Data Research, Vol. 5, Issue 1, Apr  2021, Pages 1-43; DOI:

Analysis of Basic Reproduction Number in Epidemiological Model

Data Research, Vol. 5, Issue 1, Apr  2021, Pages 1-43.


Manaye Chanie 1* , Demeke Fisseha 2

1 Department of Mathematics, Jinka University, Jinka, Ethiopia

2 Department of Mathematics, Debre Markos University, Debre Marqos, Ethiopia

Received: 3 October 2020; Accepted: 4 March 2021; Published: 6 April 2021

Download PDF | Views 27 | Download 16


In this project we present analysis of basic reproduction number in epidemiological model. Epidemiological analysis and mathematical models are now essential tools in understanding the dynamics of infectious diseases and in designing public health strategies. The threshold for many epidemiological models is the basic reproduction number R_0. It reflects the average number of secondary infections produced by one infected individual put into completely susceptible host society. It is a threshold quantity which determines whether the epidemic will occur or not. We will discuss the general SIS, SIR and SEIR model with vital dynamics for the mathematical modeling of diseases. The reproduction number and the stabilities of both the disease-free and the endemic equilibrium will be calculated. Finally numerical simulations using Matlab Software will be conducted for SIR model with vital dynamics.


Mathematical Model, Reproduction Number, Deterministic (SIS/SIR/SEIR) Model, Disease-Free Equilibrium, Endemic Equilibrium, Simulation


© 2017 by the authors. Licensee International Technology and Science Press Limited. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


[1] Anderson, R.M.May, R.M. Infectious diseases of humans: dynamics and control. Oxford: Oxford University press. 1991.

[2] Bockh, R. 1886 Statistisches Fahrbuch der Stadt Berlin, Zwolfter Jahrgang.  Statistik des Jahres 1884, pp. 30 31.  Berlin: P. Stankiewicz Co.Pte. Ltd., 2009.

[3] Delaware Heath and Social Services., Direct and indirect disease transmission, (2011).

[4] Diekmann, O.Heesterbeek, J.A.P. Mathematical epidemiology of infectious  diseases: model building, analysis and interpretation. New York: Wiley. 2000.

[5] Diekmann, O.Heesterbeek, J.A.P.Metz, J.A.J. On the definition and the  computation of the basic reproduction ratio R0 in models for infectious diseases. J. Math. Biol. 199035, 503–522.

[6] Dietz, K.  Transmission and control of arbovirus diseases. In Epidemiology (ed. D. Ludwig & K.  L. Cooke), Philadelphia: Society for Industrial and Applied Mathematics, 1975; pp. 104-121.

[7] Dietz, K. The estimation of the basic reproduction number for infectious diseases.  Stat. Methods Med.  Res. 19932, 23-41.

[8] Dublin, L.I.; Lotka, A. J. On the true rate of natural increase of a population. J.  Am. Stat. Assoc. 192520, 305-339.

[9] Awawdeh, F.Adawi, A.MustafaZSolutions of the SIR model of epidemics using HAM. Chaos, Solitons and Fractals200942, 3047-3052.

[10] Heesterbeek, J.A.P.Dietz, K. The concept of R0 in epidemic theory. Stat. Neerl. 199650, 89-110.

[11] Heesterbeek, J.A.P. A brief history of R0 and a recipe for its calculation. Acta  Biotheret. 200250, 189-204.

[12] HerbertW. Hethcote Applied Mathematical Ecology, ed. S. A. Levin, T.G. Hallam, and L.J. Gross, Springer-Verlag, 1989; pp. 119-144.

[13] Hethcote, H.W. Mathematical models for the spread of infectious diseases. In Epidemiology (ed. D. Ludwig & K. L. Cooke), Philadelphia: Society for Industrial and Applied Mathematics1975; pp. 122-131.

[14] Hethcote, H.W. The mathematics of infectious diseases. SIAM Rev. 200042, 599-653.

[15] Kermack, W.O.McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc.  R.  Soc. 1927115, 700-721.

[16] Kuczynski, R.R. The balance of births and deaths, New York: Macmillan. 1928vol. 1.

[17] Lloyd, A.L.May, R.M. Spatial heterogeneity in epidemic models. J. Theor. Biol. 1996179, 1-11

[18] KeelingM.J.RohaniP. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, 2008

[19] MacDonald, G. The analysis of equilibrium in malaria. Trop. Dis. Bull. 195249,  813-829.

[20] Mollison, D. (ed.) Epidemic models: their structure and relation to data. Cambridge: Cambridge University Press1995b.

[21] Nasell, I. The threshold concept in stochastic epidemic and endemic models. In Epidemic models:  their structure and relations to data (ed. D. Mollison), Cambridge: Cambridge University Press. 1995; pp. 71-83.

[22] Ross, R.  The prevention of malaria.  London: John Murray. 1911.

[23] Sharp, F.R.Lotka, A.J. A problem in age distribution. Phil.  Mag.  19116, 435-438.

[24] Smith, D.Keyfitz, N. Mathematical demography: selected papers. Biomathematics, Berlin: Springer1977 vol. 6.

[25] Van den Driessche, P.Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002 180, 29-48.

[26] KermackW.O.McKendrick A.G. A Contribution to the Mathematical Theory of EpidemicsProc. Roy. Soc. London A, 1927115, 700-721.

Related Articles