doi: 10.3934/dcdsb.2021262
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Threshold of a stochastic SIQS epidemic model with isolation

1. 

School of Natural Sciences Education, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam

2. 

HUS High School for Gifted Student, Hanoi National University, 182 Luong The Vinh, Thanh Xuan, Hanoi, Vietnam

3. 

Department of Mathematics, Mechanics and Informatics, , Hanoi National University, , 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

* Corresponding author: Nguyen Thanh Dieu

Received  February 2021 Revised  July 2021 Early access October 2021

The aim of this paper is to give sufficient conditions, very close to the necessary one, to classify the stochastic permanence of SIQS epidemic model with isolation via a threshold value $ \widehat R $. Precisely, we show that if $ \widehat R<1 $ then the stochastic SIQS system goes to the disease free case in sense the density of infected $ I_z(t) $ and quarantined $ Q_z(t) $ classes extincts to $ 0 $ at exponential rate and the density of susceptible class $ S_z(t) $ converges almost surely at exponential rate to the solution of boundary equation. In the case $ \widehat R>1 $, the model is permanent. We show the existence of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to this invariant measure. Some numerical examples are also provided to illustrate our findings.

Citation: Nguyen Thanh Dieu, Vu Hai Sam, Nguyen Huu Du. Threshold of a stochastic SIQS epidemic model with isolation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021262
References:
[1]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar

[2]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[3]

Y. ChenB. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Phys. A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.  Google Scholar

[4]

N. H. Du and N. T. Dieu, Long-time behavior of an SIR model with perturbed disease transmission coefficient, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3429-3440.  doi: 10.3934/dcdsb.2016105.  Google Scholar

[5]

N. T. DieuD. H. NguyenN. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dyn. Syst., 15 (2016), 1062-1084.  doi: 10.1137/15M1043315.  Google Scholar

[6]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.  Google Scholar

[7]

A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347.  Google Scholar

[8]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47.  doi: 10.1007/BF00276034.  Google Scholar

[9]

H. HerbertM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.  Google Scholar

[10]

M. IannelliF. A. Milner and A. Pugliese, Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal., 23 (1992), 662-688.  doi: 10.1137/0523034.  Google Scholar

[11]

K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrsch. Verw. Gebiete, 30 (1974), 235–254, Corrections in 39 (1977), 81–84. doi: 10.1007/BF00533476.  Google Scholar

[12]

D. Q. JiangJ. J. YuC. Y. Ji and N. Z. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Modell., 54 (2011), 221-232.  doi: 10.1016/j.mcm.2011.02.004.  Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, (part I), Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar

[14]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, (part II), Proc. Roy. Sot. Ser. A, 138 (1932), 55-83.  doi: 10.1098/rspa.1932.0171.  Google Scholar

[15]

R. Z. Khas'minskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations, Theory Probab. Appl., 5 (1960), 179-196.  doi: 10.1137/1105016.  Google Scholar

[16]

W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15 (1987), 690-707.  doi: 10.1214/aop/1176992166.  Google Scholar

[17]

D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477.  doi: 10.1137/16M1059357.  Google Scholar

[18]

D. H. NguyenG. Yin and C. Zhu, Long-term analysis of a stochastic SIRS model with general incidence rates, SIAM J. Appl. Math., 80 (2020), 814-838.  doi: 10.1137/19M1246973.  Google Scholar

[19]

M. NunoZ. FengM. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X.  Google Scholar

[20]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Providence, RI: Amer. Math. Soc., 1989. doi: 10.1090/mmono/078.  Google Scholar

[21]

L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report, Brown University, Providence, (1986), 86–16, Available from: https://www.amazon.co.uk/existence-uniqueness-invariant-continuous-processes/dp/B000722C66 Google Scholar

[22]

T. D. TuongD. H. NguyenN. T. Dieu and T. Ky, Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate, Nonlinear Anal. Hybrid Syst., 34 (2019), 121-130.  doi: 10.1016/j.nahs.2019.05.008.  Google Scholar

[23]

F. J. S. Wang, Asymptotic behavior of some deterministic epidemic models, SIAM J. Math. Anal., 9 (1978), 529-534.  doi: 10.1137/0509034.  Google Scholar

[24]

X. ZhangH. HuoH. Xiang and X. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence, Appl. Math. Comput., 243 (2014), 546-558.  doi: 10.1016/j.amc.2014.05.136.  Google Scholar

[25]

X. ZhangH. HuoH. XiangQ. Shi and D. Li, The threshold of a stochastic SIQS epidemic model, Phys. A, 482 (2017), 362-374.  doi: 10.1016/j.physa.2017.04.100.  Google Scholar

[26]

X. B. Zhang and X. H. Zhang, The threshold of a deterministic and a stochastic SIQS epidemic model with varying total population size, Appl. Math. Model., 91 (2021), 749-767.  doi: 10.1016/j.apm.2020.09.050.  Google Scholar

show all references

References:
[1]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar

[2]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[3]

Y. ChenB. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Phys. A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.  Google Scholar

[4]

