3.1. Epidemic characteristics
3.1.1. Time distribution
The total number of reported cases of varicella in Dalian from 2009 to 2019 was 37223. The incidence rate decreased and then increased on the whole. The lowest incidence rate was occurred in 2014(see Table1,Figure1). Varicella occurred throughout the year. The number of cases was increasing from February to May and August to December. From April to June, and November to January of the following year, there were more cases(see Figure2). Varicella cases showed obvious seasonality.
Table 1
The cases and incidence rate of varicella in Dalian from 2009 to 2019
Year
|
Cases
|
Incidence rate(1/100000)
|
2009
|
3502
|
57.96
|
2010
|
3371
|
55.78
|
2011
|
3709
|
55.44
|
2012
|
2922
|
43.25
|
2013
|
2684
|
39.93
|
2014
|
2455
|
36.05
|
2015
|
3139
|
46.09
|
2016
|
2897
|
42.33
|
2017
|
3895
|
55.75
|
2018
|
4296
|
61.65
|
2019
|
4353
|
62.68
|
3.1.2. Population distribution
3.1.2.1. Age distribution
Varicella occurred any at age. The cases aged from 0 to 25 accounted for 92.46% of the total cases (see Table2). More cases aged from 5 to 10 (see Figure3).
Table 2
The varicella cases age distribution and proportion in Dalian from 2009 to 2019
Age group
|
Incidence
number
|
Composition
Ratio (%)
|
Age
group
|
Incidence
number
|
Composition
Ratio (%)
|
0-
|
1144
|
3.07
|
10-
|
5953
|
15.99
|
1-
|
769
|
2.07
|
15-
|
5733
|
15.4
|
2-
|
639
|
1.72
|
20-
|
5144
|
13.82
|
3-
|
1092
|
2.93
|
25-
|
3165
|
8.5
|
4-
|
1601
|
4.3
|
30-
|
1788
|
4.8
|
5-
|
2264
|
6.08
|
35-
|
716
|
1.92
|
6-
|
1931
|
5.19
|
40-
|
165
|
0.44
|
7-
|
1679
|
4.51
|
45-
|
57
|
0.15
|
8-
|
1712
|
4.6
|
50-
|
22
|
0.06
|
9-
|
1596
|
4.29
|
55-
|
53
|
0.14
|
3.1.2.2. Gender distribution
The total cases of varicella in Dalian from 2009 to 2019 was 37,223, of which 20043 cases were males (53.85%) and 17,180 cases were females (46.15%). The number of males was higher than that of females, and the ratio of total male to female was 1.17:1. (Figure4)
3.2. GM(1,1) model
Using R software for GM(1,1) model prediction. The incidence for 2019 was 51.3/100000(Table3),the posterior difference ratio (C) was 0.987188,the probability of small error (P) was 0.3333.α=-0.0114 ,u=45.3757.Then the GM (1, 1) model was established as follows:
Table 3
GM(1,1) model prediction results with R software
Year
|
Actual value
(1/100000)
|
R software
prediction value(1/100000)
|
2009
|
57.96
|
57.96
|
2010
|
55.78
|
46.3
|
2011
|
55.44
|
46.83
|
2012
|
43.25
|
47.37
|
2013
|
39.93
|
47.91
|
2014
|
36.05
|
48.46
|
2015
|
46.09
|
49.01
|
2016
|
42.33
|
49.58
|
2017
|
55.75
|
50.15
|
2018
|
61.65
|
50.72
|
2019
|
62.68
|
51.3
|
3.3. Markov model
3.3.1. State division and state transition probability matrix::
Due to the amount of data was small, to ensure that each state had enough data, it was divided into 3 states, K-mean cluster was used to cluster data. The data distribution of each group could be determined. State division was: E1[36.05,41.13], E2[41.13,50.765], E3[50.765,61.65].
