# Dynamics of COVID-19 Epidemic Model with Asymptomatic Infection, Quarantine, Protection and Vaccination

## Authors

• Raqqasyi Rahmatullah Musafir Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Jl. Veteran Malang 65145, Indonesia
• Agus Suryanto Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Jl. Veteran Malang 65145, Indonesia
• Isnani Darti Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Jl. Veteran Malang 65145, Indonesia

## Keywords:

COVID-19 epidemic model, asymptomatic infection, quarantine, protection, vaccination, Lyapunov Function, stability analysis

## Abstract

We discuss the dynamics of new COVID-19 epidemic model by considering asymptomatic infections and the policies such as quarantine, protection (adherence to health protocols), and vaccination. The proposed model contains nine subpopulations: susceptible (S), exposed (E), symptomatic infected (I), asymptomatic infected (A), recovered (R), death (D), protected (P), quarantined (Q), and vaccinated (V ). We first show the non-negativity and boundedness of solutions. The equilibrium points, basic reproduction number, and stability of equilibrium points, both locally and globally, are also investigated analytically. The proposed model has disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable if basic reproduction number is less than one. The endemic equilibrium point exists uniquely and is globally asymptotically stable if the basic reproduction number is greater than one. These properties have been confirmed by numerical simulations using the fourth order Runge-Kutta method. Numerical simulations show that the disease transmission rate of asymptomatic infection, quarantine rates, protection rate, and vaccination rates affect the basic reproduction number and hence also influence the stability of equilibrium points.

## References

Adeyemi, M. O., Oluyo, T. O. and Oladejo, J. K., Modelling the transmission and control dynamics of coronavirus disease with social distancing and contact tracing, International Journal of Innovative Science and Research Technology, 5(5), pp. 948-964, 2020.

Aguiar, M. and Stollenwerk, N., SHAR and effective SIR models: from dengue fever toy models to a COVID-19 fully parametrized SHARUCD framework, Communication in Biomathematical Sciences, 3(1), pp. 60-89, 2020.

Ahmad, S., Ullah, A., Al-Mdallal, Q. M., Khan, H., Shah, K. and Khan, A., Fractional order mathematical modeling of COVID-19 transmission, Chaos, Solitons & Fractals, 139, pp. 110256, 2020.

Ahmad, W., Sarwar, M., Shah, K., Ahmadian, A. and Salahshour, S., Fractional order mathematical modeling of novel corona virus (COVID-19), Mathematical Methods in the Applied Sciences, In press, 2021.

Aldila, D., Analyzing the impact of the media campaign and rapid testing for COVID-19 as an optimal control problem in East Java, Indonesia, Chaos, Solitons & Fractals, 141, pp. 110364, 2020.

Aldila, D., Ndii, M. Z. and Samiadji, B. M.. Optimal control on COVID-19 eradication program in indonesia under the effect of community awareness, Mathematical Biosciences and Engineering, 17(6), pp. 6355-6389, 2020.

Asamoah, J. K. K., Owusu, M. A., Jin, Z., Oduro, F., Abidemi, A. and Gyasi, E. O., Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana, Chaos, Solitons & Fractals, 140, pp. 110103, 2020.

Baba, I. A., Nasidi, B. A. and Baleanu, D., Optimal control model for the transmission of novel COVID-19, Computers, Materials, & Continua, pp. 3089-3106, 2021.

Bahloul, M. A., Chahid, A. and Laleg-Kirati, T.-M., Fractional-order seiqrdp model for simulating the dynamics of COVID-19 epidemic, IEEE Open Journal of Engineering in Medicine and Biology, 1, pp. 249-256, 2020.

Boyce, W. E. and DiPrima, R. C., Elementary differential equations and boundary value problems, USA: John Wiley & Sons, Inc. All rights reserved., 2012.

Bugalia, S., Bajiya, V. P., Tripathi, J. P., Li, M.-T. and Sun, G.-Q., Mathematical modeling of COVID-19 transmission: the roles of intervention strategies and lockdown, Mathematical Biosciences and Engineering, 17(5), pp. 5961-5986, 2020.

