Vaccination strategies in a stochastic $ \mathscr{SIVR} $ epidemic model

Shah Hussain; Naveed Iqbal; Elissa Nadia Madi; Thoraya N. Alharthi; Ilyas Khan; Shah Hussain; Naveed Iqbal; Elissa Nadia Madi; Thoraya N. Alharthi; Ilyas Khan

doi:10.3934/math.2025204

AIMS Mathematics

2025, Volume 10, Issue 2: 4441-4456. doi: 10.3934/math.2025204

Previous Article Next Article

Research article Special Issues

Vaccination strategies in a stochastic $ \mathscr{SIVR} $ epidemic model

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin (UniSZA), Besut Campus, Terengganu, Malaysia
3.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia
4.
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia
5.
Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan

Received: 02 January 2025 Revised: 16 February 2025 Accepted: 24 February 2025 Published: 28 February 2025
MSC : 26A33, 34A08, 35R11

Effective disease control measures are essential for mitigating epidemic risks. This study introduces a novel stochastic susceptible-infected-vaccinated-recovered $ \mathscr{SIVR} $ epidemic model that incorporates white noise in vaccination dynamics. Unlike traditional deterministic models, our stochastic framework accounts for the inherent randomness in real-world disease transmission and the effectiveness of interventions. We rigorously establish the existence and uniqueness of global positive solutions using Lyapunov functions and derive conditions for disease extinction and persistence under stochastic perturbations. A key contribution is the introduction of a stochastic reproduction number $R^*_0$, which refines classical epidemic thresholds by integrating randomness. Through numerical simulations, we illustrate the impact of stochasticity on disease dynamics, demonstrating that noise can drive disease extinction even in scenarios where deterministic models predict persistence. This study provides a more realistic epidemiological framework for optimizing vaccination strategies under uncertainty, offering significant advances in epidemic modeling and public health policy.
- $ \mathscr{SIVR} $ model,
- public health risks,
- noise vaccination,
- Lyapunov functions
Citation: Shah Hussain, Naveed Iqbal, Elissa Nadia Madi, Thoraya N. Alharthi, Ilyas Khan. Vaccination strategies in a stochastic $ \mathscr{SIVR} $ epidemic model[J]. AIMS Mathematics, 2025, 10(2): 4441-4456. doi: 10.3934/math.2025204

Related Papers:

Abstract

Effective disease control measures are essential for mitigating epidemic risks. This study introduces a novel stochastic susceptible-infected-vaccinated-recovered $ \mathscr{SIVR} $ epidemic model that incorporates white noise in vaccination dynamics. Unlike traditional deterministic models, our stochastic framework accounts for the inherent randomness in real-world disease transmission and the effectiveness of interventions. We rigorously establish the existence and uniqueness of global positive solutions using Lyapunov functions and derive conditions for disease extinction and persistence under stochastic perturbations. A key contribution is the introduction of a stochastic reproduction number $R^*_0$, which refines classical epidemic thresholds by integrating randomness. Through numerical simulations, we illustrate the impact of stochasticity on disease dynamics, demonstrating that noise can drive disease extinction even in scenarios where deterministic models predict persistence. This study provides a more realistic epidemiological framework for optimizing vaccination strategies under uncertainty, offering significant advances in epidemic modeling and public health policy.

