- Email: [email protected]
Probabilistic temporal prediction of the deaths caused by traffic in Colombia. Mortality caused by traffic prediction
Accident Analysis and Prevention 135 (2020) 105332
Contents lists available at ScienceDirect
Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap
Probabilistic temporal prediction of the deaths caused by traffic in Colombia. Mortality caused by traffic prediction
T
Javier Rodríguez*, Jairo Jattin, Yolanda Soracipa Insight Group, Asociación Colombiana de Neurocirugía, Cra. 79B N° 51-16 Sur. Int. 5, Apt. 102, Kennedy, Bogotá D.C., Colombia
A R T I C LE I N FO
A B S T R A C T
Keywords: Probability Mortality Random walk Traffic
Background: from probability theory and probabilistic random walk, predictions about the quantity of cases of a given phenomenon for certain year, such as epidemics of dengue, have been previously obtained with results close to 100% in precision. Objective: To confirm the applicability of a methodology based on probability and probabilistic random walk to predict the dynamics of deaths from road traffic injuries in Colombia for 2010. Methodology: through the development of a total probability space that analyses the probabilistic behaviour of augments and decreases observed in the variation of the lengths of the death rates caused by traffic in Colombia from 2004 to 2009, the most likely event for 2010 was established for predicting the rate of deaths for that year. Results: The predicted rate of deaths caused by traffic injuries in Colombia for 2010 was 14.88 with the methodology. When this value is compared with the value reported by national statistics, which was a rate of 12.9, a precision of 86.6% with the prediction was achieved. Conclusions: the applicability of the developed methodology to predict the dynamic behaviour of deaths caused by traffic injuries in Colombia for 2010 by means of a probabilistic random walk was confirmed with a good precision, suggesting that this methodology could be useful to verify the efficacy of national road safety strategies implemented to reduce mortality rates.
1. Introduction The mathematical calculation that determines the possibility that a certain experiment occurs in the future is found through probability. All the possible results of an experiment are contained in probabilistic or sample spaces. Frequently, the sample space that provides more information within the whole possible space must be determined in order to conduct more precise predictions (Spiegel et al., 2003; Feynman et al., 1964). Different experiments that obey mathematical laws to exactly determine the moment in which a given event will happen have been developed, however, when it is wanted to establish or estimate the most likely event, calculations of probability are used. Among the different experiments that are reported, the probabilistic theoretical model of Wiener can be mentioned, highlighting that through random walks, phenomena whose behaviour in time is complex and irregular, can be represented and studied in order to determine which would be the future evolution in a trajectory (Feynman et al., 1964; Wiener, 1958). Similarly, the root mean square determines how far from an initial point a given measure is, in the context of an experiment that repeats along time as well as it also determines how likely it is for the events to
⁎
happen, that is, that they are equiprobable. For 2010, the World Health Organization estimated that approximately 1,24 million deaths took place in the different roads of the world. For Colombia, in that year, a rate of 15,6 per 100.000 general population was estimated (WHO, 2004). Nevertheless, for that time, globally, only 28 countries presented safety road integral laws in the five main risk factors, which are: drinking and driving, driving at high speeds, not wearing seat belts, not using child restraint systems and not wearing helmets, in the case of motorcycles. The anterior, made necessary to redact an inform that serves as guideline for the Decade of Action of Safety Road 2011–2020, declared by the General Assembly of the United Nations (WHO, 2004). Not only these events generate a significant mortality, but also generate an important cause of disability worldwide (PAHO, 2017). In the light of these findings, the necessity of developing studies that account over this problem has arisen. This is important since countries need to estimate not only the social and epidemiological effects, but also the economic implications due to direct and indirect costs as well as losses in productivity (García et al., 2013; Ang et al., 2017; Balakrishnan and Karuppanagounder, 2019; Carozzi et al., 2017;
Corresponding author at: Cra. 79B N° 51-16 Sur. Int. 5. Apt. 102, Kennedy, Bogotá, D.C., 110861, Colombia. E-mail address: [email protected] (J. Rodríguez).
https://doi.org/10.1016/j.aap.2019.105332 Received 8 April 2019; Received in revised form 2 July 2019; Accepted 15 October 2019 Available online 12 December 2019 0001-4575/ © 2019 Elsevier Ltd. All rights reserved.
