3.1. Climate and Malaria Cases of Mopani and Vhembe
Focussing on the study period (1 January 1998 to December 2017), results show that the daily maximum (black line) and minimum (pink line) temperature of Mopani fluctuates between 20–40 °C and 4–24 °C respectively, except for one day in January 2011 which is slightly above 40 °C (
Figure 2a). Vhembe maximum and minimum temperature mainly fall between 20–40 °C and 3–24 °C respectively, except few days in January 2009, 2010 and 2011 which went above 40 °C for maximum and below 3 °C in June 2012 for minimum temperature (
Figure 2b). The daily average temperature of Mopani falls within 15–30 °C with some variations of this range. For instance, the average temperature as high as 33 °C is observed around January of 2004, 2007, and 2016, and as low as 13 °C around July of 2007, 2010, 2011, 2012, 2014 and 2015 (
Figure 3a). The daily average temperature of Vhembe fluctuates between 10 and 32 °C. Mopani rainfall is generally below 150 mm per day except in January of 2012 and 2013, which went up to roughly 420 mm and 400 mm per day respectively (
Figure 3b). The rainfall pattern of Vhembe has decreased with time (
Figure 3b). It was higher around 1999–2002 and lower from 2011–2017 except for some days in January 2013, which went as far as 300 mm/day from <100 mm/day on other days. New reported malaria cases over Mopani were normally below 100 per day but exceptionally high in 2017 (
Figure 3c). The maximum cases of about 367 were recorded on the 4th of May 2017 as the early days of the month maintains 100 cases above. Malaria cases in Vhembe were also found below 100 cases/day except in 2017 that went far above this limit in April–May (
Figure 3c) as the maximum cases hit 243 on the 26th of April 2017. The Mann–Kendall test can be employed to statistically assess if there is an upward or downward trend of average temperature and rain amount in the two districts over time. Using Mann–Kendall test, rainfall shows a statistically significant decreasing trends (
p-value < 2.22 × 10
−16) in both Mopani and Vhembe districts while daily temperature shows a significant decreasing trends (
p-value < 2.22 × 10
−16) in Mopani district and a non-significant decreasing trends (
p-value = 0.92016) in Vhembe district over the study period.
Comparing the two study regions, Vhembe (in most days) seems hotter during the summer (December, January and February) months and cooler during the winter (June, July and August) months than Mopani although not statistically significant (
p-value = 0.132). For instance, the black line (indicating Vhembe daily average temperature) is seen overlapping the green line (indicating Mopani daily average temperature) in most of the days (
Figure 3a). However, the summer of Mopani was hotter than that of Vhembe in 2003 (
p-value = 0.0033). The rainfall pattern shows that Vhembe generally experiences more rainfall than Mopani especially between 1998 and 2010 (
p-value = 2.766 × 10
−6) (
Figure 3b). However, more rainfall is observed in Mopani than Vhembe between 2010 and 2014 (
p-value = 3.614 × 10
−11). Although similar patterns of malaria cases are observed over the two regions, the cases are more noticeable over Vhembe than Mopani (
Figure 3c). One reason traceable to this could be that the climate variables of Vhembe are more conducive for malaria transmission than that of Mopani [
4]. Malaria cases in both regions are also higher throughout 2017 compared to previous years, but the cases are slightly higher in Mopani than Vhembe in May 2017. The total malaria cases over the study period in Mopani and Vhembe are about 28,811 and 55,037 respectively. Following the 2011 census [
33,
34], the incident rate per 100,000 people in Mopani is calculated to be approximately 2637.15, while that of Vhembe is 4250.87. The Wilcoxon rank sum test with continuity correction is applied to test if daily rain amount, as well as daily average temperature and simulated daily mosquito abundance, of Mopani and Vhembe districts, is significantly different. The daily rain amount of Mopani and Vhembe districts are not statistically significantly different (
p-value = 0.8803) over the study period. The daily average temperature of Mopani and Vhembe districts are not statistically significantly different (
p-value = 0.6754) over the study period. The simulated daily mosquito abundance of Vhembe district is statistically higher than that of Mopani district (
p-value = 0.0002) over the study period.
Findings from the zero-inflated negative binomial regression model show that Mopani and Vhembe malaria incidence data are over-dispersed (
Figure 4). This is because Mopani malaria count data has its variance (206.0995) greater than mean (4.0464). Similarly, Vhembe malaria count data has variance (201.0317) greater than mean (7.5342). Moreover, zero over-inflation of the malaria counts in both locations is evident in the figure as the number of days with no malaria count exceed the number of days with positive malaria count in each of the districts.
3.2. Analysis over Mopani District Municipality
A stepwise model selection procedure based on Akaike information criterion (AIC) was applied to drop models with highest AIC values in the fitted zero-inflated negative binomial model. The root mean square error (RMSE) of the full model for Mopani district, which is a measure of the deviation of observed malaria count from the fitted value, is 13.9049 while RMSE of the reduced model is 13.9137. The AIC value for the full model is 31,597.14, while the AIC value for the reduced model is 31,542.55. As a result, the reduced model is preferred for Mopani district.
