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

• * Corresponding author: Glenn Webb
• 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.

Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25, 92C60.

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

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

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

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 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  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  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

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 $10^{-2}$ $km^2/day$
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