Background: The basic reproduction number () is a key metric in epidemiology, representing the expected number of secondary infections from a single case in a fully susceptible population. Despite its widespread application,
is often misinterpreted due to its dependence on model assumptions and population dynamics. Understanding its calculation, applications, and limitations is crucial for refining epidemic models and enhancing disease control measures. Objectives: This study examines the mathematical foundations of
, its estimation methods, applications in disease modeling, and limitations. Additionally, it explores the effective reproduction number (
) and its role in assessing intervention impacts. Methods: A systematic review of mathematical models, including the SIR, SIRD, and modified SIRD models, was conducted to evaluate various approaches for estimating . The study also highlights variations in
and the effective reproduction number (
) across different infectious diseases, such as measles, influenza, and COVID-19. Results: Findings indicate that
is highly dependent on disease-specific factors, population dynamics, and intervention strategies. While
serves as a useful threshold indicator for disease outbreak potential,
provides a more practical assessment of ongoing transmission dynamics. The
References
[1]
Patil, V.C., Ali, G.S., Nashte, A., Rautdesai, R., Garud, S.K. and Sable, N.P. (2023) Public Health Policy and Infectious Disease Control: Lessons from Recent Outbreaks. South East European Journal of Public Health, XXI, 162-170. https://www.seejph.com/index.php/seejph/article/view/449
[2]
Peddinti, S. and Sabbani, Y. (2024) Mathematical Modeling of Infectious Disease Spread Using Differential Equations and Epidemiological Insights. Journal of Nonlinear Analysis and Optimization, 15, 114-122.
[3]
Anderson, R.M. and May, R.M. (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
[4]
Amburi, S., Jalan, V. and Saravanan, S. (2021) Transforming Native Epidemic Models by Using the Machine Learning Approach. Annals of the Romanian Society for Cell Biology, 25, 2891-2899.
[5]
Delamater, P.L., Street, E.J., Leslie, T.F., Yang, Y.T. and Jacobsen, K.H. (2019) Complexity of the Basic Reproduction Number (R0). Emerging Infectious Diseases, 25, 1-4. https://doi.org/10.3201/eid2501.171901
[6]
Heffernan, J.M., Smith, R.J. and Wahl, L.M. (2005) Perspectives on the Basic Reproductive Ratio. Journal of The Royal Society Interface, 2, 281-293. https://doi.org/10.1098/rsif.2005.0042
[7]
Roberts, M.G. (2007) The Pluses and Minuses of 0. Journal of The Royal Society Interface, 4, 949-961. https://doi.org/10.1098/rsif.2007.1031
[8]
Pellis, L., Ball, F. and Trapman, P. (2012) Reproduction Numbers for Epidemic Models with Households and Other Social Structures. I. Definition and Calculation of R0. Mathematical Biosciences, 235, 85-97. https://doi.org/10.1016/j.mbs.2011.10.009
[9]
Dembek, Z.F., Chekol, T. and Wu, A. (2018) Best Practice Assessment of Disease Modelling for Infectious Disease Outbreaks. Epidemiology and Infection, 146, 1207-1215. https://doi.org/10.1017/s095026881800119x
[10]
Corteel, M. (2025) Shaping Epidemic Dynamics: An Historical Epistemology Study of the SIR Model. History of the Human Sciences. https://doi.org/10.1177/09526951251317104
[11]
AlQadi, H. and Bani-Yaghoub, M. (2022) Incorporating Global Dynamics to Improve the Accuracy of Disease Models: Example of a COVID-19 SIR Model. PLOS ONE, 17, e0265815. https://doi.org/10.1371/journal.pone.0265815
[12]
