Optimal control and analysis of a stochastic SEIR epidemic model with nonlinear incidence and treatment

Jinji Du; Chuangliang Qin; Yuanxian Hui; Jinji Du; Chuangliang Qin; Yuanxian Hui

doi:10.3934/math.20241600

AIMS Mathematics

2024, Volume 9, Issue 12: 33532-33550. doi: 10.3934/math.20241600

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Research article

Optimal control and analysis of a stochastic SEIR epidemic model with nonlinear incidence and treatment

1.
School of Mathematics and Statistics, Xinyang College, Xinyang 464000, China
2.
School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China

Received: 25 September 2024 Revised: 12 November 2024 Accepted: 19 November 2024 Published: 26 November 2024
MSC : 34D05, 34K20, 92B05, 92D25

In this paper, we represented the optimal control and dynamics of a stochastic SEIR epidemic model with nonlinear incidence and treatment rate. By using the Lyapunov function method, the existence and uniqueness of the global positive solution of the model were proved. The dynamic analysis of the stochastic model was studied and we found that the model has an ergodic stationary distribution when $ R_{0}^{s} > 1 $. The disease was extinct when $ R_{0}^{e} < 1 $. The optimal solution of the disease was obtained by using the stochastic control theory. The numerical simulation of our conclusion was carried out. The results showed that the disease decreased with the increase of the control variables.
- stochastic SEIR model,
- stationary distribution,
- extinction,
- stochastic control
Citation: Jinji Du, Chuangliang Qin, Yuanxian Hui. Optimal control and analysis of a stochastic SEIR epidemic model with nonlinear incidence and treatment[J]. AIMS Mathematics, 2024, 9(12): 33532-33550. doi: 10.3934/math.20241600

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Abstract

In this paper, we represented the optimal control and dynamics of a stochastic SEIR epidemic model with nonlinear incidence and treatment rate. By using the Lyapunov function method, the existence and uniqueness of the global positive solution of the model were proved. The dynamic analysis of the stochastic model was studied and we found that the model has an ergodic stationary distribution when $ R_{0}^{s} > 1 $. The disease was extinct when $ R_{0}^{e} < 1 $. The optimal solution of the disease was obtained by using the stochastic control theory. The numerical simulation of our conclusion was carried out. The results showed that the disease decreased with the increase of the control variables.

