Regression models for predicting the die-off rate of E. coli in solar water disinfection

Solar water disinfection (SODIS) purifies water containing microbial pathogens by storing water in small transparent containers (0.5–2 L) and exposing them to direct sunlight for a day or two, depending on the degree of cloud cover (Luzi et al. 2016). SODIS is recognized as an integral element of a broader strategy of water quality improvement and diarrhoea prevention (Clasen et al. 2015). It is one of the methods recommended by the World Health Organization (WHO) for point of use water treatment, especially where there is no other method of obtaining safe drinking water. More than 5 million people distributed across 55 countries were using SODIS for their daily water treatment as of 2016 (Luzi et al. 2016).

The most important process parameter that control pathogen removal efficiency in SODIS are UV radiation, water temperature, and water turbidity. Effective SODIS treatment requires a radiation intensity of more than 45 W/m² for 6 h or 54 W/m² for 5 h (Luzi et al. 2016). Invariably, this amount of sunlight will contain the requisite UV threshold (>270 W/m²) and/or temperature (>45 °C) required for complete bacterial inactivation. The UV portion of sunlight that reaches the earth's surface is made up of UVA (315–400 nm) and UVB (280–315 nm). UV causes cell death by either the direct attack of UVB on cell DNA which disrupts both transcription and replication (Sinha & Häder 2002; Pigeot-Rémy et al. 2012), or by the indirect mediation of UVA to produce reactive oxygen species (ROS) that attack the major cellular components, leading to loss of membrane integrity, increased ion permeability, protein fragmentation, disruption of the intracellular oxygen transport system, and attack on DNA and RNA (Sinha & Häder 2002; Pigeot-Rémy et al. 2012). Production of ROS can be enhanced by increasing the dissolved oxygen concentration of SODIS water through vigorous shaking of partially filled containers before topping to full volume (Reed 1997). Temperature kills microorganisms by its pasteurizing effect that denatures cell proteins and inhibit DNA repair mechanisms (McGuigan et al. 1998; Samoili et al. 2022). SODIS does not work well in highly turbid water (>30 NTU) because the colloidal particles that cause turbidity in water shield cells from direct solar exposure. Several studies have shown that the die-off rate of pathogens in SODIS decreased with increasing water turbidity (McGuigan et al. 1998; Gómez-Couso et al. 2009). SODIS guidelines (Luzi et al. 2016) recommend pretreatments to reduce water turbidity below 30 NTU before the application of SODIS.

Several models have been proposed to demonstrate the inactivation kinetics of UV light-based disinfection. These models are mostly empirical based on the classical Chick's first-order kinetics which has been variously modified to either include other parameters of interest or fit different inactivation curves (Dalrymple et al. 2010). Stochastic models (Jensen 2010; Fan et al. 2012) have also been contemplated based on contact time and activation sites with no consideration of the intracellular interactions and biological processes that control the rate of cell death. In an attempt to include biological information, Castro-Alférez et al. (2017a) developed a mechanistic model that considered the complex intracellular processes that lead to bacterial inactivation in SODIS based on the generation of ROS and inactivation of enzymes in non-turbid water. Castro-Alférez and coworkers proposed another mechanistic model (Castro-Alférez et al. 2017b) that incorporated the actions of solar UV and the pasteurizing effect of mild heat to capture the synergy between UV and temperature in clear water and found excellent fitting between model results and experimental observations. A more recent study evaluated the effect of turbidity and dissolved organic matter on the inactivation rate kinetics of E. coli under simulated sunlight intensities (Samoili et al. 2022). These models can only be used to describe the effect of diurnal variation in incident UV and water temperature on SODIS treatment efficiency. No model has investigated the day-to-day variability in SODIS efficiency with respect to UV, water temperature, and water turbidity despite the role they could play in judging the applicability of SODIS and selecting SODIS exposure period in any environment.

Therefore, the objective of this study was to determine the amount of variation in day-to-day pathogen die-off rate that can be explained by UV radiation intensity, maximum water temperature, and water turbidity using regression models. Regression analysis is one of the most widely used statistical techniques for exploring the functional relationship among variables (Chatterjee & Hadi 2006). It provides a theoretically simple method for analyzing multifactor data.

A value greater than 5 indicates is dependent on other predictor variables (Montgomery et al. 2012).

