August 2018, 23(6): 2415-2431. doi: 10.3934/dcdsb.2018057

A stochastic SIRI epidemic model with Lévy noise

1. 

Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, BP 133, Kénitra, Morocco

2. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, 41012-Sevilla, Spain

3. 

Department of Mathematics, Linnaeus University, 351 95 Växjö, Sweden

Received  April 2017 Revised  July 2017 Published  February 2018

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

Citation: Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
References:
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E. Allen, Modeling with Itô Stochastic Differential Equations, Published by Springer, P. O. Box 17,3300 AA Dordrecht, The Netherlands, 2007.

[2]

R. M. Anderson and R. M. May, Infectious Diesases of Humans, Oxford University Press, 1992.

[3]

R. M. Anderson and R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0.

[4]

J. BenedettiL. Corey and R. Ashley, Recurrence rates in genital herpes after symptomatic first-episode infection, Ann. Int. Med., 121 (1994), 847-854. doi: 10.7326/0003-4819-121-11-199412010-00004.

[5]

N. D. Barlow, Non-linear transmission and simple modeld for bovine tuberculosis, J. Anim. Ecol., 69 (2000), 703-713.

[6]

S. Blower, Modelling the genital herpes epidemic, Herpes, 11 (2004), 138A-146A.

[7]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.

[8]

C. Castillo-Chavez, S. Blower, P. Van den Driessche, D. Kirsschen and A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseses: Models, Methods and Theory, Springer, New York, 2002.

[9]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in: C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner (Eds. ), Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part I: An Introduction to Models, Methods and Theory, Springer-Verlag, Berlin, (2002), 229-250.

[10]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.

[12]

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.

[13]

H. F. Huo and G. M. Qiu, Stability of a mathematical model of malaria transmission with relapse, Abstract and Applied Analysis, 2014 (2014), Art. ID 289349, 9pp.

[14]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model, 38 (2014), 5067-5079. doi: 10.1016/j.apm.2014.03.037.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part I, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.

[16]

A. Y. Kim, J. Schulze zur Wiesch, T. Kuntzen, J. Timm, D. E Kaufmann, J. E. Duncan, A. M. Jones, A. G. Wurcel, B. T. Davis, R. T. Gandhi, G. K. Robbins, T. M. Allen, R. T. Chung, G. M. Lauer and B. D. Walker, Impaired hepatitis C virus-specific T cell responses and recurrent hepatitis C virus in HIV coinfection, PLoS Me 3 (2006), e492. doi: 10.1371/journal.pmed.0030492.

[17]

D. W. Kimberlin and D. J. Rouse, Genital herpes, N. Engl. J. Med, 350 (2004), 1970-1977. doi: 10.1056/NEJMcp023065.

[18]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.

[19]

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

[20]

A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10-19. doi: 10.1016/j.amc.2014.01.158.

[21]

M. L. LambertE. HaskerA. Van DeunD. RoberfroidM. Boelaert and S. P. Van der, Recurrence in tuberculosis: Relapse or reinfection?, Lancet Infect. Dis., 3 (2003), 282-287. doi: 10.1016/S1473-3099(03)00607-8.

[22]

Q. Lei and Z. Yang, Dynamical behaviors of a stochastic SIRI epidemic model, Appl. Anal., 96 (2017), 2758-2770. doi: 10.1080/00036811.2016.1240365.

[23]

B. LiS. Yuan and W. G. Zhang, Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate, Int. J. Biomath., 4 (2011), 227-239. doi: 10.1142/S1793524511001374.

[24]

Y. LinD. Jiang and P. Xia, Long-time behavior of a stochastic SIR model, Appl. Math. Comput., 236 (2014), 1-9. doi: 10.1016/j.amc.2014.03.035.

[25]

L. LiuJ. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real World Appl., 24 (2015), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[26]

S. LiuS. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Analysis: Real World Applications, 12 (2011), 119-127. doi: 10.1016/j.nonrwa.2010.06.001.

[27]

F. A. MahamatY. A. HindJ. B. PhilippC. LisaL. PetraL. MirjamC. Nakul and Z. Jakob, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, (2017), 1-17.

[28]

A. MarzanoS. Gaia and V. Ghisetti, Viral load at the time of liver transplantation and risk of hepatitis B virus recurrence, Liver Transplant, 11 (2005), 402-409. doi: 10.1002/lt.20402.

[29]

H. N. Moreira and Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496-502. doi: 10.1137/S0036144595295879.

[30]

L. F. OlsenG. L. Truty and W. M. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark.Theoret. Population Biol., 33 (1988), 344-370. doi: 10.1016/0040-5809(88)90019-6.

[31]

E. TornatoreS. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A: Stat. Mech.Appl., 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[32]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139. doi: 10.1137/1032003.

[33]

P. Van den DriesscheL. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[34]

P. Van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosc., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[35]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[36]

C. Vargas-De-León, On the global stability of infectious disease models with relapse, Abstraction & Application, 9 (2013), 50-61.

