Article Data

  • Views 732
  • Dowloads 149

Original Research

Open Access

Mathematical models applied to the prediction of doping in male athletes

  • Christine Gabriela Viscotel1,2
  • Mariana Rotariu3
  • Marius Turnea3,*,
  • Dragos Arotaritei3
  • Claudiu Mereuta2
  • Mihai Ilea3
  • Iustina Condurache3
  • Andrei Gheorghita3

1Romanian National Anti-Doping Agency, 022103 Bucharest, Romania

2Faculty of Physical Education and Sport “Dunărea de Jos”, University of Galati, 800003 Galati, Romania

3Department of Biomedical Sciences, Faculty of Medical Bioengineering, University of Medicine and Pharmacy “Grigore T. Popa”, 700454 Iasi, Romania

DOI: 10.22514/jomh.2023.061 Vol.19,Issue 7,July 2023 pp.93-100

Submitted: 22 March 2023 Accepted: 25 April 2023

Published: 30 July 2023

*Corresponding Author(s): Marius Turnea E-mail:


The compartmental model is a mathematical model (usually described by a set of differential equations) that describes how individuals from different compartments (or groups) that represent a population, interacts. The model is suitable especially for epidemic model, modeling spread of disease but also in simulation of interaction among social groups. The compartmental model has few assumptions to be feasible: “the infection/contamination rate” can be a function of many parameters (seasonality, epidemic waves, dependence of social distancing, policy of awareness, policy, and so one). The main assumption is that the population is homogeneous but, in reality, the heterogeneity of population (spatial localization, seasonal, demography) plays an important role in accuracy of models. Our approach was based on another method that has been used in the last years, the inclusion of a temporal function including heterogeneity in the influence that conduct to doping similar to rate of infection from epidemic models. In this paper, a new model is proposed for quantitative analysis of doping in a particular selected sport. Almost all the models in doping use the biological markers and effect of doping in declared by athletes involved in use of banned substances in a quantitative analysis over a group of high-performance athletes. The proposed compartmental model SEDRS (Susceptible-Exposed-Doped-Recovered-Susceptible) includes the heterogeneity shaped by awareness, due to social interaction that transmit the anti-doping policy. These measures are patterned by social interaction, especially during competitions and training, and this approach is included in system of integrodifferential equations. A heterogeneous (SEDRS) model is numerically solved and the solutions show how the social factor can contribute to decay of doping phenomenon of male athletes and the quantifiable influence in a healthier atmosphere in sport. The scope of the paper is the prediction of doping cases based on SEDRS model.


Anti-doping policy; Compartmental models; Probability distribution; Heterogenous models

Cite and Share

Christine Gabriela Viscotel,Mariana Rotariu,Marius Turnea,Dragos Arotaritei,Claudiu Mereuta,Mihai Ilea,Iustina Condurache,Andrei Gheorghita. Mathematical models applied to the prediction of doping in male athletes. Journal of Men's Health. 2023. 19(7);93-100.


[1] Doping Control Laboratory of Romania. Brief history. 2017. Available at: (Accessed: 10 February 2023).

[2] World Anti-Doping Agency. 2019 anti-doping rule violations (ADRVs) report. 2019. Available at: external_final_12_december_2021_0_0.pdf (Accessed: 10 February 2023).

[3] Hanstad DV, Smith A, Waddington I. The establishment of the world anti-doping agency. International Review for the Sociology of Sport. 2008; 43: 227–249.

[4] Dimeo P, Hunt TM, Horbury R. The individual and the state: a social historical analysis of the East German ‘doping system’. Sport in History. 2011; 31: 218–237.

[5] Exploring topics in sports: why do athletes risk using performance enhancing drugs? Available at: https://sps.northwestern. edu/stories/news-stories/why-do-athletes-risk-using-PEDs.php (Accessed: 14 July 2015).

[6] Geeraets V. Ideology, doping and the spirit of sport. Sport, Ethics and Philosophy. 2018; 12: 255–271.