The first block in
Table 1 contains the count model coefficient and their standard error,
z-score and
p-value for each of the variables. The second block corresponds to the inflation model. The inflation model contains logit coefficients for predicting excess zeroes and the corresponding standard errors,
z-scores and
p-values for the coefficients.
Table 1 presents the estimates of the zero-inflated negative binomial model (reduced model) for Mopani district. The coefficient of daily average temperature at lag 18 in the negative binomial regression part predicting the malaria count is statistically significant at 5% level of significance. The coefficients of daily rain amount at lag 9 and lag 16, daily average temperature at lag 9, lag 10, lag 12, lag 15 and lag 18, simulated daily mosquito population at lag 9, lag 10 and lag 20 in the logit model part predicting excessive zeroes are statistically significant. Other predictor variables are not statistically significant and are, therefore, excluded in the model. It is desirable to know whether zero-inflated negative binomial regression model fits the data statistically better than usual negative binomial regression model. The Vuong test [
35] is employed to determine whether the formulated model (zero-inflated negative binomial regression model) fits the data better than the usual negative binomial regression model. The Vuong test is the likelihood-ratio-based test for model selection using the Kullback–Leibler information criterion. The test suggests that the zero-inflated negative binomial model is a significant improvement over a standard negative binomial model. The Vuong statistic tests the null hypothesis that the formulated zero-inflated negative binomial model and the negative binomial model are equally close to the true data generating process, against the alternative that the formulated zero-inflated negative binomial model is closer. The Vuong test is asymptotically distributed as a standard normal distribution (that is, N (0,1)) under the null hypothesis that the models are equivalent. The test rejects the null hypothesis at 5% level of significance (
p-value < 2.22 × 10
−16) and suggests that zero-inflated negative binomial model with lagged predictors fits the data better than the usual negative binomial regression model.
The number of malaria cases decreases by a factor of 0.9742 for a one-unit increase in daily average temperature at lag 18 when other variables are held constant. This implies that it is much likely to have any malaria cases as the daily average temperature at lag 9, lag 12 and lag 14 increase. The odds of being an excessive zero would decrease by 0.9404, 0.9335, 0.8481, 0.8872, 0.8668, 0.9242, 0.8729 and 0.9455 for every one-unit increase in daily rain amount at lag 9 and lag 16, daily average temperature at lag 9, lag 10, lag 12, lag 15 and lag 18, and simulated daily mosquito at lag 9 respectively. Increase in the odds of being an excessive zero means that it is less likely that there will be malaria cases. This implies that the likelihood that daily malaria count would be zero in Mopani district municipality decreases with an increase in daily rain amount at lag 9 and lag 16, daily average temperature at lag 9, lag 10, lag 12, lag 15 and lag 18, and simulated daily mosquito at lag 9. Moreover, the log odds of being an excessive zero would increase by 1.0366 and 1.0195 for every one-unit increase in the simulated daily mosquito at lag 10 and lag 20, respectively.
3.3. Analysis over Vhembe District Municipality
A stepwise model selection procedure based on Akaike information criterion (AIC) was applied to drop models with highest AIC values in the fitted zero-inflated negative binomial model for Vhembe district. The RMSE of the full model for Vhembe district is 13.7776 while RMSE of the reduced model is 13.79789. The AIC value for the full model is 42,218.47, while the AIC value for the reduced model is 42,232.6. As a result, the reduced model is preferred for Vhembe district.
Table 2 presents the estimates of the zero-inflated negative binomial model (reduced model) for Vhembe district. The coefficients of daily average temperature at lag 9, lag 12 and lag 14, simulated daily mosquito population at lag 20 in the count model predicting daily malaria count are statistically significant at 5% level of significance. The coefficients of daily average temperature at lag 10, lag 12 and lag 14, and simulated daily mosquito population at lag 9 and lag 15 in the logit model part predicting excessive zeroes are statistically significant. Other predictors are not statistically significant and are therefore excluded from the model. The Vuong test is also employed to determine whether a negative binomial regression model fits the Vhembe district malaria data statistically better than the formulated zero-inflated negative binomial regression model. The test rejects the null hypothesis at 5% level of significance (
p-value < 2.22 × 10
−16) and suggests that zero-inflated negative binomial regression model fits the data better than the negative binomial regression model.
