# American Institute of Mathematical Sciences

December  2020, 13(12): 3535-3550. doi: 10.3934/dcdss.2020237

## Modeling epidemic outbreaks in geographical regions: Seasonal influenza in Puerto Rico

 1 Institut de Mathematique de Bordeaux, University of Bordeaux, Talence, 33400, France 2 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

* Corresponding author: Glenn Webb

Received  October 2018 Revised  March 2019 Published  December 2020 Early access  January 2020

We develop a model for the spatial spread of epidemic outbreak in a geographical region. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The model consists of a system of partial differential equations, which indirectly describes the disease transmission caused by the disease pathogen. The model is compared to data for the seasonal influenza epidemics in Puerto Rico for 2015-2016.

Citation: Pierre Magal, Ahmed Noussair, Glenn Webb, Yixiang Wu. Modeling epidemic outbreaks in geographical regions: Seasonal influenza in Puerto Rico. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3535-3550. doi: 10.3934/dcdss.2020237
##### 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] L. Bastos, D. A. Villela, L. M. Carvaldo, et al, Zika in Rio de Janeiro: assessment of basic reproductive number and its comparison with dengue, bioRxiv 2016; 055475. [3] F. Carrat, E. Vergu, N.M. Ferguson, M. Lemaitre, S. Cauchemez, S. Leach and A.-J. Valleron, Time lines of infection and disease in human influenza: A review of volunteer challenge studies, Amer. J. Epid., 167 (2008), 775-785.  doi: 10.1093/aje/kwm375. [4] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Diff. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045. [5] K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864. [6] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [7] 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. [8] 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, Theoretical Biology and Medical Modelling, 14 (2017). [9] D. G. Kendall, Deterministic and stochastic epidemics in closed populations, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. IV, 149–165. University of California Press, Berkeley and Los Angeles, 1956. [10] D. G. Kendall, Mathematical Models of the Spread of Infection, Mathematics and Computer Science in Biology and Medicine, H.M.S.O, London, 1965. [11] X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [12] H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Diff. Equ., 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044. [13] E. Lofgren et al., Influenza seasonality: Underlying causes and modeling theories, J Virol., 81 (2007), 5429-5436. [14] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [15] P. Magal and G. 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. [16] P. Magal, G. F. Webb and Y. Wu, Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico, Discrete & Continuous Dynamical Systems - B, 2019, arXiv: 1801.01856. doi: 10.3934/dcdsb.2019223. [17] P. Magal and S. Ruan, Theory and Application of Abstract Semilinear Cauchy Problems, With a foreword by Glenn Webb. Applied Mathematical Sciences, 201. Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0. [18] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44. doi: 10.2307/2001590. [19] M. Moorthy et al., Deviations in influenza seasonality: Odd coincidence or obscure consequence?, Clin Microbiol Infect., 18 (2012), 955-962. [20] R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451. [21] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine, 97–122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007. [22] N. K. Vaidya, F.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete and Continuous Dynamical Systems B, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829. [23] F.-B. Wang, J. Shi and X. Zou, Dynamics of a host-pathogen system on a bounded spatial domain, Communications on Pure and Applied Analysis, 14 (2015), 2535-2560.  doi: 10.3934/cpaa.2015.14.2535. [24] X. Wang, D. Posny and J. Wang, A reaction-convection-diffusion model for Cholera spatial dynamics, Discrete and Continuous Dynamical Systems B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073. [25] G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6. [26] X. Yu and X.-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Diff. Equ., 261 (2016), 340-372.  doi: 10.1016/j.jde.2016.03.014. [27] [28] [29]

