3.1. Dataset
The panel dataset covered the period from January 2012 to May 2019 for 17 counties and islets, including surveillance data on dengue case, entomological indices, and data on meteorological and population density. Below is a brief description of the dataset:
Dengue case surveillance data: In Taiwan, dengue fever is classified as a notifiable infectious disease and suspected cases must be reported to a clinic for diagnosis within 24 hours. Probable cases are defined as patients with body temperature above 38 °C and with least two of the following dengue-related clinical symptoms: rash, retro-orbital pain, leukopenia, myalgia, arthralgia, and hemorrhagic manifestations. In Taiwan, the dengue case surveillance system includes active surveillance (e.g., fever infrared thermal screening at airports) and passive surveillance (e.g., hospital-based reporting systems) for the comprehensive and effective surveillance of dengue infection. Epidemiological surveys of confirmed cases are conducted by local primary health centers. Suspected cases are confirmed by detecting dengue virus and differentiating virus serotypes using laboratory diagnosis. Nucleic acid identification of the dengue virus is identified by reverse-transcriptase (RT) polymerase chain reaction (PCM) (one-step real-time RT-PCR), serological testing on single or paired serum samples by dengue-specific envelope and membrane specific immunoglobulin M (IgM) and IgG antibody-capture enzyme-linked immunosorbent assay (with the exclusion of Japanese encephalitis virus infection), or virus isolation [
45]. In this study, we used weekly confirmed dengue fever cases, which included indigenous cases and imported cases, obtained from the web-based National Infectious Disease Statistics System [
29] under the Notifiable Disease Surveillance System (NDSS) of the Taiwan Center for Disease Control.
Entomological surveillance data: Since the
Stegomyia indices are considered one of the most important measurements for the monitoring of dengue vector populations [
17], we used the Breteau index (BI), which is the number of containers positive for
Ae. aegypti larvae and pupae per 100 houses, as the dependent variable in our model. Surveillance of dengue-fever-carrying mosquito populations has been set up since the dengue outbreak in the southern counties of Taiwan in 1988. For all counties and cities, vector surveillance activities including mosquito species distinction, mosquito habitat recognition, and vector sampling are conducted by trained staff following the guidelines recommended by the World Health Organization [
1]. Each community is considered a surveying unit (e.g., one unit in Kaohsiung County including 50–100 households was randomly selected for inspection [
46]). Based on the household density and the average number of households positive for
Ae. aegypti (from historical entomological data), the risk level of each surveying unit is determined. Furthermore, larval, pupae, and adult vector surveys for monitoring vector density, distribution, and breeding habitats are conducted for both indoor and outdoor areas. Depending on whether the surveying unit is classified as a high-, medium-, or low-risk area, the according inspection frequency will be weekly, monthly, or bi-monthly, respectively. The relevant surveillance data were retrieved from Taiwan National Infectious Disease Statistics System, Centers for Disease Control [
47]. As the inspection date and time varied across cities and townships in Taiwan, the data extracted included a huge combination of daily average BI values. We reorganized the county-level BI dataset and converted daily data into weekly data to ensure the consistency of start-day and end-day of each week in the dengue surveillance data.
Meteorological data: The meteorological variables included daily mean temperature, daily mean accumulative rainfall, and daily mean relative humidity. The data were systematically retrieved from the Central Weather Bureau (CWB), Taiwan [
48]. We calculated the average values of available weather data from different stations in each county. Daily weather data were then aggregated to weekly data.
Others: Other explanatory variables included population density data, regional total population, and total area data, which were retrieved from the Department of House Registration, Ministry of Interior (MOI), Taiwan.
Descriptive statistics of these variables are shown in
Table 1.
3.2. Estimating Temperature Thresholds
Temperature is widely considered the most important climatic factor for dengue incidence prediction, as it plays a more critical role in dengue transmission than other meteorological factors [
13,
39,
49]. Therefore, we focused on examining the non-linear relationship between dengue vector index and temperature in this study. To search for two or more regimes endogenously, Hansen’s [
25] threshold model was employed to test whether or not there exist threshold effects between BI and temperature.
Following Hansen (1999), the structure of the single-panel threshold model used was as follows:
where the data are from a balanced panel;
i and
t denote indices of the individual
and time
, respectively;
yit and the threshold variable
qit are scalars;
xit is a
k vector of explanatory variables;
is an indicator function;
is the fixed effect (or heterogeneity of individuals); and the error term
is assumed to be independent and identically distributed,
. Equation (1) can be rewritten as follows.
where
.
The data were separated into two regimes, whereby the threshold variable was less than or greater than the threshold value . The two regimes had different regression slopes and , respectively.
Averaging Equation (2) over time led to
where
.
Subtracting Equation (3) from (2) led to
or, in vector form
We stacked the data over individuals into
Y*,
X*, and
e*, and then derived Equation (5) to estimate threshold effects.
Ordinary least-squares (OLS) method was used to estimate
for a given
.
The vector of regression residuals was
which was minimized for SSE to estimate
:
where
The estimated slope coefficient was
, the vector of residuals was
, and the estimated variance of the residuals was
Supposing a single threshold effect was found between temperature and entomological surveillance index BI, the empirical panel threshold model was as follows.
For a balanced panel, and denote province and time (week), is entomological surveillance (Breteau) index, is average temperature (), is average precipitation (mm), Humidit is relative humidity (%), τ represents the time lag, and is the error term. Since there is about 2 weeks from laying to hatching of eggs, and eggs will hatch into larvae within 24 to 48 hours, a 2 week lag (τ = 2) was chosen to estimate the effects of meteorological factors on mosquito larval index BI.