[37]

P. Wildy, H. J. Field and A. A. Nash, Classical herpes latency revisited, in: B. W. J. Mahy, A. C. Minson, G. K. Darby (Eds. ), Virus Persistence Symposium, vol. 33, Cambridge University Press, Cambridge, (1982), 133{168.

[38]

F. WangX. Wang and S. Zhang, On pulse vaccine strategy in a periodic stochastic SIR epidemic model, Chaos Solitons Fractals, 66 (2014), 127-135. doi: 10.1016/j.chaos.2014.06.003.

[39]

L. Wang, Y. Li and L. Pang, Dynamics analysis of an epidemiological model with media impact and two delays, Math. Probl. Eng. , 2016 (2016), Art. ID 1598932, 9 pp.

[40]

R. Xu, Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Analysis: Modelling and Control, 18 (2013), 250-263.

[41]

R. Xu, Global dynamics of an SEIRI epidemiological model with time delay, Appl. Math. Comput., 232 (2014), 436-444. doi: 10.1016/j.amc.2014.01.100.

[42]

X. ZhangF. ChenK. Wang and H. Du, Stochastic SIRS model driven by Lévy noise, Acta Mathematica Scientia, 36 (2016), 740-752. doi: 10.1016/S0252-9602(16)30036-4.

[43]

X. Zhang and K. Wang, Stochastic model for spread of AIDS driven by Lévy noise, J. Dyn. Diff. Equat., 27 (2015), 215-236. doi: 10.1007/s10884-015-9459-5.

[44]

X. Zhang and K. Wang, Stochastic SEIR model with jumps, Appl. Math. Comput., 239 (2014), 133-143. doi: 10.1016/j.amc.2014.04.061.

[45]

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.

[46]

Y. ZhaoD. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A: Stat. Mech. Appl., 392 (2013), 4916-4927. doi: 10.1016/j.physa.2013.06.009.

[47]

Y. Zhao and D. Jiang, Dynamics of stochastically perturbed SIS epidemic model with vaccination. Abstract Appl. Anal. 2013 (2013), Art. ID 517439, 12 pp.

[48]

Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718-727. doi: 10.1016/j.amc.2014.05.124.

[49]

Y. ZhouW. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 244 (2014), 118-131. doi: 10.1016/j.amc.2014.06.100.

show all references

References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Published by Springer, P. O. Box 17,3300 AA Dordrecht, The Netherlands, 2007.

[2]

R. M. Anderson and R. M. May, Infectious Diesases of Humans, Oxford University Press, 1992.

[3]

R. M. Anderson and R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0.

[4]

J. BenedettiL. Corey and R. Ashley, Recurrence rates in genital herpes after symptomatic first-episode infection, Ann. Int. Med., 121 (1994), 847-854. doi: 10.7326/0003-4819-121-11-199412010-00004.

[5]

N. D. Barlow, Non-linear transmission and simple modeld for bovine tuberculosis, J. Anim. Ecol., 69 (2000), 703-713.

[6]

S. Blower, Modelling the genital herpes epidemic, Herpes, 11 (2004), 138A-146A.

[7]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.

[8]

C. Castillo-Chavez, S. Blower, P. Van den Driessche, D. Kirsschen and A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseses: Models, Methods and Theory, Springer, New York, 2002.

[9]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in: C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner (Eds. ), Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part I: An Introduction to Models, Methods and Theory, Springer-Verlag, Berlin, (2002), 229-250.

[10]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.

[12]

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.

[13]

H. F. Huo and G. M. Qiu, Stability of a mathematical model of malaria transmission with relapse, Abstract and Applied Analysis, 2014 (2014), Art. ID 289349, 9pp.

[14]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model, 38 (2014), 5067-5079. doi: 10.1016/j.apm.2014.03.037.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part I, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.

[16]

A. Y. Kim, J. Schulze zur Wiesch, T. Kuntzen, J. Timm, D. E Kaufmann, J. E. Duncan, A. M. Jones, A. G. Wurcel, B. T. Davis, R. T. Gandhi, G. K. Robbins, T. M. Allen, R. T. Chung, G. M. Lauer and B. D. Walker, Impaired hepatitis C virus-specific T cell responses and recurrent hepatitis C virus in HIV coinfection, PLoS Me 3 (2006), e492. doi: 10.1371/journal.pmed.0030492.

[17]

D. W. Kimberlin and D. J. Rouse, Genital herpes, N. Engl. J. Med, 350 (2004), 1970-1977. doi: 10.1056/NEJMcp023065.

[18]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.

[19]

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

[20]

A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10-19. doi: 10.1016/j.amc.2014.01.158.

[21]

M. L. LambertE. HaskerA. Van DeunD. RoberfroidM. Boelaert and S. P. Van der, Recurrence in tuberculosis: Relapse or reinfection?, Lancet Infect. Dis., 3 (2003), 282-287. doi: 10.1016/S1473-3099(03)00607-8.

