Dynamics of a stochastic SEIQR model: stationary distribution and disease extinction with quarantine measures

Year 2025, Volume: 5 Issue: 1, 172 - 197, 31.03.2025

S. Saravanan C. Monica

https://doi.org/10.53391/mmnsa.1572436

Abstract

This paper investigates the dynamics of a stochastic $\mathcal{SEIQR}$ epidemic model, which integrates quarantine measures and a saturated incidence rate to more accurately reflect real-world disease transmission. The model is based on the classical $\mathcal{SEIR}$ framework, with the addition of a quarantined compartment, offering insights into the impact of quarantine on epidemic control. The saturated incidence rate accounts for the diminishing rate of new infections as the susceptible population grows, addressing the limitations of traditional bilinear incidence rates in modeling epidemic spread under high disease prevalence. We first establish the basic reproductive number, $\mathcal{R}_0$, for the deterministic model, which serves as a threshold parameter for disease persistence. Through the stochastic Lyapunov function method, we identify the necessary conditions for the existence of a stationary distribution, focusing on the case where $\mathcal{R}_0^* > 1$, signals the potential long-term persistence of the disease in the population. Furthermore, we derive sufficient conditions for disease extinction, particularly when $\mathcal{R}_S^* < 1$, indicating that the disease will eventually die out despite the inherent randomness in disease transmission. Numerical simulations confirm that environmental noise and quarantine rates shape disease dynamics. Simulations show that more noise or higher quarantine rates speed up disease extinction, offering key policy insights. Our results clarify how quarantine, noise intensity, and disease dynamics interact, aiding epidemic modeling in stochastic settings.

Keywords

Stochastic epidemic model, Lyapunov function, stationary ergodic distribution, extinction

