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Controlling the Dynamical Spread of Coronavirus Disease (COVID-19) in a Population




In the paper, a model governed by a system of ordinary differential equations was considered; the whole population was divided into Susceptible individuals (S), Exposed individuals (E), Infected individuals (I), Quarantined individuals (Q) and Recovered individuals (R). The well-posedness of the model was investigated by the theory of positivity and boundedness. Analytically, the equilibrium solutions were examined. A key threshold which measures the potential spread of the Coronavirus in the population is derived using the next generation method. Bifurcation analysis and global stability of the model were carried out using centre manifold theory and Lyapunov functions respectively. The effects of some parameters such as Progression rate of exposed class to infectious class, Effective contact rate, Modification parameter, Quarantine rate of infectious class, Recovery rate of infectious class and Recovery rate of quarantined class on R0 were explored through sensitivity analysis. Numerical simulations were carried out to support the theoretical results, to reduce the burden of COVID 19 disease in the population and significant in the spread of it in the population.


Reproduction Number, Bifurcation Analysis, Lyapunov Functions


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F. Brauer, P. van den Driessche, Models for the transmission of disease with immigration of infectives, Math. Biosci. 171, 2001 143–-154.

H. Guo, M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst., Ser. B, 17(7) (2012) 2413–2430.

Garba S. Springer Nature. Sars-cov-2 and covid-19: A new virus and associated respiratory disease, 2020 . Accessed on 18th June, 2020.

F. Brauer. Mathematical epidemiology: Past, present, and future. Infectious Disease Modelling,2(2):113{127, 2017.

H. W Hethcote. The mathematics of infectious diseases. SIAM review, 42(4):599{653, 2000

C. I. Siettos and L. Russo. Mathematical modeling of infectious disease dynamics. Virulence, 4(4):295{306, 2013}.

T.M. Chen, J. Rui, Q.P. Wang, Z.Y. Zhao, J.A. Cui, and L. Yin. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infectious Diseases of Poverty, 9(1):1{8, 2020.

Y. Fang, Y. Nie, and M. Penny. Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions: A data-driven analysis. Journal of Medical Virology.

W. Jia, K. Han, Y. Song, W. Cao, S. Wang, S. Yang, J. Wang, F. Kou, P. Tai, J. Li, et al. Extended sir prediction of the epidemics trend of covid-19 in italy and compared with hunan, china.

B. Tang, N. L. Bragazzi, Q. Li, S. Tang, Y. Xiao, and J. Wu. An updated estimation of the risk of transmission of the novel coronavirus (2019-ncov). Infectious Disease Modelling, 5:248{255, 2020.}

H. Wang, Z. Wang, Y. Dong, R. Chang, C. Xu, X. Yu, S. Zhang, L. Tsamlag, M. Shang, J. Huang, et al. Phase-adjusted estimation of the number of coronavirus disease 2019 cases in wuhan, china. Cell Discovery, 6(1):1{8, 2020}.

S. Zhang, M. Y. Diao, W. Yu, L. Pei, Z. Lin, and D. Chen. Estimation of the reproductive number of novel coronavirus (covid-19) and the probable outbreak size on the diamond princess cruise ship: A data-driven analysis. International Journal of Infectious Diseases, 93:201{204, 2020.}

Elsevier. Novel coronavirus information center, 2020

Springer Nature. Sars-cov-2 and covid-19: A new virus and associated respiratory disease, 2020.

R. Xinmiao, Liu Y., Huidi C., and Meng F. Effect of delay in diagnosis on transmission of covid-19. Mathematical Biosciences and Engineering, 17(mbe-17-03-149):2725, 2020.

K. Mizumoto and G. Chowell. Estimating risk for death from 2019 novel coronavirus disease, china, january-february 2020. Emerging Infectious Diseases, 2020.

K. Roosa, Y. Lee, R. Luo, A. Kirpich, R. Rothenberg, J. M. Hyman, P. Yan, and G. Chowell. Real-time forecasts of the covid-19 epidemic in china from february 5th to february 24th, 2020. Infectious Disease Modelling, 5:256{263, 2020.