Amiri Babaei, N., Kröger, M. and Özer, T. (2024) Theoretical Analysis of a SIRD Model with Constant Amount of Alive Population and COVID-19 Applications. Applied Mathematical Modelling, 127, 237-258. https://doi.org/10.1016/j.apm.2023.12.006
[13]
Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Filippo, A., Di Matteo, A., et al. (2020) Modelling the COVID-19 Epidemic and Implementation of Population-Wide Interventions in Italy. Nature Medicine, 26, 855-860. https://doi.org/10.1038/s41591-020-0883-7
[14]
Sen, D. and Sen, D. (2021) Use of a Modified SIRD Model to Analyze COVID-19 Data. Industrial & Engineering Chemistry Research, 60, 4251-4260. https://doi.org/10.1021/acs.iecr.0c04754
[15]
Chen, Y., Liu, F., Yu, Q. and Li, T. (2021) Review of Fractional Epidemic Models. Applied Mathematical Modelling, 97, 281-307. https://doi.org/10.1016/j.apm.2021.03.044
[16]
Wang, P. and Jia, J. (2019) Stationary Distribution of a Stochastic SIRD Epidemic Model of Ebola with Double Saturated Incidence Rates and Vaccination. Advances in Difference Equations, 2019, Article No. 433. https://doi.org/10.1186/s13662-019-2352-5
[17]
Zafar, Z.U.A., Rehan, K. and Mushtaq, M. (2017) HIV/AIDS Epidemic Fractional-Order Model. Journal of Difference Equations and Applications, 23, 1298-1315. https://doi.org/10.1080/10236198.2017.1321640
[18]
Miniguano-Trujillo, A., Pearson, J.W. and Goddard, B.D. (2025) A Constrained Optimisation Framework for Parameter Identification of the SIRD Model. Mathematical Biosciences, 380, Article ID: 109379. https://doi.org/10.1016/j.mbs.2025.109379
[19]
Tsay, C., Lejarza, F., Stadtherr, M.A. and Baldea, M. (2020) Modeling, State Estimation, and Optimal Control for the US COVID-19 Outbreak. Scientific Reports, 10, Article No. 10711. https://doi.org/10.1038/s41598-020-67459-8
[20]
Afful, B.A., Safo, G.A., Marri, D., Okyere, E., Ohemeng, M.O. and Kessie, J.A. (2025) Deterministic Optimal Control Compartmental Model for COVID-19 Infection. Modeling Earth Systems and Environment, 11, Article No. 87. https://doi.org/10.1007/s40808-024-02183-0
[21]
Diekmann, O. and Heesterbeek, J.A.P. (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (Vol. 5). John Wiley & Sons.
[22]
Anastassopoulou, C., Russo, L., Tsakris, A. and Siettos, C. (2020) Data-Based Analysis, Modelling and Forecasting of the COVID-19 Outbreak. PLOS ONE, 15, e0230405. https://doi.org/10.1371/journal.pone.0230405
[23]
Boonpatcharanon, S., Heffernan, J.M. and Jankowski, H. (2022) Estimating the Basic Reproduction Number at the Beginning of an Outbreak. PLOS ONE, 17, e0269306. https://doi.org/10.1371/journal.pone.0269306
[24]
Diekmann, O., Heesterbeek, J.A.P. and Roberts, M.G. (2013) The Construction of Next-Generation Matrices for Compartmental Epidemic Models. Journal of the Royal Society Interface, 10, Article ID: 20120656.
[25]
Ferguson, N.M., Laydon, D., Nedjati-Gilani, G., Imai, N., Ainslie, K., Baguelin, M., Ghani, A.C., et al. (2020) Impact of Non-Pharmaceutical Interventions (NPIs) to Reduce COVID-19 Mortality and Healthcare Demand. Imperial College London. https://doi.org/10.25561/77482
[26]
Southall, E., Ogi-Gittins, Z., Kaye, A.R., Hart, W.S., Lovell-Read, F.A. and Thompson, R.N. (2023) A Practical Guide to Mathematical Methods for Estimating Infectious Disease Outbreak Risks. Journal of Theoretical Biology, 562, Article ID: 111417. https://doi.org/10.1016/j.jtbi.2023.111417
[27]
Fox, J.P., Elveback, L., Scott, W., Gatewood, L. and Ackerman, E. (1995) Herd Immunity: Basic Concept and Relevance to Public Health Immunization Practices1 American Journal of Epidemiology, 141, 187-197. https://doi.org/10.1093/oxfordjournals.aje.a117420
[28]