References

• [1] Bernoullid, D. A new analysis of the mortality caused by smallpox. In The History of Actuarial Science Vol VIII, pp.1-38. Paris: Routledge, (1766).
• [2] Kermack, W.O. and McKendrick, A.G. Contributions to the mathematical theory of epidemics– I. 1927. Bulletin of Mathematical Biology, 53(1-2), 33-55, (1991).
• [3] Abbey, H. An examination of the Reed-Frost theory of epidemics. Human Biology, 24(3), 201, (1952).
• [4] Bartlett, M.S. Some evolutionary stochastic processes. Journal of the Royal Statistical Society. Series B (Methodological), 11(2), 211-229, (1949).
• [5] Berger, D.W., Herkenhoff, K.F. and Mongey, S. An SEIR infectious disease model with testing and conditional quarantine. National Bureau of Economic Research, (2020).
• [6] Panigoro, H.S., Rahmi, E., Nasib, S.K., Gawa, N.P.H. and Peter, O.J. Bifurcations on a discrete–time SIS–epidemic model with saturated infection rate. Bulletin of Biomathematics, 2(2), 182–197, (2024).
• [7] Wells, C.R., Townsend, J.P., Pandey, A., Moghadas, S.M., Krieger, G., Singer, B. et al. Optimal COVID-19 quarantine and testing strategies. Nature Communications, 12, 356, (2021).
• [8] Naik, P.A., Yeolekar, B.M., Qureshi, S., Yavuz, M., Huang, Z. and Yeolekar, M. Fractional insights in tumor modeling: an interactive study between tumor carcinogenesis and macrophage activation. Advanced Theory and Simulations, 2401477, (2025).
• [9] Mustapha, U.T., Ado, A., Yusuf, A., Qureshi, S. and Musa, S.S. Mathematical dynamics for HIV infections with public awareness and viral load detectability. Mathematical Modelling and Numerical Simulation With Applications, 3(3), 256-280, (2023).
• [10] Lan, G., Yuan, S. and Song, B. The impact of hospital resources and environmental perturbations to the dynamics of SIRS model. Journal of the Franklin Institute, 358(4), 2405-2433, (2021).
• [11] Joshi, H. and Yavuz, M. Chaotic dynamics of a cancer model with singular and non-singular kernel. Discrete and Continuous Dynamical Systems-S, 18(5), 1416-1439, (2025).
• [12] Phan, T.A., Tian, J.P. and Wang, B. Dynamics of cholera epidemic models in fluctuating environments. Stochastics and Dynamics, 21(02), 2150011, (2021).
• [13] Sabbar, Y., Kiouach, D. and Rajasekar, S.P. Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Levy jumps associated with infinite measure. International Journal of Dynamics and Control, 11, 122-135, (2023).
• [14] Rauta, A.K., Rao, Y.S., Behera, J., Dihudi, B. and Panda, T.C. SIQRS epidemic modelling and stability analysis of COVID-19. In Predictive and Preventive Measures for Covid-19 Pandemic (pp. 35-50). Springer: Singapore, (2021).
• [15] Wang, K., Fan, H. and Zhu, Y. Dynamics and application of a generalized SIQR epidemic model with vaccination and treatment. Applied Mathematical Modelling, 120, 382-399, (2023).
• [16] Da¸sba¸sı, B. Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 44-55, (2021).
• [17] Dieu, N.T., Sam, V.H. and Du, N.H. Threshold of a stochastic SIQS epidemic model with isolation. Discrete & Continuous Dynamical Systems-Series B, 27(9), p5009, (2022).
• [18] Zhang, Y., Ma, X. and Din, A. Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity. AIMS Math, 6(11), 12359-12378, (2021).
• [19] Qi, H., Zhang, S., Meng, X. and Dong, H. Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems. Physica A: Statistical Mechanics and its Applications, 508, 223-241, (2018).
• [20] Ma, Y., Cui, Y. and Wang, M. Global stability and control strategies of a SIQRS epidemic model with time delay. Mathematical Methods in the Applied Sciences, 45(13), 8269-8293, (2022).
• [21] Wang, M., Hu, Y. and Wu, L. Dynamic analysis of a SIQR epidemic model considering the interaction of environmental differences. Journal of Applied Mathematics and Computing, 68, 2533–2549, (2022).
• [22] Pan, Q., Huang, J. and Wang, H. An SIRS model with nonmonotone incidence and saturated treatment in a changing environment. Journal of Mathematical Biology, 85, 23, (2022).
• [23] Yang, J., Shi, X., Song, X. and Zhao, Z. Threshold dynamics of a stochastic SIQR epidemic model with imperfect quarantine, Applied Mathematics Letters, 136, 108459, (2023).
• [24] Wang, K., Fan, H., & Zhu, Y. Dynamics and application of a generalized SIQR epidemic model with vaccination and treatment, Applied Mathematical Modeling, 120, 382-399, (2023).
• [25] Zhang, G., Li, Z. and Din, A. A stochastic SIQR epidemic model with Levy jumps and three-time delays. Applied Mathematics and Computation, 431, 127329, (2022).
• [26] Lu, C., Liu, H. and Zhang, D. Dynamics and simulations of a second order stochastically perturbed SEIQV epidemic model with saturated incidence rate. Chaos, Solitons & Fractals, 152, 111312, (2021).
• [27] Gao, S., Chen, L., Nieto, J.J. and Torres, A. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine, 24(35-36), 6037-6045, (2006).
• [28] Rajasekar, S.P. and Pitchaimani, M. Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses. Chaos, Solitons & Fractals, 118, 207-221, (2019).
• [29] Yang, J., Shi, X., Song, X. and Zhao, Z. Threshold dynamics of a stochastic SIQR epidemic model with imperfect quarantine. Applied Mathematics Letters, 136, 108459,(2023).
• [30] Selvan, T.T. and Kumar, M. Dynamics of a deterministic and a stochastic epidemic model combined with two distinct transmission mechanisms and saturated incidence rate. Physica A: Statistical Mechanics and its Applications, 619, 128741,(2023).
• [31] Mao, X. Stochastic Differential Equations and Applications. Elsevier: Oxford, (2007).
• [32] Khasminskii, R. Stochastic Stability of Differential Equations. Springer Science & Business Media: New York, (2011).
• [33] Gard, T.C. Introduction to Stochastic Differential Equations. Marcel Dekker: New York, (1988).
• [34] Zhu, C. and Yin, G. Asymptotic properties of hybrid diffusion systems. SIAM Journal on Control and Optimization, 46(4), 1155-1179, (2007).
• [35] Zhao, Y. and Jiang, D. The threshold of a stochastic SIS epidemic model with vaccination. Applied Mathematics and Computation, 243, 718-727, (2014).
• [36] Meng, X., Zhao, S., Feng, T. and Zhang, T. Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. Journal of Mathematical Analysis and Applications, 433(1), 227-242, (2016).
• [37] Higham, D.J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3), 525-546, (2001).
• [38] Wei, W., Xu, W. and Liu, J. A regime-switching stochastic SIR epidemic model with a saturated incidence and limited medical resources. International Journal of Biomathematics, 16(07), 2250124, (2023).
• [39] Goel, K. and Nilam. A mathematical and numerical study of a SIR epidemic model with time delay, nonlinear incidence and treatment rates. Theory in Bio-sciences, 138, 203-213, (2019).