Stochastic dynamics of an SIRS epidemic model with ratio-dependent incidence rate

Abstract and Applied Analysis

Volumn 2013, Issue , 2013, Pages

  Cai, Yongli ^a,b   Wang, Xixi ^a   Wang, Weiming ^a   Zhao, Min ^a  

^a WENZHOU UNIVERSITY   (China)

^b SUN YAT SEN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84879246214     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2013/172631     Document Type: Article

References (50)

1. Kermack W. O., McKendrick A. G., Contributions to the mathematical theory of epidemics-I. Bulletin of Mathematical Biology 1991 53 1-2 33 55 2-s2.0-0025985176 10. 1016/S0092-8240(05)80040-0
2. Ma Z., Zhou Y., Wu J., Modeling and Dynamics of Infectious Diseases 2009 Beijing, China Higher Education Press
3. Bain C., Applied mathematical ecology. Journal of Epidemiology and Community Health 1990 44 3 254
4. Korobeinikov A., Maini P. K., Non-linear incidence and stability of infectious disease models. Mathematical Medicine and Biology 2005 22 2 113 128 2-s2.0-25144476559 10.1093/imammb/dqi001
5. Ma J., Ma Z., Epidemic threshold conditions for seasonally forced SEIR models. Mathematical Biosciences and Engineering 2006 3 1 161 172 10.3934/mbe.2006.3.161 MR2192132 ZBL1089.92048
6. Capasso V., Serio G., A generalization of the Kermack-McKendrick deterministic epidemic model. Mathematical Biosciences 1978 42 1-2 43 61 10.1016/0025-5564(78)90006-8 MR529097 ZBL0398.92026
7. Capasso V., Mathematical Structures of Epidemic Systems 2008 97 Berlin, Germany Springer xvi+283 MR2722340
8. Liu W. M., Levin S. A., Iwasa Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. Journal of Mathematical Biology 1986 23 2 187 204 10.1007/BF00276956 MR829132 ZBL0582.92023
9. Liu W. M., Hethcote H. W., Levin S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology 1987 25 4 359 380 10.1007/BF00277162 MR908379 ZBL0621.92014
10. Jacquez J. A., O'Neill P., Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations. Mathematical Biosciences 1991 107 2 161 186 2-s2.0-0026360185 10.1016/0025-5564(91)90003-2
11. Derrick W. R., van den Driessche P., A disease transmission model in a nonconstant population. Journal of Mathematical Biology 1993 31 5 495 512 10.1007/BF00173889 MR1229446 ZBL0772.92015
12. Hethcote H. W., The mathematics of infectious diseases. SIAM Review 2000 42 4 599 653 10.1137/S0036144500371907 MR1814049 ZBL0993.92033
13. Hethcote H. W., Levin S. A., Periodicity in epidemiological models. Applied Mathematical Ecology 1989 18 193 211
14. Alexander M. E., Moghadas S. M., Periodicity in an epidemic model with a generalized non-linear incidence. Mathematical Biosciences 2004 189 1 75 96 10.1016/j.mbs.2004.01.003 MR2054956 ZBL1073.92040
15. Ruan S., Wang W., Dynamical behavior of an epidemic model with a nonlinear incidence rate. Journal of Differential Equations 2003 188 1 135 163 10.1016/S0022-0396(02)00089-X MR1954511 ZBL1028.34046
16. Xiao D., Ruan S., Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences 2007 208 2 419 429 10.1016/j.mbs.2006. 09.025 MR2348123 ZBL1119.92042
17. Li X.-Z., Li W.-S., Ghosh M., Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment. Applied Mathematics and Computation 2009 210 1 141 150 10.1016/j.amc.2008.12.085 MR2504129 ZBL1159.92036
18. Yuan S., Li B., Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate. Discrete Dynamics in Nature and Society 2009 2009 13 609306 10.1155/2009/609306 MR2563597 ZBL1177.37089
19. Li B., Yuan S., Zhang W., Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate. International Journal of Biomathematics 2011 4 2 227 239 10.1142/S1793524511001374 MR2818184
20. Beddington J. R., May R. M., Harvesting natural populations in a randomly fluctuating environment. Science 1977 197 4302 463 465 2-s2.0-0000735971
21. Allen L. J. S., Brauer F., van den Driessche P., Wu J., An introduction to stochastic epidemic models. Mathematical Epidemiology 2008 1945 Berlin, Germany Springer 81 130 10.1007/978-3-540-78911-6-3 MR2428373 ZBL1206.92022
22. Gard T. C., Introduction to Stochastic Differential Equations 1988 114 New York, NY, USA Marcel Dekker xiv+234 MR917064
23. Øksendal B., Stochastic Differential Equations 1995 4th Berlin, Germany Springer xvi+271 MR1411679
24. Mao X., Stochastic Differential Equations and Their Applications 1997 Chichester, UK Horwood Publishing Limited, xii+366 MR1475218
25. Khasminskii R. Z., Klebaner F. C., Long term behavior of solutions of the Lotka-Volterra system under small random perturbations. The Annals of Applied Probability 2001 11 3 952 963 10.1214/aoap/1015345354 MR1865029 ZBL1061.34513
26. Mao X., Marion G., Renshaw E., Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes and their Applications 2002 97 1 95 110 10.1016/S0304-4149(01)00126-0 MR1870962 ZBL1058.60046