Accident Analysis and Prevention 135 (2020) 105332
J. Rodríguez, et al.
Waters et al., 2004; Chisholm et al., 2012). The majority of studies until now are descriptive, geographical and demographical from a perspective of documentary review and, since a lot of information is compiled, usually these studies are based on statistical prospective and retrospective methodologies (Híjar, 2000; Hernández, 2013; Dultz et al., 2013; Rodríguez et al., 2017), usually using software to work with these databases. For example, in Ethiopia a study was conducted using the information provided by the local police and hospitals to corroborate the deaths caused by traffic through the application of a second method called capture-recapture (Abegaz et al., 2014; Chapman, 1951). However, the study concludes that the information provided by both institutions is not sufficiently complete for the application of these methodologies suggesting the necessity of strengthening these databases (Abegaz et al., 2014). On the other hand, in Colombia, some investigations have been developed in the frame of theoretical physics and mathematics that contributed in the area of public health with the design of new methodologies that predict the number of people infected by malaria and dengue (Rodríguez and Prieto, 2010a; Rodríguez et al., 2013a). These studies could have been developed since the epidemics presented a random walk-like behaviour, which resulted in predictions of a certain year superior to 90%. The purpose of this study is to analyse if the dynamics of deaths caused by traffic present a random walk behaviour and to confirm the reproducibility of the methodology based on probability and random walk to predict the number of deaths by traffic in Colombia for 2010 with a posterior comparison to the real values reported by the Departamento Administrativo Nacional de Estadística (DANE).
Fig. 1. Representation of a distance between two points, considering the initial coordinates as (X0 , Y0 ) and the following ones as (X1 , Y1) .
2.2.3. Probability of death rates caused by traffic a second probability space was defined from the quotient of the death rates in a year divided by the total sum of these rates that correspond to the period between 2004 to 2009 through Eq. 3:
P (N ) =
Annual rate of deaths caused by traffic Totality of death rates Totality of death rates (3)
2. Materials and methods
2.2.4. Root mean square of death rates caused by traffic used to identify if in the second probability space the probability of loaded values exists, that is, if the values found are equiprobable or not, calculated through Eq. 4:
2.1. Data source
P (Rn) =
Data was taken from the national registries of defunctions that DANE reported for 2010 (DANE, 2015) discriminating those deaths that were secondary to traffic. The year 2010 was chosen in order to rely on the official estimate that was reviewed and fixed by the DANE and to conduct a comparison with a past year to demonstrate the predictive effectivity of the methodology.
Y(f ) =
2.2.1. Lengths of death rates caused by traffic the value of the death rates caused by traffic between 2004–2009 were arranged like this: the coordinate X0 corresponds to the ordered pair that represents time (initial year) and the coordinate Y0 represents the death rates for that year; the coordinate X1 represents the coordinates of time for the following year and the coordinate Y1 represents the death rates for that year. The lengths were calculated through Eq. 1 as the calculation of the distance between two points (Fig. 1), as follows:
(Y1 − Yo )2 + (X1 − Xo )2
(4)
2.2.5. Predicon of death rates caused by traffic the prediction is established through the analysis of a third probability space in which the last three lengths found of death rates previous to the year to predict are evaluated. For this, Eq. 2 is cleared in terms of length (L) replaced on Eq. 1, until a quadratic equation is obtained in terms of Y, obtaining Eq. 5, that is:
2.2. Definitions
L=
Annual rate of deaths caused by traffic 1 ± Totality of death rates 2 N
2Y(o) ±
(−2Y(o) )2 − 4{Y 2 (o) + (Xf − Xo )2 − [(P (L)2 × (TL)2]} 2 (5)
Where f: year to predict; a: previous year; P(L): quotient of the length divided by the arithmetic mean of the probabilistic length between 2007–2009; TL: sum of the lengths between 2007–2009. Finally, a fourth probability space that contains the result of two events, augments (A) and decreases (D) of death rates with respect to the previous years, was built, in order to study the behaviour of consecutive periods of two and three years. The purpose of this space is to determine which of the two values found with Eq. 5 results most likely, and with this, a predictive value for 2010 is determined.
(1)
2.3. Procedure 2.2.2. Probabilistic length of deaths rates caused by traffic a probability space must be established through the quotient of the annual variation of the length (L) of deaths, now considered as an event, divided by the sum of the rates of these dates between 2004–2009 through Eq. 2 as follows:
P (L) =
According to the values of the death rates in Colombia comprehended between 2004–2009, the lengths of the annual variations were established (Eq. 1). Following the steps of the methodology (Rodríguez and Prieto, 2010a), the values of the coordinates in the y axis represent the death rates caused by traffic reported for those years while the variation of the values of time in the x axis represent time, and, since the annual variation remains as a constant, its value is zero. Then, the first probability space was established when each length was considered as a probabilistic event. The probability for each length
Length of annual variation of deaths caused by traffic L = Totality of lengths TL (2) 2
