# American Institute of Mathematical Sciences

September  2019, 24(9): 4827-4849. doi: 10.3934/dcdsb.2019033

## Analysis of a stochastic SIRS model with interval parameters

 1 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China 2 Department of Mathematics, University of Florida, Gainesville, FL 32611, USA 3 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

* Corresponding author: Qimin Zhang

Received  February 2018 Revised  September 2018 Published  September 2019 Early access  February 2019

Many studies of mathematical epidemiology assume that model parameters are precisely known. However, they can be imprecise due to various uncertainties. Deterministic epidemic models are also subjected to stochastic perturbations. In this paper, we analyze a stochastic SIRS model that includes interval parameters and environmental noises. We define the stochastic basic reproduction number, which is shown to govern disease extinction or persistence. When it is less than one, the disease is predicted to die out with probability one. When it is greater than one, the model admits a stationary distribution. Thus, larger stochastic noises (resulting in a smaller stochastic basic reproduction number) are able to suppress the emergence of disease outbreaks. Using numerical simulations, we also investigate the influence of parameter imprecision and susceptible response to the disease information that may change individual behavior and protect the susceptible from infection. These parameters can greatly affect the long-term behavior of the system, highlighting the importance of incorporating parameter imprecision into epidemic models and the role of information intervention in the control of infectious diseases.

Citation: Kangbo Bao, Libin Rong, Qimin Zhang. Analysis of a stochastic SIRS model with interval parameters. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4827-4849. doi: 10.3934/dcdsb.2019033
##### References:
 [1] L. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology, Lecture Notes in Mathematics, 1945 (2008), 81–130. doi: 10.1007/978-3-540-78911-6_3. [2] K. Bao and Q. Zhang, Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention, Advances in Difference Equations, 352 (2017), 19pp. doi: 10.1186/s13662-017-1406-9. [3] L. Barros, R. Bassanezi and P. Tonelli, Fuzzy modelling in population dynamics, Ecological Modelling, 128 (2000), 27-33.  doi: 10.1016/S0304-3800(99)00223-9. [4] B. Buonomo, A. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, Journal of Mathematical Analysis and Applications, 404 (2013), 385-398.  doi: 10.1016/j.jmaa.2013.02.063. [5] B. Cao, M. Shan, Q. Zhang and W. Wang, A stochastic SIS epidemic model with vaccination, Physica A, 486 (2017), 127-143.  doi: 10.1016/j.physa.2017.05.083. [6] M. Carletti, Mean-square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395-414.  doi: 10.1016/j.mbs.2007.05.009. [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, Journal of Mathematical Analysis and Applications, 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055. [8] A. Das and M. Pal, A mathematical study of an imprecise SIR epidemic model with treatment control, Journal of Applied Mathematics Computing, 56 (2018), 477-500.  doi: 10.1007/s12190-017-1083-6. [9] T. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988. [10] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71 (2011), 876-902.  doi: 10.1137/10081856X. [11] R. Has'minskii, Stochastic Stability of Differental Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [12] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [13] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, New York, 1981. [14] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. [15] W. Kermack and A. McKendric, Contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London, Series A, 115 (1927), 700–721. [16] A. Kumar, P. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevallence, Journal of Theoretical Biology, 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016. [17] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics and Probability Letters, 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021. [18] Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete and Continuous Dynamical Systems - B, 22 (2017), 2479-2500.  doi: 10.3934/dcdsb.2017127. [19] X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402. [20] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [21] R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001. [22] D. Pal, G. Mahaptra and G. Samanta, Optimal harvesting of prey-predator ststem with interval biological parameters: A bioeconomic model, Mathematical Biosciences, 241 (2013), 181-187.  doi: 10.1016/j.mbs.2012.11.007. [23] P. Panja, S. Mondal and J. Chattopadhyay, Dynamical Study in fuzzy threshold dynamics of a cholera epidemic model, Fuzzy Information and Engineering, 9 (2017), 381-401.  doi: 10.1016/j.fiae.2017.10.001. [24] G. Sahu and J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate, Applied Mathematical Modelling, 36 (2012), 908-923.  doi: 10.1016/j.apm.2011.07.044. [25] S. Sharma and G. Samanta, Optimal harvesting of a two species competition model with imprecise biological parameters, Nonlinear Dynamics, 77 (2014), 1101-1119.  doi: 10.1007/s11071-014-1354-9. [26] S. Shen, C. Mei and J. Cui, A fuzzy varying coefficient model and its estimation, Computers and Mathematics with Applications, 60 (2010), 1696-1705.  doi: 10.1016/j.camwa.2010.06.049. [27] G. Strang, Linear Algebra and Its Applications, Thomson Learing Inc, 1988. [28] E. Tornatore, S. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.1016/j.physa.2005.02.057. [29] Q. Wang, Z. Liu, X. Zhang and R. Cheke, Incorporating prey refuge into a predator-prey system with imprecise parameter estimates, Computational and Applied Mathematics, 36 (2017), 1067-1084.  doi: 10.1007/s40314-015-0282-8. [30] Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Analysis: Real World Applications, 14 (2013), 1434-1456.  doi: 10.1016/j.nonrwa.2012.10.007. [31] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [32] T. Zhang and Z. Teng, An SIRVS epidemic model with pulse vaccination strategy, Journal of Theoretical Biology, 250 (2008), 375-381.  doi: 10.1016/j.jtbi.2007.09.034. [33] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179.  doi: 10.1137/060649343.