The term is the PRESS residuals, where is the predicted response based on a model fitted without using observation. The term are the diagonals of a hat matrix ⁠. Just like the least-square ⁠, is used as a measure of the predictive capacity of the model in new observations.

The experiments were conducted in Nsukka, Enugu State, southeastern region of Nigeria. Nsukka is located within latitudes 6.86 °N and 6.83 °N of the Equator, and longitudes 7.36 °E and 7.42 °E of Greenwich Meridians. Topographically, Nsukka is located in a plateau, an escarpment region with ground elevations ranging from 280 to 530 m above sea level with a mean of about 429 m. Recent meteorological data of Nsukka Urban show a maximum rainfall of 231 mm/month with an atmospheric temperature range of 22–36 °C and a mean annual relative humidity of 77%. The region is known for its copious rainfall from June to October and an extended dry period lasting from November to March. During the dry season, the sunlight reaching Nsukka is significantly dimmed by the dust-laden airmass and aerosol pollution of the dusty northeasterly trade wind (Harmattan wind) that blows from southern Sahara to the Gulf of Guinea (Ugwuoke & Okeke 2012). An earlier study (Nwankwo & Agunwamba 2021) that investigated the pattern of seasonal variation of the solar radiation and maximum air temperature confirmed the applicability of SODIS in Nsukka. Monthly averages of 5-h midday radiation intensity and maximum air temperature indicated that only the rainy months of July, August, and September do not meet the recommended 500 W/m² required for effective application of SODIS (Nwankwo & Agunwamba 2021).

Before each experiment, the plastic bottles would be disinfected by soaking overnight in a hypochlorite solution (Hypo™) before being rinsed repeatedly with tap water and then the test water.

E. coli was adopted as the model pathogen in this study due to its status as an indicator of faecal contamination. It is also the most commonly studied species in SODIS and can resist the germicidal effect of sunlight more than most bacteria (McGuigan et al. 2012). The procedure for cultivation and enumeration of E. coli was described in Ubomba-Jaswa et al. (2010). The test water for all the experiments was drawn from a 200 L plastic water storage drum collected at once from a borehole source to avoid variation in physicochemical properties. The physicochemical properties were tested once a month. Before each experiment, the test water would be sterilized before being contaminated from a previously cultivated E. coli stock by making appropriate dilution to arrive at approximately 10⁶ CFU/mL. The bottle would be filled to two-thirds of its volume, followed by vigorous shaking to facilitate absorption of oxygen, before being topped up to full volume. This procedure consistently achieved dissolved oxygen (DO) values of 6.1±0.21 mg/L. The test water was confirmed to be free of any chlorine. Turbid water was prepared by the addition of kaolin (China clay) to the test water until the required turbidity was achieved. The turbidity of test water for the experiments was randomly varied between 0 and 30 NTU. The initial sample was taken from the reactors before solar exposure. Subsequent samples were taken using a sterilized hypodermic syringe at 30-min intervals or higher, depending on the amount of radiation that has been received since the last sample was drawn.

Ultraviolet radiation was measured using a digital UV light meter (General Tools UV513AB Digital UVA/UVB Meter, 280–400 nm). The LCD screen of the digital UV meter displays the rate of radiant energy per unit area in mW/cm² or μW/cm². The readings of the UV meter were taken using an Open Camera 1.48.3 app for android phones, which has features that allow shots to be taken repeatedly at a preset time interval. On the days of experiments, the camera would be primed and positioned to snap the LCD screen of the digital UV meter every 60 s from 10 a.m. to 4 p.m. The readings were used to estimate the daily UV profile, which was used to evaluate the daily maximum of 5-h averages of ultraviolet (UV) intensity that occurred between 10 a.m. and 4 p.m. All the experiments were conducted during the 5 months between April and August 2021.

The physicochemical properties of the test water was tested at monthly intervals during the 5 months of the experiment between April and August 2021. The tested physicochemical parameters were pH (6.8 ± 0.3), DO (4.5 ± 0.2 mg/L), total dissolved solids (285.5 ± 4.2 mg/L), nitrate (2.9 ± 1.1 mg/L), sulphate (2.6 ± 1.2 mg/L), chloride (32.5 ± 1.3 mg/L), total hardness (9.0 ± 0.8 mg/L), and turbidity (1.2 ± 0.0 NTU). All the physicochemical water quality parameters fell within WHO's drinking water guidelines (WHO 2011). It is important to note that physicochemical parameters are not the parameters of interest in this study. However, a conscientious effort was made to include them as part of the analysis for completeness of the study. All of the tested physicochemical parameters, except nitrate, are not of health concern at levels found in drinking water, but some parameters may affect the taste and aesthetic acceptability of drinking water, including the amenability of raw water to SODIS treatment.

