Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Influenza Epidemics in Puerto Rico

Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreaks that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. The model is compared with real data from seasonal influenza epidemics in Puerto Rico.


Introduction
Epidemic outbreaks evolve in geographical regions with considerable variability in spatial locations. This spatial variability is important in understanding the impact of public health policies and interventions in controlling these epidemics. A major difficulty in developing models to describe spatial variability in epidemics is accounting for the movement of people in spatial contexts. Many efforts to develop realistic descriptions of epidemics in geographical settings have used individual based models (IBM). These models employ large-scale societal data of human movement and interaction to simulate human behavior at spatial and temporal levels based on probabilistic assumptions. These models require intensive informational input, as well as intensive computational output. Our objective is to provide an alternative approach for modeling spatial epidemics based on deterministic models formulated as partial differential equations in spatial domains.
Our specific focus is upon seasonal influenza outbreaks in geographical regions. Seasonal influenza epidemics recur annually during the cold half of the year in each hemisphere. Each annual flu season is normally associated with a major influenza-virus subtype. The associated subtype changes each year, due to development of immunological resistance to a previous year's strain through exposure and vaccinations, and mutational changes in previously dormant viral strains. The beginning activity in each season varies by location, and evolves characteristically in the larger spatial domain. The exact mechanism behind the seasonal nature of influenza outbreaks is unknown. This paper is organized as follows: In Section 2 we formulate a general deterministic model for the evolution of an epidemic outbreak in a spatial domain. In Section 3 we specify the model to seasonal influenza epidemics in Puerto Rico, and simulate these epidemics in 2015-2016 and 2016-2017. In Section 4 we discuss our results and compare them to IBM formulations of spatial epidemics. In the Appendix we state and proof theorems for our deterministic model of a spatial epidemic.

A General Deterministic Spatial Epidemic Model
Partial differential equations with diffusion terms have been proposed by many authors to describe the movement of people in various applications, including [10,24,21,27,5]. In most applications, however, diffusion does not provide a valid description of the way people move in societal settings. Diffusion provides only an averaging process that cannot account for the extreme spatial and temporal heterogeneity in human movement. We argue, alternatively, that the spatial movement of the micro-organisms causing the epidemic, rather than the spatial movement of humans, is an effective way to account for epidemic spatial development. The movement of the infectious agent can be viewed indirectly, as the movement of infectious individuals, described with diffusion processes. It is clear that the contributions of local-distance and long-distance transmission are both involved in the spatial evolution of epidemics. In [13], however, it is argued that for the 2009 H1N1 influenza epidemic, local transmission was of greater importance than distant transmissions, as outbreaks in proximate communities resulted in successful infection chains, whereas, distant transmissions died out after a small number of generations. The underlying assumption is that most infections occur close to home-base of infectious individuals, which spread to nearby susceptible individuals.
Our model has the following formulation: Suppose that Ω ⊂ R 2 is a bounded domain.
Let S(t, x) and I(t, x) be the spatial densities at location x ∈ Ω and at time t of susceptible and infected individuals, respectively.

Parameterization of the Model for Puerto Rico
The parameterization of any model of a seasonal influenza epidemic presents enormous challenges, because of the incompleteness of data. In the United States, typical epidemic data consists of Morbidity and Mortality Weekly Reports (MMWR) published by the Centers for Disease Control (CDC). For seasonal influenza, this data is very incomplete, and records only a small fraction of total cases. A recent analysis argued that unreported cases and attack rates (the fractions of the total susceptible populations that become infected over the course of an epidemic) are largely underestimated [6].
In an earlier study we developed a formalism for estimating the ratio of reported to unreported cases for the seasonal influenza epidemics in Puerto Rico in 2015-2106 and 2016-9017 [19]. The estimates in [19] claimed attack rates of approximately 40% to 50%. These attack rates are higher than usually claimed for seasonal influenza epidemics. Here we have developed our parameters to reflect attack rates of approximated 30% for both epidemics, based on a comparison of the graphs of the reported cases from CDC data and the graphs of the total cases (both reported and unreported) obtained from our model simulations. The objective was to match the duration of the epidemics, the turning points, and the character of their graphs in the reported case data and the model simulations.
Based on these considerations we estimate the parameters for Puerto Rico as follows: 2. Spatial units are kilometers. The spatial region Ω is as in Figure ??.  We graph the density of the infected population at different times in Figure 5, Figure 6, and Figure 7. From the location of the initial outbreak In San Juan, the epidemic spreads west toward Arecibo, then south toward Ponce, and then west toward Mayagűez. The two peaks in the total case count arise from the spatial evolution of the epidemic, first to the regions of San Juan (population 2,350,000) and Arecibo (population 193,000), and then to the regions of Ponce (population 262,000) and Mayagűez (population 89,000). In Figure 8 we graph the total infected cases in the four major municipalities of Puerto Rico).

Simulation of the model for the 2016-2017 epidemic
A change in the outbreak location, with all other model inputs the same, simulates the data for the 2016-2017 influenza epidemic. We graph the total infected cases from the model simulation in Figure 9, and the graph (scaled) agrees with the graph for reported cases in Figure 3 for the 2016-2017 epidemic. We graph the density of the infected population at different time points in Figure 10. From the initial outbreak in Mayagűez the epidemic spreads north and east toward Arecibo, then east toward San Juan, and south toward Ponce.
In Figure 13 we graph the total infected cases in the four major municipalities of Puerto Rico.
The local evolution of the epidemic at a given outbreak location (x, y) is governed by the local basic reproduction number. If R 0 (x, y) < 1, the epidemic initially subsides, then grows.

Conclusions and discussion
The model indicates that influenza in Puerto Rico rises each season from initial small outbreak locations, and spreads through most of the island, dependent on geographic population variation. The final size of the epidemic at the end of the season depends on the initial outbreak locations, the geographic heterogeneity of the population, and the model parameters.
The model suggests a reason for the seasonality of seasonal influenza epidemics. In a general region, the epidemic lasts approximately 30 weeks, but in subregions the epidemic last approximately 6 weeks (although sometimes re-occurring). The model indicates that the epidemic duration depends strongly on the depletion of the susceptible population to a level that no longer sustains transmission. This depletion happens rapidly in local regions, while the general level of the epidemic occurs much longer in larger regions. Thus, geographic variation is important in understanding the seasonality of seasonal influenza epidemics.
The model indicates that the most effective controls are to monitor the importation of infected people into local regions, and to concentrate public health interventions in regions of high population density (where the local basic reproduction number R 0 (x, y) is highest), especially at the beginning of the season.
Future work involves the use of disease age to track infectiousness levels of infected individuals, through the incubation period, and the rise and fall of the infectious period. Particular emphasis will be given to pre-symptomatic infectiousness periods. The model will be extended to include public policy measures such as quarantine, vaccination, and school closings. Future work will extend the model to study geographic variation in other diseases, including vector-borne diseases such as zika, dengue, and malaria.
Remark .2 For the spatially independent case, the graph of I(t) can have at most one peak. Noticing 1 ≤ p ≤ q + 1, a simple calculation shows that Divide both sides of (ODE.1) by S(t) and integrate it over (0, t) to obtain Then to show that I(t) can have at most one peak, observe from (ODE.2) If I (t) = 0, then which implies I(t) is concave down wherever I (t) = 0.
Rewrite (ODE.2) as Then we can see that I(t) decreases at t = 0 if R 0 < 1 and increases at t = 0 if R 0 > 1. So the claim on I(t) follows from the fact that I(t) converges to zero and has at most one peak.
To prove lim