Dynamics and numerical simulations of a generalized mosquito-borne epidemic model using the Ornstein-Uhlenbeck process: Stability, stationary distribution, and probability density function

Wenhui Niu; Xinhong Zhang; Daqing Jiang; Wenhui Niu; Xinhong Zhang; Daqing Jiang

doi:10.3934/era.2024172

Electronic Research Archive

2024, Volume 32, Issue 6: 3777-3818. doi: 10.3934/era.2024172

Previous Article Next Article

Research article Special Issues

Dynamics and numerical simulations of a generalized mosquito-borne epidemic model using the Ornstein-Uhlenbeck process: Stability, stationary distribution, and probability density function

College of Science, China University of Petroleum (East China), Qingdao 266580, China

Received: 28 February 2024 Revised: 21 May 2024 Accepted: 27 May 2024 Published: 07 June 2024

In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number $ R_0 $ was greater than unity and the disease free equilibrium was globally asymptotically stable when $ R_0 $ was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as $ R_0^s $ and $ R_0^E $, were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model.
- mosquito-borne epidemic model,
- log-normal Ornstein-Uhlenbeck process,
- stationary distribution,
- extinction,
- density function
Citation: Wenhui Niu, Xinhong Zhang, Daqing Jiang. Dynamics and numerical simulations of a generalized mosquito-borne epidemic model using the Ornstein-Uhlenbeck process: Stability, stationary distribution, and probability density function[J]. Electronic Research Archive, 2024, 32(6): 3777-3818. doi: 10.3934/era.2024172

Related Papers:

Abstract

In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number $ R_0 $ was greater than unity and the disease free equilibrium was globally asymptotically stable when $ R_0 $ was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as $ R_0^s $ and $ R_0^E $, were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model.

