# 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 & Continuous Dynamical Systems - S, 2020, 13 (12) : 3535-3550. doi: 10.3934/dcdss.2020237
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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
]">Figure 3.  The black curve corresponds to the number of weekly reported cases of seasonal influenza in Puerto Rico in 2015-2016 [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 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
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
] and the second and fourth figures are from our simulations">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
] and the second and fourth figures are from our simulations">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 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
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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