The data used in the regression analysis and model development are presented in the Supplementary material. The regression analysis was run in three phases. Tables 1 and 2 show the results of all the analyses, including the models, the regression statistics, and the hypothesis tests. The first phase involved all the predictor variables together in a single regression model (i.e., versus ⁠, and ⁠). It can be seen that values for and are greater than 5, which indicates strong linear dependence between these variables. When a value is greater than 5, the associated regression coefficients are poorly estimated and should not be applied or interpreted further until the problem of multicollinearity is addressed (Montgomery et al. 2012). This is because when there is strong dependence among predictor variables, some predictor variables become redundant, leading to large standard error and elevated R-square since the same information is provided in more than one way (Suárez et al. 2017). Therefore, it may be superfluous to combine UV intensity and water temperature as predictor variables in a single least-square regression model.

Table 1

Regression statistics

Predictors .	Model .	.	.	.	.	.
⁠, ⁠, and		0.95	0.90	0.89	0.86	0.40
and		0.91	0.84	0.83	0.80	0.49
and		0.94	0.88	0.87	0.86	0.42
		0.91	0.84	0.84	0.82	0.49
		093	0.87	0.87	0.85	0.44
		0.17	0.03	<−0.01	−0.08	1.21

Predictors .	Model .	.	.	.	.	.
⁠, ⁠, and		0.95	0.90	0.89	0.86	0.40
and		0.91	0.84	0.83	0.80	0.49
and		0.94	0.88	0.87	0.86	0.42
		0.91	0.84	0.84	0.82	0.49
		093	0.87	0.87	0.85	0.44
		0.17	0.03	<−0.01	−0.08	1.21

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Table 2

Results of the hypothesis tests

(a) ANOVA F-test
Predictor variables: ⁠, , and
Regression	3	42.38	14.13	87.44	<0.0001
Residual	29	4.69	0.16
Total	32	47.07
Predictor variables:
Regression	2	39.72	19.86	81.00	<0.0001
Residual	30	7.35	0.24
Total	32	47.07
Predictor variables:
Regression	2	41.54	20.77	112.66	<0.0001
Residual	30	5.53	0.18
Total	32	47.07
Predictor variables:
Regression	1	39.71	39.71	167.34	<0.0001
Residual	31	7.36	0.237
Total	32	47.07
Predictor variables:
Regression	1	41.10	41.10	213.39	<0.0001
Residual	31	5.971	0.193
Total	32	47.07
Predictor variables:
Regression	1	1.421	1.42	0.96	0.3335
Residual	31	45.650	1.47
Total	32	47.071
(b) Student's t-test
Predictor variables: ⁠, ⁠, and				p-value
Intercept	−4.657	0.857	−5.434	<0.0001	N/A
	0.037	0.016	2.287	0.0297	6.75
	0.137	0.033	4.064	0.0003	6.57
	−0.009	0.009	0.331	0.3311	1.08
Predictor variables:				p-value
Intercept	−1.512	0.453	−3.336	0.0023	N/A
	0.098	0.008	12.499	<0.0001	1.03
	−0.001	0.011	−0.107	0.9151	1.03
Predictor variables:				p-value
Intercept	−5.992	0.670	−8.940	<0.0001	N/A
	0.208	0.014	14.752	<0.0001	1.01
	−0.014	0.009	−1.544	0.1330	1.01
Predictor variables:				p-value
Intercept	−1.539	0.368	−4.181	0.0002	N/A
	0.098	0.008	12.936	<0.0001	N/A
Predictor variables:				p-value
Intercept	−6.322	0.649	−9.739	<0.0001	N/A
	0.210	0.014	14.608	<0.0001	N/A
Predictor variables:				p-value
Intercept	3.542	0.502	7.062	<0.0001	N/A
	−0.025	0.026	−0.982	0.3335	N/A

