A stochastic SIRI epidemic model with relapse and media coverage

Tomás Caraballo; Mohamed El Fatini; Roger Pettersson; Regragui Taki

doi:10.3934/dcdsb.2018250

Article Contents

2018, Volume 23, Issue 8: 3483-3501. Doi: 10.3934/dcdsb.2018250

This issue Previous Article Synchronization in networks with strongly delayed couplings Next Article Horizontal patterns from finite speed directional quenching

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

Early access: August 2018

Published: September 2018

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: 92D25, 34C60, 92B05, 60J60, 60J65.

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Figure 1. Trajectories of stochastic and deterministic systems with the parameters values given in Example 6.1.

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Figure 2. Trajectories of stochastic and deterministic systems with the parameters values given in previous Example $2$

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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

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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

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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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References

	M. F. Abakar , H. Y. Azami , P. J. Bless , L. Crump , P. Lohmann , M. Laager , N. 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.
	B. Berrhazi , M. El Fatini , A. Laaribi , R. 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.
	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.
	S. Blower , Modelling the genital herpes epidemic, Herpes, 11 (2004) , Suppl 3,138A-146A.
	M. P. Brinn , K. V. Carson , A. J. Esterman , A. 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.
	B. Buonomo , A. 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.
	Y. Cai , Y. Kang , M. 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.
	J. Cui , Y. 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.
	M. El Fatini , A. Lahrouz , R. Pettersson , A. 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.
	T. C. Gard, Introduction To Stochastic Differential Equations, Marcel Dekker INC, New York, 1988.
	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.
	A. Gray , D. Greenhalgh , L. Hu , X. Mao and J. Pan , A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011) , 876-902. doi: 10.1137/10081856X.
	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.
	R. Khasminskii, Stochastic Stability of Differential Equations, Springer 2012. doi: 10.1007/978-3-642-23280-0.
	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.
	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.
	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.
	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.
	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.
	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.
	Y. Lin , D. 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.
	Y. Lin , D. 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.
	S. Liu , S. 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.
	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.
	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.
	X. Mao , G. 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.
	X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood, 2008. doi: 10.1533/9780857099402.
	H. N. Moreira and Y. Wang , Global stability in an SIRI model, SIAM Rev., 39 (1997) , 496-502. doi: 10.1137/S0036144595295879.
	G. Strang, Linear Algebra and its Applications (Fourth Edition), Thomson Learning, Inc, 2006.
	C. Sun , W. Yang , J. 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.
	J. Tchuenche , N. Dube , C. Bhunu , R. Smith and C. Bauch , The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health., 11 (2011) , 1-14.
	D. Tudor , A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990) , 136-139. doi: 10.1137/1032003.
	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.
	P. Van den Driessche , L. Wang and X. Zou , Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007) , 205-219. doi: 10.3934/mbe.2007.4.205.
	C. Vargas-De-León , On the global stability of infectious disease models with relapse, Abstraction and Application, 9 (2013) , 50-61.
	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.
	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.
	R. Xu , Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Analysis: Modelling and Control., 18 (2013) , 250-263.
	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.
	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.
	C. Zhu and G. Yin , Asymptotic properties of hybrid diffusion systems, SIAM J. Control. Optim., 46 (2007) , 1155-1179. doi: 10.1137/060649343.