A novel nonparametric mixture model for the detection pattern of COVID-19 on Diamond Princess cruise

Figures & data

Table 1. Number of tests and number of individuals testing positive for passengers and crews on the Diamond Princess cruise ship, Yokohama, Japan, February 2020 (n = 3711).

Reported	Number of	Number of individuals	The cumulative number of
date	tests	testing positive	individuals testing positive
Feb 5	31	10	10
Feb 6	71	10	20
Feb 7	171	41	61
Feb 8	6	3	64
Feb 9	57	6	70
Feb 10	103	65	135
Feb 12	53	39	174
Feb 13	221	44	218
Feb 15	217	67	285
Feb 16	289	70	355
Feb 17	504	99	454
Feb 18	681	88	542
Feb 19	607	79	621
Feb 20	52	13	634

Data on February 11 and 14 are not available.

Figure 1. Comparison of the observed detection rates $\{x_i/y_i{\}}_{i=1}^{14}$ and estimated ones if the selection bias is ignored. $x_i$ represents number of patients that were tested positive at day i, and $y_i$ is the total number of tests at day i. Scatter points are the rate of $x_i/y_i$, where red and blue colours differentiate whether $y_i>100$ or not. Black line shows the estimated detection rates $\{G_i{\}}_{i=1}^{14}$.

Table 2. Simulation results.

Parameter	True	Bias	SD	SE	CP
$F_1$	0.208	$-0.023$	0.036	0.039	95.6
$F_2$	0.208	$-0.012$	0.025	0.025	92.6
$F_3$	0.208	0.001	0.025	0.025	95.6
$F_4$	0.208	0.011	0.034	0.039	98.8
$F_5$	0.208	0.024	0.040	0.043	95.4
$F_6$	0.631	$-0.005$	0.044	0.044	95.4
$F_7$	0.696	$-0.001$	0.053	0.052	93.4
$F_8$	0.696	0.018	0.070	0.064	92.4
$F_9$	0.850	$-0.024$	0.104	0.093	91.0
$F_{10}$	0.900	$-0.019$	0.096	0.093	94.8
$F_{11}$	0.950	$-0.025$	0.089	0.085	94.6
$F_{12}$	0.950	0.002	0.085	0.074	97.2
$F_{13}$	1.000	$-0.048$	0.085	0.074	92.2
$F_{14}$	1.000	$-0.002$	0.011	0.011	95.8
N	1043	13.55	59.62	53.59	96.6
${\theta }_1$	$-0.518$	$-0.007$	0.216	0.225	96.2
${\theta }_2$	$-0.232$	0.008	0.045	0.046	95.6
${\lambda }_8$	0.321	0.001	0.039	0.041	95.8
${\lambda }_9$	0.272	0.003	0.030	0.031	96.2
${\lambda }_{10}$	0.229	0.004	0.024	0.023	96.4
${\lambda }_{11}$	0.191	0.005	0.020	0.018	95.8
${\lambda }_{12}$	0.157	0.005	0.019	0.017	96.0
${\lambda }_{13}$	0.129	0.006	0.019	0.017	95.8
${\lambda }_{14}$	0.105	0.006	0.019	0.018	95.4

Table 3. MLEs and the corresponding confidence intervals in real data analysis.

Parameter	Estimate	Standard Error	Confidence Interval
$F_1$	0.208	0.037	(0.136, 0.281)
$F_2$	0.208	0.024	(0.162, 0.255)
$F_3$	0.208	0.024	(0.162, 0.255)
$F_4$	0.208	0.032	(0.145, 0.271)
$F_5$	0.208	0.035	(0.140, 0.276)
$F_6$	0.631	0.045	(0.543, 0.719)
$F_7$	0.696	0.052	(0.587, 0.790)
$F_8$	0.696	0.052	(0.587, 0.790)
$F_9$	1.000	0.090	(0.783, 1.000)
$F_{10}$	1.000	0.079	(0.803, 1.000)
$F_{11}$	1.000	0.063	(0.837, 1.000)
$F_{12}$	1.000	0.053	(0.855, 1.000)
$F_{13}$	1.000	0.053	(0.855, 1.000)
$F_{14}$	1.000	0.010	(0.981, 1.000)
N	1064	40.91	(984, 1144)
${\theta }_1$	−0.476	0.198	(−0.864, −0.089)
${\theta }_2$	−0.231	0.042	(−0.313, −0.149)
${\lambda }_8$	0.330	0.036	(0.259, 0.402)
${\lambda }_9$	0.281	0.027	(0.229, 0.333)
${\lambda }_{10}$	0.237	0.019	(0.200, 0.274)
${\lambda }_{11}$	0.198	0.014	(0.170, 0.226)
${\lambda }_{12}$	0.164	0.013	(0.138, 0.189)
${\lambda }_{13}$	0.134	0.014	(0.107, 0.161)
${\lambda }_{14}$	0.110	0.014	(0.081, 0.138)

Figure 2. Comparison of the observed detection rates $\{x_i/y_i{\}}_{i=1}^{14}$ and fitted ones based on the proposed nonparametric mixture model. $x_i$ represents number of patients that were tested positive at day i, and $y_i$ is the total number of tests at day i. Scatter points are the rates of $x_i/y_i$, where red and blue colours differentiate whether $y_i>100$ or not. Black line shows the fitted detection rates $\{{\lambda }_i{F}_i{\}}_{i=1}^{14}$.