A stochastic SIRS epidemic model with infectious force under intervention strategies

Journal of Differential Equations

Volumn 259, Issue 12, 2015, Pages 7463-7502

  Cai, Yongli ^a   Kang, Yun ^b   Banerjee, Malay ^c   Wang, Weiming ^a  

^a HUAIYIN NORMAL UNIVERSITY   (China)

^b ARIZONA STATE UNIVERSITY   (United States)

^c INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

Author keywords

Epidemic model; Markov semigroups; Reproduction number; Stationary distribution

Indexed keywords

EID: 84943366139     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2015.08.024     Document Type: Article

References (55)

1. Kermack W.O., McKendrick A.G. Contributions to the mathematical theory of epidemics-I. Proc. R. Soc. Lond. Ser. A 1927, 115(772):701-721.
2. Wang W.D. Epidemic models with nonlinear infection forces. Math. Biosci. Eng. 2006, 3(1):267-279.
3. Xiao D., Ruan S. Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 2007, 208(2):419-429.
4. Cui J., Tao X., Zhu H. An SIS infection model incorporating media coverage. Rocky Mountain J. Math. 2008, 38(5):1323-1334.
5. Cui J., Sun Y., Zhu H. The impact of media on the control of infectious diseases. J. Dynam. Differential Equations 2008, 20(1):31-53.
6. Bauch C.T. Dynamics of an infectious disease where media coverage influences transmission. ISRN Biomath. 2012, 2012.
7. Xiao Y., Zhao T., Tang S. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Math. Biosci. Eng. 2013, 10(2):445-461.
8. Gumel A.B., Ruan S., Day T., et al. Modelling strategies for controlling SARS outbreaks. Proc. R. Soc. Lond. Ser. B 2004, 271(1554):2223-2232.
9. Zhang J., Lou J., Ma Z., Wu J. A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China. Appl. Math. Comput. 2005, 162(2):909-924.
10. Tang S., Xiao Y., Yuan L., Cheke R.A., Wu J. Campus quarantine (Fengxiao) for curbing emergent infectious diseases: lessons from mitigating a/H1N1 in Xi'an, China. J. Theoret. Biol. 2012, 295:47-58.
11. Xiao Y., Tang S., Wu J. Media impact switching surface during an infectious disease outbreak. Sci. Rep. 2015, 5:7838.
12. Tchuenche J.M., Dube N., Bhunu C.P., Bauch C.T. The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health 2011, 11(Suppl. 1).
13. Misra A.K., Sharma A., Shukla J.B. Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comput. Modelling 2011, 53(5):1221-1228.
14. Brinn M.P., Carson K.V., Esterman A.J., Chang A.B., Smith B.J. Mass media interventions for preventing smoking in young people. Cochrane Database Syst. Rev. 2010, 11.
15. Funk S., Salathé M., Jansen V.A.A. Modelling the influence of human behaviour on the spread of infectious diseases: a review. J. R. Soc. Interface 2010, 7(50):1247-1256.
16. Khanam P.A., Khuda B.e, Khane T.T., Ashraf A. Awareness of sexually transmitted disease among women and service providers in rural Bangladesh. Int. J. STD & AIDS 1997, 8(11):688-696.
17. Spencer S. Stochastic epidemic models for emerging diseases 2008, PhD thesis, University of Nottingham.
18. Beddington J.R., May R.M. Harvesting natural populations in a randomly fluctuating environment. Science 1977, 197(4302):463-465.
19. Truscott J.E., Gilligan C.A. Response of a deterministic epidemiological system to a stochastically varying environment. Proc. Natl. Acad. Sci. USA 2003, 100(15):9067-9072.
20. Mollison D. Dependence of epidemic and population velocities on basic parameters. Math. Biosci. 1991, 107(2):255-287.
21. Mao X. Stochastic Differential Equations and Their Applications 1997, Horwood, Chichester.
22. Durrett R. Stochastic spatial models. SIAM Rev. 1999, 41(4):677-718.
23. Dennis B. Allee effects in stochastic populations. Oikos 2002, 96(3):389-401.
24. Britton T. Stochastic epidemic models: a survey. Math. Biosci. 2010, 225(1):24-35.