(a) ANOVA F-test
Predictor variables: ⁠, , and
Regression	3	42.38	14.13	87.44	<0.0001
Residual	29	4.69	0.16
Total	32	47.07
Predictor variables:
Regression	2	39.72	19.86	81.00	<0.0001
Residual	30	7.35	0.24
Total	32	47.07
Predictor variables:
Regression	2	41.54	20.77	112.66	<0.0001
Residual	30	5.53	0.18
Total	32	47.07
Predictor variables:
Regression	1	39.71	39.71	167.34	<0.0001
Residual	31	7.36	0.237
Total	32	47.07
Predictor variables:
Regression	1	41.10	41.10	213.39	<0.0001
Residual	31	5.971	0.193
Total	32	47.07
Predictor variables:
Regression	1	1.421	1.42	0.96	0.3335
Residual	31	45.650	1.47
Total	32	47.071
(b) Student's t-test
Predictor variables: ⁠, ⁠, and				p-value
Intercept	−4.657	0.857	−5.434	<0.0001	N/A
	0.037	0.016	2.287	0.0297	6.75
	0.137	0.033	4.064	0.0003	6.57
	−0.009	0.009	0.331	0.3311	1.08
Predictor variables:				p-value
Intercept	−1.512	0.453	−3.336	0.0023	N/A
	0.098	0.008	12.499	<0.0001	1.03
	−0.001	0.011	−0.107	0.9151	1.03
Predictor variables:				p-value
Intercept	−5.992	0.670	−8.940	<0.0001	N/A
	0.208	0.014	14.752	<0.0001	1.01
	−0.014	0.009	−1.544	0.1330	1.01
Predictor variables:				p-value
Intercept	−1.539	0.368	−4.181	0.0002	N/A
	0.098	0.008	12.936	<0.0001	N/A
Predictor variables:				p-value
Intercept	−6.322	0.649	−9.739	<0.0001	N/A
	0.210	0.014	14.608	<0.0001	N/A
Predictor variables:				p-value
Intercept	3.542	0.502	7.062	<0.0001	N/A
	−0.025	0.026	−0.982	0.3335	N/A

N/A, Not applicable; D.F., degrees of freedom; SS, sum of squares; MS, mean sum of squares; s.e., standard error; VIF, variable inflation factor.

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The second phase of the analysis re-ran the regression for versus and and versus and ⁠, avoiding the inclusion of and in a single model. The R-square values of versus and and versus and are 0.84 and 0.88, respectively, which indicate that both models fit the data well. F-statistics for the two models are significant at 95% confidence level but the t-statistics for in the two models are not significant. The VIF values in the models are close to one, which is an indication of weak linear dependence between the predictor variables and adequate estimation of the regression coefficients. Also, the predictive R-square values versus and and versus and based on PRESS statistics suggest a strong predictive capacity when the models are used in new observations.

The third phase of the analysis developed simple regression models by regressing with ⁠, and separately. The models involving ⁠, ⁠, and produced R-square values of 0.84, 0.87, and 0.03, respectively, suggesting that the models involving and provides excellent fits to ⁠, but the model involving provides a very weak fit. Also, the t-statistics for all the regression coefficients, except the regression coefficient for ⁠, are significant. The predictive R-square values of versus and versus suggest a strong predictive capacity when the models are used in new observations.

The result of the regression model shows that there is a strong linear relationship between SODIS treatment efficiency, on the one hand, and 5-h average midday UV intensity and maximum water temperature, on the other hand. A similar degree of correlation has been reported in recent studies that regressed the cumulative die-off rate of E. coli with irradiance and water temperature (Brockliss et al. 2022; Samoili et al. 2022). However, the values of VIF show that UV intensity and water temperature cannot be successfully combined in a single least-square regression model because of the strong correlation between the parameters. This is because, in the Tropics, sunny periods are constantly hot and sunlight intensity is strongly correlated to temperature (Nwankwo & Agunwamba 2021). Weather conditions like ‘sunny and cold’ do hardly occur together in the study area. Therefore, the future least-square regression model should be cognizant of the possible interference between water temperature and UV intensity and consider eliminating one of the variables or exploring other methods for dealing with multicollinearity in regression models.

