doi: 10.3934/dcdsb.2022072
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A Markovian switching diffusion for an SIS model incorporating Lévy processes

1. 

Laboratory of mathematics and applications, FSTT, Abdelmalek Essaadi University, Tetouan, Morocco

2. 

Stochastic Modelling & Statistics group, Lab of EDP-AGS, Faculty of Sciences, Ibn Tofail University, BP 133 Kénitra, Morocco

* Corresponding author: Mohamed El Fatini

Received  October 2021 Revised  January 2022 Early access April 2022

The purpose of this work is to investigate the asymptotic properties of a stochastic version of the classical SIS epidemic model with three noises. The stochastic model studied here includes white noise, telegraph noise and Lévy noise. We established conditions for extinction in probability and in $ pth $ moment. We also showed the persistence of disease under different conditions. The presented results are demonstrated by numerical simulations.

Citation: A. Settati, A. Lahrouz, Mohamed El Fatini, A. El Haitami, M. El Jarroudi, M. Erriani. A Markovian switching diffusion for an SIS model incorporating Lévy processes. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022072
References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Prentice Hall, Upper Saddle River, NJ, 2003. doi: 10.1201/b12537.

[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. 
[3]

J. BaoB. BöttcherX. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011), 119-131.  doi: 10.1016/j.cam.2011.05.043.

[4]

J. BaoX. MaoG. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.

[5]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043.

[6]

E. BerettaV. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delay influenced by stochastic perturbations, Math. Comp. Simul., 45 (1998), 269-277.  doi: 10.1016/S0378-4754(97)00106-7.

[7]

E. Beretta and Y. Takeuchi, Global stability of a SIR epidemic model with time delay, J. Math. Biol., 33 (1995), 250-260.  doi: 10.1007/BF00169563.

[8]

N. DalalD. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055.

[9]

S. GaoaY. LiuJ. J. Nieto and H. Andrade, Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. Comput. Simulat., 81 (2011), 1855-1868.  doi: 10.1016/j.matcom.2010.10.032.

[10]

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.

[11]

A. GrayD. GreenhalghX. Mao. and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029.

[12]

Z. Han and J. Zhao, Stochastic SIRS model under regime switching, Nonlinear Anal.: Real World Appl., 14 (2013), 352-364.  doi: 10.1016/j.nonrwa.2012.06.008.

[13]

W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Mat., 115 (1927), 700-721. 

[14]

R. Z. KhasminskiiC. Zhu and G. Yin, Stability of regime-switching diffusions, Stoch. Process. Appl., 117 (2007), 1037-1051.  doi: 10.1016/j.spa.2006.12.001.

[15]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[16]

A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021.

[17]

A. LahrouzL. OmariD. Kiouach and A. Belmaati, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024.

[18]

A. LahrouzL. OmariD. Kiouach and A. Belmaati, Deterministic and stochastic stability of a mathematical model of smoking, Statist. Probab. Lett., 81 (2011), 1276-1284.  doi: 10.1016/j.spl.2011.03.029.

[19]

A. Lahrouz and A. Settati, Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput, 219 (2013), 11134-11148.  doi: 10.1016/j.amc.2013.05.019.

[20]

R. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.

[21]

M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.  doi: 10.1016/j.na.2013.02.018.

[22]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.  doi: 10.1016/j.amc.2010.08.037.

[23]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370.

[24]

Y. TakeuchiN. H. DuN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.

[25]

F. Xi, On the stability of jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018.

[26]

F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.

[27]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.

[28]

X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874.  doi: 10.1016/j.aml.2013.03.013.

[29]

X. Zhang and K. Wang, Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise, Commun. Nonlinear. Sci. Numer. Simulat, 19 (2014), 1391-1399.  doi: 10.1016/j.cnsns.2013.09.013.

[30]

J. Zhou and H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809-834.  doi: 10.1007/BF00168799.

show all references

References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Prentice Hall, Upper Saddle River, NJ, 2003. doi: 10.1201/b12537.

[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. 
[3]

J. BaoB. BöttcherX. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011), 119-131.  doi: 10.1016/j.cam.2011.05.043.

[4]

J. BaoX. MaoG. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.

[5]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043.

[6]

E. BerettaV. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delay influenced by stochastic perturbations, Math. Comp. Simul., 45 (1998), 269-277.  doi: 10.1016/S0378-4754(97)00106-7.

[7]

E. Beretta and Y. Takeuchi, Global stability of a SIR epidemic model with time delay, J. Math. Biol., 33 (1995), 250-260.  doi: 10.1007/BF00169563.

[8]

N. DalalD. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055.

[9]

S. GaoaY. LiuJ. J. Nieto and H. Andrade, Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. Comput. Simulat., 81 (2011), 1855-1868.  doi: 10.1016/j.matcom.2010.10.032.

[10]

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.

[11]

A. GrayD. GreenhalghX. Mao. and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029.

[12]

Z. Han and J. Zhao, Stochastic SIRS model under regime switching, Nonlinear Anal.: Real World Appl., 14 (2013), 352-364.  doi: 10.1016/j.nonrwa.2012.06.008.

[13]

W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Mat., 115 (1927), 700-721. 

[14]

R. Z. KhasminskiiC. Zhu and G. Yin, Stability of regime-switching diffusions, Stoch. Process. Appl., 117 (2007), 1037-1051.  doi: 10.1016/j.spa.2006.12.001.

[15]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[16]

A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021.

[17]

A. LahrouzL. OmariD. Kiouach and A. Belmaati, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024.

[18]

A. LahrouzL. OmariD. Kiouach and A. Belmaati, Deterministic and stochastic stability of a mathematical model of smoking, Statist. Probab. Lett., 81 (2011), 1276-1284.  doi: 10.1016/j.spl.2011.03.029.

[19]

A. Lahrouz and A. Settati, Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput, 219 (2013), 11134-11148.  doi: 10.1016/j.amc.2013.05.019.

[20]

R. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.

[21]

M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.  doi: 10.1016/j.na.2013.02.018.

[22]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.  doi: 10.1016/j.amc.2010.08.037.

[23]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370.

[24]

Y. TakeuchiN. H. DuN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.

[25]

F. Xi, On the stability of jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018.

[26]

F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.

[27]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.

[28]

X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874.  doi: 10.1016/j.aml.2013.03.013.

[29]

X. Zhang and K. Wang, Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise, Commun. Nonlinear. Sci. Numer. Simulat, 19 (2014), 1391-1399.  doi: 10.1016/j.cnsns.2013.09.013.

[30]

J. Zhou and H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809-834.  doi: 10.1007/BF00168799.

Figure 1.  Computer simulation of a single path of $ I(t) $ for the SDE model (2.5) with initial condition $ I(0) = 0.04 $ and its corresponding Markov chain $ r(t) $ using the parameter values of Example 6.1
Figure 2.  Results of one simulation run of SDE (2.5) with initial condition $ I(0) = 0.04 $ and its corresponding Markov chain $ r(t) $ using the parameter values of Example 6.2. Here, $ I(t) $ rises to or above $ [ 0.7426, \, 0.7556 ] $ infinitely often with probability one
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