Emergence and Subsiding of the First-wave COVID-19 Pandemic in Pakistan (2020): An Eigenvalue Analysis Based on Synergetics

First-wave COVID-19 pandemic in Pakistan


  • T.D. Frank Department of Psychological Sciences, University of Connecticut, Storrs, USA , Physics Department, University of Connecticut, Storrs, USA


COVID-19, eigenvalues, synergetics


By November 15, 2020, the COVID-19 pandemic has claimed worldwide 1,300,000 lives. Despite this tragic death toll, several countries managed to make at least their first-waves of the COVID-19 pandemic in their countries to subside to relatively low levels. Pakistan is one of those countries. An analysis of the beginning, middle stage, and ending of the pandemic in Pakistan are presented by combining standard epidemiological modeling and an eigenvalue analysis as used in synergetics. The decay of the effective contact rate throughout the three stages presumably due to intervention measures is demonstrated. Importantly, the sign-switching of the dominant eigenvalue from the beginning to the ending stage is shown that suggests that at least in August and September 2020 the diseasefree state (i.e., state free of COVID-19) was stabilized.


World Health Organization. COVID-19 Weekly Epidemiological Update 17 November 2020,

https://www.who.int/emergencies/diseases/novelcoronavirus-2019. Accessed Nov 23, 2020.

UNICEF. Pakistan COVID-19 Situation Report No 17, https://www.unicef.org/documents/pakistancovid-19-situation-report-15-september-2020. Accessed Oct 4, 2020.

F. Tanveer, A.T. Khalil, M. Ali, and Z.K. Shinwari.Ethics, pandemic and environment; looking at the future of low middle income countries. International Journal of Equity in Health 19: article 182 (2020).

World Health Organization. Novel coronavirus (2019-nCoV) Situation Report-1, https://www.who.int/emergencies/diseases/novel-coronavirus-2019.Accessed June 17, 2020.

Epicentro (Epidemiology for public health), SARSCoV-2: Infographic 15 June 2020, https://www.epicentro.iss.it/en/coronavirus. Accessed June 17,2020.

K. Rock, S. Brand, J. Moir, and M.J. Keeling.Dynamics of infectious diseases. Reports on Progress in Physics 77: article 026602 (2014).

M. Munir. Sensitivity and generalized sensitivity studies of the SIR and SEIR Models of computer virus. Proceedings of the Pakistan Academy of Sciences A 54: 167-178 (2017).

L. Pang, S. Liu, X. Zhang, T. Tian, and Z. Zhao, Transmission dynamics and control strategies of COVID-19 in Wuhan, China. Journal of Biological Systems 28: 543–561 (2020).

T.D. Frank. COVID-19 order parameters and time constants of Italy and China: a modelling approach based on synergetics. Journal of Biological Systems 28: 589-608 (2020).

C. Distante, P. Piscitelli, and A. Miani, Covid-19 outbreak progression in Italian regions: Approaching the peak by the end of March in Northern Italy and first week of April in Southern Italy. Environmental Research of Public Health 17: article 3025 (2020).

K.N. Nabi. Forecasting COVID-19 pandemic: A data-driven analysis. Chaos, Solitons, and Fractals 139: article 110046 (2020).

M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L.Carraro, R. Casagrandi, and A. Rinaldo. Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proceedings of the National Academy of Sciences USA 117: article 10484 (2020).

Z. Zhao, X. Li, F. Liu, G. Zhu, C. Ma, and L. Wang. Prediction of the COVID-19 spread in African countries and implications for prevention and control: A case study in South Africa, Egypt, Algeria, Nigeria, Senegal and Kenya. Science of the Total Environment 729: article 138959 (2020).

O. Diekmann, and J.A.P. Heesterbeek. Mathematical epidemiology of infectious diseases. John Wiley and Son, Chichester (2000).

J. Guckenheimer, and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, Berlin (1983).

H. Haken. Synergetics. An introduction. Springer, Berlin (1977).

T. Frank. Determinism and self-organization of human perception and performance. Springer,Berlin, (2019).

T. D. Frank. COVID-19 interventions in some European countries induced bifurcations stabilizing low death states against high death states: An eigenvalue analysis based on the order parameter concept of synergetics. Chaos, Solitons, and Fractals 140: article 110194 (2020).

S. Ullah, and M.A. Khan. Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study. Chaos, Solitons, and Fractals 139: article 110075 (2020).

COVID-19 tracker, Timeline data from Johns Hopkins Center for Systems Science and Engineering, https://vac-lshtm.shinyapps.io/ncov_tracker/. Accessed Oct 3, 2020.

Arab News. Pakistan to start opening schools as coronavirus cases fall, https://www.arabnews.com/node/1730846/world. Accessed Nov 24, 2020.

UNICEF. Pakistan COVID-19 Situation Report No 1, https://www.unicef.org/pakistan/reports/covid19-situation-report-1 Accessed Oct 4, 2020.

UNICEF. Pakistan COVID-19 Situation Report No 7, https://www.unicef.org/pakistan/reports/covid19-situation-report-7 Accessed Oct 4, 2020.

UNICEF. Pakistan COVID-19 Situation Report No 10, https://www.unicef.org/pakistan/reports/covid19-situation-report-10 Accessed Oct 4, 2020.

Law Library of US Congress. Smart lockdown imposed across cities of Pakistan as Covid-19 cases grow rapidly, https://www.loc.gov/law/foreignnews/article/pakistan-smart-lockdown-imposedacross-cities-of-pakistan-as-covid-19-cases-growrapidly. Accessed Nov 24, 2020.

UNICEF Pakistan COVID-19 Situation Report No 11, https://www.unicef.org/pakistan/reports/covid19-situation-report-11 Accessed Oct 4, 2020.

International Monetary Fund. Policy Responses to COVID-19, Pakistan, https://www.imf.org/en/Topics/imf-and-covid19/Polic y-Responses-toCOVID-19#P. Accessed Nov 24, 2020.




How to Cite

Frank, T. . (2021). Emergence and Subsiding of the First-wave COVID-19 Pandemic in Pakistan (2020): An Eigenvalue Analysis Based on Synergetics: First-wave COVID-19 pandemic in Pakistan . Proceedings of the Pakistan Academy of Sciences: B. Life and Environmental Sciences, 57(4), 1–7. Retrieved from http://ppaspk.org/index.php/PPAS-B/article/view/143