Herd immunity in a coronavirus disease 2019 epidemic model with consideration of vaccination and quarantine interventions

Hasan Moh; Faizal Rifky Fahreza

doi:10.59400/adecp2759

Hasan Moh Department of Mathematics, University of Jember, Jember 68121, Indonesia
Faizal Rifky Fahreza Department of Mathematics, University of Jember, Jember 68121, Indonesia

Article ID: 2759

DOI: https://doi.org/10.59400/adecp2759

Keywords: herd immunity threshold; infection rate, bifurcation diagram; basic reproduction number

Abstract

During the pandemic of COVID-19, people had reduced contact among each other. As a result of this behavior, several factors, such as economic conditions and the teaching and learning process, have been affected. Hence, it is important to identify whether the impact of COVID-19 is no longer as severe as when it was first observed. The study aimed to analyze herd immunity against COVID-19 in Indonesia according to the bifurcations and simulations of mathematical models of COVID-19 transmission. Based on the bifurcation of the disease system, whether the current pandemic was controlled with standard interventions was evaluated. The system behavior can be compared with herd immunity that should be achieved in a specific population. Thus, whether a system has resulted in the achievement of herd immunity can be evaluated. The behavior of this system can provide information on the achievement of group immunity during disease outbreaks.

References

[1]Brauer F, Castillo-Chavez C, Feng Z. Mathematical Models in Epidemiology. Springer; 2019.

[2]Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos, Solitons & Fractals. 2020; 139: 110057. doi: 10.1016/j.chaos.2020.110057

[3]Khalaf SL, Flayyih HS. Analysis, predicting, and controlling the COVID-19 pandemic in Iraq through SIR model. Results in Control and Optimization. 2023; 10: 100214. doi: 10.1016/j.rico.2023.100214

[4]Neves AGM, Guerrero G. Predicting the evolution of the COVID-19 epidemic with the A-SIR model: Lombardy, Italy and São Paulo state, Brazil. Physica D: Nonlinear Phenomena. 2020; 413: 132693. doi: 10.1016/j.physd.2020.132693

[5]Zou Y, Yang W, Lai J, et al. Vaccination and Quarantine Effect on COVID-19 Transmission Dynamics Incorporating Chinese-Spring-Festival Travel Rush: Modeling and Simulations. Bulletin of Mathematical Biology. 2022; 84(2). doi: 10.1007/s11538-021-00958-5

[6]Auger P, Moussaoui A. On the Threshold of Release of Confinement in an Epidemic SEIR Model Taking into Account the Protective Effect of Mask. Bulletin of Mathematical Biology. 2021; 83(4). doi: 10.1007/s11538-021-00858-8

[7]Negi SS, Rana PS, Sharma N, Khatri MS. A novel SEIAHR compartment model for accessing the impact of vaccination, intervention policies, and quarantine on the COVID-19 pandemic: A case study of most affected countries Brazil, India, Italy, and USA. Computational and Applied Mathematics. 2022; 41(7). doi: 10.1007/s40314-022-01993-1

[8]Yang B, Yu Z, Cai Y. The impact of vaccination on the spread of COVID-19: Studying by a mathematical model. Physica A: Statistical Mechanics and its Applications. 2022; 590: 126717. doi: 10.1016/j.physa.2021.126717

[9]Tyagi S, Martha SC, Abbas S, Debbouche A. Mathematical modeling and analysis for controlling the spread of infectious diseases. Chaos, Solitons & Fractals. 2021; 144: 110707. doi: 10.1016/j.chaos.2021.110707

[10]Mohammadi H, Rezapour S, Jajarmi A. On the fractional SIRD mathematical model and control for the transmission of COVID-19: The first and the second waves of the disease in Iran and Japan. ISA Transactions. 2022; 124: 103–114. doi: 10.1016/j.isatra.2021.04.012

[11]Caetano C, Morgado ML, Patrício P, et al. Measuring the impact of COVID-19 vaccination and immunity waning: A modelling study for Portugal. Vaccine. 2022; 40(49): 7115–7121. doi: 10.1016/j.vaccine.2022.10.007