References

[1]	Z. Hu, W. Ma, S. Ruan, Analysis of SIR epidemic models with nonlinear incidencerate and treatment, Math. Biosci., 238 (2012), 12–20. https://doi.org/10.1016/j.mbs.2012.03.010 doi: 10.1016/j.mbs.2012.03.010
[2]	M. Khan, Q. Badshah, S. Islam, I. Khan, S. Shafie, S. Khan, Global dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive vaccination, Adv. Differ. Equ., 2015 (2015), 88. https://doi.org/10.1186/s13662-015-0429-3 doi: 10.1186/s13662-015-0429-3
[3]	F. Rihan, H. Alsakaji, 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
[4]	M. Abdoon, A. Alzahrani, Comparative analysis of influenza modeling using novel fractional operators with real data, Symmetry, 16 (2024), 1126. https://doi.org/10.3390/sym16091126 doi: 10.3390/sym16091126
[5]	C. Qin, J. Du, Y. Hui, Dynamical behavior of a stochastic predator-prey model with Holling-type Ⅲ functional response and infectious predator, AIMS Mathematics, 7 (2022), 7403–7418. https://doi.org/10.3934/math.2022413 doi: 10.3934/math.2022413
[6]	T. Kar, S. Jana, Application of three controls optimally in a vector-borne disease-mathematical study, Commun. Nonlinear Sci., 18 (2013), 2868–2884. https://doi.org/10.1016/j.cnsns.2013.01.022 doi: 10.1016/j.cnsns.2013.01.022
[7]	K. Hattaf, M. Rachik, S. Saadi, Y. Tabit, N. Yousfi, Optimal control of tuberculosis with exogenous reinfection, Applied Mathematical Sciences, 3 (2009), 231–240.
[8]	F. Gumaa, M. Abdoon, A. Qazza, R. Saadeh, M. Arishi, A. Degoot, Analyzing the impact of control strategies on VisceralLeishmaniasis: a mathematical modeling perspective, Eur. J. Pure Appl. Math., 17 (2024), 1213–1227. https://doi.org/10.29020/nybg.ejpam.v17i2.5121 doi: 10.29020/nybg.ejpam.v17i2.5121
[9]	J. Tchuenche, S. Khamis, F. Agusto, S. Mpeshe, Optimal control and sensitivity analysis of an influenza model with treatment and vaccination, Acta Biotheor., 59 (2011), 1–28. https://doi.org/10.1007/s10441-010-9095-8 doi: 10.1007/s10441-010-9095-8
[10]	K. Okosun, R. Ouifki, N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, Biosystems, 106 (2011), 136–145. https://doi.org/10.1016/j.biosystems.2011.07.006 doi: 10.1016/j.biosystems.2011.07.006
[11]	H. Alsakaji, Y. El-Khatib, F. Rihan, A. Hashish, A stochastic epidemic model with time delays and unreported cases based on Markovian switching, Journal of Biosafety and Biosecurity, 6 (2024), 234–243. https://doi.org/10.1016/j.jobb.2024.08.002 doi: 10.1016/j.jobb.2024.08.002
[12]	X. Zhang, X. Liu, Backward bifurcation of an epidemic model with saturated treatment, J. Math. Anal. Appl., 348 (2008), 433–443. https://doi.org/10.1016/j.jmaa.2008.07.042 doi: 10.1016/j.jmaa.2008.07.042
[13]	S. Jana, S. Nandi, T. Kar, Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment, Acta Biotheor., 64 (2016), 65–84. https://doi.org/10.1007/s10441-015-9273-9 doi: 10.1007/s10441-015-9273-9
[14]	M. Ali, F. Guma, A. Qazza, R. Saadeh, N. Alsubaie, M. Althubyani, et al., Stochastic modeling of influenza transmission: insights into disease dynamics and epidemic management, Partial Differential Equations in Applied Mathematics, 11 (2024), 100886. https://doi.org/10.1016/j.padiff.2024.100886 doi: 10.1016/j.padiff.2024.100886
[15]	R. Upadhyay, A. Pal, S. Kumari, P. Roy, Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates, Nonlinear Dyn., 96 (2019), 2351–2368. https://doi.org/10.1007/s11071-019-04926-6 doi: 10.1007/s11071-019-04926-6
[16]	M. Ali, S. Alzahrani, R. Saadeh, M. Abdoon, A. Qazza, N. Al-kuleab, et al., Modeling COVID-19 spread and non-pharmaceutical interventions in South Africa: a stochastic approach, Scientific African, 24 (2024), e02155. https://doi.org/10.1016/j.sciaf.2024.e02155 doi: 10.1016/j.sciaf.2024.e02155
[17]	F. Rihan, H. Alsakaji, C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 502. https://doi.org/10.1186/s13662-020-02964-8 doi: 10.1186/s13662-020-02964-8
[18]	Q. Yang, D. Jiang, N. Shi, C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248–271. https://doi.org/10.1016/j.jmaa.2011.11.072 doi: 10.1016/j.jmaa.2011.11.072
[19]	S. Gani, S. Halawar, Optimal control analysis of deterministic and stochastic SIS epidemic model with vaccination, Int. J. Solids Struct., 12 (2017), 251–263.
[20]	X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110. https://doi.org/10.1016/S0304-4149(01)00126-0 doi: 10.1016/S0304-4149(01)00126-0
[21]	X. Mao, Stochastic differential equations and applications, Chichester: Horwood Publishing, 2008.
[22]	L. Wang, D. Zhou, Z. Liu, D. Xu, X. Zhang, Media alert in an SIS epidemic model with logistic growth, J. Biol. Dynam., 11 (2017), 120–137. https://doi.org/10.1080/17513758.2016.1181212 doi: 10.1080/17513758.2016.1181212
[23]	R. Khasminskii, Stochastic stability of differential equations, 2 Eds., Heidelberg: Spinger-Verlag, 2011. https://doi.org/10.1007/978-3-642-23280-0
[24]	V. Rico-Ramirez, U. Diwekar, Stochastic maximum principle for optimal control under uncertainty, Comput. Chem. Eng., 28 (2004), 2845–2849. https://doi.org/10.1016/j.compchemeng.2004.08.001 doi: 10.1016/j.compchemeng.2004.08.001
[25]	S. Khare, K. Mathur, K. Das, Optimal control of deterministic and stochastic Eco-epidemic food adulteration model, Results in Control and Optimization, 14 (2024), 100336. https://doi.org/10.1016/j.rico.2023.100336 doi: 10.1016/j.rico.2023.100336
[26]	B $\emptyset$ksendal, Stochastic differential equations: an introduction with applications, Berlin: Springer, 2003. https://doi.org/10.1007/978-3-642-14394-6
[27]	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

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