Introduction
Background
Structure of the paper
Methods
Data
Model
Traveller induced model
Infection rate induced model
Country | Population | Travellers | Transmission | |||
---|---|---|---|---|---|---|
June | July | August | \(\beta _{j,1}\) | \(\beta _{j,2}\) | ||
Decimal power/unit | 10\(^6\) | 1 | 1 | 1 | 10\(^{-1} d^{-1}\) | 10\(^{-1}d^{-1}\) |
Albania | 2.88 | 945 | 3366 | 9505 | 1.20 | 1.19 |
Austria | 8.90 | 312,364 | 636,414 | 782,818 | 1.35 | 1.36 |
Belarus | 9.45 | 1595 | 1985 | 3102 | 0.38 | 0.65 |
Belgium | 11.51 | 36,210 | 155,295 | 92,495 | 1.17 | 1.60 |
Bosnia and Herzegovina | 3.30 | 2811 | 2849 | 6702 | 1.54 | 1.10 |
Brazil | 211.05 | 6715 | 4366 | 3778 | 1.22 | 1.12 |
Bulgaria | 6.95 | 11,562 | 42,552 | 74,363 | 1.24 | 1.06 |
Canada | 37.41 | 4746 | 9778 | 8368 | 0.31 | 1.33 |
China | 1433.78 | 3711 | 5921 | 7077 | 1.55 | 1.05 |
Croatia | 4.06 | 66,029 | 84,952 | 150,790 | 0.31 | 1.10 |
Cyprus | 0.89 | 360 | 7191 | 14,049 | 0.42 | 1.89 |
Czech Republic | 10.69 | 51,518 | 130,651 | 148,353 | 0.84 | 1.41 |
Denmark | 5.81 | 48,986 | 395,924 | 571,649 | 0.77 | 1.70 |
Egypt | 100.39 | 2542 | 5134 | 7790 | 0.65 | 0.36 |
Estonia | 1.33 | 1006 | 3380 | 5967 | 0.23 | 2.00 |
Ethiopia | 112.08 | 1431 | 2089 | 2066 | 1.84 | 1.45 |
Finland | 5.53 | 4624 | 12,134 | 19,074 | 0.31 | 1.74 |
France | 67.20 | 105,905 | 326,298 | 345,913 | 0.95 | 1.96 |
Greece | 10.7 | 15,930 | 179,531 | 372,892 | 1.41 | 1.75 |
Hungary | 9.77 | 30,154 | 53,080 | 71,577 | 0.36 | 1.71 |
Iceland | 0.34 | 889 | 7892 | 13,718 | 1.66 | 1.56 |
Ireland | 4.97 | 4892 | 8965 | 9065 | 0.43 | 2.20 |
India | 1366.42 | 5168 | 8676 | 14,046 | 1.39 | 1.25 |
Israel | 88.52 | 2455 | 2693 | 797 | 1.65 | 1.23 |
Italy | 60.29 | 126,855 | 272,324 | 415,581 | 0.21 | 1.75 |
Japan | 126.86 | 1457 | 2340 | 3292 | 1.78 | 1.29 |
Kosovo | 1.72 | 586 | 7341 | 18,626 | 1.59 | 1.10 |
Latvia | 1.91 | 5936 | 12,637 | 20,798 | 0.31 | 1.52 |
Lebanon | 6.87 | 167 | 1699 | 5298 | 1.09 | 1.94 |
Lithuania | 2.79 | 1203 | 1787 | 2415 | 0.49 | 1.89 |
Luxembourg | 0.63 | 4562 | 4466 | 2946 | 2.54 | 0.51 |
Malta | 0.51 | 261 | 9338 | 16,974 | 1.16 | 1.95 |
Mexico | 127.58 | 2079 | 2726 | 2253 | 1.70 | 0.69 |
Montenegro | 0.63 | 728 | 2490 | 4118 | 3.75 | 0.54 |
Moldova | 4.04 | 972 | 1728 | 3815 | 0.85 | 1.30 |
Netherlands | 17.40 | 188,840 | 721,721 | 1,592,831 | 1.04 | 1.72 |
Northern Macedonia | 2.08 | 0 | 3486 | 9875 | 0.89 | 1.02 |
Norway | 5.37 | 8326 | 42,589 | 64,125 | 0.74 | 1.70 |
Poland | 27.94 | 95,372 | 171,127 | 268,559 | 0.52 | 1.51 |
Portugal | 10.29 | 17,659 | 63,369 | 111,867 | 0.51 | 0.81 |
Qatar | 2.83 | 6063 | 8336 | 6747 | 0.79 | 1.00 |
Romania | 19.36 | 5702 | 32,822 | 41,255 | 1.18 | 1.35 |
Russia | 145.87 | 3550 | 6017 | 7324 | 0.61 | 1.03 |
Serbia | 8.77 | 5164 | 5577 | 9672 | 1.62 | 0.61 |
Slovakia | 5.46 | 19,372 | 31,161 | 56,401 | 1.16 | 1.54 |
Slovenia | 2.07 | 3759 | 5361 | 5987 | 1.44 | 1.14 |
Spain | 47.32 | 22,209 | 331,894 | 436,624 | 1.19 | 1.86 |
Sweden | 10.32 | 9050 | 39,584 | 46,878 | 0.38 | 0.55 |
Switzerland | 8.50 | 102,698 | 272,121 | 388,971 | 1.49 | 1.35 |
Tunisia | 11.69 | 644 | 2709 | 11,292 | 1.09 | 2.12 |
Turkey | 83.43 | 36,986 | 144,350 | 343,972 | 0.77 | 1.14 |
United Kingdom | 66.43 | 17,026 | 29,925 | 32,969 | 0.92 | 1.16 |
