Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Forecasting the Pandemic COVID-19 in India: A Mathematical Approach

Journal of Applied Nonlinear Dynamics 11(3) (2022) 549--571 | DOI:10.5890/JAND.2022.09.004

Manotosh Mandal$^{1,3}$, Soovoojeet Jana$^{2}$, Suvankar Majee$^{3}$, Anupam Khatua$^{3}$, T. K. Kar$^{3}$

$^{1}$ Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk -721636, West Bengal, India

$^{2}$ Department of Mathematics, Ramsaday College, Amta-711401, Howrah, West Bengal, India

$^{3}$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India

Download Full Text PDF



Due to the unavailability of proper antiviral therapies and high disease transmission rates, the pandemic COVID-19 is still increasing at a high rate in many countries. In each of the three countries, the USA, Brazil, and India, the COVID-19 positive cases have been crossed one million. With high population density and higher percentages of migrating workers has enabled India to be vulnerable to the disease quite more than other affected countries. In this paper, we have proposed a mathematical model with the help of a system of first-order ordinary differential equations and analyzed the model in the context of the COVID-19 pandemic. We have determined the expression of the basic reproduction number and relates it to establishing the disease-free equilibrium point's asymptotic stability and endemic equilibrium point. As it has been observed that only ten states and union territories are carrying more than $70\%$ infection in India, we have predicted long-term scenarios of the COVID-19 positive cases on those $10$ states and India until the end of the year 2020.


The work of S. Jana is partially supported by Dept of Science and Technology \& Biotechnology, Govt. of West Bengal (vide memo no. 201 (Sanc.)/ST/P/S \& T/16G-12/2018 dt 19-02-2019). Research of S. Majee is financially supported by Council of Scientific and Industrial Research (CSIR) Government of India (No. 08/003(0142)/2020-EMR-I dated 18th March 2020). A. Khatua is financially supported by Department of Science and Technology-INSPIRE, Government of India (No. DST/INSPIRE Fellowship/2016/IF160667, dated: 21st September, 2016). Moreover, the authors are very much grateful to the anonymous reviewers, the associate editor Prof. Shanmuganathan Rajasekar, and the editor in chief Prof. Albert C.J. Luo for their constructive comments and useful suggestions to improve the quality and presentation of the manuscript significantly.