The number of malaria cases increases by 1.0247, 1.0189 and 1.0151 for a one-unit increase in daily average temperature at lag 9, lag 12 and lag 14, respectively, when other variables are held constant. This implies that it is more likely to have any malaria cases as the daily average temperature at lag 9, lag 12 and lag 14 increase. The number of malaria cases decreases by a factor of 0.9979 for a one-unit increase in simulated daily mosquito population at lag 20 when other variables are held constant. This implies that it is less likely to have any malaria cases as the daily average temperature at lag 18 increase. The odds of being an excessive zero would decrease by 0.7965, 0.8848, 0.8364 and 0.9541 for every one-unit increase in daily average temperature at lag 10, daily average temperature at lag 12, daily average temperature at lag 14 and simulated daily mosquito population at lag 9 respectively. This implies that the likelihood that daily malaria count would be zero in Vhembe district municipality decreases with an increase in daily average temperature at lag 10, daily average temperature at lag 12, daily average temperature at lag 14 and simulated daily mosquito population at lag 9. Moreover, the odds of being an excessive zero would increase by a factor of 1.0296 for every one-unit increase in the simulated daily mosquito population at lag 15.
The dispersion parameter
in
Table 1 and
Table 2 gives an indication if zero-inflated negative binomial model is fit for the data. If
approaches infinity, then variance equals mean and as a result, zero-inflated Poisson model will fit the data better. Additionally,
is finite implies that the variance is greater than mean. As
approaches 0, the farther the variance is from the mean. Exponentiating log(
) in
Table 1 and
Table 2, the values of
are 0.4292 and 0.6257 for Mopani and Vhembe districts. Hence, the zero-inflated negative binomial model is appropriate for the model and confirm the result in
Section 3.1.
This complements the findings of previous studies. It was argued in [
36] that a moderate transmission intensity climate is crucial to malaria transmission. Based on the findings of [
37,
38] concluded that climate predictor variables generate a better predictive power when modelling malaria incidence in areas with unstable transmission compared to areas with stable endemicity. However, [
36] shows that the development of clinical immunity buffers any effect of climate under high endemicity. In addition, [
18] showed that there is a statistically significant correspondence between malaria rates and the climate variables, mostly air temperature and precipitation. This is confirmed in the fitted models for malaria incidence in Mopani and Vhembe districts. An increase in daily average temperature and its lagged values significantly raise the chance of malaria transmission and thereby leads to an increase in malaria incidence in Vhembe district. Furthermore, an increase in rainfall amount at lags 9 and 16 increases the probability of malaria cases occurring in the Mopani district. This is in line with several other studies that have highlighted the importance of rainfall on malaria transmission and other infectious diseases in western Kenya [
39], Tanzania [
40], East Africa [
41] and Ghana [
42].
Ljung–Box test [
29] is employed to test if the residuals (
) of the zero-inflated negative binomial model (
) are correlated. The Ljung–Box test shows that residuals of a fitted model for each of Mopani district (
p-value < 2.2 × 10
−16) and Vhembe district (
p-value < 2.2 × 10
−16) are autocorrelated. This confirms the result of plots of the autocorrelation function and partial autocorrelation function in
Figure 5. The
achieves stationarity at
. The optimal models for
are ARIMA(5,1,4) and ARIMA(2,1,1) for Mopani and Vhembe district municipalities, respectively. The estimate of ARIMA(5,1,4) model for Mopani district are
,
,
,
,
,
,
,
and
. The estimates of ARIMA(2,1,1) model for Vhembe district are
,
and
.
Figure 6 presents the correlograms of the autocorrelation function and partial autocorrelation function on the residuals of ZINB+ARIMA model on malaria incidence counts. The figure shows that residuals of the fitted ZINB+ARIMA model are not correlated. The Ljung–Box test confirms that the residuals of models for Mopani district (
p-value = 0.9946) and Vhembe district (
p-value = 0.9477) are not correlated.
Figure 7 and
Figure 8 present the comparison between the observed and fitted malaria counts over Mopani and Vhembe, respectively.
3.4. Mosquito Abundance and Malaria Cases of Mopani and Vhembe
Findings further highlight the importance of mosquitoes in the transmission of malaria (
Figure 9). Results also show that abundance of
An. arabiensis is positively correlated with malaria transmission over the two study regions (
Figure 9). The measure of the correlation (Spearman’s rank correlation coefficients) between mosquito abundance and malaria count is 0.0835 (
p-value = 9.515 × 10
−13) in Mopani district while the measure of the correlation between mosquito abundance and malaria count is 0.1655 (
p-value < 2.2 ×10
−16) in Vhembe district. However, findings show that transmission is possible over the study regions even with temperate amount of
An. arabiensis. For instance, over Mopani, malaria cases maintain a steady increase from 0 to almost 250 even below estimated 60,000
An. arabiensis (
Figure 9a). Similarly, with just about 50,000
An. arabiensis, malaria cases went up to 350 in Vhembe (
Figure 9b). This is also an indication that the impact of other malaria vectors over the study regions cannot be overlooked. In other words, all control measures to eradicate malaria over these regions should target
An. arabiensis and other malaria-transmitting vectors. Although it has been established that
An. arabiensis is the primary malaria vector in South Africa [
32], the findings here suspect the presence of additional mosquito species transmitting malaria over the study regions as recently found in KwaZulu-Natal and Mpumalanga province [
32]. This is also in line with the findings of [
5] where several other mosquito species were found across five different regions in Limpopo province [
5].