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Invasion threshold in structured populations with recurrent mobility patterns - PubMed

  • ️Sun Jan 01 2012

Invasion threshold in structured populations with recurrent mobility patterns

Duygu Balcan et al. J Theor Biol. 2012.

Abstract

In this paper we develop a framework to analyze the behavior of contagion and spreading processes in complex subpopulation networks where individuals have memory of their subpopulation of origin. We introduce a metapopulation model in which subpopulations are connected through heterogeneous fluxes of individuals. The mobility process among communities takes into account the memory of residence of individuals and is incorporated with the classical susceptible-infectious-recovered epidemic model within each subpopulation. In order to gain analytical insight into the behavior of the system we use degree-block variables describing the heterogeneity of the subpopulation network and a time-scale separation technique for the dynamics of individuals. By considering the stochastic nature of the epidemic process we obtain the explicit expression of the global epidemic invasion threshold, below which the disease dies out before reaching a macroscopic fraction of the subpopulations. This threshold is not present in continuous deterministic diffusion models and explicitly depends on the disease parameters, the mobility rates, and the properties of the coupling matrices describing the mobility across subpopulations. The results presented here take a step further in offering insight into the fundamental mechanisms controlling the spreading of infectious diseases and other contagion processes across spatially structured communities.

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Figures

Figure 1
Figure 1

Illustration of commuting and subdivision of population. At any time each subpopulation is occupied by its own residents plus visitors from its neighbors. For instance, the population in subpopulation i is divided between individuals who reside and are present in the subpopulation (Nii) and those who are residents in subpopulation j but present in subpopulation i(Nji). Different classes of people move between connected subpopulations along the edges at the rates shown.

Figure 2
Figure 2

Statistical properties of commuting networks in the United States and France. Cumulative distributions of the number of connections per administrative unit and the number of commuters on each connection are displayed. The networks are highly heterogeneous in the number of connections per geographical area as well as in the flux of individuals on each connection.

Figure 3
Figure 3

Phase diagram on the σ-τ−1 plane for the case of Eq. (9) in heterogeneous and homogeneous subpopulation networks. The phase diagram separating the global invasion from the extinction regime is shown on the σ-τ−1 plane assuming Eq. (9) for commuting rates. The solid lines correspond to the solution R* = 1 of Eq. (10), above which the infection spreads at the metapopulation level as indicated by the shaded areas. We can easily see an increase of about one order of magnitude in the critical values of mobility rates as we switch from a heavy-tailed to a Poisson degree distribution. Networks have the same average degree and contain V = 104 subpopulations in which the heavy-tailed network assumes P (k) ~ k−2.1. Each subpopulation accommodates a degree-dependent population of N̄k = N̄k/〈k〉 individuals with N̄ = 104. Moreover the disease is characterized by R0 = 1.25 and μ−1 = 15 days.

Figure 4
Figure 4

Phase diagram on the σ-τ−1 plane for the case of Eq. (15) in heterogeneous and homogeneous subpopulation networks. The phase diagram separating the global invasion from the extinction regime is shown on the σ-τ−1 plane assuming Eq. (15) for commuting rates. The solid lines correspond to the solution R* = 1 of Eq. (16), above which the infection spreads at the metapopulation level as indicated by the shaded areas. The diagrams should be compared with Fig. 3.

Figure 5
Figure 5

Average global attack rate as a function of ρ and R0 in homogeneous subpopulation networks. The figure codes with color the average fraction of individuals infected by the outbreak in the space of ρ and R0. For smaller values of R0 larger values of ρ are needed for the infection to spread at the global scale. Once ρ is well below or above its critical value its precise value does not affect the attack rate. In order to vary ρ, we have fixed the return rate τ at 1/day and changed the value of the commuting rate σ. Networks are made of V = 104 subpopulations, each of which accommodates a degree-dependent population of N̄k = N̄k/〈k〉 individuals with N̄ = 104. The infectious period is set to μ−1 = 3 day.

Figure 6
Figure 6

Average epidemic size as a function of σ and τ−1 in heterogeneous (left) and homogeneous (right) subpopulation networks. Each figure shows via a color map the average percentage of subpopulations affected by the outbreak in the space of σ and τ−1. The lower the commuting rate the longer the visiting time is needed for the infection to spread to a finite fraction of subpopulations. The figure should be compared with the phase diagrams of Fig. 4. Networks are composed of V = 104 subpopulations, each of which with a degree-dependent population of N̄k = N̄k/〈k〉 residents with N̄= 104. Disease is characterized by R0 = 1.25 and μ−1 = 15 day.

Figure 7
Figure 7

Distribution of epidemic sizes in heterogeneous (left) and homogeneous (right) subpopulation networks as a function of ρ. Each color map shows the probability of observing a fraction of subpopulations affected by the outbreak as a function of commuting ratio. Each figure readily shows a critical point of ρ, below which most of the epidemics are confined to a few number of subpopulations. Above the critical point most of the realizations affect almost the entire set of subpopulations. In order to highlight the differences with respect to network topology, realizations resulting in epidemics confined to less than ten subpopulations have been excluded from the analysis of size distributions. The critical value of the commuting ratio differs more than one order of magnitude as we switch from a heavy-tailed to a Poisson degree distribution. Since the precise value of the return rate does not alter the results, we have set τ−1 = 1 day and changed σ in order to vary ρ. Both networks contain V = 105 subpopulations, each of which accommodates a degree dependent population of N̄k = N̄k/〈k〉 inhabitants with N̄ = 103. The disease is characterized by R0 = 1.5 and μ−1 = 5 day.

Figure 8
Figure 8

Probability of invasion of at least 1% of the subpopulations in heterogeneous and homogeneous networks as a function of ρ. Figure shows the fraction of realizations in which at least 1% of the subpopulations are invaded by the epidemic process as a function of commuting ratio. For clarity, the data points corresponding to zero values of probability have been excluded from the figure. We have set τ−1 = 1 day and changed σ in order to vary ρ. Both networks contain V = 105 subpopulations, each of which accommodates a degree dependent population of N̄k = N̄k/〈k〉 inhabitants with N ¯= 103. The disease is characterized by R0 = 1.5 and μ−1 = 5 day.

Figure 9
Figure 9

Average epidemic size in heterogeneous and homogeneous subpopulation networks as a function of ρ. Figure displays the average fraction of subpopulations affected by the outbreak as a function of commuting ratio. We have set τ−1 = 1 day and changed σ in order to vary ρ. Both networks contain V = 105 subpopulations, each of which accommodates a degree dependent population of N̄k = N̄k/〈k〉 inhabitants with N̄ = 103. The disease is characterized by R0 = 1.5 and μ−1 = 5 day.

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