The state change from 2009 to 2018 is showed as E3-E3-E3-E2-E1-E1-E2-E2-E3-E3, state transition status and state transition probability matrix see table 4, Table 5.State transition matrix showed below:
Table 4
State transition from 2009 to 2018
State
|
E1
|
E2
|
E3
|
E1
|
1
|
1
|
0
|
E2
|
1
|
1
|
0
|
E3
|
0
|
1
|
3
|
Table 5
State transition matrix
State
|
E1
|
E2
|
E3
|
E1
|
0.5
|
0.5
|
0
|
E2
|
0.33
|
0.33
|
0.33
|
E3
|
0
|
0.25
|
0.75
|
3.3.2. Use the state transition probability matrix to predict:
With the help of MATLAB 2015a software, using the 2018 grouping matrix (0, 0, 1) and the state transition probability matrix to predict. The result of the 2019 grouping matrix was (0, 0.25, 0.75), the 2019 Markov model prediction was more likely in E3, so the value is: 56.2075
3.4. GM(1,1)-Markov model
3.4.1. State division and state transition probability matrix:
Relative value was used to divide the state, relative value=actual value/ R software prediction value. K-mean cluster was used to cluster data. The data distribution of each group could be determined. Divided all relative values into three states: underestimated, accurate, overestimated, namely E1[1.1,1.22], E2[0.82,1.1], E3[0.74,0.82].The status changed from 2009 to 2018 is E2-E1-E1-E2-E2- E3-E2-E2-E1-E1(Table6). State transition matrix see Table7
Table 6
State transition from 2009 to 2018
Year
|
Actual value
|
Predicted value
|
Relative value
|
State division
|
2009
|
57.96
|
57.96
|
1
|
E2
|
2010
|
55.78
|
46.3
|
1.2
|
E1
|
2011
|
55.44
|
46.83
|
1.18
|
E1
|
2012
|
43.25
|
47.37
|
0.91
|
E2
|
2013
|
39.93
|
47.91
|
0.83
|
E2
|
2014
|
36.05
|
48.46
|
0.74
|
E3
|
2015
|
46.09
|
49.01
|
0.94
|
E2
|
2016
|
42.33
|
49.58
|
0.85
|
E2
|
2017
|
55.75
|
50.15
|
1.11
|
E1
|
2018
|
61.65
|
50.72
|
1.22
|
E1
|
2019
|
62.68
|
51.3
|
|
|
Table 7
State transition matrix
State
|
E1
|
E2
|
E3
|
E1
|
0.67
|
0.33
|
0
|
E2
|
0.4
|
0.4
|
0.2
|
E3
|
0
|
1
|
0
|
3.4.2. Three-step transition probability matrix to prediction varicella in 2019:
Select 2018, 2017, and 2016, the three most recent years from 2019. After one step (P1), two steps (P2), and three steps (P3), the state was transferred to 2019. Sum the column items, the maximum value was the state range, which the 2019 GM(1,1) model predicted value located in, see Table8.
Table 8
Status of the predicted values of the GM(1,1)model in 2019
Year
|
Initial state
|
transition steps
|
E1
|
E2
|
E3
|
2018
|
E1
|
1 (P1)
|
0.67
|
0.33
|
0
|
2017
|
E1
|
2 (P1)
|
0.5809
|
0.3531
|
0.066
|
2016
|
E2
|
3 (P3)
|
0.4836
|
0.418
|
0.0984
|
sum
|
|
|
1.7345
|
1.1011
|
0.1644
|
The maximum sum value was in state E1, That was the GM(1,1) model prediction value will be in the E1 state, the prediction value of 51.3 was been underestimated, and the GM(1,1)-Markov prediction value was 51.3*(1.1+1.22)/2=59.508
3.5. Models results comparison
Three models results and comparison see Table9, Figure5 and Table10. GM(1,1)-Markov model fitted actual value better and had the lowest relative error of 2019.
Table 9
Models prediction results and actual value from 2009 to 2019
Year
|
Actual value
|
GM(1,1) value
|
Markov value
|
GM(1,1)-Markov value
|
2009
|
57.96
|
57.96
|
56.2075
|
55.6416
|
2010
|
55.78
|
46.3
|
56.2075
|
53.708
|
2011
|
55.44
|
46.83
|
56.2075
|
54.3228
|
2012
|
43.25
|
47.37
|
45.9475
|
45.4752
|
2013
|
39.93
|
47.91
|
38.59
|
45.9936
|
2014
|
36.05
|
48.46
|
38.59
|
38.0411
|
2015
|
46.09
|
49.01
|
45.9475
|
47.0496
|
2016
|
42.33
|
49.58
|
45.9475
|
47.5968
|
2017
|
55.75
|
50.15
|
56.2075
|
58.174
|
2018
|
61.65
|
50.72
|
56.2075
|
58.8352
|
2019
|
62.68
|
51.3
|
56.2075
|
59.508
|
Table 10
Models results and relative errors of 2019
Year
|
Actual value
|
GM(1,1)
|
Markov
|
GM(1,1)-Markov
|
2019
|
62.68
|
53.6425
|
56.2075
|
59.508
|
Relative error(%)
|
|
14.42
|
10.31
|
5.06
|