Data Analysis and Modelling in Contemporary Epidemiology

in the frame of the International Conference “Mathematics Days in Sofia”


ver the past three years, the Covid-19 pandemic has caused a boom in the development of modern mathematical methods for the analysis and modeling of epidemiological data. Meaningful interpretation of the data requires close interaction between medical professionals (doctors, virologists, epidemiologists, immunologists) and specialists in the field of mathematical modeling. The purpose of the proposed mini-symposium is to provide a forum for these professionals to meet, where the latest results in the field can be shared and ideas can be exchanged.

Organizing Committee

  • Antoni Rangachev (Institute of Mathematics and Informatics, Bulgaria)

  • Georgi Boyadzhiev (Institute of Mathematics and Informatics, Bulgaria)

  • Ognyan Kounchev (Institute of Mathematics and Informatics, Bulgaria)

  • Radka Argirova (Acibadem City Clinic Tokuda Hospital, Bulgaria)


  • Antoni Rangachev, Institute of Mathematics and Informatics, Bulgaria

  • Dimiter Atanasov, New Bulgarian University, Bulgaria

  • George Dimitrov, Medical University of Sofia, Bulgaria

  • Georgi Boyadzhiev, Institute of Mathematics and Informatics, Bulgaria

  • Georgi Simeonov, Institute of Mathematics and Informatics, Bulgaria

  • Hristiana Batselova, Medical University of Plovdiv, Bulgaria

  • Latchezar Tomov, New Bulgarian University, Bulgaria

  • Ognyan Kounchev, Institute of Mathematics and Informatics, Bulgaria

  • Radka Argirova, Acibadem City Clinic Tokuda Hospital, Bulgaria

  • Svetoslav Markov, Institute of Mathematics and Informatics, Bulgaria

  • Trifon Valkov, Medical University of Sofia, Bulgaria

  • Tsvetomir Tsachev, Institute of Mathematics and Informatics, Bulgaria

Program and Abstracts

In this talk I will present the results of three papers (see [1]-[3] below) on the geographic and demographic impact of the COVID-19 pandemic in Bulgaria. We use standardized years of life lost measures to assess the impact of the COVID-19 pandemic which allows for comparisons between populations with different demographic structures. Bulgaria emerges as the hardest hit country in Europe exhibiting record mortality in the working age population and mortality in certain regions and demographic groups far exceeding the average for the country. I will also discuss how excess mortality as measured by years of life lost metrics correlates with the prevalence of cardiovascular disease and COVID-19 vaccination rates in Europe.


[1] A. Rangachev, G. Marinov, M. Mladenov, The demographic and geographic impact of the COVID pandemic in Bulgaria and Eastern Europe in 2020, Scientific Reports 12, Article number: 6333 (2022).

[2] G. Marinov, M. Mladenov, A. Rangachev, I. Alexiev, SARS-CoV-2 reinfectionsduring the first three major COVID-19 waves in Bulgaria, PLoS ONE 17(9): e0274509.

[3] A. Rangachev, G. Marinov, M. Mladenov, The impact and progression of the COVID-19 pandemic in Bulgaria in its first two years, Vaccines, 2022 Nov; 10(11): 1901.

In this work a statistical model for COVID-19 infection dynamics is described, using only the observed daily statistics of infected individuals. For this purpose, two special classes of branching processes without or with an immigration component are considered.

Using these models some main parameters of the infection can be estimated and the mean value of the non-observed infected population can be predicted. The serious advantage of this approach, in comparison with other more complicated models, is the requirement of only the officially reported data, which are sufficient for estimation of the model parameters. The model can be applied for different regions in the world and the corresponding parameters of the infection can be compared.

The present work is joint research with Vessela Stoimenova (Sofia University), and Nikolay M. Yanev (IMI-BAS).

The pandemic caused by the SARS-CoV-2 virus took everyone by surprise despite warnings highlighting the importance of developing programs involved in pathogen-discovery in emerging disease hotspots, associated with high risk wildlife groups, in addition to increasing health promotion for key modifiable risk factors as a strategy for pandemic preparedness [1]. Unfortunately, as almost no such measures were taken globally, the pandemic has led to countless unnecessary suffering and premature deaths [2, 3]. Efficient pandemic control is based on three main points: (I) Timely and centralized epidemiological measures; (II) Rapid and synchronized vaccination and (III) evidence-based disease severity staging and appropriate treatment approaches. Initially, there were major differences between the various European countries in all of these aspects. With time, however, most countries unified their mindsets while Bulgaria, sadly, retained its “one size fits all” mentality. Therefore, the authors, with support from the Bulgarian Medical Association, developed the first in their kind for the country, physician targeted and free to use digital vaccination, disease staging and treatment platforms which allow for the rapid individual assessment juxtaposing objective patient factors and current best evidence based medical practices. These dynamically updated tools enable the automatization and personalization of the whole decision-making process in regards to COVID-19 vaccination, disease severity stratification and management. Currently, these platforms were successfully implemented in the clinical setting with routinization, on a national level, to follow after the development of full digital accessibility.

