Random and targeted interventions for epidemic control in metapopulation models

Abstract

In general, different countries and communities respond to epidemics in accordance with their own control plans and protocols. However, owing to global human migration and mobility, strategic planning for epidemic control measures through the collaboration of relevant public health administrations is gaining importance for mitigating and containing large-scale epidemics. Here, we present a framework to evaluate the effectiveness of random (non-strategic) and targeted (strategic) epidemic interventions for spatially separated patches in metapopulation models. For a random intervention, we analytically derive the critical fraction of patches that receive epidemic interventions, above which epidemics are successfully contained. The analysis shows that the heterogeneity of patch connectivity makes it difficult to contain epidemics under the random intervention. We demonstrate that, particularly in such heterogeneously connected networks, targeted interventions are considerably effective compared to the random intervention. Our framework is useful for identifying the target areas where epidemic control measures should be focused.

Figure 1. Metapopulation model with interventions for epidemic control in local patches.

Schematic illustration of a metapopulation model consisting of high-risk patches without interventions (red open circles) and low-risk patches with interventions (blue open circles). The individuals in the patches are either susceptible (blue filled circles) or infected (red filled circles). In each patch, the disease transmission and recovery of individuals occur according to the SIS compartment model. The local basic reproduction number in the high-risk patches is given by and that in the low-risk patches by . The parameter p represents the ratio of the low-risk patches to the whole patches, corresponding to the total size of public health interventions.

Figure 2. Transition between endemic and disease-free states for random intervention.

The prevalence of infected individuals, ρ_I/ρ, is plotted against the low-risk patch ratio p. The circles and squares indicate the results for homogeneous and heterogeneous networks with N = 1000 and 〈k〉 ≈ 10, respectively. For each network type, the results for ten different network realizations are superimposed. The parameter values are set at ρ = 1, D_S = 0.1, D_I = 5, μ = 1, β_H = 1.2, and β_L = 0.2.

Figure 3. Validation of the analytical expression of the critical low-risk patch ratio for random intervention.

Phase diagrams on the (D_I, p) parameter plane, where D_I is the mobility rate of the infected individuals and p is the low-risk patch ratio. The color bar indicates the numerically obtained values of the prevalence ρ_I/ρ, averaged over ten different network realizations. The red and blue regions correspond to the endemic and disease-free states, respectively. The boundary between the two states corresponds to the critical low-risk patch ratio p_c. The crosses indicate the theoretical values of p_c obtained from equation (1) for the same ten network realizations. The parameter values are set at ρ = 1, D_S = 0.1, μ = 1, β_H = 1.2, and β_L = 0.2. (a) Homogeneous networks with N = 1000 and 〈k〉 ≈ 10. (b) Heterogeneous networks with N = 1000 and 〈k〉 ≈ 30.

Figure 4. Effects of epidemic interventions in metapopulation models with homogeneous patch connectivity.

Each panel shows the phase diagram for homogeneous networks in the (Δβ, p) parameter plane, where Δβ controls the transmission rates given by β_H = 1 + Δβ and β_L = Δβ. The color bar indicates the numerically obtained values of the prevalence ρ_I/ρ, averaged over ten different network realizations with N = 1000 and 〈k〉 ≈ 10. The red and blue regions correspond to the endemic and disease-free states, respectively. The parameter values are set at ρ = 1, D_S = 0.1, and D_I = 5. (a) Random intervention; (b) Neighborhood intervention; (c) Targeted intervention; (d) Adaptive intervention with T = 10; (e) Adaptive intervention with T = 50; and (f) Adaptive intervention with T = 100.

Figure 5. Effects of epidemic interventions in metapopulation models with heterogeneous patch connectivity.

The same as Fig. 4, but for heterogeneous networks with N = 1000 and 〈k〉 ≈ 30. The parameter values are set at ρ = 0.4, μ = 1, D_S = 0.1, and D_I = 5.

Figure 6. Effects of epidemic interventions in metapopulation models with US airline networks.

The same as Figs. 4 and 5, but for US airline networks with N = 500 and 〈k〉 ≈ 6. The parameter values are set at ρ = 1, μ = 1, D_S = 0.1, and D_I = 5.

Figure 7. Epidemic interventions in metapopulation models with the Japanese airline network.

The data of the Japanese airport network were obtained from the timetables of domestic flights of All Nippon Airway (http://www.ana.co.jp) and Japan Airline (http://www.jal.co.jp). The Japanese airport network consists of N = 74 airports and 190 flight routes. The mean, maximum, and minimum degrees are given by 〈k〉 ≈ 5.1, k_max = 50 (corresponding to the Tokyo-Haneda airport), and k_min = 1, respectively. (a) The map of the airports in Japan, which was created with Adobe Illustrator; (b) The degree distribution of the Japanese airport network; (c) The distribution of the number of passengers per flight route; and (d) The prevalence ρ_I/ρ is plotted against the low-risk patch ratio p for the random (squares), neighborhood (triangles), targeted (diamonds), and adaptive (circles) interventions. For the random and neighborhood interventions, the best result among five simulations was adopted. For adaptive intervention, the time interval for updating the low-risk patches is set at T = 10. The other parameter values are set at ρ = 0.5, μ = 1, D_S = D_I = 10, β_H = 2, and β_L = 0.1.