N. H. Du and N. T. Dieu, Long-time behavior of an SIR model with perturbed disease transmission coefficient, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3429-3440.  doi: 10.3934/dcdsb.2016105.  Google Scholar

[5]

N. T. DieuD. H. NguyenN. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dyn. Syst., 15 (2016), 1062-1084.  doi: 10.1137/15M1043315.  Google Scholar

[6]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.  Google Scholar

[7]

A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347.  Google Scholar

[8]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47.  doi: 10.1007/BF00276034.  Google Scholar

[9]

H. HerbertM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.  Google Scholar

[10]

M. IannelliF. A. Milner and A. Pugliese, Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal., 23 (1992), 662-688.  doi: 10.1137/0523034.  Google Scholar

[11]

K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrsch. Verw. Gebiete, 30 (1974), 235–254, Corrections in 39 (1977), 81–84. doi: 10.1007/BF00533476.  Google Scholar

[12]

D. Q. JiangJ. J. YuC. Y. Ji and N. Z. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Modell., 54 (2011), 221-232.  doi: 10.1016/j.mcm.2011.02.004.  Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, (part I), Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar

[14]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, (part II), Proc. Roy. Sot. Ser. A, 138 (1932), 55-83.  doi: 10.1098/rspa.1932.0171.  Google Scholar

[15]

R. Z. Khas'minskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations, Theory Probab. Appl., 5 (1960), 179-196.  doi: 10.1137/1105016.  Google Scholar

[16]

W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15 (1987), 690-707.  doi: 10.1214/aop/1176992166.  Google Scholar

[17]

D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477.  doi: 10.1137/16M1059357.  Google Scholar

[18]

D. H. NguyenG. Yin and C. Zhu, Long-term analysis of a stochastic SIRS model with general incidence rates, SIAM J. Appl. Math., 80 (2020), 814-838.  doi: 10.1137/19M1246973.  Google Scholar

[19]

M. NunoZ. FengM. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X.  Google Scholar

[20]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Providence, RI: Amer. Math. Soc., 1989. doi: 10.1090/mmono/078.  Google Scholar

[21]

L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report, Brown University, Providence, (1986), 86–16, Available from: https://www.amazon.co.uk/existence-uniqueness-invariant-continuous-processes/dp/B000722C66 Google Scholar

[22]

T. D. TuongD. H. NguyenN. T. Dieu and T. Ky, Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate, Nonlinear Anal. Hybrid Syst., 34 (2019), 121-130.  doi: 10.1016/j.nahs.2019.05.008.  Google Scholar

[23]

F. J. S. Wang, Asymptotic behavior of some deterministic epidemic models, SIAM J. Math. Anal., 9 (1978), 529-534.  doi: 10.1137/0509034.  Google Scholar

[24]

X. ZhangH. HuoH. Xiang and X. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence, Appl. Math. Comput., 243 (2014), 546-558.  doi: 10.1016/j.amc.2014.05.136.  Google Scholar

[25]

X. ZhangH. HuoH. XiangQ. Shi and D. Li, The threshold of a stochastic SIQS epidemic model, Phys. A, 482 (2017), 362-374.  doi: 10.1016/j.physa.2017.04.100.  Google Scholar

[26]

X. B. Zhang and X. H. Zhang, The threshold of a deterministic and a stochastic SIQS epidemic model with varying total population size, Appl. Math. Model., 91 (2021), 749-767.  doi: 10.1016/j.apm.2020.09.050.  Google Scholar

Figure 1.  Estimated paths of $ \frac{\ln I_z(t)}t $ (in red line), $ \frac{\ln Q_z(t)}t $ (in ping line) and $ \frac{\ln|S_z(t)- \widetilde S_u^0(t)|}{t} $ (in blue line) in Example 3.1
Figure 2.  Trajectories of $ (S_z(t), I_z(t), Q_z(t)) $ in Example 3.2
Figure 3.  Marginal one dimensional densities of $ (S_z(t), I_z(t), Q_z(t)) $
Figure 4.  Marginal two dimensional densities of $ (S_z(t), I_z(t), Q_z(t)) $
[1]

Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119

[2]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[3]

Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783

[4]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371

[5]

Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051

[6]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

[7]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

[8]

Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021208

[9]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[10]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[11]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[12]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699

[13]

Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743

[14]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041

[15]

Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123

[16]

Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219

[17]

Bum Il Hong, Nahmwoo Hahm, Sun-Ho Choi. SIR Rumor spreading model with trust rate distribution. Networks & Heterogeneous Media, 2018, 13 (3) : 515-530. doi: 10.3934/nhm.2018023

[18]

Pierre Gabriel, Hugo Martin. Steady distribution of the incremental model for bacteria proliferation. Networks & Heterogeneous Media, 2019, 14 (1) : 149-171. doi: 10.3934/nhm.2019008

[19]

Arturo Hidalgo, Lourdes Tello. On a climatological energy balance model with continents distribution. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1503-1519. doi: 10.3934/dcds.2015.35.1503

[20]

Yu Chen, Zixian Cui, Shihan Di, Peibiao Zhao. Capital asset pricing model under distribution uncertainty. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021113

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]