# American Institute of Mathematical Sciences

December  2021, 26(12): 6131-6154. doi: 10.3934/dcdsb.2021010

## Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination

 Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Liang Wang, Zhipeng Qiu

Received  June 2020 Revised  October 2020 Published  December 2021 Early access  December 2020

Fund Project: Z. Qiu is supported by the National Natural Science Foundation of China (NSFC) grants No. 12071217 and No. 11671206; L. Wang is supported by the National Science Foundation for Young Scientists of China grant No. 12001271, Natural Science Foundation of Jiangsu Province grant No. BK20200484; X. Zhao is supported by the Scholarship Foundation of China Scholarship Council grant No. 201906840072, the Postgraduate Research & Practice Innovation Program of Jiangsu Province grant No. KYCX20_0242; T. Feng is supported by the Scholarship Foundation of China Scholarship Council grant No. 201806840120, the Out-standing Chinese and Foreign Youth Exchange Program of China Association of Science and Technology, and the Fundamental Research Funds for the Central Universities grant No. 30918011339

In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of $R_0^s$ with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewise-stochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control.

Citation: 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
##### References:
 [1] F. B. Agusto and M. A. Khan, Optimal control strategies for dengue transmission in Pakistan, Mathematical Biosciences, 305 (2018), 102-121.  doi: 10.1016/j.mbs.2018.09.007. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [3] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602. [4] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review/Revue Internationale de Statistique, 62 (1994), 229-243.  doi: 10.2307/1403510. [5] X. Cao, Semi-Markov decision problems and performance sensitivity analysis, IEEE Transactions on Automatic Control, 48 (2003), 758-769.  doi: 10.1109/TAC.2003.811252. [6] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [7] F. H. Chen, A susceptible-infected epidemic model with voluntary vaccinations, Journal of Mathematical Biology, 53 (2006), 253-272.  doi: 10.1007/s00285-006-0006-1. [8] P. V. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [9] N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Applicandae Mathematicae, 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8. [10] T. Feng and Z. Qiu, Global analysis of a stochastic TB model with vaccination and treatment, Discrete & Continuous Dynamical Systems-B, 24 (2019), 2923-2939.  doi: 10.3934/dcdsb.2018292. [11] T. Feng and Z. Qiu, Analysis of an epidemiological model driven by multiple noises: Ergodicity and convergence rate, Journal of the Franklin Institute, 357 (2020), 2203-2216.  doi: 10.1016/j.jfranklin.2019.09.004. [12] I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, II[M], , Springer-Verlag, New York-Heidelberg, 1975. [13] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [14] H. W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences, 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2. [15] H. W. Hethcote and V. Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539. [16] T. K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 104 (2011), 127-135.  doi: 10.1016/j.biosystems.2011.02.001. [17] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society Of London. Series A, Containing Papers of a Mathematical and Physical Character, 115 (1927), 700-721. [18] A. Lahrouz, L. Omari and D. Kiouach, 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. [19] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics & Probability Letters, 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021. [20] J. Li and Z. Ma, Global analysis of SIS epidemic models with variable total population size, Mathematical and Computer Modelling, 39 (2004), 1231-1242.  doi: 10.1016/j.mcm.2004.06.004. [21] D. Li, M. Liu and S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bulletin of Mathematical Biology, 77 (2015), 1705-1743.  