From 2010 to 2019, 3194 people were infected with brucellosis. Of these, the highest incidence related to the year 2015 with 512 (16%), and the lowest incidence related to the year 2010 with 192 (6%). In matching the seasons with the Gregorian months, spring is related to March, April and May, summer is related to June, July and August, fall is related to September, October, and November and finally, winter is related to December, January and February. Among seasons, the summer with 961 (30.1%) and winter with 805 (25.2%) have the highest number of infected, respectively. The highest number of patients is related to temperatures 26℃ (6.6%), 25℃ (6.4%). The highest number of infected people is related to zero rainfall of 444 cases (13.9%) and the lowest number of patients is related to the average-monthly-rainfall of 17.8 (0.2%) and 64.5 (0.2%). The lowest number of patients is related to the average-wind-speed of 0.7 with 6 cases (0.2%).
Fitting MSM
The MSM was fitted with two and three states and both models were fitted with 0, -1, and -2 lags for climate variables. Temporal lag is defined as the time interval between climatic characteristics and the incidence of brucellosis. Based on a comparison between models, based on BIC, two-state MSM with a time lag of -1 is suitable. For this reason, only this model is offered to provide more results.
Age, month, rural-ratio, men-ratio, Non-pasteurized dairy, average-monthly-temperature, total-monthly-rainfall, average-wind-speed, maximum-monthly-temperature, Minimum-monthly-temperature, average-wind-speed, ratio of contact with livestock were recognized as significant variables (Table 1).
Autocorrelation and partial autocorrelation of residual and squared residual for model is confirmed lack of autocorrelation in the residual and the model seems to fit logically and there's no serial dependency on the residual.
Table 1
The fit of the two-state markov switching model with time lags of 0, -1, -1
Variables(state)
|
MSM with two states
|
Lag 0
|
Lag − 1
|
Lag − 2
|
B
|
SE
|
p-value
|
B
|
SE
|
p-value
|
B
|
SE
|
p-value
|
Intercept(1)
|
3.274
|
3.073
|
0.287
|
9.981
|
11.531
|
0.387
|
33.426
|
11.879
|
0.005
|
Intercept(2)
|
10.010
|
11.628
|
0.389
|
3.254
|
3.047
|
0.286
|
-2.162
|
14.711
|
0.883
|
Age(1)
|
-0.152
|
0.016
|
0.000
|
-0.116
|
0.118
|
0.326
|
-0.250
|
0.059
|
0.000
|
Age(2)
|
-0.116
|
0.118
|
0.327
|
-0.152
|
0.016
|
0.000
|
0.499
|
0.224
|
0.026
|
Month (1)
|
1.618
|
0.108
|
0.000
|
0.817
|
0.371
|
0.028
|
1.164
|
0.408
|
0.004
|
Month (2)
|
0.816
|
0.371
|
0.028
|
1.618
|
0.108
|
0.000
|
-0.045
|
0.585
|
0.939
|
Rural Ratio(1)
|
3.172
|
0.113
|
0.000
|
-0.287
|
0.389
|
0.461
|
2.666
|
0.513
|
0.000
|
Rural Ratio(2)
|
-0.287
|
0.389
|
0.461
|
3.173
|
0.113
|
0.000
|
0.164
|
0.365
|
0.653
|
Men Ratio(1)
|
2.920
|
0.079
|
0.000
|
-3.009
|
0.665
|
0.000
|
-3.013
|
0.608
|
0.000
|
Men Ratio (2)
|
-3.012
|
0.667
|
0.000
|
2.928
|
0.100
|
0.000
|
1.412
|
1.517
|
0.352
|
Non-pasteurized dairy (1)
|
-0.570
|
0.070
|
0.000
|
0.569
|
0.229
|
0.013
|
1.056
|
0.296
|
0.000
|
Non-pasteurized dairy (2)
|
0.570
|
0.229
|
0.0127
|
-0.571
|
0.069
|
0.000
|
-0.273
|
0.230
|
0.235
|
Average monthly temperature (1)
|
-4.223
|
0.135
|
0.000
|
0.241
|
0.820
|
0.769
|
0.935
|
0.859
|
0.276
|
Average monthly temperature (2)
|
0.239
|
0.852
|
0.779
|
-4.229
|
0.142
|
0.000
|
-2.546
|
1.017
|
0.012
|
Total monthly rainfall (1)
|
-0.038
|
0.015
|
0.012
|
-0.034
|
0.041
|
0.405
|
-0.242
|
0.048
|
0.000
|
Total monthly rainfall (2)
|
-0.034
|
0.041
|
0.405
|
-0.038
|
0.015
|
0.011
|
0.009
|
0.053
|
0.850
|
Average wind speed (1)
|
1.509
|
0.214
|
0.000
|
19.024
|
2.848
|
0.000
|
18.097
|
3.285
|
0.000
|
Average wind speed (2)
|
19.017
|
2.869
|
0.000
|
1.533
|
0.185
|
0.000
|
5.163
|
3.336
|
0.122
|
Maximum monthly temperature (1)
|
1.717
|
0.119
|
0.000
|
-0.177
|
0.567
|
0.754
|
-1.377
|
0.597
|
0.021
|
Maximum monthly temperature (2)