Cooper, I., Mondal, A. and Antonopoulos, C. G., A sir model assumption for the spread of COVID-19 in different communities, Chaos, Solitons & Fractals, 139, pp. 110057, 2020.

Darti, I., Habibah, U., Astutik, S., Kusumawinahyu, W. M., Marsudi and Suryanto, A., Comparison of phenomenological growth models in predicting cumulative number of COVID-19 cases in East Java province, Indonesia, Communications in Mathematical Biology and Neuroscience, 2021, article ID 14, 2021.

Darti, I., Suryanto, A., Panigoro, H. S. and Susanto, H., Forecasting COVID-19 epidemic in Spain and Italy using a generalized Richards model with quantified uncertainty, Communication in Biomathematical Sciences, 3(2), pp. 90-100, 2020.

Daud, A. A. M., A note on Lienard-Chipart criteria and its application to epidemic models, Mathematics and Statistics, 9(1), pp. 41-45, 2021.

Farman, M., Aslam, M., Akgul, A. and Ahmad, A., Modeling of fractional-order COVID-19 epidemic model with quarantine and social distancing, Mathematical Methods in the Applied Sciences, 44, pp. 9334-9350, 2021.

Feng, L.-X., Jing, S.-L., Hu, S.-K., Wang, D.-F. and Huo, H.-F., Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK, Mathematical Biosciences and Engineering, 17(4), pp. 3618-3636, 2020.

Fuady, A., Nuraini, N., Sukandar, K. K. and Lestari, B. W., Targeted vaccine allocation could increase the COVID-19 vaccine benefits amidst its lack of availability: A mathematical modeling study in Indonesia, Vaccines, 9(5), pp. 462, 2021.

Ghostine, R., Gharamti, M., Hassrouny, S. and Hoteit, I., An extended seir model with vaccination for forecasting the COVID-19 pandemic in Saudi Arabia using an ensemble Kalman filter, Mathematics, 9(6), pp. 636, 2021.

Harapan, H., Wagner, A. L., Yufika, A., Winardi, W., Anwar, S., Gan, A. K., Setiawan, A. M., Rajamoorthy, Y., Sofyan, H. and Mudatsir, M., Acceptance of a COVID-19 vaccine in Southeast Asia: a cross-sectional study in Indonesia, Frontiers in Public Health, 8, Article 381, 2020.

Kassa, S. M., Njagarah, J. B. and Terefe, Y. A., Analysis of the mitigation strategies for COVID-19: from mathematical modelling perspective, Chaos, Solitons & Fractals, 138, pp. 109968, 2020.

Khan, M. A., Atangana, A., Alzahrani, E. and Fatmawati., The dynamics of COVID-19 with quarantined and isolation, Advances in Difference Equations, 2020, Article No. 425, 2020.

Kim, J. H., Marks, F. and Clemens, J. D., Looking beyond COVID-19 vaccine phase 3 trials, Nature medicine, 27(2), pp. 205-211, 2021.

Koziol, K., Stanislawski, R. and Bialic, G., Fractional-order sir epidemic model for transmission prediction of COVID-19 disease, Applied Sciences, 10(23), pp. 8316, 2020.

Li, Y., Tenchov, R., Smoot, J., Liu, C., Watkins, S. and Zhou, Q., A comprehensive review of the global efforts on COVID-19 vaccine development, ACS Central Science, 7(4), pp. 512-533, 2021.

Lin, C.-Y., Social reaction toward the 2019 novel coronavirus (COVID-19), Social Health and Behavior, 3(1), p. 1, 2020.

Lopez, L. and Rodo, X., A modified SEIR model to predict the COVID-19 outbreak in Spain and Italy: simulating control scenarios and multi-scale epidemics, Results in Physics, 21, pp. 103746, 2021.

Martcheva, M., An introduction to mathematical epidemiology, Springer, 2015.

Megasari, N. L. A., Utsumi, T., Yamani, L. N., Gunawan, E., Furukawa, K., Nishimura, M., Lusida, M. I. and Mori, Y., Seroepidemiological study of sars-cov-2 infection in East Java, Indonesia. Plos One, 16(5), pp. 251234, 2021.