Greenhalgh, T., Alwan, N., Bogaert, D., Burns, S.H., Cheng, K.K., Colbourn, T., Tannock, C., et al. (2020) COVID-19: An Open Letter to the UK’s Chief Medical Officers. https://blogs.bmj.com/bmj/2020/09/21/covid-19-an-open-letter-to-the-uks-chief-medical-officers
[29]
Sanche, S., Lin, Y.T., Xu, C., Romero-Severson, E., Hengartner, N. and Ke, R. (2020) High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2. Emerging Infectious Diseases, 26, 1470-1477. https://doi.org/10.3201/eid2607.200282
[30]
Liu, Y. and Rocklöv, J. (2022) The Effective Reproductive Number of the Omicron Variant of SARS-CoV-2 Is Several Times Relative to Delta. Journal of Travel Medicine, 29, taac037. https://doi.org/10.1093/jtm/taac037
[31]
Biggerstaff, M., Cauchemez, S., Reed, C., Gambhir, M. and Finelli, L. (2014) Estimates of the Reproduction Number for Seasonal, Pandemic, and Zoonotic Influenza: A Systematic Review of the Literature. BMC Infectious Diseases, 14, Article No. 480. https://doi.org/10.1186/1471-2334-14-480
[32]
Nguyen, P., Ajisegiri, W.S., Costantino, V., Chughtai, A.A. and MacIntyre, C.R. (2021) Reemergence of Human Monkeypox and Declining Population Immunity in the Context of Urbanization, Nigeria, 2017-2020. Emerging Infectious Diseases, 27, 1007-1014. https://doi.org/10.3201/eid2704.203569
[33]
Fine, P.E.M., Jezek, Z., Grab, B. and Dixon, H. (1988) The Transmission Potential of Monkeypox Virus in Human Populations. International Journal of Epidemiology, 17, 643-650. https://doi.org/10.1093/ije/17.3.643
[34]
Grant, R., Nguyen, L.L. and Breban, R. (2020) Modelling Human-to-Human Transmission of Monkeypox. Bulletin of the World Health Organization, 98, 638-640. https://doi.org/10.2471/blt.19.242347
[35]
Althaus, C.L. (2014) Estimating the Reproduction Number of Ebola Virus (EBOV) during the 2014 Outbreak in West Africa. PLOS Currents, 6, 5 p. https://doi.org/10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288
[36]
Heesterbeek, J.A.P. (2002) A Brief History of R₀ and a Recipe for Its Calculation. Acta Biotheoretica, 50, 189-204. https://doi.org/10.1023/a:1016599411804
[37]
Adegoke, B.O., Odugbose, T. and Adeyemi, C. (2024) Data Analytics for Predicting Disease Outbreaks: A Review of Models and Tools. International Journal of Life Science Research Updates, 2, 1-9. https://doi.org/10.53430/ijlsru.2024.2.2.0023
[38]
Alimohamadi, Y., Taghdir, M. and Sepandi, M. (2020) Estimate of the Basic Reproduction Number for COVID-19: A Systematic Review and Meta-Analysis. Journal of Preventive Medicine and Public Health, 53, 151-157. https://doi.org/10.3961/jpmph.20.076
[39]
Breban, R., Vardavas, R. and Blower, S. (2007) Theory versus Data: How to Calculate R0? PLoS ONE, 2, e282. https://doi.org/10.1371/journal.pone.0000282
[40]
Cori, A., Ferguson, N.M., Fraser, C. and Cauchemez, S. (2013) A New Framework and Software to Estimate Time-Varying Reproduction Numbers during Epidemics. American Journal of Epidemiology, 178, 1505-1512. https://doi.org/10.1093/aje/kwt133
[41]
Grenfell, B.T., Pybus, O.G., Gog, J.R., Wood, J.L.N., Daly, J.M., Mumford, J.A., et al. (2004) Unifying the Epidemiological and Evolutionary Dynamics of Pathogens. Science, 303, 327-332. https://doi.org/10.1126/science.1090727
[42]
Hall, V., Foulkes, S., Insalata, F., Kirwan, P., Saei, A., Atti, A., et al. (2022) Protection against SARS-CoV-2 after Covid-19 Vaccination and Previous Infection. New England Journal of Medicine, 386, 1207-1220. https://doi.org/10.1056/nejmoa2118691
[43]
Grenfell, B.T. and Anderson, R.M. (1989) Pertussis in England and Wales: An Investigation of Transmission Dynamics and Control by Mass Vaccination. Proceedings of the Royal Society B: Biological Sciences, 236, 213-252.
