October  2018, 23(8): 3483-3501. doi: 10.3934/dcdsb.2018250

A stochastic SIRI epidemic model with relapse and media coverage

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

2. 

Ibn Tofail University, FS, Department of Mathematics, BP 133, Kénitra, Morocco

3. 

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

Received  January 2018 Published  August 2018

Fund Project: This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2015-63723-P, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492 and the Faculty of Sciences, Ibn Tofail University

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.

Citation: Tomás Caraballo, Mohamed El Fatini, Roger Pettersson, Regragui Taki. A stochastic SIRI epidemic model with relapse and media coverage. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3483-3501. doi: 10.3934/dcdsb.2018250
References:
[1]

M. F. AbakarH. Y. AzamiP. J. BlessL. CrumpP. LohmannM. LaagerN. Chitnis and J. Zinstag, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, 11 (2017), 1-17. doi: 10.1371/journal.pntd.0005214.

[2]

B. BerrhaziM. El FatiniA. LaaribiR. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68. doi: 10.1016/j.chaos.2017.10.007.

[3]

B. Berrhazi, M. El Fatini, T. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete and Continuous Dynamical Systems, Series B.

[4]

S. Blower, Modelling the genital herpes epidemic, Herpes, 11 (2004), Suppl 3,138A-146A.

[5]

M. P. BrinnK. V. CarsonA. J. EstermanA. B. Chang and B. J. Smith, Mass media interventions for preventing smoking in young people, Cochrane Data base Syst., 11 (2010), 1-47. doi: 10.1002/14651858.CD001006.pub2.

[6]

B. BuonomoA. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16. doi: 10.1016/j.mbs.2008.07.011.

[7]

Y. CaiY. KangM. Banerjee and W. Wang, A stochastic epidemic model incorporating media coverage, Communications in Mathematical Sciences, 14 (2016), 839-910. doi: 10.4310/CMS.2016.v14.n4.a1.

[8]

J. CuiY. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dynam. Differential Equations, 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0.

[9]

M. El FatiniA. LahrouzR. PetterssonA. Settati and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 316 (2018), 326-341. doi: 10.1016/j.amc.2017.08.037.

[10]

T. C. Gard, Introduction To Stochastic Differential Equations, Marcel Dekker INC, New York, 1988.

[11]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507. doi: 10.1016/j.amc.2013.02.044.

[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]

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.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer 2012. doi: 10.1007/978-3-642-23280-0.

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, 1994. doi: 10.1007/978-3-642-57913-4.

[16]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[17]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[18]

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

[19]

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.

[20]

Y. Li and J. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2353-2365. doi: 10.1016/j.cnsns.2008.06.024.

[21]

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.

[22]

Y. LinD. Jiang and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A, 394 (2014), 187-197. doi: 10.1016/j.physa.2013.10.006.

[23]

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.

[24]

Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74. doi: 10.1142/S1793524508000023.

[25]

W. Liu and Q. Zheng, A stochastic SIS epidemic model incorporating media coverage in a two patch setting, Appl. Math. Comput., 262 (2015), 160-168. doi: 10.1016/j.amc.2015.04.025.

[26]

X. MaoG. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[27]

X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood, 2008. doi: 10.1533/9780857099402.

[28]

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

[29]

G. Strang, Linear Algebra and its Applications (Fourth Edition), Thomson Learning, Inc, 2006.

[30]

C. SunW. YangJ. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95. doi: 10.1016/j.mbs.2011.01.005.

[31]

J. TchuencheN. DubeC. BhunuR. Smith and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health., 11 (2011), 1-14.

[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 and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[34]

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.

[35]

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

[36]

L. Wang and D. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, App. Math. Lett., 73 (2017), 22-28. doi: 10.1016/j.aml.2017.04.029.

[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, Cambridge University Press, Cambridge, 33 (1982), 133-168.

[38]

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

[39]

W. Zhang and X. Meng, Stochastic analysis of a novel nonautonomous periodic SIRI epidemic system with random disturbances, Physica A, 492 (2018), 1290-1301. doi: 10.1016/j.physa.2017.11.057.

[40]

M. Zhao and H. Zhao, Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage, Advances in Difference Equations, 149 (2016), 1-17. doi: 10.1186/s13662-016-0884-5.