doi: 10.1007/s11538-015-0101-9. [22] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [23] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching, Journal of Differential Equations, 266 (2019), 3973-4017.  doi: 10.1016/j.jde.2018.09.026. [24] J. Li and Z. Ma, Global analysis of SIS epidemic models with variable total population size, Mathematical and Computer Modelling, 39 (2004), 1231-1242.  doi: 10.1016/j.mcm.2004.06.004. [25] M. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5. [26] N. Limnios and G. Oprisan, Semi-Markov Processes and Reliability[M], Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0161-8. [27] H. Liu, H. Xu and J. Yu, Stability on coupling SIR epidemic model with vaccination, Journal of Applied Mathematics, 2005 (2005), 301-319.  doi: 10.1155/JAM.2005.301. [28] X. Mao, Stability of stochastic differential equations with Markovian switching, Heilongjiang Science & Technology Information, 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [29] X. Mao, Stability of stochastic differential equations with markovian switching, Stochastic Processes and their Applications, $\texttt79$ (1999), 45–67. doi: 10.1016/S0304-4149(98)00070-2. [30] X. Mao, Stochastic Differential Equations and Applications[M], Elsevier, 2007. doi: 10.1533/9780857099402. [31] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [32] H. Margolis, M. Alter and S. Hadler, Hepatitis B: Evolving epidemiology and implications for control, Seminars in Liver Disease, 11 (1991), 84-92.  doi: 10.1055/s-2008-1040427. [33] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅱ: Continuous-time processes and sampled chains, Advances in Applied Probability, 25 (1993), 487-517.  doi: 10.2307/1427521. [34] D. Mollison, Spatial contact models for ecological and epidemic spread, Journal of the Royal Statistical Society, 39 (1977), 283-326.  doi: 10.1111/j.2517-6161.1977.tb01627.x. [35] X. Mu and Q. Zhang, Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching, Mathematical Methods in the Applied Sciences, 42 (2019), 767-789.  doi: 10.1002/mma.5378. [36] C. Serra, M.D. Martinez and X. Lana, European dry spell length distributions, years 1951-2000, Theoretical and Applied Climatology, 114 (2013), 531-551.  doi: 10.1007/s00704-013-0857-5. [37] M. J. Small and D. J. Morgan, The Relationship between a continuous-time renewal model and a discrete Markov chain model of precipitation occurrence, Water Resources Research, 22 (1986), 1422-1430.  doi: 10.1029/WR022i010p01422. [38] C. Sun, Y. Hsieh and P. Georgescu, A model for HIV transmission with two interacting high-risk groups, Nonlinear Analysis: Real World Applications, 40 (2018), 170-184.  doi: 10.1016/j.nonrwa.2017.08.012. [39] A. Swishchuk and J. Wu, Evolution of Biological Systems in Random Media: Limit Theorems and Stability[M], Springer Science & Business Media, 2003. doi: 10.1007/978-94-017-1506-5. [40] E. Vergu, H. Busson and P. Ezanno, Impact of the infection period distribution on the epidemic spread in a meta population model, PloS One, 5 (2010), e9371. [41] K. Wang, Random Mathematical Biology Model,, Science Press, Beijing, 2010. [42] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism, Journal of Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028. [43] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025. [44] X. Zhang, D. Jiang and A. Alsaedi, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Applied Mathematics Letters, 59 (2016), 87-93.  doi: 10.1016/j.aml.2016.03.010. [45] Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Applied Mathematics and Computation, 243 (2014), 718-727.  doi: 10.1016/j.amc.2014.05.124. [46] Y. Zhao and D. Jiang, The threshold of a stochastic sirs epidemic model with saturated incidence, Applied Mathematics Letters, 34 (2014), 90-93.  doi: 10.1016/j.aml.2013.11.002. [47] Y. Zhao, D. Jiang and X. Mao, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 1277-1295.  doi: 10.3934/dcdsb.2015.20.1277. [48] B. Zheng, X. Liu, M. Tang and J. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, Journal of Theoretical Biology, 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010. [49] L. Zu, D. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.

