Multiple Linear Regression Model of Meningococcal Disease in Ukraine: 1992–2015

Estimating the rates of invasive meningococcal disease (IMD) from epidemiologic data remains critical for making public health decisions. In Ukraine, such estimations have not been performed. We used epidemiological data to develop a national database. These data were used to estimate the population susceptible to IMD and identify the prevalence of asymptomatic carriers of N. meningitidis using simple epidemiological models of meningococcal disease that may be used by the national policy makers. The goal was to create simple, easily understood analysis of patterns of the infection within Ukraine that would capture the major features of the infection dynamics. Studies used nationally reported data during 1992–2015. A logic model identified the prevalence of carriage and the proportion of the population susceptible to IMD as key drivers of IMD incidence. Multiple linear regression models for all ages (total population) and for children ≤14 years old were fit to national-level data. Linear models with the incidence of IMD as an outcome were highly associated with carriage and estimated susceptible population in both total population and children (R2 = 0.994 and R2 = 0.978, respectively). The susceptibility rate to IMD in the study total population averaged 0.0034 ± 0.0009% annually. At the national level, IMD can be characterized by the simple interaction between the prevalence of asymptomatic carriage and the proportion of the susceptible population. IMD association with prevalence rates of carriage and the proportion of susceptible population is sufficiently strong for national-level planning of intervention strategies for IMD.


Introduction
e global incidence of invasive meningococcal disease (IMD) ranges from 500,000 to 1,200,000 cases annually; among which 50,000 to 135,000 cases are fatal [1]. Despite the availability of efficient antibiotic therapy and vaccines against various subgroups of meningococcus, meningococcal infection (MI) remains a global health problem. Continued surveillance is needed to predict the dynamic changes in the epidemiology of the disease and to impact the recommendations for current and future vaccines or other prevention strategies [2]. Despite the fact that there are tools for the prevention of IMD, 10-1,000 cases per 100,000 population occur during epidemics in the African meningitis belt, but the incidence of this disease in Europe, North America, and Australia still ranges between 0.3 and 3 cases per 100,000 population [3,4].
Reduction in morbidity and mortality can be achieved by appropriate and effective control methods and well-developed prevention strategies informed by a high-quality surveillance system. Statistical modeling is an important tool to study the structure of the epidemic process, and multiple regression analysis of empirical data has been successfully applied in countries with different levels of incidence and mortality of IMD to address this issue [5][6][7]. e incidence of IMD is most commonly used as the dependent variable. e following independent variables (predictors) also have been included: demographic variables (e.g, total population, population density, and percentage of urban population) as well as environmental variables (e.g., percentage of surface by land use or land cover and percentage of water bodies) and atmospheric variables (e.g, wind speed, air temperature, and humidity) [8,9]. Predictors can also include asymptomatic carriers (presence or density), susceptible population size, herd immunity, smoker prevalence, poverty prevalence, and other indicators of public health [10,11].
Such analyses have not been conducted in Ukraine, which makes prevention challenging. Our research aimed to integrate epidemiological and microbiological data collected in Ukraine over a 24-year period  to estimate the proportion of the susceptible population and prevalence of asymptomatic meningococcal carriers. Our analyses were developed using the multiple linear regression model, including necessary and most prevalent risk factors [12].

Study Design.
We provide a descriptive populationbased study of the incidence of invasive meningococcal disease and the carriage of meningococcal infection, which is based on the linear regression model of the epidemic process of meningococcal infection.