Accident Analysis and Prevention 135 (2020) 105332
J. Rodríguez, et al.
was calculated from the division of the value of every length over the sum of all lengths (Eq. 2). Subsequently, a second probability space was established when the calculated values of probabilities of death rates through Eq. 3. To determine if the values of probability presented a loading towards specific values, the root mean square (Eq. 4) was established comparing this value with the expected one. When the set of values that belong to the first and second probability space was defined, a third probability space, used to predict the death rate caused by traffic for 2010, was established from the arithmetic average of the three previous lengths found between 2007 to 2009, calculating the probabilistic lengths of these values. Then, the arithmetic average of the probabilistic lengths and the sum of lengths between 2007 to 2009 as well as the value of the length for 2009 were replaced in Eq. 5, obtaining as solution two values that correspond to the ranges that determine the prediction. The establishment of a fourth probability space was developed with two events: decreases (D) and augments (A) of deaths with respect to the previous year. This space was built in order to determine which of the two obtained values with Eq. 5 results more likely in order, which will set the value for 2010. This behaviour was studied with consecutive periods of two and three years
Table 2 Frequency and probability of consecutive augments (A) and decreases (D) between 2004–2009 of the deaths rates caused by traffic.
P(L)
P(N)
RMS+
RMS-
RMS+P
RMS-P
2004 2005 2006 2007 2008 2009
15.3 14.3 14.8 15.3 14.7 14.7
15.3 1 0.5 0.5 0.6
0.8 0.06 0.02 0.02 0.03 0
0.17 0.16 0.16 0.17 0.16 0.16
0.22 0.21 0.21 0.22 0.21 0.21
0.11 0.10 0.11 0.11 0.11 0.11
0.05 0.05 0.05 0.05 0.05 0.05
−0.05 −0.05 −0.05 −0.05 −0.05 −0.05
1 2 3 4 Total
1 1 0
2 0 0 0 2
0.2 0.4 0 0 0.6
3
0.4 0 0 0 0.4
Combinations
DDD
DDA
DAD
DAA
ADD
ADA
AAD
AAA
Total
Value Probability
0 0
0 0
0 0
1 0.3
0 0
1 0.3
1 0.3
0 0
3 1
4. Discussion This is the first paper that confirms the capability of the methodology to analyse and predict the behaviour of the temporal dynamic of death rates caused by traffic in Colombia, proving the existence of an analogy when compared with the trajectory of a random walk through the establishment of probabilistic lengths and the variation of the last three consecutive years. A percentage of success of 86.6% was obtained with respect to the real value for 2010. This shows that the phenomenon is irregular and complex, but in a practical manner, the methodology is useful to complementarily perform predictions in a quick and easy way, which could be handy in public health when evaluating the impact of strategies to prevent mortality and improving road safety. The methodologies that explore issues such as predicting death rates secondary to road traffic injuries are based mainly on big data. Some of these studies incorporate artificial neural networks optimized through Genetic algorithm or time-series analysis with autoregressive integrated moving average (ARIMA) with explanatory variables (ARIMAX) and seasonal autoregressive integrated moving average (SARIMA) among others (Leveau and Ubeda, 2012; Jafari et al., 2013; Parvareh et al., 2018; Ihueze and Onwurah, 2018; Liyan and Chunfu, 2009; García et al., 2018; Deublein et al., 2015; Mehmandar et al., 2016; Jayatilleke and Jayatilleke, 2016; Zolala et al., 2016; Huang et al., 2016). However, although they are effective and provide valuable information about some cause-effect relationships, these require and rely on huge inputs collected from several years of epidemiological retrospective information, specialized software and personnel as well as the necessity for adjusting the results for each studied population. Similarly, as the aforementioned methodologies, this complementary method considers the values reported by official entities that gather information such as death rates by traffic. Due to the fact that some primary data sources under-report these events or the data is not accurate (Samuel et al., 2012), the predictive precision could be affected by this factor. On the other side, this method only requires values of annual death rates up to three previous years to develop predictions, which reduces the evaluation of methods such as endemic channels that require at least 5 years (Hernández et al., 2016). Also, it is not necessary to perform populational statistical analyses determined by epidemiology, since the physical-theoretical perspective in which this investigation
Table 1 Values of the annual deaths rates caused by traffic in Colombia between 2004–2009. L
Decreases Value
average between the two values predicted for 2010 was found, which was 14.70 per 100.000 general population, and was later compared to the real value, which was 12.9 per 100.000 general population, finding a success percentage of 86.6%.