This is a joint work with T. Valkov (Medical University of Sofia), Ognyan Kounchev (IMI-BAS), and R. Argirova (Tokuda Hospital, Sofia).


  1. Ge, X.Y., et al., Isolation and characterization of a bat SARS-like coronavirus that uses the ACE2 receptor. Nature, 2013. 503(7477): p. 535-8.
  2. Ugarte, M.P., et al., Premature mortality attributable to COVID-19: potential years of life lost in 17 countries around the world, January-August 2020. BMC Public Health, 2022. 22(1): p. 54.
  3. Collaborators, C.-N.P., Pandemic preparedness and COVID-19: an exploratory analysis of infection and fatality rates, and contextual factors associated with preparedness in 177 countries, from Jan 1, 2020, to Sept 30, 2021. Lancet, 2022.

In this talk a spatially heterogeneous deterministic Susceptible-Exposed-Infectious-Removed (SEIR) model is presented. It is suitable for the first stage of the epidemic when the rate of Infected persons is relatively low. The system of parabolic PDEs that describes the model is cooperative one, and some features as the validity of comparison principle, existence and uniqueness on the solutions are given.

The main purpose of the present research is to demonstrate the significant association between the restrictive measures called non-pharmaceutical interventions (NPIs), their degree of compliance or non-compliance, and the shape of the resulting infectious curve, for the case of the Omicron variant of SARS-Cov-2 virus.

We pay special attention to the period when the Omicron variant emerged – beginning of December 2021 to end of January 2022 – and the period after when the Omicron variant started to decay. Since early 2020, NPIs implemented at different levels of rigor and based on widely-divergent perspectives of risk tolerance, have been the primary means to control SARS-CoV-2 transmission. Due to the short time for reaction, practically about two months, NPIs were more suitable to implement in January-February 2022 in Bulgaria than vaccination (assuming boosters are excluded).

We provide a study of the Covid-19 spread in Bulgaria in the period starting December 15, 2021 until early February, 2022. In particular, we provide predictive scenarios for the peak of the pandemic. Based on these scenarios, we estimate the risks in terms of fatalities in the case no restrictive measures are imposed. The main challenge is distinguishing the influence of the Delta variant which is still dominating in December, 2021; while Omicron becomes dominant in early January, 2022.

In the present research we use the deterministic SEIR models for analysis and forecasting of the Omicron data, see Brauer (2017) and Hethcode (2000).

The present work is joint research with O. Kounchev, G. Boyadzhiev, M. Kunchev, Zh. Kuncheva.


  1. Brauer, F. (2017) Mathematical epidemiology: Past, present, and future, Infectious Disease Modelling 2 113-127.
  2. Hethcote, H. W. (2000) The mathematics of infectious diseases. SIAM Review, 41, 599-653.

Varicella is an acute infectious disease caused by Varicella-zoster virus or Human herpesvirus 3 (HHV3). It is characterized by extreme contagiousness. Chickenpox is a common disease worldwide. In the prevaccination era, chickenpox was endemic in the United States, and virtually all adults had already had the disease. As a result, about 4 million people fell ill annually, which is almost the entire birth cohort in one year. The age group under 15 was most affected (90%). In the absence of a varicella vaccine, the number of cases each year is equal to the cohort of newborns, with 52-78% of cases being children under 6 years of age and 89-95% becoming ill before the age of 12. Varicella is the most frequently registered childhood infectious disease in Bulgaria. Children under 14 are most affected. Predominance among 1-4 year olds. Epidemic outbreaks in organized collectives happen every year. For a period of 13 years (2009-2021), 317,117 cases of varicella were registered in Bulgaria. Coverage with the live attenuated vaccine is negligible. Measles is a viral infectious disease that is characterized as highly contagious. In the prevaccine era (1963, when widespread vaccination was introduced), large epidemics occurred approximately every 2-3 years, and the disease caused about 2.6 million deaths each year. According to the World Health Organization, measles is one of the most contagious diseases in the world. For the period of three years (2020-2022), coverage in Bulgaria with the first dose of the measles vaccine was below 95%, and in 2 years it was even below 90%. For the same period, children vaccinated with a second dose were less than 89%. This increases the risk of outbreaks of measles and even an epidemic in Bulgaria. Varicella and measles are infectious diseases that pose a risk to public health and the mathematical modeling of their spread is of great importance for the correct implementation of anti-epidemic measures, including immunoprophylaxis.