[7] Berbecaru CF, Stănescu M, Vâjială GE, Epuran M. Theoretical and methodological aspects on doping phenomenon in elite athletes. Procedia—Social and Behavioral Sciences. 2014; 149: 102–106.

[8] Pitsch W. “The science of doping” revisited: fallacies of the current anti-doping regime. European Journal of Sport Science. 2009; 9: 87–95.

[9] Catlin DH, Fitch KD, Ljungqvist A. Medicine and science in the fight against doping in sport. Journal of Internal Medicine. 2008; 264: 99–114.

[10] Montagna S, Hopker J. A Bayesian approach for the use of athlete performance data within anti-doping. Frontiers in Physiology. 2018; 9: 884.

[11] Faiss R, Saugy J, Zollinger A, Robinson N, Schuetz F, Saugy M, et al. Estimate of blood doping in elite track and field athletes during two major international events. Frontiers in Physiology. 2020; 11: 160.

[12] Anderson RM, May RM. Infectious disease of human: dynamics and control. Oxford University Press: New York; 1991.

[13] Hethcote HW. The mathematics of infectious diseases. SIAM Review. 2000; 42: 599–653.

[14] Pastor-Satorras R, Vespignani A. Epidemic dynamics in finite size scale-free networks. Physical Review E. 2002; 65: 035108.

[15] May RM, Lloyd AL. Infection dynamics on scale-free networks. Physical Review E. 2001; 64: 066112.

[16] Isham V, Kaczmarska J, Nekovee M. Spread of information and infection on finite random networks. Physical Review E. 2011; 83: 046128.

[17] Bansal S, Grenfell BT, Meyers LA. When individual behaviour matters: homogeneous and network models in epidemiology. Journal of the Royal Society Interface. 2007; 4: 879–891.

[18] Hofstad RVD, Janssen JEM, Leeuwaarden JSH. Critical epidemics, random graphs, and Brownian motion with parabolic drift. Advances in Applied Probability. 2010; 42: 1187–1206.

[19] Barabási A, Albert R. Emergence of scaling in random networks. Science. 1999; 286: 509–512.

[20] Estrada E, Meloni S, Sheerin M, Moreno Y. Epidemic spreading in random rectangular networks. Physical Review E. 2016; 94: 052316.

[21] Wang H, Moore JM, Small M, Wang J, Yang H, Gu C. Epidemic dynamics on higher-dimensional small world networks. Applied Mathematics and Computation. 2022; 421: 126911.

[22] Miller JC, Volz EM. Incorporating disease and population structure into models of SIR disease in contact networks. PLoS One. 2013; 8: e69162.

[23] Miller JC, Slim AC, Volz EM. Edge-based compartmental modelling for infectious disease spread. Journal of the Royal Society Interface. 2012; 9: 890–906.

[24] Hu G, Geng J. Heterogeneity learning for SIRS model: an application to the COVID-19. Statistics and Its Interface. 2021; 14: 73–81.

[25] Pastor-Satorras R, Vespignani A. Epidemic dynamics and endemic states in complex networks. Physical Review E. 2001; 63: 066117.

[26] Landry NW, Restrepo JG. The effect of heterogeneity on hypergraph contagion models. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2020; 30: 103117.

[27] Gou W, Jin Z. How heterogeneous susceptibility and recovery rates affect the spread of epidemics on networks. Infectious Disease Modelling. 2017; 2: 353–367.

[28] Beddington JR. Mutual interference between parasites or predators and its effect on searching efficiency. The Journal of Animal Ecology. 1975; 44: 331.

[29] DeAngelis DL, Goldstein RA, O’Neill RV. A model for trophic interaction. Ecology. 1975; 56: 881–892.

[30] Crowley PH, Martin EK. Functional responses and interference within and between year classes of a dragonfly population. Journal of the North American Benthological Society. 1989; 8: 211–221.

[31] Avila-Vales E, Pérez ÁGC. Dynamics of a reaction—diffusion SIRS model with general incidence rate in a heterogeneous environment. Journal of Applied Mathematics and Physics. 2022; 73: 9.

