# On The Study of Covid-19 Transmission Using Deterministic and Stochastic Models with Vaccination Treatment and Quarantine

## Authors

• Mona Zevika Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
• Anita Triska Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
• Nuning Nuraini Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
• Glenn Lahodny Jr. Department of Mathematics, University of Texas at San Antonio, Texas 78249, USA

## Keywords:

Covid-19 pandemic, SEIR model, reproduction number, probability of extinction, vaccination

## Abstract

In this study, we propose deterministic and stochastic models of the spread of Covid-19 with vaccination and quarantine programs. The model considers the facts that vaccines do not provide full protection, the efficacy of current vaccines only lasts for a limited time, and recovered people could be reinfected. The routine analysis was carried out for the deterministic model, including calculating an expression for the basic reproduction number. The stochastic formulation makes use of a Continuous-Time Markov Chain (CTMC) model. The basic reproduction number from the deterministic model relates to the stochastic model's analysis in producing a formula for the probability of extinction of Covid-19. Furthermore, numerical simulations are carried out to analyze the sensitivity of the dynamical states and the basic reproduction number to the model parameters. An expression for the probability of disease extinction in terms of the model parameters and initial conditions is given. The results of this study suggest that current conditions in Indonesia will lead to a longterm Covid-19 epidemic. One of the efforts to overcome the Covid-19 epidemic is by increasing the provision of vaccines to the susceptible population. However, the number of vaccinated people in the population is not always an ideal control for dealing with the spread of the disease. The vaccine efficacy is also important to reduce the infection. As long as the efficacy is not sufficient to give a good protection to the human population and it lasts only for a short period of time, quarantine is still needed.

## References

Aini, M., and Widjaja, G., Mandatory coronavirus disease-19 (covid-19) vaccination in Indonesia: legal aspect, Journal of Legal, Ethical and Regulatory Issues, pp. 1-15, 2021.

Alanazi, S. A., Kamruzzaman, M., Alruwaili, M.. Alshammari, N., Alqahtani, S. A., and Karime, A., Measuring and preventing covid-19 using the sir model and machine learning in smart health care, Journal of healthcare engineering, 2020, 8857346, 2020.

Allen, L. J., and Lahodny, G. E. Jr., Extinction thresholds in deterministic and stochastic epidemic models, Journal of biological dynamics, 6(2), pp. 590-611, 2012.

Allen, L. J., and Driessche, P. V. D., Relations between deterministic and stochastic thresholds for disease extinction in continuousand discrete-time infectious disease models, Mathematical biosciences, 243(1), pp. 99-108, 2013.

Annas, S., Pratama, M. I., Rifandi, M., Sanusi, W., and Side, S., Stability analysis and numerical simulation of seir model for pandemic covid-19 spread in indonesia, Chaos, Solitons Fractals, 139, p. 110072, 2020.

Athreya, KB and Ney, PE, Branching processes, Springer-Verlag; NewYork, 1972.

Chitnis, N., Hyman, J. M., and Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of mathematical biology, 70(5), p. 1272, 2008.

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, p. 110057, 2020.

Covid19 Id, 3 Vaksin COVID-19 Dalam Program Vaksinasi Nasional Saat Ini Semua Direkomendasikan WHO, WHO, https://covid19.go.id/berita/3-vaksin-covid-19-dalam-program-vaksinasi-nasional-saat-ini-semua-direkomendasikan-who, Accessed December 3, 2021.

Dan, J., Mateus, J., Kato, Y., Hastie, K., Faliti, E. Yu, C., and Crotty, S., Immunological memory to sars-cov-2 assessed for up to 8 months after infection, Science, 371(6529), p. eabf4063, 2021.

Driessche, P. V. d., and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180(1-2), pp. 29-48, 2002.

Driessche, P. V. d., and Watmough, J., Further notes on the basic reproduction number, in Mathematical epidemiology, pp. 159-178, Springer, 2008.

Dwipayana, I. D. A. P., Efforts in securing vaccine for covid-19 outbreak in indonesia, Health Notions, 4(10), pp. 313-317, 2020.

Evensen, G., Amezcua, J., Bocquet, M., Carrassi, A., Farchi, A., Fowler, A., Houtekamer, P., Jones, C. K., De Moraes, R., Pulido, M., et al., An international assessment of the covid-19 pandemic using ensemble data assimilation, medRxiv, 2020.

Feng, S., Z. Feng, Z., Ling, C., Chang, C., and Feng, Z., Prediction of the covid-19 epidemic trends based on seir and ai models, Plos one, 16(1), p. e0245101, 2021.