Analysis.
We developed a logical model of the epidemic process of the infection (see Figure 1). is model is presented as an organogram of hierarchical subordination of various causal factors in the epidemic process of IMI infection [12]. In a general sense, the numbers of IMD depended on the size of the susceptible population and the asymptomatic population, as patients displaying IMD are less likely to transmit the pathogen to the population as a whole. Similarly, we assume that asymptomatic carriers interact randomly with susceptible population members.
e sizes of the susceptible and carrier pools were estimated from retrospective, cross-sectional survey data. e model aggregated the oblast-level data to the national level as the sum of the indicators for all the administrative divisions. e size of the carrier pool was estimated from the prevalence of meningococcal carriage in the survey population. e index of "susceptibles" was derived as the proportion (%) of IMD among meningococcal carriers (approximate proportion of the population susceptible to IMD � APPSIMD) as they developed clinical disease as a result of infection and disease progression. To calculate APPSIMD, we first calculated the annual estimated number of carriers (infected people without clinical manifestations). en, we calculated the approximate amount of annual meningococcal infected persons as where AAQC, annual approximate quantity of carriers (infected people without clinical manifestations of IMD); CPR, carrier prevalence rate (the ratio of the carriers detected in the number of examinees); N, the number of population of the oblast; 365, days in a year; and D, the average duration of carriage (14 days) [14].
e proportion of susceptible population then was calculated as e IMD analysis used multiple regression [15]: where Y, incidence IMD per 100,000 population; a, Y intercept; X 1, prevalence of carriage (%); X 2, approximate proportion of susceptible population (APPSIMD); b 1 , the regression coefficient for prevalence of carriage; and b 2 , the regression coefficient for approximate proportion of the population susceptible. e predictor and output parameters were tested for normality [16]. Informativeness of the model was characterized by the coefficient of multiple correlation R. Mathematical modeling of IMD was carried out in several stages: development of a logical model (see Figure 1), development of a mathematical model, and assessment of the model quality. e main goal of the logical model development was to demonstrate the causal relationship between the various factors involved in the epidemic process of meningococcal disease.
e quality of the model was assessed by the indicators of its informativeness, adequacy, stability of Pearson correlation coefficient, and model structure. Calculations were performed using MS Excel 2003 and NCSS 2000 software.  properties of the model. e forms of both equations had IMD incidence per 100,000 rising with increases in both prevalence of carriage and with the proportion of susceptible populations for total population:

us
And for children (0-14 years of age), e model of the general population is adequate, since the calculated value of F-test (895.28) in our case is significantly higher than the F-table value (8.75). is model with a 95% probability reflects a set of properties of the epidemic process of meningococcal disease.
e stability of the model or structure of the regression equation corresponds to two basic conditions: (1) the maximum coefficient of pair correlation (r) between regressors X 1 and X 2 is less than 0.3-0.5 and is r X 1 X 2 � 0.2959; (2) the coefficients of pair correlation with Y in absolute value are much higher than the correlation coefficient between regressors (r YX 1 � 0.7659 and r YX 2 � 0.8321 more than r X 1 X 2 � 0.2959). us, this model is statistically stable and uncorrelated.
It follows from the linear regression equation of the general population that if the level of meningococcal carriage in Ukraine increases by 1%, the incidence of invasive meningococcal disease increases by 0.84 per 100,000 population. If the susceptible population increases by 1%, then the incidence of IMD increases by 455.58 per 100,000 population.
Overall, there has been a downward trend in the rates of IMD among both total population and children over the course of the study (see Tables 1 and 2). On average, the rates of IMD are about ten times more sensitive to the carriage rate in children (0-14 years of age) than in total population. Ukraine has reduced the incidence of IMD through a decreased number of carriers (X 1 ) and a decrease of susceptible population percentage (X 2 ) between 1992 and 2015. We also calculated the approximate annual number of carriers of meningococcal disease, which accounts for one confirmed IMD case (see Table 1) ranging from 17,252 (1992) to 45,967 (2015) with an average of 31,113 carriers.
It is worth noting that this indicator in the general population had a strong tendency towards increase, and among children, it tended to decrease. In our opinion, this difference is due to higher susceptibility of children under the age of 14 to meningococcal infection than the general population, and in the general population, adults who are less susceptible predominate. us, a direct benefit of the model is helping assess the efficiency of vaccine prevention of meningococcal disease. e objective criterion for the effectiveness of vaccination is the proportion of susceptible population, which can be calculated by using the parameters of the model. Ultimately, the proportion of susceptible total population occurs within the range of 0.00218% (2015) to 0.0058% (1992) indicating that a limited proportion of the population of Ukraine is susceptible to IMD (see Table 1). Calculations show that an approximate number of people who have been infected with meningococcal disease in Ukraine ranged annually from 13,884,822 to 34,746,741 persons.