Ukraine | 43.99 | 3020 | 8934 | 14,759 | 0.77 | 1.44 |
United States of America | 329.06 | 24,123 | 42,409 | 41,613 | 0.81 | 1.62 |
United Arab Emirates | 9.77 | 3231 | 9394 | 6856 | 0.59 | 1.10 |
Travellers and travel impact rate
Models, parameter bounds and initial values
\(\beta _{0/1}\) | \(\delta\) | \(\mu\) | \(\tau\) | \(\alpha _j\) | \(N_0\) | \(E_0\) | \(I_0\) | \(R_0\) | \(D_0\) |
---|---|---|---|---|---|---|---|---|---|
\(> 0.05\) | \(0.05{-}1\) | \(\le 0.1\) | \(3{-}40\) | \(>0\) | 82,846,340 | \(> 0\) | 9,\(407/\delta\) | 165,\(632/\delta\) | 8555 |
Likelihood function
Model specification
Metropolis algorithm
Param. | \(\beta _{0/1}\) | \(\delta\) | \(\mu\) | \(\tau\) | \(\alpha _j\) | \(E_0\) |
---|---|---|---|---|---|---|
Init. val. | 0.1 | 0.3 | 0.005 | 20 | 1 | \(1/2\cdot I_0\) |
Confidence intervals of the parameters
Current reproductive number
Sensitivity analysis
Time-dependent measures
Time-independent measures
Numerical results
Model A
Parameter | Mean value | \(\sigma\) of Metropolis |
---|---|---|
\(\beta _0\) | \(8.65 \times 10^{-2}\)d\(^{-1}\) | \(0.19\times 10^{-2}\)d\(^{-1}\) |
\(\beta _1\) | \(1.39\times 10^{-1}\)d\(^{-1}\) | \(0.01 \times 10^{-1}\)d\(^{-1}\) |
\(\delta\) | \(1.84\times 10^{-1}\) | \(0.03 \times 10^{-1}\) |
\(\mu\) | \(1.56\times 10^{-3}\) | \(0.06\times 10^{-3}\) |
\(\tau\) | \(2.60 \times 10^1\)d | \(0.04\times 10^1\)d |
\(E_0\) | \(3.60 \times 10^3\) | \(0.05\times 10^3\) |
Model B
Parameter | Mean value | \(\sigma\) of Metropolis | \(\psi\) of PACI | \(\psi\) of SACI |
---|---|---|---|---|
\(\beta\) | \(3.59 \times 10^{-2}\)d\(^{-1}\) | \(0.03\times 10^{-2}\)d\(^{-1}\) | \(0.002\times 10^{-2}\)d\(^{-1}\) | \(0.04\times 10^{-2}\)d\(^{-1}\) |
\(\delta\) | \(5.78\times 10^{-1}\) | \(0.18 \times 10^{-1}\) | \(0.002 \times 10^{-1}\) | \(0.03 \times 10^{-1}\) |
\(\mu\) | \(6.18\times 10^{-3}\) | \(0.19\times 10^{-3}\) | \(0.004\times 10^{-3}\) | \(0.09\times 10^{-3}\) |
\(\tau\) | \(2.59 \times 10^1\)d | \(0.04\times 10^1\)d | \(0.001\times 10^1\)d | \(0.03\times 10^1\)d |
\(E_0\) | \(2.59 \times 10^3\) | \(0.05\times 10^3\) | \(0.001\times 10^3\) | \(0.03\times 10^3\) |
\(\alpha\) | 2.97 | 0.06 | 0.002 | 0.04 |
Model C
Parameter | Mean value | \(\sigma\) of Metropolis | \(\psi\) of PACI | \(\psi\) of SACI |
---|---|---|---|---|
\(\beta\) | \(5.09 \times 10^{-2}\)d\(^{-1}\) | \(0.12\times 10^{-2}\)d\(^{-1}\) | \(0.002\times 10^{-2}\)d\(^{-1}\) | \(0.06\times 10^{-2}\)d\(^{-1}\) |
\(\delta\) | \(4.94 \times 10^{-1}\) | \(0.08 \times 10^{-1}\) | \(0.007 \times 10^{-1}\) | \(0.19\times 10^{-1}\) |
\(\mu\) | \(5.34 \times 10^{-3}\) | \(0.10\times 10^{-3}\) | \(0.004\times 10^{-3}\) | \(0.11\times 10^{-3}\) |
\(\tau\) | \(2.58 \times 10^{1}\)d | \(0.06 \times 10^{1}\) d | \(0.01 \times 10^{1}\) d | \(0.34 \times 10^{1}\) d |
\(E_0\) | \(2.45 \times 10^3\) | \(0.03\times 10^3\) | \(0.003\times 10^3\) | \(0.07\times 10^3\) |
\(\alpha _0\) | 2.23 | 0.05 | 0.002 | 0.06 |
\(\alpha _1\) | 2.45 | 0.05 | 0.008 | 0.02 |
\(\alpha _2\) | 3.14 | 0.05 | 0.004 | 0.11 |
Comparison
J(u) | # of Parameters | BIC | |
---|---|---|---|
Model A | \(4.4174 \times 10^{-5}\) | 6 | − 9087.0 |
Model B | \(4.1812 \times 10^{-1}\) | 6 | − 8538.6 |
Model C | \(4.1919 \times 10^{-1}\) | 8 | − 8529.9 |