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Dynamics of an SIS Model with an Influx of Infective Visitors or Immigrants

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Abstract

Traveling is an exciting activity where visitors can see beautiful places and learn about the culture of other countries. Before traveling to certain places, visitors are required to take vaccines—for example, a quadrivalent meningococcal vaccine for performing Umrah or Hajj in Mecca, Saudi Arabia. However, for new diseases that suddenly appear (recently, COVID-19, monkeypox, and Langya), many aspects are not yet discovered at the beginning of the outbreak such as the way of spreading and the vaccines or medications to control the spread of these diseases. Infected individuals may think it is not a severe illness but bringing a deadly disease or virus to other countries or back home. In order to study how the diseases can spread, an infectious disease model is developed consisting the susceptible and infective individuals with an influx of infective visitors or immigrants. Under the assumption of a constant flow of infective visitors or immigrants, the proposed model does not have the basic reproduction number and disease-free equilibrium point but a unique positive endemic equilibrium point. The global stability of this endemic equilibrium point is proved by constructing a suitable Lyapunov function. The model is quantitatively consistent with the COVID-19 cases in Malaysia for the data reported in May and June 2022. The sensitivity analysis of the model and the optimal control problem are also numerically solved. The normalized forward sensitivity indices at the endemic equilibrium point revealed which model inputs significantly impact the model’s long-term dynamics. Numerical simulations with the optimal control scheme were given to show the control’s effectiveness in diminishing the infective individuals.

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Acknowledgements

Thanks to the Ministry of Higher Education, Malaysia for the UTM Research Grant (Q.J130000.3854.20J20). Thanks to Universiti Teknologi Malaysia and Universiti Malaysia Terengganu for the research facilities. Thanks to the anonymous reviewers for comments that led to a clearer presentation of this manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310, Johor Bahru, Johor, Malaysia

    Noorehan Yaacob & Fuaada Mohd Siam

  2. Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030, Kuala Nerus, Terengganu, Malaysia

    Hanis Nasir

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  1. Noorehan Yaacob
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  2. Fuaada Mohd Siam
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  3. Hanis Nasir
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Correspondence to Hanis Nasir.

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Yaacob, N., Mohd Siam, F. & Nasir, H. Dynamics of an SIS Model with an Influx of Infective Visitors or Immigrants. Differ Equ Dyn Syst (2023). https://doi.org/10.1007/s12591-023-00649-8

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