American Institute of Mathematical Sciences

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

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

Pierre Magal 1, , Ahmed Noussair 1, , Glenn Webb 2,, and  Yixiang Wu 2,

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.

Keywords: Epidemic, spatial, geographical, seasonal, influenza.
Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25, 92C60.
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]

http://www.salud.gov.pr/Estadisticas-Registros-y-Publicaciones/EstadisticasInfluenza/InformeInfluenzaSemana262017.

[28]

https://en.wikipedia.org/wiki/Influenza.

[29]

http://nominatim.openstreetmap.org/.

show all references

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]

http://www.salud.gov.pr/Estadisticas-Registros-y-Publicaciones/EstadisticasInfluenza/InformeInfluenzaSemana262017.

[28]

https://en.wikipedia.org/wiki/Influenza.

[29]

http://nominatim.openstreetmap.org/.

Figure 1.  The map of Puerto Rico with at most 50 points to defined the boundary of each municipality
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The black curve corresponds to the number of weekly reported cases of seasonal influenza in Puerto Rico in 2015-2016 [27]
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Total number of weekly cases from week 52 in 2015 to week 20 of 2016 obtained by the simulation of the model
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  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 $
Table Options

Download as excel

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 $
[1]

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
[2]

Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561
[3]

Eunha Shim. Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1615-1634. doi: 10.3934/mbe.2013.10.1615
[4]

Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073
[5]

Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045
[6]

Mudassar Imran, Mohamed Ben-Romdhane, Ali R. Ansari, Helmi Temimi. Numerical study of an influenza epidemic dynamical model with diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2761-2787. doi: 10.3934/dcdss.2020168
[7]

Naveen K. Vaidya, Feng-Bin Wang, Xingfu Zou. Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2829-2848. doi: 10.3934/dcdsb.2012.17.2829
[8]

Olivia Prosper, Omar Saucedo, Doria Thompson, Griselle Torres-Garcia, Xiaohong Wang, Carlos Castillo-Chavez. Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza. Mathematical Biosciences & Engineering, 2011, 8 (1) : 141-170. doi: 10.3934/mbe.2011.8.141
[9]

Sherry Towers, Katia Vogt Geisse, Chia-Chun Tsai, Qing Han, Zhilan Feng. The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 413-430. doi: 10.3934/mbe.2012.9.413
[10]

Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1877-1911. doi: 10.3934/dcdsb.2021113
[11]

Karen R. Ríos-Soto, Baojun Song, Carlos Castillo-Chavez. Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 199-222. doi: 10.3934/mbe.2011.8.199
[12]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35
[13]

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
[14]

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
[15]

Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4045-4057. doi: 10.3934/dcdsb.2020273
[16]

Ming-Zhen Xin, Bin-Guo Wang. Spatial dynamics of an epidemic model in time almost periodic and space periodic media. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022116
[17]

Tao Zheng, Yantao Luo, Xinran Zhou, Long Zhang, Zhidong Teng. Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021154
[18]

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
[19]

Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[20]

Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 4019-4037. doi: 10.3934/dcds.2020056

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (341)
  • HTML views (385)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy