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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  January 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, doi: 10.3934/dcdsb.2018057
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, R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0.

[4]

J. Benedetti, L. Corey, 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, 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, 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. Diekmann, J. A. P. Heesterbeek, 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. Gray, D. Greenhalgh, L. Hu, X. Mao, 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, 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, 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, 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, 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, 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. Lambert, E. Hasker, A. Van Deun, D. Roberfroid, M. Boelaert, 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, Z. Yang, Dynamical behaviors of a stochastic SIRI epidemic model, Appl. Anal., 96 (2017), 2758-2770. doi: 10.1080/00036811.2016.1240365.

[23]

B. Li, S. Yuan, 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. Lin, D. Jiang, 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. Liu, J. Wang, 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. Liu, S. Wang, 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. Mahamat, Y. A. Hind, J. B. Philipp, C. Lisa, L. Petra, L. Mirjam, C. Nakul, Z. Jakob, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, (2017), 1-17.

[28]

A. Marzano, S. Gaia, 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, Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496-502. doi: 10.1137/S0036144595295879.

[30]

L. F. Olsen, G. L. Truty, 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. Tornatore, S. M. Buccellato, 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 Driessche, L. Wang, 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, 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, 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. Wang, X. Wang, 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. Zhang, F. Chen, K. Wang, 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, 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, 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, K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. doi: 10.1016/j.aml.2013.03.013.

[46]

Y. Zhao, D. Jiang, 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, 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. Zhou, W. Zhang, 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, R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0.

[4]

J. Benedetti, L. Corey, 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, 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, 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. Diekmann, J. A. P. Heesterbeek, 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. Gray, D. Greenhalgh, L. Hu, X. Mao, 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, 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, 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, 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, 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, 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. Lambert, E. Hasker, A. Van Deun, D. Roberfroid, M. Boelaert, 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, Z. Yang, Dynamical behaviors of a stochastic SIRI epidemic model, Appl. Anal., 96 (2017), 2758-2770. doi: 10.1080/00036811.2016.1240365.

[23]

B. Li, S. Yuan, 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. Lin, D. Jiang, 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. Liu, J. Wang, 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. Liu, S. Wang, 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. Mahamat, Y. A. Hind, J. B. Philipp, C. Lisa, L. Petra, L. Mirjam, C. Nakul, Z. Jakob, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, (2017), 1-17.

[28]

A. Marzano, S. Gaia, 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, Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496-502. doi: 10.1137/S0036144595295879.

[30]

L. F. Olsen, G. L. Truty, 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. Tornatore, S. M. Buccellato, 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 Driessche, L. Wang, 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, 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, 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. Wang, X. Wang, 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. Zhang, F. Chen, K. Wang, 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, 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, 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, K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. doi: 10.1016/j.aml.2013.03.013.

[46]

Y. Zhao, D. Jiang, 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, 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. Zhou, W. Zhang, 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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