Q. Lin, S. Zhao, D. Gao, Y. Lou, S. Yang, S. S. Musa, M. H. Wang, Y. Cai, W. Wang, L. Yang, et al. A conceptual model for the coronavirus disease 2019 (covid-19) outbreak in wuhan, china with individual reaction and governmental action. International journal of infectious diseases, 2020.

Leung, N. H., Chu, D. K., Shiu, E. Y., Chan, K. H., McDevitt, J. J., Hau, B. J., ... & Seto, W. H. Respiratory Virus Shedding in Exhaled Breath and Efficacy of Face Masks. Nature Medicine.

Bourouiba, L. (2020). Turbulent Gas Clouds and Respiratory Pathogen Emissions: Potential Implications for Reducing Transmission of COVID-19. JAMA.

World Health Organization. (2020). Modes of transmission of virus causing COVID-19: implications for IPC precaution recommendations: scientific brief, 27 March 2020.

Han, Q., Lin, Q., Ni, Z., & You, L. (2020). Uncertainties about the transmission routes of 2019 novel coronavirus. Influenza and Other Respiratory Viruses.

van Doremalen, N., Bushmaker, T., Morris, D. H., Holbrook, M. G., Gamble, A., Williamson, B. N., ... & Lloyd-Smith, J. O. (2020). Aerosol and surface stability of SARS-CoV-2 as compared with SARS-CoV-1. New England Journal of Medicine.

Ong, S. W. X., Tan, Y. K., Chia, P. Y., Lee, T. H., Ng, O. T., Wong, M. S. Y., & Marimuthu, K. (2020). Air, surface environmental, and personal protective equipment contamination by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) from a symptomatic patient. JAMA.

Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., & Shaman, J. (2020). Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). Science.

Wu, J. T., Leung, K., & Leung, G. M. (2020). 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), 689-697.

Tang, B., Bragazzi, N. L., Li, Q., Tang, S., Xiao, Y., & Wu, J. (2020). An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov). Infectious Disease Modelling, 5, 248–255.

Kucharski, A. J., Russell, T. W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., ... & Davies, N. (2020). Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The Lancet Infectious Diseases.

Calafiore, G. C., Novara, C., & Possieri, C. (2020). A Modified SIR Model for the COVID-19 Contagion in Italy.

Simha, A., Prasad, R. V., & Narayana, S. (2020). A simple Stochastic SIR model for COVID 19 Infection Dynamics for Karnataka: Learning from Europe.

Dehning, J., Zierenberg, J., Spitzner, F. P., Wibral M., Neto, J. P., Wilczek, M., & Priesemann, V. (2020). Inferring COVID-19 spreading rates and potential change points for case number forecasts.

Nesteruk, I. (2020). Statistics based predictions of coronavirus 2019-nCoV spreading in mainland China.

Zhang, Y., Yu, X., Sun, H., Tick, G. R., Wei, W., & Jin, B. (2020). COVID-19 infection and recovery in various countries: Modeling the dynamics and evaluating the non-pharmaceutical mitigation scenarios.

Anastassopoulou, C., Russo, L., Tsakris, A., & Siettos, C. (2020). Data-Based Analysis, Modelling and Forecasting of the novel Coronavirus (2019-nCoV) outbreak. medRxiv

Moore, S. E., & Okyere, E. Controlling the transmission dynamics of COVID-19. arXiv preprint arXiv:2004.00443.

Ferguson, N., Laydon, D., Nedjati Gilani, G., Imai, N., Ainslie, K., Baguelin, M., ... & Dighe, A. (2020). Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand.

Wilder, B., Charpignon, M., Killian, J. A., Ou, H. C., Mate, A., Jabbari, S., ... & Majumder, M. S. (2020). The Role of Age Distribution and Family Structure on COVID-19 Dynamics: A Preliminary Modeling Assessment for Hubei and Lombardy.

Biswas, K., Khaleque, A., & Sen, P. (2020). Covid-19 spread: Reproduction of data and prediction using a SIR model on Euclidean network.