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##### References:
 [1] F. B. Agusto and M. A. Khan, Optimal control strategies for dengue transmission in Pakistan, Mathematical Biosciences, 305 (2018), 102-121.  doi: 10.1016/j.mbs.2018.09.007. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [3] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602. [4] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review/Revue Internationale de Statistique, 62 (1994), 229-243.  doi: 10.2307/1403510. [5] X. Cao, Semi-Markov decision problems and performance sensitivity analysis, IEEE Transactions on Automatic Control, 48 (2003), 758-769.  doi: 10.1109/TAC.2003.811252. [6] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [7] F. H. Chen, A susceptible-infected epidemic model with voluntary vaccinations, Journal of Mathematical Biology, 53 (2006), 253-272.  doi: 10.1007/s00285-006-0006-1. [8] P. V. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [9] N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Applicandae Mathematicae, 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8. [10] T. Feng and Z. Qiu, Global analysis of a stochastic TB model with vaccination and treatment, Discrete & Continuous Dynamical Systems-B, 24 (2019), 2923-2939.  doi: 10.3934/dcdsb.2018292. [11] T. Feng and Z. Qiu, Analysis of an epidemiological model driven by multiple noises: Ergodicity and convergence rate, Journal of the Franklin Institute, 357 (2020), 2203-2216.  doi: 10.1016/j.jfranklin.2019.09.004. [12] I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, II[M], , Springer-Verlag, New York-Heidelberg, 1975. [13] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [14] H. W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences, 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2. [15] H. W. Hethcote and V. Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539. [16] T. K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 104 (2011), 127-135.  doi: 10.1016/j.biosystems.2011.02.001. [17] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society Of London. Series A, Containing Papers of a Mathematical and Physical Character, 115 (1927), 700-721. [18] A. Lahrouz, L. Omari and D. Kiouach, 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. [19] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics & Probability Letters, 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021. [20] J. Li and Z. Ma, Global analysis of SIS epidemic models with variable total population size, Mathematical and Computer Modelling, 39 (2004), 1231-1242.  doi: 10.1016/j.mcm.2004.06.004. [21] D. Li, M. Liu and S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bulletin of Mathematical Biology, 77 (2015), 1705-1743.  doi: 10.1007/s11538-015-0101-9. [22] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [23] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching, Journal of Differential Equations, 266 (2019), 3973-4017.  doi: 10.1016/j.jde.2018.09.026. [24] J. Li and Z. Ma, Global analysis of SIS epidemic models with variable total population size, Mathematical and Computer Modelling, 39 (2004), 1231-1242.  doi: 10.1016/j.mcm.2004.06.004. [25] M. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5. [26] N. Limnios and G. Oprisan, Semi-Markov Processes and Reliability[M], Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0161-8. [27] H. Liu, H. Xu and J. Yu, Stability on coupling SIR epidemic model with vaccination, Journal of Applied Mathematics, 2005 (2005), 301-319.  doi: 10.1155/JAM.2005.301. [28] X. Mao, Stability of stochastic differential equations with Markovian switching, Heilongjiang Science & Technology Information, 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [29] X. Mao, Stability of stochastic differential equations with markovian switching, Stochastic Processes and their Applications, $\texttt79$ (1999), 45–67. doi: 10.1016/S0304-4149(98)00070-2. [30] X. Mao, Stochastic Differential Equations and Applications[M], Elsevier, 2007. doi: 10.1533/9780857099402. [31] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [32] H. Margolis, M. Alter and S. Hadler, Hepatitis B: Evolving epidemiology and implications for control, Seminars in Liver Disease, 11 (1991), 84-92.  