# Mathematical Modelling and Control of COVID-19 Transmission in the Presence of Exposed Immigrants

## Authors

• Reuben Iortyer Gweryina Department of Mathematics, Joseph Sarwuan Tarka University, P.M.B 2373, Makurdi, Nigeria
• Chinwendu Emilian Madubueze Department of Mathematics, Joseph Sarwuan Tarka University, P.M.B 2373, Makurdi, Nigeria
• Martins Afam Nwaokolo Department of Mathematics and Statistics, Federal University Wukari, Nigeria

## Keywords:

COVID-19, exposed immigrants, doubling time, transcritical bifurcation

## Abstract

In this paper, a mathematical model for COVID-19 pandemic that spreads through horizontal transmission in the presence of exposed immigrants is studied. The model has equilibrium points, notably, COVID-19-free equilibrium and COVID-19-endemic equilibrium points. The model exhibits a basic reproduction number, R0 which determines the elimination and persistence of the disease. It was found that when R0 < 1, then the equilibrium becomes locally asymptotically stable and endemic equilibrium does not exists. However, when R0 > 1, the equilibrium is found to be stable globally. This implies that continuous mixing of exposed immigrants with the susceptible population will make the eradication of COVID-19 difficult and endemic in the community. The system is also proved qualitatively to experience transcritical bifurcation close to the COVID-19-free equilibrium at the point R0 = 1. Numerically, the model is used to investigate the impact of certain other relevant parameters on the spread of COVID-19 and how to curtail their effect.

## References

Luo, X., Zhou, G., Zhang, Z., Peng, L., Zou, L., Yeng, Y., Coronaviruses and gastrointestinal diseases, Military Medical Research, 7(2020), pp. 1-49, 2020.

Ye, Z., Yuan, S., Yuen, K., Fung, S., Chan, C., and Jin, D., Zoonotic origins of human Coronaviruses, International Journal of Biological Sciences, 16(10), pp. 1686-1697, 2020.

WHO(a)., Coronavirus disease 2019 (COVID-19) Situation Report-101. Accessed from https://www.who.int.docs on May 1, 2021.

WHO(b)., Coronavirus disease 2019 (COVID-19) Situation Report-51, Accessed from https://reliefweb.int/report/china/coronavirus-disease-2019-covid-19-situation-report-51-11-march-2020 on May 15, 2021.

Huang, C., y. Wang, V., Li, X., Ren, L., Zhao., J., and Hu, Y.,Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, Lancet, 395(2020), pp. 497-506, 2020.

Song, W. Y., Zang, P., Ding, Z. X., Fang, X. Y., Zhu, L. G., Bao, C., Chen, F., Wu, M., Peng, Z. H., Massive migration promotes the early spread of COVID-19 in China: a study based on a scale-free network, Infectious Diseases of Poverty, 9(1), pp. 1-8, 2020.

Indseth,N., Grosland, M., Arnesen, T., Skyrud, K., Klovstad, H., Lamprini, V., Telle, K and Kjollesdal, M., COVID-19 among immigrants in Norway, notified infections, related hospitalization and associated mortality: A register-based study, Scandinavian Journal of Public Health, 49(1), pp. 48-56, 2021.

Wang, L., Wang, X., Influence of temporary migration on the transmission of infectious diseases in a migrant home village, Journal of Theoretical Biology, 300, pp. 100-109, 2012.

He, X. et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nature Medicine, 26(5), pp. 672-675, 2020.

Mahmood, M., Iiyas, N., Khan, M. F., Hasrat, M. N.and Richwagen, N. Transmission frequency of COVID-19 through presymptomatic and asymptomatic patients in AJK: a report of 201 cases, Virology Journal, 18, pp. 138, 2021. https://doi.org/10.1186/s12985-021-01609-w.

Ahmed, I., Modu, G. U., Yusuf, A., Kumam, P and Yusuf, I., A mathematical model of Coronavirus disease (COVID-19) containing asymptomatic and symptomatic classes, Results in Physics, 21, p. 103776, 2021.

Deressa, C. T., Duressa, G. F., Modelling and optimal analysis of transmission dynamics of COVID-19: The case of Ethiopia, Alexandria Engineering Journal, 60(1), pp. 719-732, 2021.

Kim, B. N., Kim, E., Lee, S., and Oh, C., Mathematical model of COVID-19 transmission dynamics in South Korea: The impacts of travel restrictions, social distancing and early detection, Processes, 8(10), p. 1304, 2020.

Makhoul, M., Ayoub, H. H.,Chemaitelly, H.. Seedat, S., Mumtaz, G. R., Al-Omari, S., Abu-Raddad, L. J., Epidemiological impact of SARS-COV-2 vaccination: Mathematical modelling Analyzes, Vaccines, 8(4), p. 668, 2020.

Ayana, M., Hailleqiorgis, T., and Getnet, K., The impact of infective immigrants and self isolation on the dynamics and spread of COVID-19 pandemic: A mathematical modelling study, Pure and Applied Mathematics Journal, 9(6), pp. 109-117, 2020.

Adeniyi, M. O., Ekum, M. I., Iluno, C., Ogunsanya, A. S., Akinyemi, J. A., Oke, S. I. and Matadi, M. B., Dynamic model of COVID-19 disease with exploratory data analysis, 9, pp. e00477, 2020.

Agaba, G. O., Modelling the spread of COVID-19 with impact of awareness and medical assistance, Mathematical Theory and Modelling, 10(4), pp. 20-28, 2020.

Ndam, J. N., Modelling the impacts of lockdown and isolation on the eradication of COVID-19, BIOMATH, 9(2), p. 2009107, 2020.

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.

Centers for Disease Control and Prevention (CDC), Guidance for correctional and detention facilities, Retrieved on Wednesday 10, 2021 from https://www.cdc.gov/coronavirus/2019-ncov/community/correction-detention/guidance-correctional-detention.html, 2021.

Alshammari, F. S., A mathematical model to investigate the transmission of COVID-19 in the Kingdom of Saudi Arabian, Computational and Mathematical Methods in Medicine, 9136157, pp. 1-13, 2020.

Kibona, I., Mahera, W., Makinde, D., and Mango, J., A deterministic model of HIV/AIDS with vertical transmission in the presence of infected immigrants, International Journal of the Physical Sciences, 6(23), pp. 5383-5398, 2011.

Patel, P. B., Doubling Time and its Interpretation for COVID-19 Cases, National Journal of Community Medicine, 11(3), pp. 141-143, 2020.

Galvani, A. P., lEI, X. and Jewell, N. P., Severe Acute Respiratory Syndrome: Temporal Stability and Geographic Variation in Case-Fatality Rates and Doubling Times, Emerging Infectious Diseases, 9(8), pp. 991-994, 2003.

Martcheva, M., An Introduction to Mathematical Epidemiology, Springer Science and Business Media, 61, pp. 100-101, 2015.

Alkhudhari, Z., Al-Asheikh, S. and Al-Tuwairqi, S., Stability analysis of a giving up smoking model, International Journal of Applied Mathematical Research, 3(2), pp. 168-177, 2014.

La Salle, J. P., The Stability of Dynamical Systems, Hamilton Press, Berlin, New Jersey, USA, pp. 1-70, 1976.

Naji, R. K., HussienR. M., The dynamics of epidemic model with two types of infectious diseases and vertical transmission, Journal of Applied Mathematics, 16, p. 4907964, 2016.