This is a joint work with George Dimitrov (Medical University of Sofia), Trifon Valkov (Medical University of Sofia), Ognyan Kounchev (IMI-BAS), Georgi Momekov (Medical University of Sofia), Georgi Boyadjiev (IMI-BAS), Radka Argirova (Acibadem City Clinic Tokuda Hospital, Sofia).

Classical epidemiologic modeling consists of deterministic mathematical models, relying on the S-I-R model as base. For simple pandemics with clear seasonality and permanent immunity, and without pathogen evolution, they can be used even for prediction after some initial delay, needed to estimate the characteristics of the pathogen. Covid pandemic, however, is the most complex pandemic humanity has had so far. Seasonality is much weaker; waves are up to five a year and thousands of different variants appeared. Evolution of variants from epidemiological perspective is unpredictable – neither the time of arrival, whether they will become dominant, how virulent are, et cetera. A different approach is needed We predict new cases automatically by the combination of linear birth-death process and change point analysis – a reactive approach for precise short-term prediction (the software is installed on the Avitohol supercomputer). Our approach can be extended to predict cases by age, or to detect changes in mortality. We use time series analysis (regression with Arima errors) approach to predict deaths and to test hypothesis, related to the patterns of spread across age groups and the factors that influence it. We don’t need to wait for estimation of the characteristics of the virus at the price of relatively short-term predictions, with a semi-automated process.

Latchezar Tomov is grantee of the European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria, project № BG-RRP-2.004-0008-C01.

In the talk we discuss the relation between the SEI (Susceptible-Exposed-Infections) epidemiological model and the classical Logistic Model. Using Reaction Network Theory we demonstrate that under certain conditions a SEI reaction network can be “approximated” by a single-step reaction, which is either of logistic autocatalytic type, or, of a first-order exponential non-catalytic type. We thus show that the time evolution graph of the growing species I changes its shape between a sigmoidal logistic-type and a concave first-order exponential-type, depending on the ratio of the two rate parameters involved. More precisely, we show that the Logistic Model is a limit case of the SEI model when the rate of species E tends to infinity. Therefore, we can conclude that the SEI reaction network is a generalization of the logistic reaction network.

Acknowledgments. The second named author is gratefully supported by project KP-06-N52/1 of 2021 at Bulgarian NSF (National Science Fund).


[1] Anguelov R., Borisov M., Iliev A., Kyurkchiev N., S. Markov, On the chemical meaning of some growth models possessing Gompertzian-type property. Math. Meth. Appl. Sci. 2017; 1—2,

[2] Borisov, M., S. Markov, The two-step exponential decay reaction network: analysis of the solutions and relation to epidemiological SIR models with logistic and Gompertz type infection contact patterns, Journal of Mathematical Chemistry, 59(5) (2021), 1283–1315, DOI: 10.1007/s10910-021-01240-8

[3] Borisov, M., Markov, S. On the Numerical Simulation of Exponential Decay and Outbreak Data Sets Involving Uncertainties. Numerical Methods and Applications. NMA 2022. Lecture Notes in Computer Science, vol 13858. Springer, Cham. (2023)

[4] Svetoslav Markov, Reaction networks reveal new links between Gompertz and Verhulst growth functions, Biomath 8 (2019), 1904167,

[5] Svetoslav M. Markov, The Gompertz model revisited and modified using reaction networks: Mathematical analysis, Biomath 10 (2021), 2110023,

The main purpose of the present lecture is to tell about the analysis of the hidden cases of Covid-19 spread in Bulgaria by means of the SEIR model, in the period December 15, 2021 – until March 2022. As is well known from the sequenced data, after mid-January 2021, the dominant variant of SARS-Cov-2 virus was already Omicron.

The numerical experiments which have been carried out are based on the hidden cases till mid- January, provide a forecast of the peak of the infectious curve about February 8, 2022, and reaching the maximum of about 90,000 cases. The forecast shows that end of the epidemics (in Bulgaria) would be in mid-May, 2022. On the other hand, this forecast is compared to the previous forecasts based on the same data excluding hidden cases; they have shown, that the peak will be about February 28, reaching about 75,000 cases. and will vanish until end of May.

Our forecast of the end of the particular wave of epidemics shows just two weeks difference with what happened in real life – the wave ended in the first week of June. It is curious (although quite clear) that though the peak of the curve in the hidden cases approach has shifted to February 8, the cumulative number of cases in both approaches is nearly the same. More details on the methods used in the present research and the non-hidden cases study, are provided as well.

In the present research we use deterministic SEIR models, see Brauer (2017) and Hethcode (2000).

The present work is joint research with G. Boyadzhiev, G. Simeonov, M. Kunchev, Zh. Kuncheva.