Chang, S. L., Harding, N., Zachreson, C., Cliff, O. M., & Prokopenko, M. (2020). Modelling transmission and control of the COVID-19 pandemic in Australia.

Ruiz Estrada, M. A., & Koutronas, E. (2020). The Networks Infection Contagious Diseases Positioning System (NICDP-System): The Case of Wuhan-COVID-19. Available at SSRN 3548413.

Castillo-Chavez C, Song B. Dynamical models of tuberculosis and their applications. Math Biosci Eng. 2004;1:361–404.

Akanni J.O. and Akinpelu F.O. (2016) An HIV/AIDs model with vertical transmission, treatment and progression rate. Asian Research Journal of Mathematics, 1(4):1-17, Article no.ARJOM.28549

J. A. Lewnard, M. L. Ndeffo Mbah, J. A. Alfaro-Murillo et al.,“Dynamics and control of Ebola virus transmission inMontserrado, Liberia: a mathematical modelling analysis,” The Lancet Infectious Diseases, vol. 14, no. 12, pp. 1189–1195, 2014.

J. Astacio, D. Briere, M. Guillen, J. Martinez, F. Rodriguez, N. Valenzuela-Campos, “Mathematical models to study the outbreaks of Ebola,” Biometrics Unit Technical Report, Numebr BU-1365-M, Cornell University, 1996.

A. Plüddemann and C. D. H. Parry. Methamphetamine use and associated problems among adolescents in the Western Cape province of South Africa. MRC South Africa,


LaSalle JP (1976) The stability of dynamical systems. In: Regional conference series in applied mathematics. SIAM, Philadelphia, Pa.

Chitnis, N.; Cushing, J. M.; Hyman, J. M.; Bifurcation Analysis of a Mathematical model for malaria transmission. SIAM J. Appl. Math. 67 (1) (2006), 24–45.

J. Arino, C. C. McCluskey, and P. van den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,” SIAM Journal on Applied Mathematics, vol. 64, no. 1, pp. 260–276, 2003

S. Olaniyi and O.S. Obabiyi, Qualitative analysis of malaria dynamics with nonlinear incidence function, Applied Mathematical Sciences, 8, No 78 (2014), 3889–3904.

Garba S, Gumel A, Bakar M. Backward bifurcation in dengue transmission dynamics.Math Biosci. 2008;215:11–25.

A. D. Adediipo, J. O. Akanni, O. M. Shangodare (2020), Bifurcation and Stability Analysis of the Dynamics of Gonorrhea Disease in the Population, World Scientific News 143 (2020) 139-154

A. Assiri, A. McGeer, T. M. Perl, C. S. Price, A. A. Al Rabeeah, D. A. T. Cummings, Z. N. Alabdullatif, M. Assad, A. Almulhim, H. Makhdoom, H. Madani, R. Alhakeem, J. A. Al-Tawfiq, M. Cotten, S. J. Watson, P. Kellam, A. I. Zumla,and Z. A. Memish, Hospital outbreak of Middle East Respiratory Syndrome Coronavirus, N Engl J Med 369 (5) (2013) 407–416.

Steffen E. Eikenberry , Marina Mancuso, Enahoro Iboi, Tin Phan, Keenan Eikenberry, Yang Kuang, Eric Kostelich, and Abba B. Gumel, (2020). To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic. Infectious disease modellingj, vol 5, 2020, 293-308.

S. Usaini, A. S. Hassan, S. M. Garba & JM-S. Lubuma (2019): Modeling the transmission dynamics of the Middle East Respiratory Syndrome Coronavirus (MERS-CoV) withlatent immigrants, Journal of Interdisciplinary Mathematics, DOI: 10.1080/09720502.2019.1692429 S. USAINI, A. S. HASSAN, S. M. GARBA AND JM-S. LUBUMA

Moore, S. E., & Okyere, E. Controlling the transmission dynamics of COVID-19. arXiv preprint arXiv:2004.00443.