doi: 10.1055/s-2008-1040427. [33] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅱ: Continuous-time processes and sampled chains, Advances in Applied Probability, 25 (1993), 487-517.  doi: 10.2307/1427521. [34] D. Mollison, Spatial contact models for ecological and epidemic spread, Journal of the Royal Statistical Society, 39 (1977), 283-326.  doi: 10.1111/j.2517-6161.1977.tb01627.x. [35] X. Mu and Q. Zhang, Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching, Mathematical Methods in the Applied Sciences, 42 (2019), 767-789.  doi: 10.1002/mma.5378. [36] C. Serra, M.D. Martinez and X. Lana, European dry spell length distributions, years 1951-2000, Theoretical and Applied Climatology, 114 (2013), 531-551.  doi: 10.1007/s00704-013-0857-5. [37] M. J. Small and D. J. Morgan, The Relationship between a continuous-time renewal model and a discrete Markov chain model of precipitation occurrence, Water Resources Research, 22 (1986), 1422-1430.  doi: 10.1029/WR022i010p01422. [38] C. Sun, Y. Hsieh and P. Georgescu, A model for HIV transmission with two interacting high-risk groups, Nonlinear Analysis: Real World Applications, 40 (2018), 170-184.  doi: 10.1016/j.nonrwa.2017.08.012. [39] A. Swishchuk and J. Wu, Evolution of Biological Systems in Random Media: Limit Theorems and Stability[M], Springer Science & Business Media, 2003. doi: 10.1007/978-94-017-1506-5. [40] E. Vergu, H. Busson and P. Ezanno, Impact of the infection period distribution on the epidemic spread in a meta population model, PloS One, 5 (2010), e9371. [41] K. Wang, Random Mathematical Biology Model,, Science Press, Beijing, 2010. [42] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism, Journal of Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028. [43] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025. [44] X. Zhang, D. Jiang and A. Alsaedi, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Applied Mathematics Letters, 59 (2016), 87-93.  doi: 10.1016/j.aml.2016.03.010. [45] Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Applied Mathematics and Computation, 243 (2014), 718-727.  doi: 10.1016/j.amc.2014.05.124. [46] Y. Zhao and D. Jiang, The threshold of a stochastic sirs epidemic model with saturated incidence, Applied Mathematics Letters, 34 (2014), 90-93.  doi: 10.1016/j.aml.2013.11.002. [47] Y. Zhao, D. Jiang and X. Mao, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 1277-1295.  doi: 10.3934/dcdsb.2015.20.1277. [48] B. Zheng, X. Liu, M. Tang and J. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, Journal of Theoretical Biology, 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010. [49] L. Zu, D. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.
Simulations of $(S(t), I(t), R(t))$ with initial values $(5, 1, 0)$, the distribution function of each state of semi-Markov chain is Hyper-exponential distribution. (a) Sample of $(S(t), I(t), R(t), r(t))$ of system (2) with initial values $(5, 1, 0)$, $r(0) = 1$ and $\theta(0) = 0$, the corresponding $m_1 = 0.3750$, $m_2 = 0.7833$ and $R_0^s = 1.3181>1$; (b) Sample of $(S_1(t), I_1(t), R_1(t))$ of subsystem in system (2) under state $\textbf{1}$ with $\beta_1 = 0.0056$, the disease $I_1(t)$ is persistent; (c) Sample of $(S_2(t), I_2(t), R_2(t))$ of subsystem in system (2) under state $\textbf{2}$ with $\beta_2 = 0.0013$, the disease $I_2(t)$ is extinct
Simulations of $(S(t), I(t), R(t))$ by using the parameter values in Table 2 with initial values $(5, 1, 0)$, the distribution function of each state of semi-Markov chain as the Gamma distribution. (a) Sample of $(S(t), I(t), R(t), r(t))$ of system (2) with initial values $(5, 1, 0)$, $r(0) = 1$ and $\theta(0) = 0$, the corresponding $m_1 = 0.3333$, $m_2 = 3$ and $R_0^s = 0.8471<1$. (b) Sample of $(S_1(t), I_1(t), R_1(t))$ of subsystem in system (2) under state $\textbf{1}$ with $\beta_1 = 0.0056$. (c) Sample of $(S_2(t), I_2(t), R_2(t))$ of subsystem in system (2) under state $\textbf{2}$ with $\beta_2 = 0.0013$