  1. [1]  COVID-19 Coronavirus Pandemic,, (accessed 26th July 2020).
  2. [2]  Bernoulli, D. (1760), Essai daeune nouvelle analyse de la mortalite causee par la petite verole, Mem. math. phy. acad. roy. sci. paris. 145.
  3. [3]  Ross, R. (1916), An application of the theory of probabilities to the study of a priori pathometry: Part I, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 92(638), 204-226.
  4. [4]  Dodd, P.J., Gardiner, E., Coghlan, R., and Seddon, J.A. (2014), Burden of childhood tuberculosis in 22 high-burden countries: a mathematical modelling study, The Lancet Global Health, 2(8), e453-e459.
  5. [5]  De, A., Maity, K., Jana, S., and Maiti, M. (2016), Application of various control strategies to Japanese encephalitic: A mathematical study with human, pig and mosquito, Mathematical Biosciences, 282, 46-60.
  6. [6]  Hamer, W.H. (1906), Epidemic disease in England, Lancet, 1, 733-739.
  7. [7]  Kermack, W.O. and McKendrick, A.G. (1927), Contributions to the mathematical theory of epidemics-I, Proceedings of the Royal Society, 115A, 700-721.
  8. [8]  Keeling, M.J. and Rohani, P. (2008), Modelling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, New Jersey.
  9. [9]  Wang, W. and Zhao, X.Q. (2004), An epidemic model in a patchy environment, Math Biosci, 190, 97-112.
  10. [10]  Buonomo, B., d-Onofrio, A., and Lacitignola, D. (2008), Global stability of an SIR epidemic model with information dependent vaccination, Math Biosci, 216, 9-16.
  11. [11]  Zhou, Y., Yang, K., Zhou, K., and Liang, Y. (2014), Optimal vaccination policies for an SIR model with limited resources, Acta Biotheor, 62, 171-181.
  12. [12]  Jana, S., Nandi, S.K., and Kar, T.K. (2016), Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment, Acta Biotheor, 64, 65-84.
  13. [13]  Jana, S., Mandal, M., and Kar, T.K. (2020), Population dispersal and optimal control of an SEIR epidemic model, International Journal of Modelling, Identification and Control, 34(4), 379-395.
  14. [14]  Mandal, M., Jana, S., and Kar, T.K. (2021), Complex dynamics of an epidemic model with optimal vaccination and treatment in the presence of population dispersal, Discontinuity, Nonlinearity, and Complexity, 10(3), 471-497.
  15. [15]  Ndariou, F., Area, I., Nieto, J.J., and Torres, D.F. (2020), Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons $\&$ Fractals, DOI: j.chaos.2020.109846.
  16. [16]  Kucharski, A.J., Russell, T.W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., and Eggo, R.M. (2020), Early dynamics of transmission and control of COVID-19: a mathematical modelling study, The Lancet Infectious Diseases.
  17. [17]  Prem, K., Liu, Y., Russell, T.W., Kucharski, A.J., Eggo, R.M., and Davies, N. (2020), The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study, The Lancet Public Health.
  18. [18]  Fanelli, D. and Piazza, F. (2020), Analysis and forecast of COVID-19 spreading in China, Italy and France, Chaos, Solitons $\&$ Fractals, 134(2020), 109761.
  19. [19]  Mizumoto, K. and Chowell, G. (2020), Transmission potential of the novel coronavirus (COVID-19) onboard the diamond Princess Cruises Ship, Infectious Disease Modelling, 5, 264-270.
  20. [20]  Hellewell, J., Abbott, S., Gimma, A., Bosse, N.T., Jarvis, C.I., Russell, T.W., Munday, J.D., Kucharski, A.J., and Edmunds, W.J. (2020), Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8, 488-496.
  21. [21]  Liu, Y., Gayle, A.A., Wilder-Smith, A., and Rockl\"{o}v, J. (2020), The reproductive number of COVID-19 is higher compared to SARS coronavirus, Journal of Travel Medicine, 1-4.
  22. [22]  Ribeiro, M.H.D.M., Silva, R.G., Mariani, V.C., and Coelho, L.S. (2020), Short-term forecasting COVID-19 cumu-lative confirmed cases: Perspectives for Brazil, Chaos, Solitons $\&$ Fractals, DOI: 10.1016/j.chaos.2020.109853.
  23. [23]  Mandal, M., Jana, S., Nandi, S.K., Khatua, A., Adak, S., and Kar, T.K. (2020), A model based study on the dynamics of COVID-19: Prediction and control, Chaos, Solitons and Fractals, 136, 109889.
  24. [24]  Mandal, M., Jana, S., Khatua, A., and Kar, T.K. (2020), Modeling and control of COVID-19: A short-term forecasting in the context of India, Chaos: An interdisciplinary, Journal of Nonlinear Science, 30(11), 113119.