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##### References:
 [1] L. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology, Lecture Notes in Mathematics, 1945 (2008), 81–130. doi: 10.1007/978-3-540-78911-6_3. [2] K. Bao and Q. Zhang, Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention, Advances in Difference Equations, 352 (2017), 19pp. doi: 10.1186/s13662-017-1406-9. [3] L. Barros, R. Bassanezi and P. Tonelli, Fuzzy modelling in population dynamics, Ecological Modelling, 128 (2000), 27-33.  doi: 10.1016/S0304-3800(99)00223-9. [4] B. Buonomo, A. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, Journal of Mathematical Analysis and Applications, 404 (2013), 385-398.  doi: 10.1016/j.jmaa.2013.02.063. [5] B. Cao, M. Shan, Q. Zhang and W. Wang, A stochastic SIS epidemic model with vaccination, Physica A, 486 (2017), 127-143.  doi: 10.1016/j.physa.2017.05.083. [6] M. Carletti, Mean-square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395-414.  doi: 10.1016/j.mbs.2007.05.009. [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, Journal of Mathematical Analysis and Applications, 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055. [8] A. Das and M. Pal, A mathematical study of an imprecise SIR epidemic model with treatment control, Journal of Applied Mathematics Computing, 56 (2018), 477-500.  doi: 10.1007/s12190-017-1083-6. [9] T. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988. [10] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71 (2011), 876-902.  doi: 10.1137/10081856X. [11] R. Has'minskii, Stochastic Stability of Differental Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [12] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [13] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, New York, 1981. [14] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. [15] W. Kermack and A. McKendric, Contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London, Series A, 115 (1927), 700–721. [16] A. Kumar, P. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevallence, Journal of Theoretical Biology, 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016. [17] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics and Probability Letters, 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021. [18] Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete and Continuous Dynamical Systems - B, 22 (2017), 2479-2500.  doi: 10.3934/dcdsb.2017127. [19] X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402. [20] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [21] R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001. [22] D. Pal, G. Mahaptra and G. Samanta, Optimal harvesting of prey-predator ststem with interval biological parameters: A bioeconomic model, Mathematical Biosciences, 241 (2013), 181-187.  doi: 10.1016/j.mbs.2012.11.007. [23] P. Panja, S. Mondal and J. Chattopadhyay, Dynamical Study in fuzzy threshold dynamics of a cholera epidemic model, Fuzzy Information and Engineering, 9 (2017), 381-401.  doi: 10.1016/j.fiae.2017.10.001. [24] G. Sahu and J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate, Applied Mathematical Modelling, 36 (2012), 908-923.  doi: 10.1016/j.apm.2011.07.044. [25] S. Sharma and G. Samanta, Optimal harvesting of a two species competition model with imprecise biological parameters, Nonlinear Dynamics, 77 (2014), 1101-1119.  doi: 10.1007/s11071-014-1354-9. [26] S. Shen, C. Mei and J. Cui, A fuzzy varying coefficient model and its estimation, Computers and Mathematics with Applications, 60 (2010), 1696-1705.  doi: 10.1016/j.camwa.2010.06.049. [27] G. Strang, Linear Algebra and Its Applications, Thomson Learing Inc, 1988. [28] E. Tornatore, S. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.1016/j.physa.2005.02.057. [29] Q. Wang, Z. Liu, X. Zhang and R. Cheke, Incorporating prey refuge into a predator-prey system with imprecise parameter estimates, Computational and Applied Mathematics, 36 (2017), 1067-1084.  doi: 10.1007/s40314-015-0282-8. [30] Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Analysis: Real World Applications, 14 (2013), 1434-1456.  doi: 10.1016/j.nonrwa.2012.10.007. [31] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [32] T. Zhang and Z. Teng, An SIRVS epidemic model with pulse vaccination strategy, Journal of Theoretical Biology, 250 (2008), 375-381.  doi: 10.1016/j.jtbi.2007.09.034. [33] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179.  doi: 10.1137/060649343.
The path of $S(t)$, $I(t)$, $R(t)$ and the histogram of the probability density function of $I(150)$ assuming p = 0.1, $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ under different noise intensities
The path of $S(t)$, $I(t)$, $R(t)$ assuming p = 0.1, $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ with different noise intensities
Variation of $\mathscr{R}_0$ and $\mathscr{R}_s$ as $p$ varies
The path of $S(t)$, $I(t)$, $R(t)$ with initial $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ for p = 0.2, p = 0.4 and p = 0.6, respectively
The path of $I(t)$ with initial $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ under different noise intensities and imprecise parameter p
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