PRCC values for system (47), using the basic reproduction number $R_0^s$ in (49) as response functions. By using the same parameter values in Table 2, we get almost the same picture: (a) Analysis of $R_0^s$ in example under Hyper-exponential distribution with $m_1 = 0.3750$, $m_2 = 0.7833$; (b) Analysis of $R_0^s$ in counterexample under Gamma distribution with $m_1 = 0.3333$, $m_2 = 3$
(a) The value of $R_0^s$ when we set $m_1 \in [0, 10]$ and $m_2 \in [0, 10]$; (b) The relation between $R_0^s$ and $m_1$ when we fixed $m_2 = 0.1, 0.5, 1$; (c) The relation between $R_0^s$ and $m_2$ when we fixed $m_1 = 0.1, 0.5, 1$
In this example we set $F_i(t)$ as the Hyper-exponential distribution. In order to make the picture clear, we adopted $\Delta t = \frac{T}{10}$. One can use the smaller $\Delta t$ to ensure accuracy
We use a new interval $\Delta \ddot{t} = {\Delta t} /2$, and assume $r(t) = L(\tilde{t}_i )$ if $t \in [\tilde{t}_i , \tilde{t}_{i+1}]$. Obviously, the smaller $\Delta \mathfrak{t}$ we choose, the more accurately this discrete semi-Markov chain is simulated
The biological significance of each parameter for stochastic system (2)
 Natation Biological meanings $S(t)$ Number of susceptibles at time $t$ $I(t)$ Number of infective individuals at time $t$ $R(t)$ Number of recovered individuals at time $t$ $B(t)$ Standard Brownian motion in one dimension ${\sigma}$ The intensity of $B(t)$ $\Lambda$ The recruitment rate of the population $p$ The proportion of population that is vaccinated $\mu$ The death rates of susceptibles, infectives, and recovered individuals $\beta(\cdot)$ The infection coefficient $\alpha$ The death rate of infected individuals from disease-related causes $\delta$ The recovery rate of the infective individuals $\lambda$ The recovered individuals immunity lose rate
 Natation Biological meanings $S(t)$ Number of susceptibles at time $t$ $I(t)$ Number of infective individuals at time $t$ $R(t)$ Number of recovered individuals at time $t$ $B(t)$ Standard Brownian motion in one dimension ${\sigma}$ The intensity of $B(t)$ $\Lambda$ The recruitment rate of the population $p$ The proportion of population that is vaccinated $\mu$ The death rates of susceptibles, infectives, and recovered individuals $\beta(\cdot)$ The infection coefficient $\alpha$ The death rate of infected individuals from disease-related causes $\delta$ The recovery rate of the infective individuals $\lambda$ The recovered individuals immunity lose rate
Parameters for the each subsystem
 Parameters Value Source p 0.833 [26] $\Lambda$ 0.33 $days^{-1}$ [11,25] $\mu$ 0.006 $days^{-1}$ [11,25] $\alpha$ 0.06 $days^{-1}$ [11,25] $\beta_1$ 0.0056 $days^{-1}$ [11,25] $\beta_2$ 0.0013 $days^{-1}$ [11,25] $\lambda$ 0.021 $days^{-1}$ [11,25] $\delta$ 0.01 $days^{-1}$ [11,25] $a_1$ 0.001 [11,25] $a_2$ 0.001 [11,25] $\sigma$ 0.04 Estimated
 Parameters Value Source p 0.833 [26] $\Lambda$ 0.33 $days^{-1}$ [11,25] $\mu$ 0.006 $days^{-1}$ [11,25] $\alpha$ 0.06 $days^{-1}$ [11,25] $\beta_1$ 0.0056 $days^{-1}$ [11,25] $\beta_2$ 0.0013 $days^{-1}$ [11,25] $\lambda$ 0.021 $days^{-1}$ [11,25] $\delta$ 0.01 $days^{-1}$ [11,25] $a_1$ 0.001 [11,25] $a_2$ 0.001 [11,25] $\sigma$ 0.04 Estimated
Symbols used in simulation
 Symbols Definition Value $T$ Total simulation time 10 $\tilde{t}_i (i=0, 1, \cdots)$ Every checkpoint $\{0, 1, \cdots, 10 \}$ $t_s$ Accumulation of time of per cycle - - $\Delta t$ Time interval between each jump 1 $s_i(i=1, 2)$ The states of the semi-Markov switching {1, 2} $L(t)$ The state of the semi-Markov chain in $t$ 1 or 2 $r(t)$ The state of the discrete chain in $t$ 1 or 2 $M$ The transition probability matrix - -
 Symbols Definition Value $T$ Total simulation time 10 $\tilde{t}_i (i=0, 1, \cdots)$ Every checkpoint $\{0, 1, \cdots, 10 \}$ $t_s$ Accumulation of time of per cycle - - $\Delta t$ Time interval between each jump 1 $s_i(i=1, 2)$ The states of the semi-Markov switching {1, 2} $L(t)$ The state of the semi-Markov chain in $t$ 1 or 2 $r(t)$ The state of the discrete chain in $t$ 1 or 2 $M$ The transition probability matrix - -
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