# American Institute of Mathematical Sciences

June  2020, 25(6): 2185-2202. doi: 10.3934/dcdsb.2019223

## 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  June 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (6) : 2185-2202. doi: 10.3934/dcdsb.2019223
##### References:
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Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.  doi: 10.1137/0135022. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [25] T. C. Germann, K. Kadau, I. 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Katriel, et al., Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel, PLOS One, 7 (2012). [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. [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. [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. [34] J. P. LaSalle, Some extensions of Liapunov's second method., IRE Transactions on Circuit Theory, 7 (1960), 520-527. [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. [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. [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. [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. [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. [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. [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. [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. [43] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. [44] N. Masuda and P. Holme, Temporal Network Epidemiology, Theoretical Biology, Springer Nature Singapore, 2017. doi: 10.1007/978-981-10-5287-3. [45] S. Merler and M. Ajelli, The role of population heterogeneity and human mobility in the spread of pandemic influenza, Proc. R. Soc. 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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.  doi: 10.1137/0135022. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [34] J. P. LaSalle, Some extensions of Liapunov's second method., IRE Transactions on Circuit Theory, 7 (1960), 520-527. [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. [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. [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. [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. [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. [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. [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. [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. [43] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. [44] N. Masuda and P. Holme, Temporal Network Epidemiology, Theoretical Biology, Springer Nature Singapore, 2017. doi: 10.1007/978-981-10-5287-3. [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. [46] M. Moorthy, D. Castronovo and A. Abraham, et al., Deviations in influenza seasonality: Odd coincidence or obscure consequence?, Clin. Microbiol. Infec., 18 (2012), 955-962. doi: 10.1111/j.1469-0691.2012.03959.x. [47] J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. [48] L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, 102, American Mathematical Society, 2003. [49] C. Reed, F. J. Angulo, D. L. Swerdlow, M. Lipsitch, M. I. Meltzer, D. Jernigan and L. Finelli, Estimates of the prevalence of pandemic (H1N1) 2009, United States, April-July 2009, Emerg. Infect. Dis., 15 (2009), . doi: 10.3201/eid1512.091413. [50] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [51] S. 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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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