Discussion
e advantage of regression statistical models for epidemiological assessments is the ability to estimate the morbidity and evaluate the effectiveness of vaccine prophylaxis from relatively simple parameters and assumptions. Research on forecasting of meningococcal disease and the efficacy of meningococcal vaccines via regression analysis has been conducted primarily in the United Kingdom and countries of the European Union, North America, New Zealand, and the countries of the meningitis belt [5][6][7].
In Ukraine, such studies have not been conducted until now. Our regression model is aimed at quantifying the main factors that form the incidence of IMD in Ukraine.
A positive feature of the model is the ability to quantitatively produce a representation (within a given statistical confidence interval) of the complex causal relationships between risk factors and the IMD epidemic process. e latter allows determining the specific weight (importance) of the effect of individual factors on the epidemic process of IMD. e quality of our models largely depends on the quality of the accessible data. e model used aggregated data from survey values of IMD morbidity and carriage among 26 regions as territorial units. In our paradigm, the susceptible and carrier pools are the only risk factors of IMD emergence and spread in the human population. ese variables at least have some potential of being monitored as part of ongoing public health surveillance. All other possible factors that may affect IMD prevalence are acting indirectly through risk of infection (RI) or risk of contamination (number of carriers or percentage of carriers) and risk of susceptibility (RS) or percentage of people who became ill when infected (see Figure 1).
We have made many assumptions and simplifications for this analysis in characterizing the transmission of the pathogen.
ese include that (1) the level of carriage of meningococcal pathogens does not change within one year; (2) the intensity of the transmission mechanism is relatively stable within one year; (3) the risk of contagion for all members of the population (the total population of Ukraine) is uniform. Such assumptions and simplifications may be acceptable because the size of the susceptible and infected pools is at any given time, a negligible proportion of the entire population.
Our models are further limited by using aggregated passive data from a survey, so that formal residual analysis is limited. It should be stressed that the proposed model does not forecast epidemics, as the quantitative values of the input parameters of the model and the severity of disease for each individual and the period of time are retrospectively identified.
Taking into account the prevalence of asymptomatic meningococcal carriage to construct a regression model of meningococcal disease appears to be one of the most important model variables. Carrier state research can make a significant contribution to our understanding of the epidemiology and pathogenesis of diseases caused by N. meningitidis [17].
Another important factor that can negatively affect the quality of our data is the sensitivity of bacteriological tests. e average annual prevalence of meningococcal carriage in Ukraine was reported as 1%. Additionally, in other European countries, 10% of the total population appear to be carriers of N. meningitidis [7,17], suggesting that the prevalence of carriage of meningococcus in Ukraine may be underestimated.
Another important indicator for the construction of the model is the duration of meningococcal carriage. In accordance with our observations of meningococci in carriers with repeated bacteriological examination, the average duration of carriage appears to be 14 days. is value is empirically derived as twice the average incubation period [18]. If the sensitivity of the bacteriological diagnostic of meningococcal infection in Ukraine is less accurate than in other European countries, the duration of the carriage may appear longer than 14 days. Some researchers believe that the duration of meningococcal carriage occurs within the range of 1 to 9 months [6], but other researchers believe that the carriage lasts from several days to months [19].

Conclusions
e present model can be used as a prototype for the construction of models of those infections that have similar epidemiological patterns (i.e., aerosol transmission, asymptomatic clinical forms, <1% of the population susceptible, and high mortality rate) including pneumococcal, Haemophilus influenzae type b (Hib) infection, streptococcal, staphylococcal, or diphtheria among others. e analysis suggests that the nature of MI epidemic process strongly correlates with the prevalence of meningococcal carriage as well as the size/density of susceptible populations, both representing factors of immediate risk of 8,120 * IMD/100,000 total population; † X 1 is expressed as a percentage; ‡ APPSIMD is expressed as an approximate proportion of the total population susceptible to invasive meningococcal disease, where APPSIMD � (IMD/AAQC) × 100%; § predicted IMD rate/100,000 total population. ¶ AAQC � annual approximate quantity of carriers (number of infected people without clinical manifestations of invasive meningococcal disease); # SD � standard deviation.
IMD infection and spread. Altogether the present and past surveillance of bacterial meningitis in Ukraine provides a unique source for a comprehensive understanding of the disease dynamics and, most importantly, allows for the development of tools and strategies for disease control and prevention. us, our model of the epidemic process of IMD shows a very small and stable proportion of the total population (an average of 0.00343%), which is susceptible to meningococcal infection; therefore, in Ukraine, the change in the incidence of IMD depends mainly on the level of healthy carriage of meningococci.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest. ; † X 1 is expressed as a percentage; ‡ APPSIMD � (IMD/AAQC) × 100%; § predicted IMD rate/100,000 children. ¶ AAQC � annual approximate quantity of carriers (i.e., # infected children displaying no IMD); # SD � standard deviation. 6 Computational and Mathematical Methods in Medicine