Scholar [31] A. Kallen, P. Arcuri and and J. D. Murray, A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393. doi: 10.1016/S0022-5193(85)80276-9. Google Scholar [32] I. Z. Kiss, J. C. Miller and P. L. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Interdisciplinary Applied Mathematics, 46, Springer Nature, 2017. doi: 10.1007/978-3-319-50806-1. Google Scholar [33] T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960. doi: 10.1080/00036811.2016.1199796. Google Scholar [34] J. P. LaSalle, Some extensions of Liapunov's second method., IRE Transactions on Circuit Theory, 7 (1960), 520-527. Google Scholar [35] H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044. Google Scholar [36] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. Google Scholar [37] E. T. Lofgren, J. B. Wenger and N. H. Fefferman, et al., Disproportional effects in populations of concern for pandemic influenza: insights from seasonal epidemics in Wisconsin, 1967-2004, Influenza Other Resp., 4 (2010), 205-212. doi: 10.1111/j.1750-2659.2010.00137.x. Google Scholar [38] I. M. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. Cummings and M. E. Halloran, Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. Google Scholar [39] P. Magal, O. Seydi and G. F. Webb, Final size of an epidemic for a two-group SIR model, SIAM J. Appl. Math., 76 (2016), 2042-2059. doi: 10.1137/16M1065392. Google Scholar [40] P. Magal, O. Seydi and G. F. Webb, Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission, Math. Biosci., 301 (2018), 59-67. doi: 10.1016/j.mbs.2018.03.020. Google Scholar [41] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. Google Scholar [42] P. Magal and G. F. Webb, The parameter identification problem for SIR epidemic models: identifying unreported cases, J. Math. Biol., 77 (2018), 1629-1648. doi: 10.1007/s00285-017-1203-9. Google Scholar [43] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. Google Scholar [44] N. Masuda and P. Holme, Temporal Network Epidemiology, Theoretical Biology, Springer Nature Singapore, 2017. doi: 10.1007/978-981-10-5287-3. Google Scholar [45] S. Merler and M. Ajelli, The role of population heterogeneity and human mobility in the spread of pandemic influenza, Proc. R. Soc. B, 277 (2010), 557-565. doi: 10.1098/rspb.2009.1605. Google Scholar [46] M. Moorthy, D. Castronovo and A. 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Top. The 76 municipalities in Puerto Rico (wikipedia.org). Bottom. The population density of Puerto Rico (wikipedia.org)
The population density of the initial susceptible population $S_0({\bf x})$
The geographical mesh with 552 nodes. In the simulations, 23772 nodes are used. The spatial units are kilometers
(top) Reported cases of seasonal influenza Puerto Rico in 2015-2016 (yellow graph) and 2016-2017 (black graph); (bottom) Total cases from the model simulation for 2016-2017
Estimated reported case data (per 100,000 inhabitants) for four municipalities Mayaqűez, Arecibo, San Juan, and Ponce in the 2016-2017 seasonal influenza epidemic in Puerto Rico. The epidemic arises in Mayaqűez, spreads to Arecibo and San Juan, and last to Ponce
Model simulation of total cases for four municipalities in the seasonal influenza 2016-2017 epidemic in Puerto Rico
Simulation of spatial spread of 2016-2017 influenza outbreak in Puerto Rico. The population density of Puerto Rico is set as the initial value of the susceptible population. The initial size of the infected population is assumed to be 30, concentrated in the northwest
Model simulation of the infected population densities (number of cases per 100,000 people) in the 2016-2017 seasonal influenza epidemic in Puerto Rico in all municipalities for weeks 4 (top left), 6 (top right), 10 (bottom left), and 18 (bottom right)
The total number of reported cases of influenza strain subtypes in 2015-2016. An outbreak of type B strain peaks at week 21 in 2016 (Departamento de Salud, Puerto Rico)
Estimated reported case data (per 100,000 inhabitants) from Departamento de Salud for four municipalities San Juan, Arecibo, Ponce, and Mayaqűez in the 2015-2016 seasonal influenza epidemic in Puerto Rico. The late second peak is present in all four municipalities
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