https://doi.org/10.1140/epjp/s13360-025-06108-3
Regular Article
Dynamics and optimal control of an extended SIQR model with protected human class and public awareness
1
Department of Mathematics, Asansol Girls’ College, 713304, Asansol, West Bengal, India
2
Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam bin Abdulaziz University, Al Kharj, Saudi Arabia
3
Saveetha School of Engineering, SIMATS, Chennai, India
4
Mathematics Department, Faculty of Science, Umm Al-Qura University, 21955, Makkah, Saudi Arabia
5
Department of Mathematics, K. L. S. College, Magadh University, 805110, Nawada, Bodh Gaya, Bihar, India
Received:
5
December
2024
Accepted:
9
February
2025
Published online:
22
February
2025
In this article, we have developed the SIQR type mathematical model including the protected human population and public awareness in the model for the dynamics of an epidemic outbreak. The “level of awareness”, due to awareness campaign, is taken as a separate model variable. Both local (information sharing from local area, relatives) and global awareness (information sharing from social media, Radio, TV, etc.) can increase the level of awareness. We have also included the impact of treatment for recovery from the infection. Also, we have assumed infection transmission as a decreasing function of media awareness. The existence of equilibria of the model and their stability nature have been studied with qualitative theory. The disease-free equilibrium is stable when 
and unstable when . A unique endemic equilibrium exists when 
, and it shows a Hopf bifurcation if the infection rate crosses its critical value. The unstable endemic system becomes stable when the global awareness rate is increased. To obtain crucial insights into disease management strategies, sensitivity analysis is performed to examine the link between model parameters and the basic reproduction number 
. Finally, we formulate an optimal control problem including three control parameters and solved using Pontryagin’s maximum principle. Numerical simulations are executed on the basis of analytical results. The regions of stability of the disease-free equilibrium are identified in different parameter planes. We have determined the optimal profiles of the three control functions to make the disease management process economically viable. This study concluded that the transmission dynamics of the pandemic depend on the rate of infection, the rate of global awareness, and the rate of awareness-based treatments. The proposed awareness-induced mathematical model that includes an optimal control approach is applicable to cost-effectively manage an epidemic outbreak.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