It was found that the values of the lengths for death rates caused by traffic in Colombia between 2004–2009 were among 0.5–15.3 per 100.000 general population. Probabilistic lengths varied between 0 and 0.8 while the values of the proportion of distances with regards to the minimal distance found were between 0 and 0.3 (data not shown). The probability of the annual death rates varied between 0.16 to 0.172 while the values of root mean square varied in a range of 0.1 and 0.2 finding that a difference between the last ones and the expected values varied in a range of 0,05 to -0,05 (Table 1). Indeed, the difference among the values of the root mean square obtained for the death rates show that the behaviour of this dynamic is not equiprobable, but it shows a loading of probability, allowing the establishment of predictions (Table 1). Based on the analysis performed to the probability space of the last three years and the application of Eq. 5, two values of the prediction were determined for the death rates, which were 15.07 and 14.33 per general 100.000 population (data not shown). The determination of the most likely event for 2010 was made through the study of the behaviour of the frequency and probability of consecutive augments (A) and decreases (D) between 2004–2009 (Table 2). The analysis of this probability space showed that for two or more consecutive years, the number of consecutive A and D varied between 1 and 4, finding that the augments for two years have a probability of 0.4 (Table 3). When evaluating the possible combinations of A and D for a period of three consecutive years it was found that there is a higher probability of finding ADA, not finding the same possibility for those cases of possible decreases (Table 3). Then, to establish the most likely event for 2010, the result of the arithmetic
D.R
Augments Value
Table 3 Number of possible combinations of augments (A) and decreases (D) for a period of three consecutive years of the deaths rates caused by traffic in Colombia.
3. Results
Year
Consecutive years
DR: death rates; L: length of deaths rates caused by traffic; P(L): probabilistic length of the deaths rates caused by traffic; RMS: root mean square. 3
Accident Analysis and Prevention 135 (2020) 105332
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was framed, that is, following the paths of the logical-inductive reasoning independent of causal factors, establishes that generalizations are applicable to any particular case in the universe and no adjustments are required for variables such as sex, age or any other, which simplifies the analysis the phenomenon by excluding the analysis of other variables that may be confusing. Finally, no specialized software or personnel are required to reproduce the method. It is worth noting that this method does not consider the most likely locations of road traffic events that result in mortality or disability as outcomes and the most likely populations involved are as well no predicted, which are considered in other studies. However, future applications of this methodology could be oriented to predict the specific areas of cities where deaths are result of road traffic injuries. In the context of probability theory, random walk allows to mathematically predict the phenomena and allows to study the efficacy of the campaigns oriented to improve road safety since the methodology provides a reliably expected value of mortality. This approach, from the abstractions of the phenomena, allows to generally apply methodologies from theoretical physics and mathematical laws in the scenarios of clinical practice and public health with good results. For example, generalizations have been performed in the field of arterial re-stenosis (Rodríguez et al., 2010b), the prediction of the binding of peptides to HLA class II (Rodríguez, 2008), as well as CD4+ lymphocytes count relying on blood count (Rodríguez et al., 2013b) and the evolution of cardiac dynamics (Rodríguez et al., 2018).
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Inmunologa 27, 151–166. https://doi.org/10.1016/ S0213-9626(08)70064-7. Rodríguez, J., Oliveros, D., Rodríguez, D., Sosa, J., Prieto, S., Correa, C., 2018. Diagnostic methodology of cardiac dynamics based on the Zipf-Mandelbrot law: evaluation with 50 patients. Rev. Mex. Cardiol. 29, 83–89. Rodríguez, J., Prieto, P., Correa, C., Arnold, Y., Álvarez, L., Bernal, P., 2013a. Dinámica de la epidemia del dengue en Colombia: Predicciones de la trayectoria de la epidemia. Rev. Fac. Med 21, 38–45. Rodríguez, J., Prieto, S., 2010a. Dinámica de la epidemia de malaria. Predicciones de su trayectoria. Rev. Fac. Med. 18, 12–20. Rodríguez, J., Prieto, S., Correa, C., Bernal, P., Puerta, G., Vitery, S., 2010b. Theoretical generalization of normal and sick coronary arteries with fractal dimensions and the arterial intrinsic mathematical harmony. BMC Med. Phys. 10, 1–6. https://doi.org/ 10.1186/1756-6649-10-1. Rodríguez, J., Prieto, S., Correa, C., Pérez, C., Mora, J., Bravo, J., 2013b. Predictions of CD4 lymphocytes’ count in HIV patients from complete blood count. BMC Med. Phys. 13, 3. https://doi.org/10.1186/1756-6649-13-3. Rodríguez, J.M., Camelo, F.A., Chaparro, P.E., 2017. Seguridad vial en Colombia en la década de la seguridad vial: resultados parciales 2010-2015. Salud 49https://doi. org/10.18273/revsal.v49n2-2017001. 