|
-0.176
|
0.585
|
0.763
|
1.718
|
0.118
|
0.000
|
0.722
|
0.629
|
0.251
|
Minimum monthly temperature (1)
|
3.203
|
0.096
|
0.000
|
0.231
|
0.408
|
0.572
|
0.345
|
0.403
|
0.393
|
Minimum monthly temperature (2)
|
0.231
|
0.417
|
0.579
|
3.207
|
0.101
|
0.000
|
2.289
|
0.599
|
0.000
|
Wind speed (1)
|
0.944
|
0.072
|
0.000
|
-0.698
|
0.301
|
0.020
|
-0.433
|
0.250
|
0.083
|
Wind speed (2)
|
-0.699
|
0.302
|
0.020
|
0.944
|
0.072
|
0.000
|
0.469
|
0.288
|
0.103
|
Ratio of contact with livestock (1)
|
-0.387
|
0.059
|
0.000
|
0.904
|
0.280
|
0.001
|
0.283
|
0.221
|
0.200
|
Ratio of contact with livestock (2)
|
0.904
|
0.280
|
0.001
|
-0.387
|
0.059
|
0.000
|
1.212
|
0.561
|
0.031
|
Figure 2 shows the smoothed and filtered probabilities for state of one and two. Smoothed probabilities are used to determine peaks and depressions and 0.5 is determined as the cut-off value for 1 and 2 states. The filtered probabilities are calculated using the first observation up to t and the smoothed probabilities are calculated using the total observations.
The Q-Q plot shows where the normality hypothesis is questionable for series. Transition probability matrix in MSM as follows:
Where the probability of non-outbreak state in both t and t+1 periods is 0.72, the probability of changing series from non-outbreak state in period t to outbreak state in period t+1 is 0.28 and the probability of the series changing from the outbreak state in period t to non-outbreak state in period t+1 is 0.65. When we are in a non-outbreak state (0.72), the process tends to stay the same state and the process is transferred to the outbreak state with a probability of 0.28.
The probability of an outbreak in t+1 is as follows:
Since the data is up to the second month of 2019, the probability of an outbreak for the third month of 2019 (one month later) is very low and is equal to 0.30%.
The biggest difference between the coefficients of the variables in two states is related to the average-wind-speed. Therefore, the average-wind-speed is the most important factor in incidence brucellosis.
Month, average-wind-speed and minimum-temperature coefficients are positive which indicate a positive effect on the number of brucellosis. The age and total-monthly-rainfall coefficients are negative, indicating a negative effect on the number of brucellosis.
The temporal changes of the observed cases of brucellosis and the values estimated by the MSM are illustrated in Fig 3. The model has a relatively good performance in identifying peaks incidence of brucellosis.
Where the probability of non-outbreak state in both t and t+1 periods is 0.72, the probability of changing series from non-outbreak state in period t to outbreak state in period t+1 is 0.28 and the probability of the series changing from the outbreak state in period t to non-outbreak state in period t+1 is 0.65. When we are in a non-outbreak state (0.72), the process tends to stay the same state and the process is transferred to the outbreak state with a probability of 0.28.
The probability of an outbreak in t+1 is as follows:
Since the data is up to the second month of 2019, the probability of an outbreak for the third month of 2019 (one month later) is very low and is equal to 0.30%.
The biggest difference between the coefficients of the variables in two states is related to the average-wind-speed. Therefore, the average-wind-speed is the most important factor in incidence brucellosis.
Month, average-wind-speed and minimum-temperature coefficients are positive which indicate a positive effect on the number of brucellosis. The age and total-monthly-rainfall coefficients are negative, indicating a negative effect on the number of brucellosis.
The temporal changes of the observed cases of brucellosis and the values estimated by the MSM are illustrated in Fig 3. The model has a relatively good performance in identifying peaks incidence of brucellosis.