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Modeling human mobility responses to the large-scale spreading of infectious diseases - PubMed

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Modeling human mobility responses to the large-scale spreading of infectious diseases

Sandro Meloni et al. Sci Rep. 2011.

Abstract

Current modeling of infectious diseases allows for the study of realistic scenarios that include population heterogeneity, social structures, and mobility processes down to the individual level. The advances in the realism of epidemic description call for the explicit modeling of individual behavioral responses to the presence of disease within modeling frameworks. Here we formulate and analyze a metapopulation model that incorporates several scenarios of self-initiated behavioral changes into the mobility patterns of individuals. We find that prevalence-based travel limitations do not alter the epidemic invasion threshold. Strikingly, we observe in both synthetic and data-driven numerical simulations that when travelers decide to avoid locations with high levels of prevalence, this self-initiated behavioral change may enhance disease spreading. Our results point out that the real-time availability of information on the disease and the ensuing behavioral changes in the population may produce a negative impact on disease containment and mitigation.

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Figures

Figure 1
Figure 1. Schematic representation of the Metapopulation System.

A population of individuals is divided into V subpopulations connected with each other following a heterogeneous network. Within each subpopulation, individuals are classified according to their dynamical status as Susceptible (S), Infected (I) and Removed (R). In absence of behavioral changes (blue arrows), individuals move from a subpopulation to another at a rate λ following the shortest path connecting both subpopulations. The discontinuous arrows represent the second mechanism of behavioral reaction in which people travel avoiding places with high prevalence at the cost of larger diffusion paths.

Figure 2
Figure 2. Mobility Threshold.

To compare the analytical insights with numerical results here we represent the number of diseased subpopulations D/V as a function of the mobility rate λ. The analytical value (Eq.(5)) is indicated by the arrow and the green triangle. Full circles are results from numerical simulations and represent the average over at least 100 stochastic runs (the line is a visual guide). The value of α has been approximated by formula image . The substrate topology is an uncorrelated scale-free network generated according to the uncorrelated configuration model with γ = 2.5, V = 3000 subpopulations and N = 3×106 individuals. Other parameters are indicated in the figure.

Figure 3
Figure 3. Effects of behavioral changes in synthetic networks.

The figure compares the fraction of diseased subpopulations D/V when behavioral reaction mechanisms are active with the situation in which such behavioral responses are not taken into account (null model). (A) We show the dependency of D/V with the mobility rate λ (A) for random scale-free networks generated according to the uncorrelated configuration model. Symbols represent the results obtained when individuals do not react to the presence of the disease (error bars are smaller than symbol sizes). The rest of the results correspond to the mechanisms of behavioral changes: “DP” stands for “departure probability” and represents the mechanism in which individuals decide whether or not to travel; “RR” (rerouting) corresponds to the case in which people travel while trying to minimize the risk of infection avoiding subpopulations with high prevalence at the cost of long travel paths. The results confirm that the invasion threshold is independent of behavioral changes and that the latter has a significant impact on the invasion dynamics of the metapopulation. The points are the averages among at least 100 stochastic runs and we consider µ = 0.04 and h = 0.1. See the main text for further details. (B) we report the relative difference of subpopulations experiencing an outbreak in the RR and baseline scenarios as a function of λ. It is possible to see the non-linear behavior that first induces a decrease – close to the invasion threshold – and then a sharp increase in the number of affected subpopulations.

Figure 4
Figure 4. Effects of behavioral changes in data-driven scenarios.

Comparison of the fraction of diseased subpopulations D/V for the full (behavioral reaction mechanisms are active) and null (behavioral responses are not taken into account) limits of the metapopulation system. We plot D/V as a function of the mobility rate λ. The results confirm even in this case that the invasion threshold is independent of behavioral changes. Moreover, as for synthetic networks, epidemic awareness enhances the disease spreading as given by the increase in the number of subpopulations affected by the disease. The averages were taken over at least 100 stochastic realizations and we fix µ = 0.04 and h = 0.1. See the main text for further details.

Figure 5
Figure 5. Invasion tree.

Invasion tree describing the air transportation network inside the USA of an epidemic starting in New York. The invasion tree specifies the disease progression by defining a directed link i → j from the infecting to the infected subpopulation. In panel (A) we show the invasion tree for the null model when no behavioral reactions are considered. In panel (B) we show the invasion tree starting from the same initial conditions but consider both mechanisms of behavioral reaction to be active. In order to provide a clear representation we consider in both cases just the first 100 infected nodes among the total 425 in a single run, respectively. The color scale is a measure of time and is the same for both cases. At the time step in which the first subpopulation is infected from the seed it is yellow. At the time step in which the last subpopulation (among the first 100 in both cases) is infected it is red. All the other time steps are in the gradient between these two limits. Panel (A) shows clearly that in the absence of behavioral changes the infection tree is heterogeneous and contains several hubs that are infected first and that determine the time scale of the spreading infection to smaller airports. This is not the case in the presence of behavioral changes where the entire tree originates and grows much faster from the initially infected subpopulation as shown in panel (B). In both cases, we fix the parameters to R0 = 2, µ = 0.04 and λ = 10−6.

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