Fitting the die-off rate constant with UV intensity and water temperature separately showed that maximum water temperature provided a better fit than maximum 5-h UV intensity even though both yielded very strong R-square values. This suggests that, in this region, maximum water temperature is a better predictor of SODIS efficiency than maximum 5-h UV intensity. An earlier study found that daily maximum water temperature could contain some dosimetric information (Nwankwo & Agunwamba 2021). For example, complete inactivation was recorded in all the experiments whose water temperature exceeded 45 °C. Maximum water temperature in the Tropics usually occurs between 1 and 3 p.m. (Nwankwo et al. 2019). Moreover, temperature data is cheaper to obtain and may require only a simple mercury-in-glass thermometer that costs less than a dollar compared to the costly, ‘high-tech’ devices required to measure the intensity of UV light.

Water turbidity is an extremely important consideration in the application of SODIS. Previous studies have largely shown that the disinfection effectiveness of SODIS decreases significantly with increasing water turbidity (McGuigan et al. 1998; Gómez-Couso et al. 2009; Amirsoleimani & Brion 2021). For example, increasing turbidity from 0 to 200 NTU was found to decrease the die-off rate of E. coli from 5 to 1 log (Amirsoleimani & Brion 2021). This is because the depth of light penetration is significantly reduced in highly turbid water which interferes with the SODIS processes (Marques et al. 2013). However, these studies included water of turbidity well outside the range tested in this study. Over time, the turbidity threshold of 30 NTU emerged as the turbidity criterion for selecting SODIS water. A simple test for determining if the turbidity of water is less than 30 NTU was described in Luzi et al. (2016). Pretreatment to reduce turbidity levels below this threshold is recommended as part of the SODIS protocol. The results demonstrated a lack of evidence that water turbidity in the range of 0–30 NTU has any effect on the pathogen removal efficiency. The turbidity variable was found not to be significant when regressed alone and when regressed together with other predictor variables. Another study (Brockliss et al. 2022) that used unprotected water sources with turbidities lower than 30 NTU reported a weak correlation between the die-off rate of E. coli and turbidity.

Note that it is only the direct effect UVB has on the major cellular components of bacterial pathogens at the inner layer of SODIS water that is shut out by the suspended particles in turbid water; the absorbed and scattered portions of UVA in water are not lost and can still generate ROS which can migrate to inner water layers and kill pathogens (Luzi et al. 2016). This may well explain the reason complete disinfection could still be achieved in turbid water even when most cells do not receive direct sunlight. Another factor that aids disinfection in turbid water is water temperature which is usually higher due to the capacity of suspended particles to trap radiant heat emitted by the sun as infrared (IR). Amirsoleimani & Brion (2021) reported a statistically significant difference between the temperatures of 0 and 30 NTU SODIS water exposed to the same conditions in which SODIS containers with 30 NTU water were found to be much hotter. Improved efficiency has been reported in SODIS treating moderately turbid water (38 NTU) when compared with SODIS efficiency treating water of low turbidities (<5 NTU) (Meera & Ahammed 2008). Complete inactivation can be achieved in SODIS water of higher turbidities provided temperature values of 55 °C and above are reached (Joyce et al. 1996).

Despite the temperature advantage, high turbidity (>30 NTU) is not a desirable quality in SODIS water. SODIS water is more durable (no bacteria regrowth) when clear water is used provided the requisite UV dose and complete disinfection are reached (Vivar & Fuentes 2016; Keogh et al. 2017). Again, the aesthetics and user acceptance of SODIS water could be considerably reduced for highly turbid water.

Regression models/analyses can be successfully employed to establish a functional relationship between SODIS treatment efficiency and treatment conditions and explain the day-to-day variability in the die-off rate constant of pathogens. Understanding the dynamics of this relationship is crucial for the accurate prediction of SODIS applicability, exposure period, and the effectiveness of SODIS treatment in different regions of the world. In regions where sunny days are always hot, there could be strong dependence among sunlight intensity and water temperature so that these two parameters cannot be successfully combined in a single least-square regression model due to the problems of collinearity. The daily maximum water temperature, which is cheaper to obtain, could be a better predictor of pathogen removal rate in SODIS when compared with the maximum 5-h UV intensity. Water turbidities in the range of 1–30 NTU do not have any significant effect on the die-off rate of bacterial pathogens. Using collinearity diagnostic tools, the degree of dependence between these parameters should be checked and addressed before they can be reasonably combined to predict SODIS applicability/efficiency in regression models.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.