280-209. Samuel, J., Sankhulani, E., Qureshi, J., Baloyi, P., Thupi, C., Lee, C., Miller, W., Cairns, B., Charles, A., 2012. Under-reporting of road traffic mortality in developing countries: application of a capture-recapture statistical model to refine mortality estimates. PLoS One 7, e31091. Spiegel, M., Schiller, J., Srinivasan, R., 2003. Probability and Statistics, first ed. McGrawHill, Bogotá. Waters, H.R., Hyder, A.A., Phillips, T.L., 2004. Economic evaluation of interventions to reduce road traffic injuries–a review of the literature with applications to low and middle-income countries. Asia. J. 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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We thank Asociación Colombiana de Neurocirugía, specially to its president, Doctor Marco Fonseca and Doctor Germán Forero, research director, for their support to our investigations. References Abegaz, T., Berhane, Y., Alemayehu, W., Assrat, A., Assefa, A., 2014. Road traffic deaths and injuries are under-reported in Ethiopia: a capture-recapture method. PLoS One 9, e103001. https://doi.org/10.1371/journal.pone.0103001. Ang, B.H., Chen, W.S., Lee, S.W.H., 2017. Global burden of road traffic accidents in older adults: a systematic review and meta-regression analysis. Arch. Gerontol. Geriatr. 72, 32–38. https://doi.org/10.1016/j.archger.2017.05.004. Balakrishnan, S., Karuppanagounder, K., 2019. Cost of two-wheeler road accidents in India. Int. J. Inj. Contr. Saf. Promot. 1–7. https://doi.org/10.1080/17457300.2019. 1576208. Carozzi, S., Elorza, M.E., Moscoso, N.S., Ripari, N.V., 2017. Methodologies for estimating the indirect costs of traffic accidents. Rev. Med. Inst. Mex. Seguro Soc. 55, 441–451. Chapman, D.G., 1951. Some properties of the hypergeometric distribution with applications to zoological censuses. Univ. Calif. Publ. Stat 1, 131–160. Chisholm, D., Naci, H., Hyder, A.A., Tran, N.T., Peden, M., 2012. Cost effectiveness of strategies to combat road traffic injuries in sub-Saharan Africa and South East Asia: mathematical modelling study. BMJ 344, e612. https://doi.org/10.1136/bmj.e612. DANE, 2015. Estadísticas Vitales. http://microdatos.dane.gov.co/index.php/catalog/ MICRODATOS/about_collection/22/5. Deublein, M., Schubert, M., Adey, B.T., García, B., 2015. A Bayesian network model to predict accidents on Swiss highways. Infrastruct. Asset Manage. 2, 145–158. https:// doi.org/10.1680/jinam.15.00008. Dultz, L., Foltin, G., Simon, R., Wall, S., Levine, D., Bholat, O., Dekeya, S.L., Jacko, S., Marr, M., Glass, N., Pachter, H., Frangos, S.G., 2013. Vulnerable roadway users struck by motor vehicles at the center of the safest, large US city. J. Trauma Acute Care Surg. 74, 1138–1145. https://doi.org/10.1097/TA.0b013e31827ab722. Feynman, R.P., Leighton, R.B., Sands, M., 1964. Probabilidad. In: Feynman, R.P., Leighton, R.B., Sands, M. (Eds.), Physics. Addison-Wesley Iberoamericana,
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Contents lists available at ScienceDirect
Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap
Probabilistic temporal prediction of the deaths caused by traffic in Colombia. Mortality caused by traffic prediction
T
Javier Rodríguez*, Jairo Jattin, Yolanda Soracipa Insight Group, Asociación Colombiana de Neurocirugía, Cra. 79B N° 51-16 Sur. Int. 5, Apt. 102, Kennedy, Bogotá D.C., Colombia
A R T I C LE I N FO
A B S T R A C T
Keywords: Probability Mortality Random walk Traffic
Background: from probability theory and probabilistic random walk, predictions about the quantity of cases of a given phenomenon for certain year, such as epidemics of dengue, have been previously obtained with results close to 100% in precision. Objective: To confirm the applicability of a methodology based on probability and probabilistic random walk to predict the dynamics of deaths from road traffic injuries in Colombia for 2010. Methodology: through the development of a total probability space that analyses the probabilistic behaviour of augments and decreases observed in the variation of the lengths of the death rates caused by traffic in Colombia from 2004 to 2009, the most likely event for 2010 was established for predicting the rate of deaths for that year. Results: The predicted rate of deaths caused by traffic injuries in Colombia for 2010 was 14.88 with the methodology. When this value is compared with the value reported by national statistics, which was a rate of 12.9, a precision of 86.6% with the prediction was achieved. Conclusions: the applicability of the developed methodology to predict the dynamic behaviour of deaths caused by traffic injuries in Colombia for 2010 by means of a probabilistic random walk was confirmed with a good precision, suggesting that this methodology could be useful to verify the efficacy of national road safety strategies implemented to reduce mortality rates.