[44]
Chemaitelly, H., Tang, P., Hasan, M.R., AlMukdad, S., Yassine, H.M., Benslimane, F.M., et al. (2021) Waning of BNT162b2 Vaccine Protection against SARS-CoV-2 Infection in Qatar. New England Journal of Medicine, 385, e83. https://doi.org/10.1056/nejmoa2114114
[45]
Burki, T.K. (2022) Omicron Variant and Booster COVID-19 Vaccines. The Lancet Respiratory Medicine, 10, e17. https://doi.org/10.1016/s2213-2600(21)00559-2
[46]
Troiano, G. and Nardi, A. (2021) Vaccine Hesitancy in the Era of Covid-19. Public Health, 194, 245-251. https://doi.org/10.1016/j.puhe.2021.02.025
[47]
Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. PLOS Medicine, 5, e74. https://doi.org/10.1371/journal.pmed.0050074
[48]
Robertson, D., Heriot, G. and Jamrozik, E. (2024) Herd Immunity to Endemic Diseases: Historical Concepts and Implications for Public Health Policy. Journal of Evaluation in Clinical Practice, 30, 625-631. https://doi.org/10.1111/jep.13983
[49]
Rocklöv, J. and Sjödin, H. (2020) High Population Densities Catalyse the Spread of Covid-19. Journal of Travel Medicine, 27, taaa038. https://doi.org/10.1093/jtm/taaa038
[50]
Colizza, V., Barrat, A., Barthélemy, M. and Vespignani, A. (2007) Predictability and Epidemic Pathways in Global Outbreaks of Infectious Diseases: The SARS Case Study. BMC Medicine, 5, Article No. 34. https://doi.org/10.1186/1741-7015-5-34
[51]
Kraemer, M.U.G., Yang, C., Gutierrez, B., Wu, C., Klein, B., Pigott, D.M., et al. (2020) The Effect of Human Mobility and Control Measures on the COVID-19 Epidemic in China. Science, 368, 493-497. https://doi.org/10.1126/science.abb4218
[52]
Lloyd-Smith, J.O., Schreiber, S.J., Kopp, P.E. and Getz, W.M. (2005) Superspreading and the Effect of Individual Variation on Disease Emergence. Nature, 438, 355-359. https://doi.org/10.1038/nature04153
[53]
Danon, L., Ford, A.P., House, T., Jewell, C.P., Keeling, M.J., Roberts, G.O., et al. (2011) Networks and the Epidemiology of Infectious Disease. Interdisciplinary Perspectives on Infectious Diseases, 2011, Article ID: 284909. https://doi.org/10.1155/2011/284909
[54]
Brauner, J.M., Mindermann, S., Sharma, M., Johnston, D., Salvatier, J., Gavenčiak, T., et al. (2021) Inferring the Effectiveness of Government Interventions against Covid-19. Science, 371, eabd9338. https://doi.org/10.1126/science.abd9338
[55]
Pastor-Satorras, R., Castellano, C., Van Mieghem, P. and Vespignani, A. (2015) Epidemic Processes in Complex Networks. Reviews of Modern Physics, 87, 925-979. https://doi.org/10.1103/revmodphys.87.925
[56]
Ball, F. and Neal, P. (2002) A General Model for Stochastic SIR Epidemics with Two Levels of Mixing. Mathematical Biosciences, 180, 73-102. https://doi.org/10.1016/s0025-5564(02)00125-6
[57]
Keeling, M.J. and Rohani, P. (2008) Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
[58]
Kiss, I.Z., Miller, J.C. and Simon, P.L. (2017) Mathematics of Epidemics on Networks: From Exact to Approximate Models. Springer. https://doi.org/10.1007/978-3-319-50806-1_1
[59]
Bansal, S., Grenfell, B.T. and Meyers, L.A. (2007) When Individual Behaviour Matters: Homogeneous and Network Models in Epidemiology. Journal of The Royal Society Interface, 4, 879-891. https://doi.org/10.1098/rsif.2007.1100
[60]
Wallinga, J. and Lipsitch, M. (2006) How Generation Intervals Shape the Relationship between Growth Rates and Reproductive Numbers. Proceedings of the Royal Society B: Biological Sciences, 274, 599-604. https://doi.org/10.1098/rspb.2006.3754
[61]
Wallinga, J. (2004) Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures. American Journal of Epidemiology, 160, 509-516. https://doi.org/10.1093/aje/kwh255