[12]Law KB, Peariasamy KM, Mohd Ibrahim H, Abdullah NH. Modelling infectious diseases with herd immunity in a randomly mixed population. Scientific Reports. 2021; 11(1). doi: 10.1038/s41598-021-00013-2

[13]MacIntyre CR, Costantino V, Trent M. Modelling of COVID-19 vaccination strategies and herd immunity, in scenarios of limited and full vaccine supply in NSW, Australia. Vaccine. 2022; 40(17): 2506–2513. doi: 10.1016/j.vaccine.2021.04.042

[14]Yuan R, Jiang W, Wang Y. Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator-prey model with time-delay and prey harvesting. Journal of Mathematical Analysis and Applications. 2015; 422(2): 1072–1090. doi: 10.1016/j.jmaa.2014.09.037

[15]Lynch S. Dynamical Systems with Applications using Python. Springer International Publishing; 2018.

[16]Castillo-Chavez C, Song B. Dynamical Models of Tuberculosis and Their Applications. Mathematical Biosciences and Engineering. 2004; 1(2): 361–404. doi: 10.3934/mbe.2004.1.361

[17]Wangari IM, Stone L. Backward bifurcation and hysteresis in models of recurrent tuberculosis. PLoS One. 2018; 13(3): e0194256. doi: 10.1371/journal.pone.0194256

[18]Asamoah JKK, Nyabadza F, Jin Z, et al. Backward bifurcation and sensitivity analysis for bacterial meningitis transmission dynamics with a nonlinear recovery rate. Chaos, Solitons & Fractals. 2020; 140: 110237. doi: 10.1016/j.chaos.2020.110237

[19]Nadim SS, Chattopadhyay J. Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19. Chaos, Solitons & Fractals. 2020; 140: 110163. doi: 10.1016/j.chaos.2020.110163

[20]Omame A, Abbas M, Onyenegecha CP. Backward bifurcation and optimal control in a co-infection model for SARS-CoV-2 and ZIKV. Results in Physics. 2022; 37: 105481. doi: 10.1016/j.rinp.2022.105481

[21]Nudee K, Chinviriyasit S, Chinviriyasit W. The effect of backward bifurcation in controlling measles transmission by vaccination. Chaos, Solitons & Fractals. 2019; 123: 400–412. doi: 10.1016/j.chaos.2019.04.026

[22]World Health Organization. Coronavirus disease (COVID-19) pandemic. Available online: https://covid19.who.int/region/searo/country/id (accessed on 5 February 2024).

[23]Toharudin T, Pontoh RS, Caraka RE, et al. National Vaccination and Local Intervention Impacts on COVID-19 Cases. Sustainability. 2021; 13(15): 8282. doi: 10.3390/su13158282

[24]Djalante R, Lassa J, Setiamarga D, et al. Review and analysis of current responses to COVID-19 in Indonesia: Period of January to March 2020. Progress in Disaster Science. 2020; 6: 100091. doi: 10.1016/j.pdisas.2020.100091

[25]Fahreza FR, Hasan M, Kusbudiono K. Model and Simulation of COVID-19 Transmission with Vaccination and Quarantine Interventions in Jember. InPrime: Indonesian Journal of Pure and Applied Mathematics. 2023; 5(1): 1–21. doi: 10.15408/inprime.v5i1.27192

[26]Mao Y, Wang W, Ma J, et al. Reinfection rates among patients previously infected by SARS-CoV-2: Systematic review and meta-analysis. Chinese Medical Journal. 2021; 135(2): 145–152. doi: 10.1097/cm9.0000000000001892

[27]Siqueira PC, Cola JP, Comerio T, et al. Herd immunity threshold for SARS-CoV-2 and vaccination effectiveness in Brazil. Jornal Brasileiro de Pneumologia. 2022; 48(2): e20210401. doi: 10.36416/1806-3756/e20210401

[28]Suryawanshi YN, Biswas DA. Biswas DA. Herd Immunity to Fight Against COVID-19: A Narrative Review. Cureus. 2023; 15(1): e33575. doi: 10.7759/cureus.33575

[29]Kyurkchiev N, Kyurkchiev V, Iliev A, Rahnev A. A New Modifications of the sir/seir models with “intervention polynomial factor”. methodological aspects. International Journal of Differential Equations and Applications. 2021; 20(1): 15–30.