References

[1]	R. Anderson, R. May, Infectious diseases of humans dynamics and control, 1 Eds., Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198545996.001.0001
[2]	H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
[3]	F. Brauer, C. C. Chavez, Mathematical models in population biology and epidemiology, 2 Eds., New York: Springer, 2012. http://dx.doi.org/10.1007/978-1-4614-1686-9
[4]	F. Rihan, J. Alsakran, Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE, Results Phys., 28 (2021), 104658. https://doi.org/10.1016/j.rinp.2021.104658 doi: 10.1016/j.rinp.2021.104658
[5]	M. Keeling, P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2011. https://doi.org/10.2307/j.ctvcm4gk0
[6]	A. Alkhazzan, J. Wang, Y. Nie, H. Khan, J. Alzabut, A stochastic Susceptible-Vaccinees-Infected-Recovered epidemic model with three types of noises, Math. Method. Appl. Sci., 47 (2024), 8748–8770. https://doi.org/10.1002/mma.10042 doi: 10.1002/mma.10042
[7]	A. Alkhazzan, J. Wang, Y. Nie, H. Khan, J. Alzabut, An effective transport-related SVIR stochastic epidemic model with media coverage and Levy noise, Chaos Soliton. Fract., 175 (2023), 113953. https://doi.org/10.1016/j.chaos.2023.113953 doi: 10.1016/j.chaos.2023.113953
[8]	J. Jacquez, P. O'Neill, Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations, Math. Biosci., 107 (1991), 161–186. https://doi.org/10.1016/0025-5564(91)90003-2 doi: 10.1016/0025-5564(91)90003-2
[9]	H. Alsakjia, A. Rihan, A. Hashish, Dynamics of a stochastic epidemic model with vaccination and multiple time-delays for COVID-19 in the UAE, Complexity, 1 (2022), 4247800. https://doi.org/10.1155/2022/4247800 doi: 10.1155/2022/4247800
[10]	A. Din, Y. Li, A. Omame, A stochastic stability analysis of an HBV-COVID-19 coinfection model in resource limitation settings, Wave. Random Complex, 2022. https://doi.org/10.1080/17455030.2022.2147598
[11]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[12]	S. Li, W. Li, Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns, PLoS One, 19 (2024), e0310175. https://doi.org/10.1371/journal.pone.0310175 doi: 10.1371/journal.pone.0310175
[13]	O. Ogunmiloro, F. O. Abedo, H. A. Kareem, Numerical and stability analysis of the transmission dynamics of the SVIR epidemic model with standard incidence rate, Malays. J. Comput., 4 (2019), 349–361. https://doi.org/10.24191/mjoc.v4i2.5828 doi: 10.24191/mjoc.v4i2.5828
[14]	S. Hussain, E. N. Madi, H. Khan, S. Etemad, S. Rezapour, T. Sitthiwirattham, et al., Investigation of the stochastic modeling of COVID-19 with environmental noise from the analytical and numerical point of view, Mathematics, 9 (2021), 3122. https://doi.org/10.3390/math9233122 doi: 10.3390/math9233122
[15]	F. Brauer, C. C. Chavez, Mathematical models in epidemiology, 1 Eds., New York: Springer, 2019. https://doi.org/10.1007/978-1-4939-9828-9
[16]	M. Barman, N. Mishra, Hopf bifurcation in a networked delay SIR epidemic model, J. Math. Anal. Appl., 525 (2023), 127131. https://doi.org/10.1016/j.jmaa.2023.127131 doi: 10.1016/j.jmaa.2023.127131
[17]	L. Zhu, H. Hu, A stochastic SIR epidemic model with density-dependent birth rate, Adv. Differential Equ., 330 (2015), 1–2. https://doi.org/10.1186/s13662-015-0669-2 doi: 10.1186/s13662-015-0669-2
[18]	L. Allen, An introduction to stochastic epidemic models, 1 Eds., Berlin: Springer-Verlag, 2008, 81–130. https://doi.org/10.1007/978-3-540-78911-6-3
[19]	H. Alsakaji, Y. El Kahtib, F. Rihan, A. Hashish, A stochastic epidemic model with time delays and unreported cases based on Markovian switching, J. Biol. Dynam., 6 (2024), 234–243. https://doi.org/10.1016/j.jobb.2024.08.002 doi: 10.1016/j.jobb.2024.08.002
[20]	F. Rao, W. Wang, Z. Li, Stability analysis of an epidemic model with diffusion and stochastic perturbation, Commun. Nonlinear Sci., 7 (2012), 2551–2563. https://doi.org/10.1016/j.cnsns.2011.10.005 doi: 10.1016/j.cnsns.2011.10.005
[21]	N. Dalal, D. Greenhalgh, X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36–53. https://doi.org/10.1016/j.jmaa.2006.01.055 doi: 10.1016/j.jmaa.2006.01.055
[22]	C. Ji, D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067–5079. https://doi.org/10.1016/j.apm.2014.03.037 doi: 10.1016/j.apm.2014.03.037
[23]	D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)