Murray, J. D., Mathematical Biology I. An Introduction, volume 17 of Interdisciplinary Applied Mathematics, ed. 2, Berlin: Springer Science & Business Media, 2002.

Ndii, M. Z., Hadisoemarto, P., Agustian, D. and Supriatna, A., An analysis of COVID-19 transmission in Indonesia and Saudi Arabia, Communication in Biomathematical Sciences, 3(1), pp. 19-27, 2020.

Nuraini, N., Khairudin, K. and Apri, M., Modeling simulation of COVID-19 in Indonesia based on early endemic data. Communication in Biomathematical Sciences, 3(1), pp. 1-8, 2020.

Oud, M. A. A., Ali, A., Alrabaiah, H., Ullah, S., Khan, M. A. and Islam, S., A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load, Advances in Difference Equations, 2021, Article No. 106, 2021.

Polack, F. P., Thomas, S. J., Kitchin, N., Absalon, J., Gurtman, A., Lockhart, S., Perez, J. L., Marc, G. P., Moreira, E. D., Zerbini, C., Bailey, R., Swanson, K. A., Roychoudhury, S., Koury, K., Li, P., Kalina, W. V., Cooper, D., Frenck, R. W., Hammitt, L. L., Tureci, O., Nell, H., Schaefer, A. , Unal, S., Tresnan, D. B., Mather, S., Dormitzer, P. R., Sahin, U., Jansen, K. U. and Gruber, W. C., Safety and efficacy of the bnt162b2 mrna COVID-19 vaccine, New England Journal of Medicine, 2020.

Rajagopal, K., Hasanzadeh, N., Parastesh, F. , Hamarash, I. I., Jafari, S. and Hussain, I., A fractional-order model for the novel coronavirus (COVID-19) outbreak, Nonlinear Dynamics, 101, pp. 711-718, 2020.

Rezapour, S., Mohammadi, H. and Samei, M. E., SEIR epidemic model for COVID-19 transmission by Caputo derivative of fractional order, Advances in Difference Equations, 2020, Article No. 490, 2020.

Riyapan, P., Shuaib, S. E. and Intarasit, A., A mathematical model of COVID-19 pandemic: A case study of Bangkok, Thailand, Computational and Mathematical Methods in Medicine, 2021, Article ID 6664483, 2021.

Shakhany, M. Q. and Salimifard, K., Predicting the dynamical behavior of COVID-19 epidemic and the effect of control strategies, Chaos, Solitons & Fractals, 146, p. 110823, 2021.

Soewono, E., On the analysis of COVID-19 transmission in Wuhan, Diamond Princess and Jakarta-cluster, Communication in Biomathematical Sciences, 3(1), pp. 9-18, 2020.

Varghese, V., Bhoyar, S. and Nisar, K. S., Analysis of fractional-order model of COVID-19 pandemics with a nonlinear incidence rate, Innovative Biosystems and Bioengineering, 4(3), pp. 160-167, 2020.

Wu, K., Darcet, D., Wang, Q. and Sornette, D., Generalized logistic growth modeling of the COVID-19 outbreak: comparing the dynamics in the 29 provinces in China and in the rest of the world, Nonlinear dynamics, 101, pp. 1561-1581, 2020.

Zhai, S., Luo, G., Huang, T., Wang, X., Tao, J. and Zhou, P., Vaccination control of an epidemic model with time delay and its application to COVID-19, Nonlinear Dynamics, 106, pp. 1279-1292, 2021.

Zhang, Z., Gul, R. and Zeb, A., Global sensitivity analysis of COVID-19 mathematical model, Alexandria Engineering Journal, 60(1), pp. 565-572, 2021.

Zhang, Z., Zeb, A., Egbelowo, O. F. and Erturk, V. S., Dynamics of a fractional order mathematical model for COVID-19 epidemic, Advances in Difference Equations, 2020(1), pp. 1-16, 2020.

Zhu, C.-C. and Zhu, J., Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method, Chaos, Solitons & Fractals, 143, pp. 110546, 2021.