25. Gray A., Greenhalgh D., Hu L., Mao X., Pan J. A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 2011, 71(3):876-902.
26. Mao X., Sabanis S., Renshaw E. Asymptotic behaviour of the stochastic Lotka-Volterra model. J. Math. Anal. Appl. 2003, 287(1):141-156.
27. Ji C., Jiang D., Shi N. Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 2009, 359(2):482-498.
28. Zhu C., Yin G. On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 2009, 357(1):154-170.
29. Mandal P.S., Banerjee M. Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model. Phys. A 2012, 391(4):1216-1233.
30. Imhof L., Walcher S. Exclusion and persistence in deterministic and stochastic chemostat models. J. Differential Equations 2005, 217(1):26-53.
31. Mao X., Marion G., Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Process. Appl. 2002, 97(1):95-110.
32. Allen L.J.S., Allen E.J. A comparison of three different stochastic population models with regard to persistence time. Theor. Popul. Biol. 2003, 64(4):439-449.
33. Allen L.J.S., Kirupaharan N. Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens. Int. J. Numer. Anal. Model. 2005, 2(3):329-344.
34. Allen L.J.S., Driessche P. Stochastic epidemic models with a backward bifurcation. Math. Biosci. Eng. 2006, 3(3):445-458.
35. Allen L.J.S. An introduction to stochastic epidemic models. Math. Epidem 2008, 81-130. Springer.
36. Greenhalgh D., Khan Q.J.A., Lewis F.I. Hopf bifurcation in two SIRS density dependent epidemic models. Math. Comput. Modelling 2004, 39(11):1261-1283.
37. Tang Y., Huang D., Ruan S., Zhang W. Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate. SIAM J. Appl. Math. 2008, 69(2):621-639.
38. Ruan S., Wang W. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differential Equations 2003, 188(1):135-163.
39. Komorowski T., Tyrcha J. Asymptotic properties of some Markov operators. Bull. Pol. Acad. Sci. Math. 1989, 37(1-6):221-228.
40. Lasota A., Mackey M.C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics 1994, vol. 97. Springer.
41. Rudnicki R. On asymptotic stability and sweeping for Markov operators. Bull. Pol. Acad. Sci. Math. 1995, 43(3):245-262.
42. Kadanoff L.P. Statistical Physics: Statics, Dynamics, and Renormalization 2000, World Scientific.
43. Rudnicki R. Markov operators: applications to diffusion processes and population dynamics. Appl. Math. 2000, 27(1):67-79.
44. Rudnicki R., Pichor K., Tyran-Kaminska M. Markov Semigroups and Their Applications. Lecture Notes in Phys. 2002, vol. 597:215-238.
45. Rudnicki R. Long-time behaviour of a stochastic prey-predator model. Stochastic Process. Appl. 2003, 108(1):93-107.
46. LaSalle J.P. The Stability of Dynamical Systems 1987, vol. 25. Society for Industrial and Applied Mathematics.
47. Lyapunov A.M. The general problem of the stability of motion. Internat. J. Control 1992, 55(3):531-534.
48. Bell D.R. The Malliavin Calculus 2006, Dover Publications.
49. Stroock D.W., Varadhan S.R.S. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 1972, vol. 3:333-359.
50. Arous G.B., Léandre R. Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Related Fields 1991, 90(3):377-402.
51. Pichór K., Rudnicki R. Stability of Markov semigroups and applications to parabolic systems. J. Math. Anal. Appl. 1997, 215(1):56-74.
52. Higham D.J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 2001, 43(3):525-546.
53. Murray J.D., Stanley E.A., Brown D.L. On the spatial spread of rabies among foxes. Proc. R. Soc. Lond. Ser. B 1986, 229(1255):111-150.
54. Scott D.W. On optimal and data-based histograms. Biometrika 1979, 66:605-610.
55. Silverman B.W. Density Estimation for Statistics and Data Analysis 1986, Chapman and Hall, London.