1. Introduction The mathematical calculation that determines the possibility that a certain experiment occurs in the future is found through probability. All the possible results of an experiment are contained in probabilistic or sample spaces. Frequently, the sample space that provides more information within the whole possible space must be determined in order to conduct more precise predictions (Spiegel et al., 2003; Feynman et al., 1964). Different experiments that obey mathematical laws to exactly determine the moment in which a given event will happen have been developed, however, when it is wanted to establish or estimate the most likely event, calculations of probability are used. Among the different experiments that are reported, the probabilistic theoretical model of Wiener can be mentioned, highlighting that through random walks, phenomena whose behaviour in time is complex and irregular, can be represented and studied in order to determine which would be the future evolution in a trajectory (Feynman et al., 1964; Wiener, 1958). Similarly, the root mean square determines how far from an initial point a given measure is, in the context of an experiment that repeats along time as well as it also determines how likely it is for the events to
⁎
happen, that is, that they are equiprobable. For 2010, the World Health Organization estimated that approximately 1,24 million deaths took place in the different roads of the world. For Colombia, in that year, a rate of 15,6 per 100.000 general population was estimated (WHO, 2004). Nevertheless, for that time, globally, only 28 countries presented safety road integral laws in the five main risk factors, which are: drinking and driving, driving at high speeds, not wearing seat belts, not using child restraint systems and not wearing helmets, in the case of motorcycles. The anterior, made necessary to redact an inform that serves as guideline for the Decade of Action of Safety Road 2011–2020, declared by the General Assembly of the United Nations (WHO, 2004). Not only these events generate a significant mortality, but also generate an important cause of disability worldwide (PAHO, 2017). In the light of these findings, the necessity of developing studies that account over this problem has arisen. This is important since countries need to estimate not only the social and epidemiological effects, but also the economic implications due to direct and indirect costs as well as losses in productivity (García et al., 2013; Ang et al., 2017; Balakrishnan and Karuppanagounder, 2019; Carozzi et al., 2017;
Corresponding author at: Cra. 79B N° 51-16 Sur. Int. 5. Apt. 102, Kennedy, Bogotá, D.C., 110861, Colombia. E-mail address: [email protected] (J. Rodríguez).
https://doi.org/10.1016/j.aap.2019.105332 Received 8 April 2019; Received in revised form 2 July 2019; Accepted 15 October 2019 Available online 12 December 2019 0001-4575/ © 2019 Elsevier Ltd. All rights reserved.
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Waters et al., 2004; Chisholm et al., 2012). The majority of studies until now are descriptive, geographical and demographical from a perspective of documentary review and, since a lot of information is compiled, usually these studies are based on statistical prospective and retrospective methodologies (Híjar, 2000; Hernández, 2013; Dultz et al., 2013; Rodríguez et al., 2017), usually using software to work with these databases. For example, in Ethiopia a study was conducted using the information provided by the local police and hospitals to corroborate the deaths caused by traffic through the application of a second method called capture-recapture (Abegaz et al., 2014; Chapman, 1951). However, the study concludes that the information provided by both institutions is not sufficiently complete for the application of these methodologies suggesting the necessity of strengthening these databases (Abegaz et al., 2014). On the other hand, in Colombia, some investigations have been developed in the frame of theoretical physics and mathematics that contributed in the area of public health with the design of new methodologies that predict the number of people infected by malaria and dengue (Rodríguez and Prieto, 2010a; Rodríguez et al., 2013a). These studies could have been developed since the epidemics presented a random walk-like behaviour, which resulted in predictions of a certain year superior to 90%. The purpose of this study is to analyse if the dynamics of deaths caused by traffic present a random walk behaviour and to confirm the reproducibility of the methodology based on probability and random walk to predict the number of deaths by traffic in Colombia for 2010 with a posterior comparison to the real values reported by the Departamento Administrativo Nacional de Estadística (DANE).
Fig. 1. Representation of a distance between two points, considering the initial coordinates as (X0 , Y0 ) and the following ones as (X1 , Y1) .
2.2.3. Probability of death rates caused by traffic a second probability space was defined from the quotient of the death rates in a year divided by the total sum of these rates that correspond to the period between 2004 to 2009 through Eq. 3:
P (N ) =
Annual rate of deaths caused by traffic Totality of death rates Totality of death rates (3)
2. Materials and methods
2.2.4. Root mean square of death rates caused by traffic used to identify if in the second probability space the probability of loaded values exists, that is, if the values found are equiprobable or not, calculated through Eq. 4:
2.1. Data source
P (Rn) =
Data was taken from the national registries of defunctions that DANE reported for 2010 (DANE, 2015) discriminating those deaths that were secondary to traffic. The year 2010 was chosen in order to rely on the official estimate that was reviewed and fixed by the DANE and to conduct a comparison with a past year to demonstrate the predictive effectivity of the methodology.