Before estimating Equation (11), we applied panel unit root tests to examine whether the variables were stationary or not, and the results indicated they were stationary (please refer to
Appendix A). Equation (11) was estimated to see whether there were one, two, or three thresholds.
Table 2 displays the results of the threshold effect tests, including the test statistics F
1, F
2, F
3 and their corresponding bootstrap
p-values.
According to the results of the nationwide model, a single threshold effect F
1 was statistically significant at about the 5% level (
p-value = 0.06), while the tests for a double F
2 and a triple threshold F
3 were not significant. This indicated that a single temperature threshold exists at the nationwide level. Based on the results of the Southern model, a single threshold F
1 was statistically significant at the 1% level and a double threshold F
2 was also statistically significant at the 1% level, while the test result of triple thresholds F
3 was not significant. We concluded that there are two temperature thresholds in the southern region of Taiwan. The estimates of the temperature thresholds are reported in
Table 3.
Table 3 shows that the estimated single threshold was 27.21 °C at national level. For the double threshold effects of temperature in the Southern region, the estimated values were 27.27 and 30.17 °C. The likelihood ratio (LR) statistics are plotted in
Figure 1 to display the confidence interval construction for the nationwide model (a) and the southern region model (b).
3.3. Estimating Threshold Effects of Meteorological Factors on Breteau Index
After estimation of the temperature thresholds of the BI, this study went one step further to examine the threshold effects of weather factors on the dengue vector index. In this stage, count data regression models were employed because the dependent variable (
BIit) was a non-negative integer random variable. Most previous studies have applied Poisson regression models to estimate the relationship between ecological factors and dengue. However, the Poisson model has a strong restriction in that the variance and mean are equal, an assumption which is often violated in real datasets. When the conditional variance exceeds the conditional mean, the count data are over-dispersed. As a consequence, hypotheses on the Poisson regression parameters may be rejected more often than they should be [
50,
51], since the estimation also includes underestimated standard errors of parameter estimates. To resolve this issue, we first described dengue cases by the corresponding BI, so the means of dengue cases within each level of BI were lower than the variances within each level. In other words, the conditional means were lower than the conditional variances. We then fit two regression models specifically developed for count outcomes, Poisson and negative binomial (NB), and then compared these two models using the likelihood ratio (LR) test.
The LR test performs a test of the null hypothesis that the parameter vector of a statistical model satisfies some smooth constraints. To conduct the test, both the unrestricted and the restricted models must be fit using the maximum likelihood. Let L
0 and L
1 be the log-likelihood values associated with the full (NB) and constrained (Poisson) models, respectively. The test statistic of the likelihood ratio test is LR = −2(L
1 -L
0). If the constrained model is true, LR is approximately χ
2 distributed with
d0 – d1 degrees of freedom, where
d0 and
d1 are the degrees of freedom associated with the full and constrained models, respectively [
52]. The LR test statistic is approximately distributed as chi-squared, and was separately established for the nationwide model and the southern region model.
As can be seen in
Table 4, the results of the test statistic allowed us to reject the constrained model hypothesis for both the nationwide model and the southern region model.
Table 4 also includes the model selection indices including Akaike’s information criterion (AIC) and the Bayesian information criterion (BIC). The results indicated that the NB regression model was favored over the Poisson for estimation in both the nationwide and southern region models.
To estimate the threshold effects of weather factors on BI dengue vector index using panel NB regression models, we decomposed data into separate sets according to the single temperature threshold in the nationwide model and double temperature thresholds in the southern region model. The effects of meteorological factors on BI were estimated as follows.
For the nationwide model:
For the southern region model:
The NB parameter was assumed to follow a Gamma distribution. The NB model was considered a generalization of Poisson model, since it had the same mean structure as Poisson regression and it had an extra parameter to model the over-dispersion (i.e., error term
allowed the conditional variance of y to exceed the conditional mean). We also computed marginal effects by multiplying the estimated coefficients with the exponential of expected value of the dependent variable BI. The estimation results are displayed in
Table 5.
At the national level, when the weekly average temperature was less than 27.21 °C, a 1 °C increase in average temperature caused the expected value of BI to increase by 0.09 unit. When the weekly average temperature was higher than 27.21°C, a 1 °C increase in temperature led to a 0.26 unit increase in the expected value of BI. For the southern regions of Taiwan, the impacts of all weather factors on dengue vector index were stronger than those at the national level. Specifically, when the weekly average temperature was less than 27.27 °C, a 1 °C increase in temperature led to a 0.29 unit increase in the expected value of BI. When the weekly average temperature was between 27.27 and 30.17 °C, the expected value of BI increased by 0.63 units for every additional degree Celsius increase. When the weekly average temperature was above 30.17 °C, a 1 °C increase in temperature caused the expected value of BI to increase by 1.49 units.
The empirical results also indicated that the impacts of precipitation and relative humidity on dengue vector index vary under different regimes of weekly average temperature. At the national level, relative humidity had a negative effect on BI when weekly average temperature was less than 27.21 °C. However, this effect became positive when weekly average temperature exceeded 27.21 °C. For the southern region of Taiwan, the effects of relative humidity on the expected value of BI became stronger when temperature increased. There were significant positive effects of precipitation on the expected value of BI when weekly average temperature was below 30.2 °C. However, the effects of precipitation on the expected value of BI became insignificant when the weekly average temperature rose above 30.2 °C.