Mathematical model for an infectious disease with quarantine

Document Type : Research Article

Authors

Fatemeh Iranzad

Hamidreza Marasi

Hossein Kheiri

Azizeh Jabbari

Faculty of Mathematics, Statistics, and Computer Science, University of Tabriz, Tabriz, Iran.

10.22067/ijnao.2025.93898.1661

Abstract

This article presents a novel compartmental mathematical model for an infectious disease that uniquely distinguishes susceptible individuals into quarantined and non-quarantined groups, while explicitly capturing dynamic transitions between these compartments. Unlike traditional models, ours integrates the combined effects of quarantine effectiveness and population mobility on disease transmission dynamics. We rigorously define the biologically feasible region and prove its positive invariance, ensuring model consistency. Using the next-generation matrix method, we derive a precise expression for the basic reproduction number ($\mathcal{R}_0$). Our analytical contributions include establishing both local and global stability of the disease-free equilibrium for $\mathcal{R}_0 < 1$, and, notably, developing a novel Lyapunov function to prove the global asymptotic stability of the endemic equilibrium when $\mathcal{R}_0 > 1$. Through comprehensive numerical simulations, we validate the theoretical results and provide new insights into the critical role of quarantine measures in epidemic control. These findings offer valuable guidance for designing more effective public health intervention strategies.

Keywords

Infectious diseases‎

‎Local stability‎

Global stability‎

‎Reproduction number‎

‎Lyapunov function

Subjects

• Modeling and Simulation

---

References

1] Aakash, M., Gunasundari, C., Athithan, S., Sharmila, N.B., Kumar, G.S. and Guefaifia, R.
Mathematical modeling of COVID-19 with the effects of quarantine and detection, Partial
Differ. Equ. Appl. Math., 9, (2024) 100609.
[2] Al-Yahyai, M., Al-Musalhi, F., Elmojtaba, I. and Al-Salti, N. Mathematical analysis of a
COVID-19 model with different types of quarantine and isolation, Math. Biosci. Eng., 20
(2022), 1344–1375.
[3] Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J. On the definition and computation of
the basic reproduction ratio R0 in models for infectious disease in heterogeneous population,
J. Math. Biol. 28 (1990), 365–382.
[4] Ega, T.T. and Ngeleja, R.C. Mathematical Model Formulation and Analysis for COVID-
19 Transmission with Virus Transfer Media and Quarantine on Arrival, Comput. Math.
Methods, 1 (2022), 1–16.
[5] Kheiri, H. and Jafari, M. Fractional optimal control of an HIV/AIDS epidemic model with
random testing and contact tracing, J. Appl. Math. Comput., 60 (2019), 387–411.
[6] Kiseleva, O., Yakovlev, S., Prytomanova, O. and Kuzenkov, O. Mathematical modeling of
regional infectious disease dynamics based on extended compartmental models, Computation,
13, (2025) 187. 14.
[7] LaSalle, J.P. The stability of dynamical systems, Society for Industrial and Applied Mathe-
matics, Philadelphia, 1976.
[8] Lotfi, M., Jabbari, A. and Kheiri, H. A mathematical analysis of a tuberculosis epidemic
model with two treatments and exogenous re-infection, Int. J. Biomath., 13 (08) (2020),
2050082.
[9] Lynch, S. Dynamical systems with applications using MATLAB, Springer Science & Business
Media, 20
[10] Ndaïrou, F., Area, I., Nieto, J.J. and Torres, D.F., Mathematical modeling of COVID-19
transmission dynamics with a case study of Wuhan, Chaos Solit. Fract., 135 (2020), 1–6.
[11] Safdar, S., Ngonghala, C.N. and Gumel, A.B. Mathematical assessment of the role of waning
and boosting immunity against the BA.1 Omicron variant in the United States, Math. Biosci.
Eng., 20 (2023), 179–212.
[12] Shuai Z. and van den Driessche, P. Global stability of infectious disease models using Lya-
punov function, SIAM J. Appl. Math., 73 (2013), 1513–1532.
[13] Van den Driessche, P. and Watmough, J. Reproduction numbers and subthreshold endemic
equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002),
29–48.