# American Institute of Mathematical Sciences

May  2021, 26(5): 2693-2719. doi: 10.3934/dcdsb.2020201

## Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  January 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The second author is supported by NSFC (Grant No. 11671123)

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold $\lambda$, it is proved that if $\lambda>0$, the disease is permanent and there is a stationary distribution. And when $\lambda<0$, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

Citation: Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
##### References:
 [1] J. H. Bao and J. H. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM Journal on Mathematical Analysis, 48 (2016), 725-739.  doi: 10.1137/15M1024512. [2] S. Y. Cai, Y. M. Cai and X. R. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039. [3] Y. L. Chen, B. Y. Wen and Z. D. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085. [4] N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Appl. Math., 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8. [5] N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 64 (2017), 223-230.  doi: 10.1016/j.aml.2016.09.012. [6] J. P. Gao and S. J. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28 pp doi: 10.1142/S0218127420500741. [7] 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. [8] A. Gray, D. Greenhalgh, X. R. Mao and J. F. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [9] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [10] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981. [11] C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037. [12] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [13] A. Lahrouz, L. Omari and D. Kioach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [14] A. Lahrouz, L. Omari, D. Kioach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Applied Mathematics and Computation, 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024. [15] G. J. Lan, Y. L. Huang, C. J. Wei and S. W. Zhang, A stochastic SIS epidemic model with saturating contact rate, Physica A, 529 (2019), 121504, 14 pp. doi: 10.1016/j.physa.2019.121504. [16] S. Z. Li and S. J. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164. doi: 10.1016/j.physa.2020.124164. [17] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376 (1) (2011), 11-28.  doi: 10.1016/j.lmaa.2010.10.053. [18] Q. Liu and Q. M. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Applied Mathematics and Computation, 282 (2016), 155-166.  doi: 10.1016/j.amc.2016.02.022. [19] Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate, Physica A, 526 (2019), 120946. doi: 10.1016/j.physa.2019.04.182. [20] X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [21] D. H. Nguyen, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4. [22] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [23] H. H. Qiu, S. J. Guo and S. Z. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224. [24] J. H. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electronic Journal of Probability, 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018. [25] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. [26] B. Sounvoravong, S. J. Guo and Y. Z. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), 16 pp. [27] Z. D. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084. [28] B. Y. Wen, R. Rifhat and Z. D. Teng, The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence, Physica A, 524 (2019), 258-271.  doi: 10.1016/j.physa.2019.04.049. [29] C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, J. Math. Anal. Appl., 447 (2017), 736-757.  doi: 10.1016/j.jmaa.2016.10.041. [30] Y. Zhang, Y. Li, Q. L. Zhang and A. H. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187.  doi: 10.1016/j.physa.2018.02.191. [31] X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644. [32] Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution in a stochastic SIS model, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 592821, 12 pp. doi: 10.1155/2013/592821. [33] Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100. [34] C. J. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012. [35] R. Zou and S. J. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, (2020). doi: 10.3934/dcdsb.2020093.

show all references

##### References:
 [1] J. H. Bao and J. H. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM Journal on Mathematical Analysis, 48 (2016), 725-739.  doi: 10.1137/15M1024512. [2] S. Y. Cai, Y. M. Cai and X. R. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039. [3] Y. L. Chen, B. Y. Wen and Z. D. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085. [4] N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Appl. Math., 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8. [5] N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 64 (2017), 223-230.  doi: 10.1016/j.aml.2016.09.012. [6] J. P. Gao and S. J. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28 pp doi: 10.1142/S0218127420500741. [7] 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. [8] A. Gray, D. Greenhalgh, X. R. Mao and J. F. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [9] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [10] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981. [11] C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037. [12] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [13] A. Lahrouz, L. Omari and D. Kioach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [14] A. Lahrouz, L. Omari, D. Kioach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Applied Mathematics and Computation, 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024. [15] G. J. Lan, Y. L. Huang, C. J. Wei and S. W. Zhang, A stochastic SIS epidemic model with saturating contact rate, Physica A, 529 (2019), 121504, 14 pp. doi: 10.1016/j.physa.2019.121504. [16] S. Z. Li and S. J. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164. doi: 10.1016/j.physa.2020.124164. [17] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376 (1) (2011), 11-28.  doi: 10.1016/j.lmaa.2010.10.053. [18] Q. Liu and Q. M. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Applied Mathematics and Computation, 282 (2016), 155-166.  doi: 10.1016/j.amc.2016.02.022. [19] Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate, Physica A, 526 (2019), 120946. doi: 10.1016/j.physa.2019.04.182. [20] X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [21] D. H. Nguyen, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4. [22] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [23] H. H. Qiu, S. J. Guo and S. Z. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224. [24] J. H. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electronic Journal of Probability, 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018. [25] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. [26] B. Sounvoravong, S. J. Guo and Y. Z. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), 16 pp. [27] Z. D. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084. [28] B. Y. Wen, R. Rifhat and Z. D. Teng, The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence, Physica A, 524 (2019), 258-271.  doi: 10.1016/j.physa.2019.04.049. [29] C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, J. Math. Anal. Appl., 447 (2017), 736-757.  doi: 10.1016/j.jmaa.2016.10.041. [30] Y. Zhang, Y. Li, Q. L. Zhang and A. H. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187.  doi: 10.1016/j.physa.2018.02.191. [31] X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644. [32] Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution in a stochastic SIS model, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 592821, 12 pp. doi: 10.1155/2013/592821. [33] Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100. [34] C. J. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012. [35] R. Zou and S. J. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, (2020). doi: 10.3934/dcdsb.2020093.
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = b = c = 0.1$
Stationary distribution of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = b = c = 0.1$: (a) the graph of the relative frequency densities of $S$ and $I$; (b) the joint density distribution of solution $(S,I)$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 1$, $b = 0.1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 2$, $b = 0.1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.31$
Numerical simulations of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.31$: (a) Trajectories of the solution $S(t)$ of (26) and the solution $\varphi$ of (10) shows the convergence of $S$ to a boundary distribution; (b) The dynamics of $S(t)$ and $I(t)$ in time average
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 1$, $b = 0.1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 0.1$, $c = 1$
 [1] Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371 [2] Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335 [3] Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6173-6184. doi: 10.3934/dcdsb.2021012 [4] Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783 [5] Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169 [6] Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119 [7] Nguyen Thanh Dieu, Vu Hai Sam, Nguyen Huu Du. Threshold of a stochastic SIQS epidemic model with isolation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021262 [8] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [9] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4887-4905. doi: 10.3934/dcdsb.2020317 [10] Adel Settati, Aadil Lahrouz, Mustapha El Jarroudi, Mohamed El Fatini, Kai Wang. On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1985-1997. doi: 10.3934/dcdsb.2020012 [11] Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki, Aziz Laaribi. A stochastic threshold for an epidemic model with isolation and a non linear incidence. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2513-2531. doi: 10.3934/cpaa.2020110 [12] Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277 [13] Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 [14] Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069 [15] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010 [16] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [17] Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3831-3844. doi: 10.3934/dcdsb.2021208 [18] Xinli Hu. Threshold dynamics for a Tuberculosis model with seasonality. Mathematical Biosciences & Engineering, 2012, 9 (1) : 111-122. doi: 10.3934/mbe.2012.9.111 [19] Zachary P. Kilpatrick. Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2211-2231. doi: 10.3934/dcdsb.2016044 [20] Toru Sasaki. The effect of local prevention in an SIS model with diffusion. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 739-746. doi: 10.3934/dcdsb.2004.4.739

2020 Impact Factor: 1.327