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.  Google Scholar

[2]

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

[3]

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

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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.  Google Scholar

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

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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