Y(f ) =
2.2.1. Lengths of death rates caused by traffic the value of the death rates caused by traffic between 2004–2009 were arranged like this: the coordinate X0 corresponds to the ordered pair that represents time (initial year) and the coordinate Y0 represents the death rates for that year; the coordinate X1 represents the coordinates of time for the following year and the coordinate Y1 represents the death rates for that year. The lengths were calculated through Eq. 1 as the calculation of the distance between two points (Fig. 1), as follows:
(Y1 − Yo )2 + (X1 − Xo )2
(4)
2.2.5. Predicon of death rates caused by traffic the prediction is established through the analysis of a third probability space in which the last three lengths found of death rates previous to the year to predict are evaluated. For this, Eq. 2 is cleared in terms of length (L) replaced on Eq. 1, until a quadratic equation is obtained in terms of Y, obtaining Eq. 5, that is:
2.2. Definitions
L=
Annual rate of deaths caused by traffic 1 ± Totality of death rates 2 N
2Y(o) ±
(−2Y(o) )2 − 4{Y 2 (o) + (Xf − Xo )2 − [(P (L)2 × (TL)2]} 2 (5)
Where f: year to predict; a: previous year; P(L): quotient of the length divided by the arithmetic mean of the probabilistic length between 2007–2009; TL: sum of the lengths between 2007–2009. Finally, a fourth probability space that contains the result of two events, augments (A) and decreases (D) of death rates with respect to the previous years, was built, in order to study the behaviour of consecutive periods of two and three years. The purpose of this space is to determine which of the two values found with Eq. 5 results most likely, and with this, a predictive value for 2010 is determined.
(1)
2.3. Procedure 2.2.2. Probabilistic length of deaths rates caused by traffic a probability space must be established through the quotient of the annual variation of the length (L) of deaths, now considered as an event, divided by the sum of the rates of these dates between 2004–2009 through Eq. 2 as follows:
P (L) =
According to the values of the death rates in Colombia comprehended between 2004–2009, the lengths of the annual variations were established (Eq. 1). Following the steps of the methodology (Rodríguez and Prieto, 2010a), the values of the coordinates in the y axis represent the death rates caused by traffic reported for those years while the variation of the values of time in the x axis represent time, and, since the annual variation remains as a constant, its value is zero. Then, the first probability space was established when each length was considered as a probabilistic event. The probability for each length
Length of annual variation of deaths caused by traffic L = Totality of lengths TL (2) 2
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was calculated from the division of the value of every length over the sum of all lengths (Eq. 2). Subsequently, a second probability space was established when the calculated values of probabilities of death rates through Eq. 3. To determine if the values of probability presented a loading towards specific values, the root mean square (Eq. 4) was established comparing this value with the expected one. When the set of values that belong to the first and second probability space was defined, a third probability space, used to predict the death rate caused by traffic for 2010, was established from the arithmetic average of the three previous lengths found between 2007 to 2009, calculating the probabilistic lengths of these values. Then, the arithmetic average of the probabilistic lengths and the sum of lengths between 2007 to 2009 as well as the value of the length for 2009 were replaced in Eq. 5, obtaining as solution two values that correspond to the ranges that determine the prediction. The establishment of a fourth probability space was developed with two events: decreases (D) and augments (A) of deaths with respect to the previous year. This space was built in order to determine which of the two obtained values with Eq. 5 results more likely in order, which will set the value for 2010. This behaviour was studied with consecutive periods of two and three years
Table 2 Frequency and probability of consecutive augments (A) and decreases (D) between 2004–2009 of the deaths rates caused by traffic.
P(L)
P(N)
RMS+
RMS-
RMS+P
RMS-P
2004 2005 2006 2007 2008 2009
15.3 14.3 14.8 15.3 14.7 14.7
15.3 1 0.5 0.5 0.6
0.8 0.06 0.02 0.02 0.03 0
0.17 0.16 0.16 0.17 0.16 0.16
0.22 0.21 0.21 0.22 0.21 0.21
0.11 0.10 0.11 0.11 0.11 0.11
0.05 0.05 0.05 0.05 0.05 0.05
−0.05 −0.05 −0.05 −0.05 −0.05 −0.05
1 2 3 4 Total
1 1 0
2 0 0 0 2
0.2 0.4 0 0 0.6
3
0.4 0 0 0 0.4
Combinations
DDD
DDA
DAD
DAA
ADD
ADA
AAD
AAA
Total
Value Probability
0 0
0 0
0 0
1 0.3
0 0
1 0.3
1 0.3
0 0
3 1
4. Discussion This is the first paper that confirms the capability of the methodology to analyse and predict the behaviour of the temporal dynamic of death rates caused by traffic in Colombia, proving the existence of an analogy when compared with the trajectory of a random walk through the establishment of probabilistic lengths and the variation of the last three consecutive years. A percentage of success of 86.6% was obtained with respect to the real value for 2010. This shows that the phenomenon is irregular and complex, but in a practical manner, the methodology is useful to complementarily perform predictions in a quick and easy way, which could be handy in public health when evaluating the impact of strategies to prevent mortality and improving road safety. The methodologies that explore issues such as predicting death rates secondary to road traffic injuries are based mainly on big data. Some of these studies incorporate artificial neural networks optimized through Genetic algorithm or time-series analysis with autoregressive integrated moving