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doi: 10.3934/dcdsb.2021262
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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[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

[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.

[23]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[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

[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.

[23]

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

[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.

[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.

[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.

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)) $
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