  25. [25]  Das, D.K., Khatua, A., Kar, T.K., and Jana, S. (2020), The effectiveness of contact tracing in mitigating COVID-19 outbreak: A model-based analysis in the context of India, Applied Mathematics and Computation, 404, 126207.
  26. [26]  Birkoff, G. and Rota, G.C. (1982), Ordinary Differential Equations, Ginn, Boston.
  27. [27]  Diekmann, O., Metz, J.A.P., and Heesterbeek, J.A.P. (1990), On the definition on the computation of the basic reproduction number ratio $R_0$ in models for infectious diseases in heterogeneous population, J Math Biol, 28(1990) 365-382.
  28. [28]  Fulford, G.R., Roberts, M.G., and Heesterbeek, J.A.P. (2002), The meta population dynamics of an infectious disease: tuberculosis in possums, J Theor Biol, 6115-29.
  29. [29]  Kar, T.K., Nandi, S.K., Jana, S., and Mandal, M.(2019), Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos, Solitons $\& $ Fractals, 120, 188-199.
  30. [30]  Van den Driessche, P. and Watmough, J. (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(1-2), 29-48.
  31. [31]  Castillo, C., Feng, Z., and Huang, W. (2002), On the computation of $R_{0}$ and its role on global stability, Mathematical Approaches for Emerging and Reemerging Infectious Disease: Models, Methods, and Theory, Springer-Verlag, Berlin, 125, 229-241.
  32. [32]  Guckenheimer. G. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag: New York.
  33. [33]  Kar, T.K. and Jana, S. (2013), A theoretical study on mathematical modeling of an infectious disease with application of optimal control, BioSystems, 111(1), 37-50.
  34. [34]  Jana, S., Haldar, P., and Kar, T.K. (2016), Mathematical analysis of an epidemic model with isolation and optimal controls, International Journal of Computer Mathematics, 94(2017), 1318-1336.
  35. [35]  Official Updates Coronavirus, COVID-19 in India, Government of India,, (accessed 26th July 2020).
  36. [36]  Government of Maharashtra Public Health Department,, (accessed 26th July 2020).
  37. [37]  Government of National Capital Territory of Delhi,, (accessed 26th July 2020).
  38. [38]  Health \& Family Welfare Department, Government of Gujarat,, (accessed 26th July 2020).
  39. [39]  Health \& Family Welfare Department, Government of Tamil Nadu,, (accessed 26th July 2020).
  40. [40]  Directorate of Medical \& Health Services, Government of Uttar Pradesh,, (accessed 26th July 2020).
  41. [41]  Department of Health, Medical and Family Welfare, Government of Andhra Pradesh,, (accessed 26th July 2020).
  42. [42]  Government of Telangana,, (accessed 26th July 2020).
  43. [43]  Department of Health, Medical \& Family Welfare, Government of Rajasthan,, (accessed 26th July 2020).
  44. [44]  Health \& Family Welfare, Government of Karnataka,, (accessed 26th July 2020).
  45. [45]  Health \& Family Welfare, Government of West Bengal,, (accessed 26th July 2020).
  46. [46]  Ministry of Health and Welfare, Government of India,, (accessed 26th July 2020).
  47. [47]  National Institution for Transforming India (NITI Aayog), Government of India,, (accessed 26th July 2020).
  48. [48]  Indian Council of Medical Research (ICMR), Government of India,, (accessed 26th July 2020).
  49. [49]  Castillo-Chavez, C. and Song, B. (2004), Dynamical models of tuberculosis and their applications, Mathematical Biosciences $\&$ Engineering, 1(2), 361.
  50. [50]  Das, D.K., Khajanchi, S., and Kar T.K. (2020), The impact of the media awareness and optimal strategy on the prevalence of tuberculosis, Applied Mathematics and Computation, 366, Article id 124732.
  51. [51]  Zhang, X. and Liu, X. (2008), Backward bifurcation of an epidemic model with saturated treatment, J Math Anal Appl, 348, 433-443.
  52. [52]  Acuna-Zegarra, M.A., Santana-Cibrian, M., and Velasco-Hernandez, J.X. (2020), Modeling behavioral change and COVID-19 containment in Mexico: A trade-off betweenlockdown and compliance, Mathematical Biosciences, DOI:
  53. [53]  Ngonghala, C.N., Iboi, E., Eikenberry, S., Scotch, M., Raina, C., Intyre, M., Bonds, M.H., and Gumel, A.B. (2020), Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus, Mathematical Biosciences, 325, Article id 108364.