  1. Brauer, F. (2017) Mathematical epidemiology: Past, present, and future, Infectious Disease Modelling 2 113-127.
  2. Hethcote, H. W. (2000) The mathematics of infectious diseases. SIAM Review, 41, 599-653.
  3. Kounchev O., G. Boyadzhiev, G. Simeonov, M. Kunchev, Zh. Kuncheva, Implementation of hidden infections in the SEIR model Analysis of the Omicron variant spread in Bulgaria, C.R. acad. Bulg. Des Sci. 2023, to appear

Bulgaria ranks as one of the countries with the highest mortality rate and lowest number of COVID-19 vaccinated individuals in Europe. At the same time, the country is listed amongst one of the highest in regards to vaccine hesitancy and prevalence of chronic diseases. A nation-wide retrospective analysis was conducted to assess COVID-19 outcome in both fully vaccinated and non-vaccinated, high-risk patients harboring a single socially significant chronic condition (i.e., cardiovascular, solid malignancy or a chronic pulmonary disease).

A total of 1,126,946 confirmed COVID-19 patients were retrospectively analyzed between March 2020 and April 2022. We focused on cohorts harboring a single documented chronic comorbidity: Cardiovascular pathology, an oncological disease or a chronic pulmonary disorder, comparing the outcomes in vaccinated and non-vaccinated populations classified by gender and age groups in the ambulatory, hospital and ICU settings.

Out of all the confirmed 247,441 hospitalized COVID-19 cases, 57,846 (23.3%) had a documented cardiovascular disease, 2140 (0.9%) had a confirmed solid malignancy (regardless of stage) and 3243 (1.3%) had an established chronic pulmonary disorder as their only chronic pathology. The cumulative deaths in each subgroup were 10,105 (in-hospital = 5812 and ICU = 4353); 344 (in-hospital = 196 and ICU = 148); and 494 (in-hospital = 287 and ICU = 207), resp. Our findings suggested alarming increasing popularity of anti-vaccine attitudes in society, including among people suffering from cardiovascular, oncological or chronic pulmonary diseases. Statistical significance (p<0.001) was obtained in favor of reduced ambulatory, hospitalization and both in-hospital and ICU related mortality, favoring the vaccinated cohorts, and BNT162b2 as the most effective, at preventing mortality vaccine in all age groups. As a reason for that situation we consider and analyze a range of communication, political, health professional and personal problems – all of them possible to be overcome by a systemic continuous positive information campaign headed by professionals but not by politicians.

This is the joint talk with G. Dimitrov (Medical University of Sofia), T. Valkov (Medical University of Sofia), H. Batselova (Medical University of Plovdiv), O. Kounchev (IMI-BAS), G. Momekov (Medical University of Sofia)

The current pandemic caused by SARS-CoV-2 has faced humanity with the most serious health crisis of the last century. According to WHO data, by 30.04.2023, the total number of people infected with the causative agent on a global scale exceeds 765,000,000 people, and the number of deaths exceeds 20,000,000.

One of the main problems faced by doctors serving such patients was related to the possibility of correct and timely diagnosis and the implementation of adequate therapy. While rapid diagnostic tests, as well as highly sensitive and specific PCR-tests, were created relatively quickly, the problem of correct therapeutic procedures, often requiring an individual approach to each patient, remained unsolved for a long time. This was contributed by:

  1. the appearance of a new strain of the virus, unknown until now;
  2. the lack of adequate, specific and effective antiviral therapy;
  3. the need to apply an individual approach to each individual patient, in view of the large number of accompanying diseases, often showing signs of decompensation against the background of viral infection.

In this Handbook, the authors have included the protocols for diagnosis and treatment of a total of 4 countries (USA, Germany, England and Israel), indicated as leaders in the success rate related to the treatment of this disease. The information in the Handbook is based solely on proven medical practices and does not include subjective and individual approaches not based on such.

We use a SEIR model of spread of an infectious disease endowed with a control parameter (function) to model the spread of the COVID-19 infection in Bulgaria and the attempts of the Bulgarian government to counter it. The control function takes values in [0, 1] and will represent the severity of the restrictions imposed on social life which, in turn, influence the speed of the infection spread. In our setting 𝑢(𝑡)=0 means that no restrictions are imposed at moment 𝑡 (relaxed regime) and 𝑢(𝑡)=1 means that the most severe restrictions (like the ones imposed in Bulgaria in March-April 2020) are imposed at moment 𝑡 (strict regime). The control 𝑢(·) is incorporated in the transmission rate coefficient, as well as in the removal rate coefficient, turning them into functions of time. The objective is to minimize the losses for the society, consisting of losses due to subdued economic activity and losses due to loss of labour. The optimal control problem is with free (not specified in advance) terminal time and there is a terminal condition: the aim is to achieve a “herd (or community) immunity” at the (not prespecified) end of the time horizon.

The present work is based on a joint research with G. Boyadzhiev and O. Kounchev.

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