Background
Methods
Mathematical model
Distribution of arrival times of infected individuals in countries seeded by originating country
Likelihood formulation
Simulation testing
Performance at different levels of reproduction numbers and travel rates
Robustness against a decreasing trend of travel
Sensitivity analysis of parameter assumptions
-
1/γ ~ Gamma distribution with a mean of 3 days and 1 day SD
-
θ ~ Uniform distribution ranged from 10 to 50%
-
d ~ Uniform distribution ranged from 1 day to 7 days
Numerical study: 2009 influenza pandemic a/H1N1
Country | Number of passengersa
| Arrival dates | Illness onset date | Additional referenceb
|
---|---|---|---|---|
United States | 2,474,897 | - | March 28 | - |
Canada | 101,313 | April 8 | April 11 | - |
El Salvador | 15,090 | April 19 | - | - |
Colombia | 24,535 | - | April 14 | [26] |
United Kingdom | 20,513 | April 21 | April 24 | - |
Spain | 65,724 | April 22 | April 25 | - |
France | 61,960 | - | April 23 | [18] |
Costa Rica | 16,950 | April 25 | April 25 | - |
Argentina | 24,609 | April 25 | April 27 | [27] |
Cuba | 42,802 | April 25 | - | - |
The Netherlands | 27,640 | April 27 | - | - |
Germany | 35,772 | - | April 28 | - |
Hong Kong | 35,706 | April 30 | April 30 | [30] |
Italy | 12,060 | April 29 | May 3 | [31] |
Guatemala | 39,460 | - | May 1 | - |
Results
Simulations
Low travel volumes | High travel volumes | ||||||||
---|---|---|---|---|---|---|---|---|---|
n
| Mean | SD | SE | 95% CrI | Mean | SD | SE | 95% CrI | |
R
0
= 1.2 | 4 | 1.21 | 0.02 | 0.01 | (1.18, 1.25) | 1.21 | 0.02 | 0.02 | (1.17, 1.25) |
8 | 1.21 | 0.01 | 0.01 | (1.18, 1.23) | 1.21 | 0.01 | 0.01 | (1.18, 1.23) | |
12 | 1.21 | 0.01 | 0.01 | (1.19, 1.22) | 1.20 | 0.01 | 0.01 | (1.18, 1.22) | |
16 | 1.21 | 0.01 | 0.01 | (1.19, 1.22) | 1.20 | 0.01 | 0.01 | (1.19, 1.22) | |
20 | 1.21 | 0.01 | 0.01 | (1.19, 1.22) | 1.20 | 0.01 | 0.01 | (1.19, 1.22) | |
R
0
= 1.7 | 4 | 1.72 | 0.05 | 0.05 | (1.64, 1.82) | 1.72 | 0.06 | 0.05 | (1.63, 1.85) |
8 | 1.71 | 0.04 | 0.03 | (1.65, 1.79) | 1.71 | 0.04 | 0.04 | (1.64, 1.80) | |
12 | 1.71 | 0.03 | 0.03 | (1.66, 1.76) | 1.71 | 0.03 | 0.03 | (1.65, 1.78) | |
16 | 1.71 | 0.02 | 0.02 | (1.66, 1.75) | 1.71 | 0.03 | 0.03 | (1.66, 1.77) | |
20 | 1.70 | 0.02 | 0.02 | (1.67, 1.74) | 1.70 | 0.02 | 0.02 | (1.66, 1.75) | |
R
0
= 2.2 | 4 | 2.22 | 0.09 | 0.08 | (2.08, 2.43) | 2.23 | 0.11 | 0.09 | (2.04, 2.50) |
8 | 2.21 | 0.06 | 0.06 | (2.11, 2.35) | 2.21 | 0.07 | 0.07 | (2.08, 2.38) | |
12 | 2.21 | 0.05 | 0.05 | (2.11, 2.32) | 2.21 | 0.05 | 0.05 | (2.12, 2.32) | |
16 | 2.21 | 0.04 | 0.04 | (2.13, 2.29) | 2.21 | 0.05 | 0.05 | (2.12, 2.31) | |