[32] Cui J, Sun Y, Zhu H. The impact of media on the control of infectious diseases. Journal of Dynamics and Differential Equations. 2008; 20: 31–53.

[33] Agrawal A, Tenguria A, Modi G. Stability analysis of an SIR epidemic model with specific non-linear incidence rate. Mathematical Theory and Modeling. 2016; 6: 45–51.

[34] Zhou Y, Xiao D, Li Y. Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action. Chaos, Solitons & Fractals. 2007; 32: 1903–1915.

[35] Kong L, Wang J, Han W, Cao Z. Modeling heterogeneity in direct infectious disease transmission in a compartmental model. International Journal of Environmental Research and Public Health. 2016; 13: 253.

[36] Kong L, Wang J, Li Z, Lai S, Wu H, Yang W. Modeling the heterogeneity of dengue transmission in a city. International Journal of Environmental Research and Public Health. 2018; 15: 1128.

[37] Kong L, Wang J, Han W, Cao Z. Modeling heterogeneity in direct infectious disease transmission in a compartmental model. International Journal of Environmental Research and Public Health. 2016; 13: 253.

[38] Bury K. Statistical distributions in engineering. 1st edn. Cambridge University Press: Cambridge. 1999.

[39] Vynnycky E, White R. An introduction to infectious disease modelling. Oxford University Press: New York. 2010.

[40] Ghanbari B. On forecasting the spread of the COVID-19 in Iran: the second wave. Chaos, Solitons & Fractals. 2020; 140: 110176.

[41] Shuai Z, van den Driessche P. Imapct of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences and Engineering. 2012; 9: 393–411.

[42] Gerasimov A, Lebedev G, Lebedev M, Semenycheva I. COVID-19 dynamics: a heterogeneous model. Frontiers in Public Health. 2021; 8: 558368.

[43] Kong L, Wang J, Han W, Cao, Z. Modeling heterogeneity in direct infectious disease. International Journal of Environmental Research and Public Health. 2016; 13: 253.

[44] Bamberger M, Yaeger D. Over the edge: aware that drug testing is a sham, athletes seem to rely more than ever on banned performance enhancers. Sports Illustrated. 1997; 86: 60–70.

[45] Statista. Share of non-Olympic sport doping tests which reported a prohibited substance worldwide from 2012 to 2021. 2023. Available at: (Accessed: 12 January 2023).

Abstracted / indexed in

Science Citation Index Expanded (SciSearch) Created as SCI in 1964, Science Citation Index Expanded now indexes over 9,200 of the world’s most impactful journals across 178 scientific disciplines. More than 53 million records and 1.18 billion cited references date back from 1900 to present.

Journal Citation Reports/Science Edition Journal Citation Reports/Science Edition aims to evaluate a journal’s value from multiple perspectives including the journal impact factor, descriptive data about a journal’s open access content as well as contributing authors, and provide readers a transparent and publisher-neutral data & statistics information about the journal.

Directory of Open Access Journals (DOAJ) DOAJ is a unique and extensive index of diverse open access journals from around the world, driven by a growing community, committed to ensuring quality content is freely available online for everyone.

SCImago The SCImago Journal & Country Rank is a publicly available portal that includes the journals and country scientific indicators developed from the information contained in the Scopus® database (Elsevier B.V.)

Publication Forum - JUFO (Federation of Finnish Learned Societies) Publication Forum is a classification of publication channels created by the Finnish scientific community to support the quality assessment of academic research.

Scopus: CiteScore 0.7 (2022) Scopus is Elsevier's abstract and citation database launched in 2004. Scopus covers nearly 36,377 titles (22,794 active titles and 13,583 Inactive titles) from approximately 11,678 publishers, of which 34,346 are peer-reviewed journals in top-level subject fields: life sciences, social sciences, physical sciences and health sciences.

Norwegian Register for Scientific Journals, Series and Publishers Search for publication channels (journals, series and publishers) in the Norwegian Register for Scientific Journals, Series and Publishers to see if they are considered as scientific. (

Submission Turnaround Time