Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico

Pierre Magal; Glenn F. Webb; Yixiang Wu

doi:10.3934/dcdsb.2019223

Article Contents

2020, Volume 25, Issue 6: 2185-2202. Doi: 10.3934/dcdsb.2019223

This issue Previous Article Spreading speeds of rabies with territorial and diffusing rabid foxes Next Article Density dependent replicator-mutator models in directed evolution

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

Early access: September 2019

Published: June 2020

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Abstract

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.

Keywords:

Kermack-McKendrick reaction-diffusion system,
spatial spread of epidemic,
non homogeneous environment,
numerical simulations.

Mathematics Subject Classification: Primary: 35K55, 35B40, 92D30.

Citation:

Full Text(HTML)

Figure 1. Top. The 76 municipalities in Puerto Rico (wikipedia.org). Bottom. The population density of Puerto Rico (wikipedia.org)

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Figure 2. The population density of the initial susceptible population $ S_0({\bf x}) $

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Figure 3. The geographical mesh with 552 nodes. In the simulations, 23772 nodes are used. The spatial units are kilometers

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Figure 4. (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

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Figure 5. 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

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Figure 6. Model simulation of total cases for four municipalities in the seasonal influenza 2016-2017 epidemic in Puerto Rico

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Figure 7. 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

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Figure 8. 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)

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Figure 9. 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)

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Figure 10. 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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