References

[1]	H. Lee, S. Halverson, N. Ezinwa, Mosquito-borne diseases, Primary Care: Clin. Off. Pract., 45 (2018), 393–407. https://doi.org/10.1016/j.pop.2018.05.001
[2]	M. A. Tolle, Mosquito-borne diseases, Curr. Probl. Pediatr. Adolesc. Health Care, 39 (2009), 97–140. https://doi.org/10.1016/j.cppeds.2009.01.001 doi: 10.1016/j.cppeds.2009.01.001
[3]	J. Oliveira-Ferreira, M. V. G. Lacerda, P. Brasil, J. L. Ladislau, P. L. Tauil, C. T. Daniel-Ribeiro, Malaria in Brazil: an overview, Malar. J., 9 (2010), 1–15. https://doi.org/10.1186/1475-2875-9-115 doi: 10.1186/1475-2875-9-115
[4]	V. Wiwanitkit, Dengue fever: diagnosis and treatment, Expert Rev. Anti-Infect. Ther., 8 (2010), 841–845. https://doi.org/10.1586/eri.10.53 doi: 10.1586/eri.10.53
[5]	E. D. Barnett, Yellow fever: epidemiology and prevention, Clin. Infect. Dis., 44 (2007), 850–856. https://doi.org/10.1086/511869 doi: 10.1086/511869
[6]	E. B. Hayes, J. J. Sejvar, S. R. Zaki, R. S. Lanciotti, A. V. Bode, Virology, pathology, and clinical manifestations of West Nile virus disease, Emerging Infect. Dis., 11 (2005), 1174. https://doi.org/10.3201/eid1108.050289b doi: 10.3201/eid1108.050289b
[7]	L. Esteva, H. M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132–147. https://doi.org/10.1016/j.mbs.2005.06.004 doi: 10.1016/j.mbs.2005.06.004
[8]	E. A. Newton, P. Reiter, A model of the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on dengue epidemics, Am. J. Trop. Med. Hyg., 47 (1992), 709–720. https://doi.org/10.4269/ajtmh.1992.47.709 doi: 10.4269/ajtmh.1992.47.709
[9]	H. R. Pandey, G. R. Phaijoo, D. B. Gurung, Analysis of dengue infection transmission dynamics in Nepal using fractional order mathematicalmodeling, Chaos, Solitons Fractals: X, 11 (2023), 100098. https://doi.org/10.1016/j.csfx.2023.100098 doi: 10.1016/j.csfx.2023.100098
[10]	R. Shi, H. Zhao, S. Tang, Global dynamic analysis of a vector-borne plant disease model, Adv. Differ. Equations, 59 (2014), 1–16. https://doi.org/10.1186/1687-1847-2014-59
[11]	N. S. Chong, J. M. Tchuenche, R. J. Smith, A mathematical model of avian influenza with half-saturated incidence, Theory Biosci., 133 (2014), 23–38. https://doi.org/10.1007/s12064-013-0183-6 doi: 10.1007/s12064-013-0183-6
[12]	Y. Li, F. Haq, K. Shah, G. Rahman, M. Shahzad, Numerical analysis of fractional order pine wilt disease model with bilinear incident rate, J. Math. Comput. Sci., 17 (2017), 420–428. https://doi.org/10.22436/jmcs.017.03.07 doi: 10.22436/jmcs.017.03.07
[13]	X. Yu, S. Yuan, T. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 359–374. https://doi.org/10.1016/j.cnsns.2017.11.028 doi: 10.1016/j.cnsns.2017.11.028
[14]	X. Lv, X. Meng, X. Wang, Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation, Chaos, Solitons Fractals, 110 (2018), 273–279. https://doi.org/10.1016/j.chaos.2018.03.038 doi: 10.1016/j.chaos.2018.03.038
[15]	T. Su, Q. Yang, X. Zhang, D. Jiang, Stationary distribution, extinction and probability density function of a stochastic SEIV epidemic model with general incidence and Ornstein-Uhlenbeck process, Physica A, 615 (2023), 128605. https://doi.org/10.1016/j.physa.2023.128605 doi: 10.1016/j.physa.2023.128605
[16]	B. Zhou, D. Jiang, B. Han, T. Hayat, Threshold dynamics and density function of a stochastic epidemic model with media coverage and mean-reverting Ornstein-Uhlenbeck process, Math. Comput. Simul., 196 (2022), 15–44. https://doi.org/10.1016/j.matcom.2022.01.014 doi: 10.1016/j.matcom.2022.01.014
[17]	X. Mu, D. Jiang, T. Hayat, A. Alsaedi, Y. Liao, A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations, Nonlinear Dyn., 107 (2022), 2805–2817. https://doi.org/10.1007/s11071-021-07093-9 doi: 10.1007/s11071-021-07093-9
[18]	E. Allen, Environmental variability and mean-reverting processes, Discrete Contin. Dyn. Syst. - Ser. B, 21 (2016), 2073–2089. https://doi.org/10.3934/dcdsb.2016037 doi: 10.3934/dcdsb.2016037
[19]	X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Woodhead Publishing, 2011.
[20]	Z. Ma, Y. Zhou, C. Li, Qualitative and Stability Methods for Ordinary Differential Equations, Science Press, Beijing, 2001.
[21]	B. Zhou, D. Jiang, Y. Dai, T. Hayat, Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity, Nonlinear Dyn., 105 (2021), 931–955. https://doi.org/10.1007/s11071-020-06151-y doi: 10.1007/s11071-020-06151-y
[22]	S. P. Meyn, R. L. Tweedie, Stability of markovian processes III: foster-lyapunov criteria for continuous-time processes, Adv. Appl. Probab., 25 (1993), 518–548. https://doi.org/10.1017/s0001867800025532 doi: 10.1017/s0001867800025532
[23]	N. T. Dieu, Asymptotic properties of a stochastic SIR epidemic model with Beddington-DeAngelis incidence rate, J. Dyn. Partial Differ. Equations, 30 (2018), 93–106. https://doi.org/10.1007/s10884-016-9532-8 doi: 10.1007/s10884-016-9532-8
[24]	J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mt. J. Math., 20 (1990), 857–872. https://doi.org/10.1216/rmjm/1181073047 doi: 10.1216/rmjm/1181073047
[25]	Z. Shi, D. Jiang, Dynamical behaviors of a stochastic HTLV-I infection model with general infection form and Ornstein-Uhlenbeck process, Chaos, Solitons Fractals, 165 (2022), 112789. https://doi.org/10.1016/j.chaos.2022.112789 doi: 10.1016/j.chaos.2022.112789
[26]	C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Berlin, 1985. https://doi.org/10.1007/978-3-662-02452-2
[27]	D. J. 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
[28]	M. Aguiar, V. Anam, K. B. Blyuss, C. D. S. Estadilla, B. V. Guerrero, D. Knopoff, et al., Mathematical models for dengue fever epidemiology: A 10-year systematic review, Phys. Life Rev., 40 (2022), 65–92. https://doi.org/10.1016/j.plrev.2022.02.001 doi: 10.1016/j.plrev.2022.02.001
[29]	J. K. K. Asamoah, E. Yankson, E. Okyere, G. Sun, Z. Jin, R. Jan, et al., Optimal control and cost-effectiveness analysis for dengue fever model with asymptomatic and partial immune individuals, Results Phys., 31 (2021), 104919. https://doi.org/10.1016/j.rinp.2021.104919 doi: 10.1016/j.rinp.2021.104919
[30]	J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, 2015.
[31]	A. Yang, H. Wang, T. Zhang, S. Yuan, Stochastic switches of eutrophication and oligotrophication: Modeling extreme weather via non-Gaussian Lévy noise, Chaos, 32 (2022), 043116. https://doi.org/10.1063/5.0085560 doi: 10.1063/5.0085560
[32]	S. Khajanchi, S. Bera, T. K. Roy, Mathematical analysis of the global dynamics of a HTLV-I infection model, considering the role of cytotoxic T-lymphocytes, Math. Comput. Simul., 180 (2021), 354–378. https://doi.org/10.1016/j.matcom.2020.09.009 doi: 10.1016/j.matcom.2020.09.009
[33]	B. Zhou, X. Zhang, D. Jiang, Dynamics and density function analysis of a stochastic SVI epidemic model with half saturated incidence rate, Chaos, Solitons Fractals, 137 (2020), 109865. https://doi.org/10.1016/j.chaos.2020.109865 doi: 10.1016/j.chaos.2020.109865
[34]	X. Mu, D. Jiang, A. Alsaedi, Analysis of a stochastic phytoplankton–zooplankton model under non-degenerate and degenerate diffusions, J. Nonlinear Sci., 32 (2022), 35. https://doi.org/10.1007/s00332-022-09787-9 doi: 10.1007/s00332-022-09787-9
[35]	Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210–226. https://doi.org/10.1016/j.amc.2018.02.009 doi: 10.1016/j.amc.2018.02.009
[36]	Q. Yang, X. Zhang, D. Jiang, Dynamical behaviors of a stochastic food chain system with Ornstein-Uhlenbeck process, J. Nonlinear Sci., 32 (2022), 34. https://doi.org/10.1007/s00332-022-09796-8 doi: 10.1007/s00332-022-09796-8
[37]	B. Zhou, D. Jiang, Y. Dai, T. Hayat, Threshold dynamics and probability density function of a stochastic avian influenza epidemic model with nonlinear incidence rate and psychological effect, J. Nonlinear Sci., 33 (2023), 1–52. https://doi.org/10.1007/s00332-022-09885-8 doi: 10.1007/s00332-022-09885-8

Reader Comments

Your name:*

Email:*
© 2024 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)