average (ARIMA) with explanatory variables (ARIMAX) and seasonal autoregressive integrated moving average (SARIMA) among others (Leveau and Ubeda, 2012; Jafari et al., 2013; Parvareh et al., 2018; Ihueze and Onwurah, 2018; Liyan and Chunfu, 2009; García et al., 2018; Deublein et al., 2015; Mehmandar et al., 2016; Jayatilleke and Jayatilleke, 2016; Zolala et al., 2016; Huang et al., 2016). However, although they are effective and provide valuable information about some cause-effect relationships, these require and rely on huge inputs collected from several years of epidemiological retrospective information, specialized software and personnel as well as the necessity for adjusting the results for each studied population. Similarly, as the aforementioned methodologies, this complementary method considers the values reported by official entities that gather information such as death rates by traffic. Due to the fact that some primary data sources under-report these events or the data is not accurate (Samuel et al., 2012), the predictive precision could be affected by this factor. On the other side, this method only requires values of annual death rates up to three previous years to develop predictions, which reduces the evaluation of methods such as endemic channels that require at least 5 years (Hernández et al., 2016). Also, it is not necessary to perform populational statistical analyses determined by epidemiology, since the physical-theoretical perspective in which this investigation
Table 1 Values of the annual deaths rates caused by traffic in Colombia between 2004–2009. L
Decreases Value
average between the two values predicted for 2010 was found, which was 14.70 per 100.000 general population, and was later compared to the real value, which was 12.9 per 100.000 general population, finding a success percentage of 86.6%.
It was found that the values of the lengths for death rates caused by traffic in Colombia between 2004–2009 were among 0.5–15.3 per 100.000 general population. Probabilistic lengths varied between 0 and 0.8 while the values of the proportion of distances with regards to the minimal distance found were between 0 and 0.3 (data not shown). The probability of the annual death rates varied between 0.16 to 0.172 while the values of root mean square varied in a range of 0.1 and 0.2 finding that a difference between the last ones and the expected values varied in a range of 0,05 to -0,05 (Table 1). Indeed, the difference among the values of the root mean square obtained for the death rates show that the behaviour of this dynamic is not equiprobable, but it shows a loading of probability, allowing the establishment of predictions (Table 1). Based on the analysis performed to the probability space of the last three years and the application of Eq. 5, two values of the prediction were determined for the death rates, which were 15.07 and 14.33 per general 100.000 population (data not shown). The determination of the most likely event for 2010 was made through the study of the behaviour of the frequency and probability of consecutive augments (A) and decreases (D) between 2004–2009 (Table 2). The analysis of this probability space showed that for two or more consecutive years, the number of consecutive A and D varied between 1 and 4, finding that the augments for two years have a probability of 0.4 (Table 3). When evaluating the possible combinations of A and D for a period of three consecutive years it was found that there is a higher probability of finding ADA, not finding the same possibility for those cases of possible decreases (Table 3). Then, to establish the most likely event for 2010, the result of the arithmetic
D.R
Augments Value
Table 3 Number of possible combinations of augments (A) and decreases (D) for a period of three consecutive years of the deaths rates caused by traffic in Colombia.
3. Results
Year
Consecutive years
DR: death rates; L: length of deaths rates caused by traffic; P(L): probabilistic length of the deaths rates caused by traffic; RMS: root mean square. 3
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was framed, that is, following the paths of the logical-inductive reasoning independent of causal factors, establishes that generalizations are applicable to any particular case in the universe and no adjustments are required for variables such as sex, age or any other, which simplifies the analysis the phenomenon by excluding the analysis of other variables that may be confusing. Finally, no specialized software or personnel are required to reproduce the method. It is worth noting that this method does not consider the most likely locations of road traffic events that result in mortality or disability as outcomes and the most likely populations involved are as well no predicted, which are considered in other studies. However, future applications of this methodology could be oriented to predict the specific areas of cities where deaths are result of road traffic injuries. In the context of probability theory, random walk allows to mathematically predict the phenomena and allows to study the efficacy of the campaigns oriented to improve road safety since the methodology provides a reliably expected value of mortality. This approach, from the abstractions of the phenomena, allows to generally apply methodologies from theoretical physics and mathematical laws in the scenarios of clinical practice and public health with good results. For example, generalizations have been performed in the field of arterial re-stenosis (Rodríguez et al., 2010b), the prediction of the binding of peptides to HLA class II (Rodríguez, 2008), as well as CD4+ lymphocytes count relying on blood count (Rodríguez et al., 2013b) and the evolution of cardiac dynamics (Rodríguez et al., 2018).
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