Epidemiological Prognosis of Pertussis Incidence in Bulgaria

Introduction: Epidemiological forecasting facilitates scientifically sound solutions to upcoming theoretical and practical issues, in the development of public health management, in particular of infectious diseases. Aim: To critically analyze the most recent scientific advances in the biosocial nature and methodology of epidemiological forecasting to present a real-life example of pertussis, a disease with shifting epidemiology. Materials and methods: For the prediction of pertussis morbidity the autoregressive integrated moving average (ARIMA) the model was established by utilizing the method of time series analysis to construct a model of overall morbidity using Time series modeller in SPSS v.25. To model pertussis morbidity we obtained official data from the Ministry of Health and the National Center for Infectious and Parasitic Diseases, since the beginning of disease registration from 1903 until 2018. We also analyzed the shifting epidemiology of pertussis. Results: The proper identification procedures we applied indicated ARIMA (3,0,0) model to best fit our original time series of the annual whooping cough morbidity for the 1921-2018 period. The model predicts better morbidity in a one-step forecast. The incidence rate is expected to be stable at about 1.35 per 100,000 in the next three years, which is close to the 2016 level and lower than those in 2017-2018. Conclusion: The ARIMA (3,0,0) model in our study is an adequate tool for presenting the pertussis morbidity trend and is suitable to forecast near-future disease dynamics, with acceptable error tolerance.


INTRODUCTION
In the 1940s, one of the founders of epidemiology of infectious diseases, LV Gromashevsky, said: "The scientific thought has always sought not only to accumulate facts but also to systematize them and to explain them in a single theoretical system". 1 An epidemiological prognosis provides an opportunity for scientifically sound solutions to the-oretical and practical issues in the development of science and public health management, in particular of infectious diseases at future stages. The first attempts for a discussion in Bulgaria date from the late 1960s 2 and the early 1970s. In 1976, the book "Epidemiological prognosis", based on the biosocial nature of the epidemic process of infectious diseases, was published by Emil Kamenov, professor of epidemiology. 3 Science and practice in Bulgaria are in debt to the basic methods of epidemiological prognosis and mathematical modeling of infectious diseases. The other descriptive, analytical and experimental methods are broadly in use. At present, the molecular epidemiology and a variety of mathematical techniques are reported in the literature dealing with epidemiologic prognosis to be in the focus of interest of scientists. [4][5][6] Prognostic studies allow better understanding of the natural history of an epidemic process, facilitate the selection of appropriate optimal solutions for good medical practice, as well as more accurately predict the outcome of the disease. 7 The evaluation of the results of an appropriate statistical analysis allows us to draw conclusions that can influence the epidemic process.

AIM
The aim of the present study was to analyze the most recent scientific advances in the biosocial nature and methodology of epidemiological prognosis in order to present a real-life example of pertussis, a disease with shifting epidemiology.

MATERIALS AND METHODS
We obtained official data from the Ministry of Health and the National Center for Infectious and Parasitic Diseases, Sofia, for pertussis morbidity since the beginning of its registration in 1903 until 2018.
In the article we analyzed the shifting epidemiology of pertussis by means of a chronological overview. For prediction of pertussis morbidity, the autoregressive integrated moving average (ARIMA) model was established by utilizing the method of time series analysis to construct a time series model of overall morbidity using Time series modeller in SPSS v.25. The ARIMA modeling was based on the yearly morbidity pertussis rates (per 100,000 population) from 1903 to 2018 (the whole period) and from 1921 to 2018 (due to missing data from 1908 to 1920) in Bulgaria. The decomposition of pertussis morbidity time series into a stochastic trend and a stationary component was reasoned based on the model defined. In addition, we assessed the importance of the short-term and immediate effect of one shock. 4