[41]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control. Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

show all references

References:
[1]

M. F. AbakarH. Y. AzamiP. J. BlessL. CrumpP. LohmannM. LaagerN. Chitnis and J. Zinstag, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Tropical Diseases, 11 (2017), 1-17. doi: 10.1371/journal.pntd.0005214.

[2]

B. BerrhaziM. El FatiniA. LaaribiR. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68. doi: 10.1016/j.chaos.2017.10.007.

[3]

B. Berrhazi, M. El Fatini, T. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete and Continuous Dynamical Systems, Series B.

[4]

S. Blower, Modelling the genital herpes epidemic, Herpes, 11 (2004), Suppl 3,138A-146A.

[5]

M. P. BrinnK. V. CarsonA. J. EstermanA. B. Chang and B. J. Smith, Mass media interventions for preventing smoking in young people, Cochrane Data base Syst., 11 (2010), 1-47. doi: 10.1002/14651858.CD001006.pub2.

[6]

B. BuonomoA. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16. doi: 10.1016/j.mbs.2008.07.011.

[7]

Y. CaiY. KangM. Banerjee and W. Wang, A stochastic epidemic model incorporating media coverage, Communications in Mathematical Sciences, 14 (2016), 839-910. doi: 10.4310/CMS.2016.v14.n4.a1.

[8]

J. CuiY. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dynam. Differential Equations, 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0.

[9]

M. El FatiniA. LahrouzR. PetterssonA. Settati and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 316 (2018), 326-341. doi: 10.1016/j.amc.2017.08.037.

[10]

T. C. Gard, Introduction To Stochastic Differential Equations, Marcel Dekker INC, New York, 1988.

[11]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507. doi: 10.1016/j.amc.2013.02.044.

[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]

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.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer 2012. doi: 10.1007/978-3-642-23280-0.

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, 1994. doi: 10.1007/978-3-642-57913-4.

[16]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[17]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[18]

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

[19]

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.

[20]

Y. Li and J. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2353-2365. doi: 10.1016/j.cnsns.2008.06.024.

[21]

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.

[22]

Y. LinD. Jiang and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A, 394 (2014), 187-197. doi: 10.1016/j.physa.2013.10.006.

[23]

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.

[24]

Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74. doi: 10.1142/S1793524508000023.

[25]

W. Liu and Q. Zheng, A stochastic SIS epidemic model incorporating media coverage in a two patch setting, Appl. Math. Comput., 262 (2015), 160-168. doi: 10.1016/j.amc.2015.04.025.

[26]

X. MaoG. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[27]

X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood, 2008. doi: 10.1533/9780857099402.

[28]

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

[29]

G. Strang, Linear Algebra and its Applications (Fourth Edition), Thomson Learning, Inc, 2006.

[30]

C. SunW. YangJ. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95. doi: 10.1016/j.mbs.2011.01.005.

[31]

J. TchuencheN. DubeC. BhunuR. Smith and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health., 11 (2011), 1-14.

[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 and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[34]

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.

[35]

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

[36]

L. Wang and D. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, App. Math. Lett., 73 (2017), 22-28. doi: 10.1016/j.aml.2017.04.029.

[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, Cambridge University Press, Cambridge, 33 (1982), 133-168.

[38]

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

[39]

W. Zhang and X. Meng, Stochastic analysis of a novel nonautonomous periodic SIRI epidemic system with random disturbances, Physica A, 492 (2018), 1290-1301. doi: 10.1016/j.physa.2017.11.057.

[40]

M. Zhao and H. Zhao, Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage, Advances in Difference Equations, 149 (2016), 1-17. doi: 10.1186/s13662-016-0884-5.

[41]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control. Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

Figure 1.  Trajectories of stochastic and deterministic systems with the parameters values given in Example 6.1.
Figure 2.  Trajectories of stochastic and deterministic systems with the parameters values given in previous Example $2$
Figure 3.  The kernel density function estimations of $S(t)$, $I(t)$ and $R(t)$ of stochastic system (1.2) at time $t$ = 9000, based on 10000 stochastic simulation with the parameters values given in Example 3
Figure 4.  The kernel density function estimations of $S(t)$, $I(t)$ and $R(t)$ of stochastic system (1.2) at time $t$ = 9500, based on 10000 stochastic simulation with the parameters values given in example 3
Figure 5.  Paths simulations of $I(t)$ for stochastic model with the parameters values as in Example 4 and $\beta_2 = 0.01, \; 0.1, \; 0.15$ respectively.
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