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Attractors for a class of perturbed nonclassical diffusion equations with memory
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 |
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.
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. Chen, B. 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. Dieu, D. H. Nguyen, N. 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. Gray, D. Greenhalgh, L. Hu, X. 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. Herbert, M. 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. Iannelli, F. 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. Jiang, J. J. Yu, C. 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. Nguyen, G. 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. Nuno, Z. Feng, M. 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. Tuong, D. H. Nguyen, N. 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. Zhang, H. Huo, H. 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. Zhang, H. Huo, H. Xiang, Q. 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. Chen, B. 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. Dieu, D. H. Nguyen, N. 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. Gray, D. Greenhalgh, L. Hu, X. 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. Herbert, M. 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. Iannelli, F. 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. Jiang, J. J. Yu, C. 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. Nguyen, G. 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. Nuno, Z. Feng, M. 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. Tuong, D. H. Nguyen, N. 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. Zhang, H. Huo, H. 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. Zhang, H. Huo, H. Xiang, Q. 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. |




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