# How Many Can You Infect? Simple (and Naive) Methods of Estimating the Reproduction Number

## DOI:

https://doi.org/10.5614/cbms.2020.3.1.4## Keywords:

Reproduction number, infectious disease, compartment model, COVID-19## Abstract

This is a pedagogical paper on estimating the number of people that can be infected by one infectious person during an epidemic outbreak, known as the reproduction number. Knowing the number is crucial for developing policy responses. There are generally two types of such a number, i.e., basic and effective (or instantaneous). While basic reproduction number is the average expected number of cases directly generated by one case in a population where all individuals are susceptible, effective reproduction number is the number of cases generated in the current state of a population. In this paper, we exploit the deterministic susceptibleinfected-removed (SIR) model to estimate them through three different numerical approximations. We apply the methods to the pandemic COVID-19 in Italy to provide insights into the spread of the disease in the country. We see that the effect of the national lockdown in slowing down the disease exponential growth appeared

about two weeks after the implementation date. We also discuss available improvements to the simple (and naive) methods that have been made by researchers in the field. Authors of this paper are members of the SimcovID (Simulasi dan Pemodelan COVID-19 Indonesia) collaboration.

## References

The 2019-nCoV Outbreak Joint Field Epidemiology Investigation Team, Qun Li. An Outbreak of NCIP (2019-nCoV) Infection in China - Wuhan, Hubei Province, 2019-2020. China CDC Weekly, 2(5), 79-80, 2020. doi: 10.46234/ccdcw2020.022

Callaway, E., Time to use the p-word? Coronavirus enter dangerous new phase, Nature, 579, 12, 2020.

Wu, J.T., Leung, K. and Leung, G.M., Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, The Lancet, 395(10225), pp. 689-697, 2020.

Kucharski, A.J., Russell, T.W., Diamond, C., Liu, Y., Edmunds, J., Funk, S. and Eggo, R.M., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, The Lancet Infectious Diseases, 20(5), pp. 553-558, 2020.

National responses to the COVID-19 pandemic, Wikipedia, 2020. Available at: https://en.wikipedia.org/wiki/National responses to the COVID-19 pandemic, Accessed on May 8, 2020.

Ma, J., Estimating epidemic exponential growth rate and basic reproduction number, Infectious Disease Modelling, 5, pp. 129-141, 2020.

Chowell, G. and Viboud, C., Is it growing exponentially fast? { Impact of assuming exponential growth for characterizing and forecasting epidemics with initial near-exponential growth dynamics, Infectious Disease Modelling, 1(1), pp. 71-8, 2016.

Taslim Ali, S.K., A study on stochastic epidemic models with the optimal control policies, PhD Thesis, Karnatak University, 2014. Available at http://hdl.handle.net/10603/98827.

Gumel, A.B., Ruan, S., Day, T., Watmough, J., Brauer, F., Van den Driessche, P., Gabrielson, D., Bowman, C., Alexander, M.E., Ardal, S. and Wu, J., Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London, Series B: Biological Sciences, 271(1554), pp. 2223-2232, 2004.

Yang, Z., Zeng, Z., Wang, K., Wong, S.S., Liang, W., Zanin, M., Liu, P., Cao, X., Gao, Z., Mai, Z. and Liang, J., Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions, Journal of Thoracic Disease, 12(3), p. 165, 2020.

Fang, Y., Nie, Y., Penny, M., Transmission dynamics of the COVID-19 outbreak and effectiveness of government interventions: A data-driven analysis, Journal of Medical Virology, 92(6), pp. 645-59, 2020.

Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, and J. Shaman. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science 368(6490), pp. 489-493, 2020.

Fanelli, D. and Piazza, F., Analysis and forecast of COVID-19 spreading in China, Italy and France., Chaos, Solitons & Fractals, 134:109761, 2020.

Dietz, K., The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research, 2(1), pp. 23-41, 1993.

Chowell, G. and Brauer, F., The Basic Reproduction Number of Infectious Diseases: Computation and Estimation Using Compartmental Epidemic Models, In Mathematical and Statistical Estimation Approaches in Epidemiology, pp. 1-30, Springer, Dordrecht, 2009.

Nishiura, H. and Chowell, G., The effective reproduction number as a prelude to statistical estimation of time-dependent epidemic trends, In Mathematical and Statistical Estimation Approaches in Epidemiology, pp. 103-121, Springer, Dordrecht, 2009.

van den Driessche, P., Reproduction numbers of infectious disease models, Infectious Disease Modelling, 2(3), pp. 288-303, 2017.

Ridenhour, B., Kowalik, J.M. and Shay, D.K., Unraveling R0: Considerations for Public Health Applications, American Journal of Public Health, 108(S6), pp. S445-S454, 2018.

Delamater, P.L., Street, E.J., Leslie, T.F., Yang, Y. and Jacobsen, K.H., Complexity of the Basic Reproduction Number (R0), Emerging Infectious Diseases, 25(1), pp. 1-4, 2019.

Cintr'on-Arias, A., Castillo-Ch'avez, C., Bettencourt, L.M.A., Lloyd, A.L., and Banks, H.T., The estimation of the effective reproductive number from disease outbreak data, Mathematical Biosciences and Engineering, 6, pp. 261-283, 2009.

Chen, Y-C, Lu, P-E., Chang, C-S. and Liu, T-H. A time-dependent SIR model for COVID-19 with undetectable infected persons. arXiv:2003.00122 [q-bio.PE]

Bettencourt, L.M.A. and Ribeiro, R.M., Real Time Bayesian Estimation of the Epidemic Potential of Emerging Infectious Diseases, PLoS ONE, 3(5): e2185, 2008.

Chowell, G., Nishiura, H. and Bettencourt, L.M.A., Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, Journal of The Royal Society Interface, 4, pp. 155-166, 2007.

Nuraini, N., Khairudin, K. and Apri, M., Modeling Simulation of COVID-19 in Indonesia based on Early Endemic Data, Communication in Biomathematical Sciences, pp. 3(1), pp. 1-8, 2020.

Novel coronavirus (COVID-19) cases, provided by Johns Hopkins University Center for Systems Science and Engineering (JHU CCSE). https://github.com/CSSEGISandData/COVID-19

Li, J., Blakeley, D. and Smith, R.J., The Failure of R0, Computational and Mathematical Methods in Medicine 2011, Article ID 527610.

Cori, A., Ferguson, N.M., Fraser, C., Cauchemez, S., A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics, American Journal of Epidemiology, 178(9), pp. 1505-1512, 2013.

Wallinga, J. and Teunis, P., Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures, American Journal of Epidemiology, 160(6), pp. 509-516, 2004.

Obadia, T., Haneef, R. and Boelle, P-Y., The R0 package: a toolbox to estimate reproduction numbers for epidemic outbreaks, BMC Medical Informatics and Decision Making, 12, p. 147, 2012.