[22]

Q. Lei and Z. Yang, Dynamical behaviors of a stochastic SIRI epidemic model, Appl. Anal., 96 (2017), 2758-2770. doi: 10.1080/00036811.2016.1240365.

[23]

B. LiS. Yuan and W. G. Zhang, Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate, Int. J. Biomath., 4 (2011), 227-239. doi: 10.1142/S1793524511001374.

[24]

Y. LinD. Jiang and P. Xia, Long-time behavior of a stochastic SIR model, Appl. Math. Comput., 236 (2014), 1-9. doi: 10.1016/j.amc.2014.03.035.

[25]

L. LiuJ. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real World Appl., 24 (2015), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[26]

S. LiuS. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Analysis: Real World Applications, 12 (2011), 119-127. doi: 10.1016/j.nonrwa.2010.06.001.

[27]

F. A. MahamatY. A. HindJ. B. PhilippC. LisaL. PetraL. MirjamC. Nakul and Z. Jakob, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, (2017), 1-17.

[28]

A. MarzanoS. Gaia and V. Ghisetti, Viral load at the time of liver transplantation and risk of hepatitis B virus recurrence, Liver Transplant, 11 (2005), 402-409. doi: 10.1002/lt.20402.

[29]

H. N. Moreira and Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496-502. doi: 10.1137/S0036144595295879.

[30]

L. F. OlsenG. L. Truty and W. M. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark.Theoret. Population Biol., 33 (1988), 344-370. doi: 10.1016/0040-5809(88)90019-6.

[31]

E. TornatoreS. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A: Stat. Mech.Appl., 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[32]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139. doi: 10.1137/1032003.

[33]

P. Van den DriesscheL. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[34]

P. Van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosc., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[35]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[36]

C. Vargas-De-León, On the global stability of infectious disease models with relapse, Abstraction & Application, 9 (2013), 50-61.

[37]

P. Wildy, H. J. Field and A. A. Nash, Classical herpes latency revisited, in: B. W. J. Mahy, A. C. Minson, G. K. Darby (Eds. ), Virus Persistence Symposium, vol. 33, Cambridge University Press, Cambridge, (1982), 133{168.

[38]

F. WangX. Wang and S. Zhang, On pulse vaccine strategy in a periodic stochastic SIR epidemic model, Chaos Solitons Fractals, 66 (2014), 127-135. doi: 10.1016/j.chaos.2014.06.003.

[39]

L. Wang, Y. Li and L. Pang, Dynamics analysis of an epidemiological model with media impact and two delays, Math. Probl. Eng. , 2016 (2016), Art. ID 1598932, 9 pp.

[40]

R. Xu, Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Analysis: Modelling and Control, 18 (2013), 250-263.

[41]

R. Xu, Global dynamics of an SEIRI epidemiological model with time delay, Appl. Math. Comput., 232 (2014), 436-444. doi: 10.1016/j.amc.2014.01.100.

[42]

X. ZhangF. ChenK. Wang and H. Du, Stochastic SIRS model driven by Lévy noise, Acta Mathematica Scientia, 36 (2016), 740-752. doi: 10.1016/S0252-9602(16)30036-4.

[43]

X. Zhang and K. Wang, Stochastic model for spread of AIDS driven by Lévy noise, J. Dyn. Diff. Equat., 27 (2015), 215-236. doi: 10.1007/s10884-015-9459-5.

[44]

X. Zhang and K. Wang, Stochastic SEIR model with jumps, Appl. Math. Comput., 239 (2014), 133-143. doi: 10.1016/j.amc.2014.04.061.

[45]

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.

[46]

Y. ZhaoD. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A: Stat. Mech. Appl., 392 (2013), 4916-4927. doi: 10.1016/j.physa.2013.06.009.

[47]

Y. Zhao and D. Jiang, Dynamics of stochastically perturbed SIS epidemic model with vaccination. Abstract Appl. Anal. 2013 (2013), Art. ID 517439, 12 pp.

[48]

Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718-727. doi: 10.1016/j.amc.2014.05.124.

[49]

Y. ZhouW. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 244 (2014), 118-131. doi: 10.1016/j.amc.2014.06.100.

Figure 1.  rajectories of the solutions to the systems (1.1) and (1.3) for Moroccan zoonotic tuberculosis with $\mathcal{R}_0\leq 1$.
Figure 2.  Trajectories of the solutions to the systems (1.1) and (1.3) for bovine tuberculosis [5] with $\mathcal{R}_0>1$, $N = \mu = 0.1,\;\beta = 0.6\;and\;\gamma = \delta = 0.5$.
Figure 3.  Trajectories of the solutions to the systems (1.1) and (1.3) with various relapse rate $\delta$: $0.14, 0.2, 0.4$.
Figure 4.  Trajectories of the solutions to the systems (1.1) and (1.3) with a various recovery rate $\gamma$: $0.09, 0.18, 0.22$
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