27. Bandyopadhyay M., Chattopadhyay J., Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity 2005 18 2 913 936 10.1088/0951-7715/18/2/022 MR2122690 ZBL1078.34035
28. Jiang D., Shi N., A note on nonautonomous logistic equation with random perturbation. Journal of Mathematical Analysis and Applications 2005 303 1 164 172 10.1016/j.jmaa.2004.08.027 MR2113874 ZBL1076.34062
29. Luo Q., Mao X., Stochastic population dynamics under regime switching. Journal of Mathematical Analysis and Applications 2007 334 1 69 84 10.1016/j.jmaa.2006.12.032 MR2332539 ZBL1113.92052
30. Chatterjee S., Isaia M., Bona F., Badino G., Venturino E., Modelling environmental influences on wanderer spiders in the Langhe region (Piemonte-NW Italy). Journal of Numerical Analysis, Industrial and Applied Mathematics 2008 3 3-4 193 209 MR2478623 ZBL1166.92037
31. Li X., Mao X., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems Series A 2009 24 2 523 545 10.3934/dcds.2009.24.523 MR2486589 ZBL1161.92048
32. Liu M., Wang K., Survival analysis of stochastic single-species population models in polluted environments. Ecological Modelling 2009 220 9-10 1347 1357 2-s2.0-63549091562 10.1016/j.ecolmodel.2009.03.001
33. Lv J., Wang K., Asymptotic properties of a stochastic predator-prey system with Holling II functional response. Communications in Nonlinear Science and Numerical Simulation 2011 16 10 4037 4048 10.1016/j.cnsns.2011.01.015 MR2802710 ZBL1218.92072
34. Wang X., Huang H., Cai Y., Wang W., The complex dynamics of a stochastic predator-prey model. Abstract and Applied Analysis 2012 2012 24 10.1155/2012/401031 401031 MR2970012 ZBL1253.93142
35. Beretta E., Kolmanovskii V., Shaikhet L., Stability of epidemic model with time delays influenced by stochastic perturbations. Mathematics and Computers in Simulation 1998 45 3-4 269 277 10.1016/S0378-4754(97)00106-7 MR1622405 ZBL1017.92504
36. Carletti M., On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. Mathematical Biosciences 2002 175 2 117 131 10.1016/S0025-5564(01)00089-X MR1882285 ZBL0987.92027
37. Nåsell I., Stochastic models of some endemic infections. Mathematical Biosciences 2002 179 1 1 19 10.1016/S0025-5564(02)00098-6 MR1908734 ZBL0991.92026
38. Tornatore E., Buccellato S. M., Vetro P., Stability of a stochastic SIR system. Physica A 2005 354 1-4 111 126 2-s2.0-19944381861 10.1016/j.physa.2005. 02.057
39. Dalal N., Greenhalgh D., Mao X., A stochastic model of AIDS and condom use. Journal of Mathematical Analysis and Applications 2007 325 1 36 53 10.1016/j.jmaa.2006.01.055 MR2273027 ZBL1101.92037
40. Dalal N., Greenhalgh D., Mao X., A stochastic model for internal HIV dynamics. Journal of Mathematical Analysis and Applications 2008 341 2 1084 1101 10.1016/j.jmaa.2007.11.005 MR2398271 ZBL1132.92015
41. Yu J., Jiang D., Shi N., Global stability of two-group SIR model with random perturbation. Journal of Mathematical Analysis and Applications 2009 360 1 235 244 10.1016/j.jmaa.2009.06.050 MR2548379 ZBL1184.34064
42. Britton T., Stochastic epidemic models: a survey. Mathematical Biosciences 2010 225 1 24 35 10.1016/j.mbs.2010.01.006 MR2642269 ZBL1188.92031
43. Ball F., Sirl D., Trapman P., Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Mathematical Biosciences 2010 224 2 53 73 10.1016/j.mbs.2009.12.003 MR2655798 ZBL1192.92037
44. Jiang D., Ji C., Shi N., Yu J., The long time behavior of DI SIR epidemic model with stochastic perturbation. Journal of Mathematical Analysis and Applications 2010 372 1 162 180 10.1016/j.jmaa.2010.06.003 MR2672517 ZBL1194.92053
45. Jiang D., Yu J., Ji C., Shi N., Asymptotic behavior of global positive solution to a stochastic SIR model. Mathematical and Computer Modelling 2011 54 1-2 221 232 10.1016/j.mcm.2011.02.004 MR2801881 ZBL1225.60114
46. Gray A., Greenhalgh D., Hu L., Mao X., Pan J., A stochastic differential equation SIS epidemic model. SIAM Journal on Applied Mathematics 2011 71 3 876 902 10.1137/10081856X MR2821582
47. Liu Z., Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates. Nonlinear Analysis. Real World Applications 2013 14 3 1286 1299 10.1016/j.nonrwa.2012.09.016 MR3004500 ZBL1261.34040
48. Imhof L., Walcher S., Exclusion and persistence in deterministic and stochastic chemostat models. Journal of Differential Equations 2005 217 1 26 53 10.1016/j.jde.2005.06.017 MR2170527 ZBL1089.34041
49. Higham D. J., An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review 2001 43 3 525 546 10.1137/S0036144500378302 MR1872387 ZBL0979.65007
50. Afanasiev V. N., Kolmanovskii V. B., Nosov V. R., Mathematical Theory of Control Systems Design 1996 341 Dordrecht, The Netherlands Kluwer Academic Publishers Group xxiv+668 MR1381691