RESULTS
We selected the Auto-Regressive Integrated Moving Average (ARIMA), a forecast model pointed out by many researchers as having high accuracy for short-term forecasting. It was proposed by Box and Jenkins in the early 1970s. It includes three components: ' AR' -autoregression, 'I' -integration, 'MA' -moving average. ARIMA is an epidemiological method for predicting future values based on previously observed values and is one of the most effective forecasting for time series, i.e. time series have been increasingly used in recent years for epidemiological research for decision-making purposes. Extracting meaningful statistics and data characteristics is tantamount to dynamic time series analysis.
Both models 1 and 2 have a similar coefficient of determination (R 2 ) of 64% and therefore both have a good approximation to the observed values ( Table 1). The one-step  forecast for Model 1А differs from the real data for 2018 (1.62 per 100 000) by 0.83 per 100,000, while in Model 2А the difference is 0.69 per 100 000. Model 1 approximates a period of 115 years, with linear time series (d=1) and eliminated by subtraction from the data. There is a correlation between the observations at dY and the correlation coefficient is φ 2 (p=2). There is also a relationship between the current value and the random deviation at lag 2 (q=2) (i.e. two steps back in the row), with the correlation coefficient θ3 representing the magnitude of this relationship. Model 1 represents the approximation of a more complex time series having simultaneously three components in the ARIMA model, i.e. a mixed modelwith a trend: relationship between each current and several previous row values; and dependence also between the deviations with different lags. Due to data gap Model 2 approximates a period of 97 years with the time series being stationary (d=0) and missing the moving average component (q=0). The number of autoregressive components (p) is 3, therefore there is a correlation between the observations in d3Y and the correlation coefficient is φ3. Model 2A is closer to Model 2 in forecast than Model 1A to 1.
The benefits of these models make them optimally suited for pertussis epidemiological modelling, a disease with shifting epidemiology. Additionally, for the period 2012-2018 we have analyzed the shifting epidemiology of pertussis by focusing on the age distribution of cases. The registered cases of pertussis are 606 (average 86 cases per year) and they vary between 35 (2015) and 116 (2017). Children under 4 years of age are with the highest proportion-60.4%, followed by children between 5-9 years -13.86% (Fig. 2). Almost 15% of all pertussis cases are in children above 10 years of age (10-14 yrs. -10.07% and 15-19 yrs.-4.95%).
fusion of molecular biology with the epidemic process to further characterize the epidemiological discipline). Molecular techniques make the study more comprehensible and indicate: reliably to the highest degree the causes of the occurrence and spread of infectious diseases; unknown aspects of infectious diseases; solutions to more effective prevention and control strategies at an individual and population levels; ideas for future directions for research in this field; necessity to train competent professionals. 8 Looking back, we have found that in Bulgaria the epidemiological prognosis is based on the long-standing and currently complex biosocial nature of the epidemic process.
Unfortunately, in our country, for a long period, the mathematical and statistical modelling methods are insufficiently developed and slightly used, even underestimated. In the 1970's they are defined as vehemence. 3 9 Looking ahead the following categories are defined according to the literature: epidemiological prognosis model; the modeling process and its logical sequence; the available and evolving science-based and refined accurate prognostic models such as ARIMA, etc. In this regard, prognostic studies require the selection of variables or risk factors that assess their impact on the outcome of the epidemic process. Many variables have a confounding effect on other parameters and this should be minimized. The conditions under which the prognostic study is conducted should correlate with the local practices of the institutions in which the finding might apply. The evaluation of the prognostic studies is based on determining the internal validity of the design and assessing the impact of systemic errors or biases. 6 The ARIMA (p, d, q) models are applied when the time series are stationary, i.e. characterizes the change of a random process in which there is no dynamics (there is no trend of change). Most epidemiological time series are non-stationary and before the application of ARIMA models, they need to be transformed into stationary ones.
The proper identification procedures, we applied, indicated ARIMA (3,0,0) model to best fit our original time series of the annual whooping cough morbidity for the 1921-2018 period. In trend-stationary models, characterized by given common properties, the time series returns to the trend, in particular, at stationarity -to the average. 4 As noted Model 2А better predicts morbidity in one-step forecast. The incidence rate is expected to be stable at about 1.35 per 100,000 in the next three years, which is close to 2016 levels and lower than in 2017-2018.
The accurate simulation and prognosis of the cases of infectious diseases indicates the resources and the methods for planning of future epidemics. However, in some diseases, the cases cannot be easily identified or managed. Such an example is whooping cough, which is similar to other

DISCUSSION
The biosocial nature of infectious diseases is based on a branch of epidemiology -molecular epidemiology (the respiratory diseases which have a mild or atypical course. In addition, there is a shift in morbidity, which is affected and limited by several reasons. First is the shifting trends in morbidity -in countries with mass vaccination the morbidity has decreased by 100 times since 1953 10,11 and in Bulgaria the reduction is by 150 times. 12 Second is the periodic changes (the incidence has increased in the last decade in the EU). Globally pertussis is endemic, it never completely disappears, making it an important public health problem. [13][14][15] For an example, with 86% global immunization coverage in 2016, 131,991 have been infected and 89,000 have died, 95% of them in developing countries. 14 Third is the accidental disorders due to lack of or inaccurate diagnosis, lack of awareness, etc. Some effective and accurate models also need to be identified for the prediction of the disease based on historical data. 12 Analysis of the shifting epidemiology of whooping cough has revealed some important facts: 1. Despite the high vaccination coverage among young children worldwide, many countries have reported an increase in the cases of whooping cough in adolescents and adults. 16,17 Our study identified 91 (15.02%) of a total of 606 cases in children over 10 years of age which is in favour of a booster dose of the vaccine, for an example at 12 years.
2. The age distribution of the laboratory-confirmed 4341 cases of whooping cough in England for 2017 reveals: 81% of the cases are in children over 14 years of age, 10% -0-9 years of age, 9% -10-14 years. 18,19 For 2017 our results show a much lower proportion of those over 14 years old -17.23%, but extremely high proportions -74.14% in children 0-9 years and 8.62% for 10-14 years old.
3. Natural infection and vaccination do not provide lifelong protection against whooping cough. The change during the years in the estimates of the duration of protection after natural infection is as follows: for life in 1951, for 20 years in 1995, for 7-10 years in 1997 and for 3.5-12 years in 2002. The duration of protection after vaccination with whole-cell vaccines is estimated to be 4-12 years, and with acellular vaccines 5-6 years. 14,20,21