20 | 2.20 | 0.04 | 0.04 | (2.14, 2.28) | 2.21 | 0.04 | 0.04 | (2.13, 2.31) |
r = 0.4% |
r = 0.8% | ||||||||
---|---|---|---|---|---|---|---|---|---|
n
| Mean | SD | SE | 95% CrI | Mean | SD | SE | 95% CrI | |
R
0
= 1.2 | 4 | 1.20 | 0.04 | 0.02 | (1.16, 1.30) | 1.20 | 0.04 | 0.03 | (1.14, 1.28) |
8 | 1.20 | 0.02 | 0.02 | (1.16, 1.24) | 1.20 | 0.02 | 0.02 | (1.14, 1.25) | |
12 | 1.19 | 0.01 | 0.01 | (1.17, 1.23) | 1.18 | 0.02 | 0.01 | (1.15, 1.22) | |
16 | 1.19 | 0.01 | 0.01 | (1.17, 1.22) | 1.18 | 0.01 | 0.01 | (1.15, 1.22) | |
20 | 1.19 | 0.01 | 0.01 | (1.17, 1.21) | 1.18 | 0.01 | 0.01 | (1.16, 1.21) | |
R
0
= 1.7 | 4 | 1.74 | 0.09 | 0.07 | (1.61, 1.95) | 1.75 | 0.1 | 0.07 | (1.62, 1.97) |
8 | 1.72 | 0.06 | 0.05 | (1.62, 1.83) | 1.73 | 0.06 | 0.05 | (1.63, 1.88) | |
12 | 1.72 | 0.05 | 0.04 | (1.64, 1.82) | 1.73 | 0.05 | 0.04 | (1.65, 1.83) | |
16 | 1.71 | 0.04 | 0.04 | (1.64, 1.79) | 1.72 | 0.04 | 0.04 | (1.66, 1.81) | |
20 | 1.71 | 0.04 | 0.03 | (1.66, 1.79) | 1.72 | 0.03 | 0.03 | (1.66, 1.79) | |
R
0
= 2.2 | 4 | 2.27 | 0.14 | 0.13 | (2.03, 2.60) | 2.29 | 0.16 | 0.13 | (2.04, 2.65) |
8 | 2.25 | 0.11 | 0.09 | (2.08, 2.49) | 2.27 | 0.1 | 0.09 | (2.10, 2.48) | |
12 | 2.23 | 0.08 | 0.07 | (2.09, 2.42) | 2.25 | 0.08 | 0.07 | (2.13, 2.42) | |
16 | 2.23 | 0.07 | 0.06 | (2.11, 2.37) | 2.26 | 0.07 | 0.06 | (2.13, 2.41) | |
20 | 2.23 | 0.06 | 0.06 | (2.12, 2.36) | 2.26 | 0.06 | 0.06 | (2.15, 2.39) |
Numerical study
1/γ = 3 days | 1/γ = 2 days | 1/γ = 4 days | ||||||
---|---|---|---|---|---|---|---|---|
Settings |
\( {\widehat{R}}_0 \)
| 95% CI |
\( {\widehat{R}}_0 \)
| 95% CI |
\( {\widehat{R}}_0 \)
| 95% CI | ||
Uniformly travels |
d = 3 days |
θ = 30% | 1.69 | (1.65, 1.73) | 1.45 | (1.42, 1.48) | 1.94 | (1.88, 1.99) |
θ = 10% | 1.67 | (1.63, 1.70) | 1.44 | (1.41, 1.46) | 1.91 | (1.86, 1.95) | ||
θ = 50% | 1.73 | (1.69, 1.78) | 1.48 | (1.44, 1.51) | 1.99 | (1.93, 2.06) | ||
d = 5 days |
θ = 30% | 1.66 | (1.62, 1.70) | 1.43 | (1.40, 1.45) | 1.90 | (1.84, 1.95) | |
θ = 10% | 1.64 | (1.60, 1.67) | 1.41 | (1.39, 1.44) | 1.86 | (1.81, 1.91) | ||
θ = 50% | 1.70 | (1.65, 1.74) | 1.45 | (1.42, 1.48) | 1.95 | (1.88, 2.01) | ||
Decreasing travels |
d = 3 days |
θ = 30% | 1.70 | (1.66, 1.74) | 1.46 | (1.43, 1.48) | 1.95 | (1.89, 2.00) |
θ = 10% | 1.68 | (1.64, 1.71) | 1.44 | (1.41, 1.46) | 1.92 | (1.87, 1.96) | ||
θ = 50% | 1.74 | (1.69, 1.79) | 1.48 | (1.45, 1.51) | 2.00 | (1.93, 2.06) | ||
d = 5 days |
θ = 30% | 1.68 | (1.63, 1.71) | 1.43 | (1.40, 1.46) | 1.90 | (1.85, 1.96) | |
θ = 10% | 1.64 | (1.60, 1.68) | 1.42 | (1.39, 1.44) | 1.87 | (1.82, 1.92) | ||
θ = 50% | 1.70 | (1.66, 1.75) | 1.46 | (1.43, 1.49) | 1.95 | (1.89, 2.01) |