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), p. 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), p. 636, 2021.

Harris, T. E., et al., The theory of branching processes, Springer Berlin, 1963.

Healtdata, COVID-19 vaccine efficacy summary, IHME, http://www.healthdata.org/covid/covid-19-vaccine-efficacy-summary, Accessed December 4, 2021.

Karlin, S., A first course in stochastic processes, Academic press, 2014.

Khumaeroh, M. S., Soewono, E., and Nuraini, N., A dynamical model of 'invisible wall' in mosquito control, Commun. Biomath. Sci, 1(2), pp. 88-99, 2018.

Machado, B., Antunes, L., Caetano, C., Pereira, J. F., Nunes, B., Patricio, P., and Morgado, M. L., The impact of vaccination on the evolution of covid-19 in portugal, Mathematical Biosciences and Engineering, 19(1), pp. 936-952, 2022.

Ndairou, F., Area, I., Nieto, J. J., and Torres, D. F., Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan, Chaos, Solitons Fractals, 135, p. 109846, 2020.

Omae, Y., Kakimoto, Y., Sasaki, M., Toyotani, J., Hara, K., Gon, Y., and Takahashi, H., Sirvvd model-based verification of the effect of first and second doses of covid-19/sars-cov-2 vaccination in japan, Mathematical Biosciences and Engineering, 19(1), pp. 1026-1040, 2022.

Our World in Data, Daily share of the population receiving a first COVID-19 vaccine dose, Oxford Martin School, https://ourworldindata.org/explorers/coronavirus-data-explorer?facet=none&Metric=People+vaccinated&Interval=7-day+rolling+average&Relative+to+Population=true&Align+outbreaks=false&country=?IDN, Accessed December 27, 2021.

Penisson, S., Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis, PhD thesis, Universitat Potsdam, Universita Paris Sud-Paris XI, 2010.

Sarkar, K., Khajanchi, S., and Nieto, J. J., Modeling and forecasting the covid-19 pandemic in India, Chaos, Solitons Fractals, 139, p. 110049, 2020.

Savasan, A., Bilgen, K., Gokbulut,N., Hincal, E., and Yoldascan, E., Sensitivity analysis of covid-19 in mediterranean island, Computer Modeling in Engineering Sciences, pp. 133-148, 2022.

Suandi, D., Wijaya, K. P., Apri, M., Sidarto, K. A., Syafruddin, D., Gotz, T., and Soewono, E., A one-locus model describing the evolutionary dynamics of resistance against insecticide in anopheles mosquitoes, Applied Mathematics and Computation, 359, pp. 90-106, 2019.

Soewono, E., and Lahodny, G., On the effect of postponing pregnancy in a zika transmission model, Advances in Difference Equations, 2021(1), pp. 1-14, 2021.

Suryani, D. R., Zevika, M., and Nuraini, N., Deterministic and stochastic models for middle east respiratory syndrome (mers), in AIP Conference Proceedings, 1937, p. 020017, AIP Publishing LLC, 2018.

Susanto, H.,Tjahjono, V., Hasan, H., Kasim, M., Nuraini, N., Putri, E., Kusdiantara, R., and Kurniawan, H., How many can you infect? simple (and naive) methods of estimating the reproduction number, arXiv preprint arXiv:2006.15706, 2020.

WHO, Coronavirus Disease (COVID-19) Outbreak Situation, WHO, https://www.who.int/emergencies/diseases/novel-coronavirus-2019, Accessed May 12, 2021.

WHO, What are the symptoms of COVID-19?, WHO, https://www.who.int/emergencies/diseases/novel-coronavirus-2019/question-and-answers-hub/q-a-detail/coronavirus-disease-covid-19, Accessed June 2, 2021.

WHO, Vaccine efficacy effectiveness and protection, WHO, https://www.who.int/news-room/feature-stories/detail/ vaccine-efficacy-effectiveness-and-protection, Accessed December 27, 2021.

Zevika, M., and Soewono, E., Deterministic and stochastic ctmc models from zika disease transmission, in AIP Conference Proceedings, 1937, p. 020023, AIP Publishing LLC, 2018.

2022-04-15

## How to Cite

Zevika, M., Triska, A., Nuraini, N., & Lahodny Jr., G. (2022). On The Study of Covid-19 Transmission Using Deterministic and Stochastic Models with Vaccination Treatment and Quarantine. Communication in Biomathematical Sciences, 5(1), 1-19. https://doi.org/10.5614/cbms.2022.5.1.1

Articles