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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] L. Bastos, D. A. Villela, L. M. Carvaldo, et al, Zika in Rio de Janeiro: assessment of basic reproductive number and its comparison with dengue, bioRxiv 2016; 055475. [3] F. Carrat, E. Vergu, N.M. Ferguson, M. Lemaitre, S. Cauchemez, S. Leach and A.-J. Valleron, Time lines of infection and disease in human influenza: A review of volunteer challenge studies, Amer. J. Epid., 167 (2008), 775-785.  doi: 10.1093/aje/kwm375. [4] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Diff. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045. [5] K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864. [6] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [7] 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. [8] 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, Theoretical Biology and Medical Modelling, 14 (2017). [9] D. G. Kendall, Deterministic and stochastic epidemics in closed populations, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. IV, 149–165. University of California Press, Berkeley and Los Angeles, 1956. [10] D. G. Kendall, Mathematical Models of the Spread of Infection, Mathematics and Computer Science in Biology and Medicine, H.M.S.O, London, 1965. [11] X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [12] H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Diff. Equ., 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044. [13] E. Lofgren et al., Influenza seasonality: Underlying causes and modeling theories, J Virol., 81 (2007), 5429-5436. [14] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [15] P. Magal and G. 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. [16] P. Magal, G. F. Webb and Y. Wu, Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico, Discrete & Continuous Dynamical Systems - B, 2019, arXiv: 1801.01856. doi: 10.3934/dcdsb.2019223. [17] P. Magal and S. Ruan, Theory and Application of Abstract Semilinear Cauchy Problems, With a foreword by Glenn Webb. Applied Mathematical Sciences, 201. Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0. [18] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44. doi: 10.2307/2001590. [19] M. Moorthy et al., Deviations in influenza seasonality: Odd coincidence or obscure consequence?, Clin Microbiol Infect., 18 (2012), 955-962. [20] R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451. [21] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine, 97–122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007. [22] N. K. Vaidya, F.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete and Continuous Dynamical Systems B, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829. [23] F.-B. Wang, J. Shi and X. Zou, Dynamics of a host-pathogen system on a bounded spatial domain, Communications on Pure and Applied Analysis, 14 (2015), 2535-2560.  doi: 10.3934/cpaa.2015.14.2535. [24] X. Wang, D. Posny and J. Wang, A reaction-convection-diffusion model for Cholera spatial dynamics, Discrete and Continuous Dynamical Systems B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073. [25] G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6. [26] X. Yu and X.-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Diff. Equ., 261 (2016), 340-372.  doi: 10.1016/j.jde.2016.03.014. [27] [28] [29]
The map of Puerto Rico with at most 50 points to defined the boundary of each municipality
On the top we plot the mesh used for the simulation. On the bottom we graph the Puerto Rico municipalities and their corresponding coding number
The black curve corresponds to the number of weekly reported cases of seasonal influenza in Puerto Rico in 2015-2016 [27]
On the top we plot the density of the infected population for Puerto Rico at week 52 in 2015, obtained from reported case data [27]. On the bottom we plot $b(t, x)$ with $\varepsilon = 0.01$. The larger $\varepsilon$ is, the more spread out is the infection around an original location of an infected individual
Population density of the municipalities of Puerto Rico in 2016 (US Census Bureau). In the model the distribution corresponds to $n(0, x) = s(0, x)+i(0, x)+e(0, x)+r(0, x)$
Total number of weekly cases from week 52 in 2015 to week 20 of 2016 obtained by the simulation of the model
The number of weekly cases from week 52 in 2015 to week 20 in 2016. The figures (a) (b) (c) and (d) correspond, respectively, to the model simulation of cases for the municipalities of San Juan, Arecibo, Ponce and Mayaguez, respectively
Density of Infected population at weeks 1 (first two) and 5 (last two). The first and third figures are based on reported cases data [27] and the second and fourth figures are from our simulations
Density of Infected population at weeks 1 (first two) and 5 (last two). The first and third figures are based on reported cases data [27] and the second and fourth figures are from our simulations
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, which may account for the small second peak in total reported cases graphed in Figure 6
List of parameters used for the simulations
 Symbol Description Value Units $\beta$ Transmission rate $0.002$ $\gamma$ Recorvering rate $1/5$ 1/Day $r$ Incubation period $2$ Days $\kappa$ $10^{-4}$ $p$ $1$ real $q$ $2$ real $\epsilon$ diffusion rate $10^{-2}$ $km^2/day$
 Symbol Description Value Units $\beta$ Transmission rate $0.002$ $\gamma$ Recorvering rate $1/5$ 1/Day $r$ Incubation period $2$ Days $\kappa$ $10^{-4}$ $p$ $1$ real $q$ $2$ real $\epsilon$ diffusion rate $10^{-2}$ $km^2/day$
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