The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate

Bulletin of Mathematical Biology

Volumn 77, Issue 9, 2015, Pages 1705-1743

  Li, Dan ^a   Cui, Jing’an ^b   Liu, Meng ^c   Liu, Shengqiang ^a  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

^b BEIJING UNIVERSITY OF CIVIL ENGINEERING AND ARCHITECTURE   (China)

^c HUAIYIN NORMAL UNIVERSITY   (China)

Author keywords

Environmental noise; Extinction; Influenza A; Stationary distribution; Stochastic SIRS model

Indexed keywords

BIOLOGICAL MODEL; COMPUTER SIMULATION; EPIDEMIC; HUMAN; INCIDENCE; INFLUENZA A VIRUS; INFLUENZA, HUMAN; MARKOV CHAIN; MATHEMATICAL PHENOMENA; NONLINEAR SYSTEM; STATISTICS AND NUMERICAL DATA; TRANSMISSION;

COMPUTER SIMULATION; EPIDEMICS; HUMANS; INCIDENCE; INFLUENZA A VIRUS; INFLUENZA, HUMAN; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS; STOCHASTIC PROCESSES;

EID: 84947491559     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-015-0101-9     Document Type: Article

References (80)

1. Anderson RM, May RM (1991) Infectious diseases in humans: dynamics and control. Oxford University Press, Oxford
2. Andreasen V, Sasaki A (2006) Shaping the phylogenetic tree of influenza by cross-immunity. Popul Biol 70:164
3. Arenas AJ, Parra GG, Moraño JA (2009) Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain. Biosystems 96:206
4. Arino J, Brauer F, van den Driessche P, Watmoughd J, Wu J (2008) A model for influenza with vaccination and antiviral treatment. Theor Popul Biol 253:118
5. Cai Y, Kang Y, Banerjee M, Wang W (2015a) A stochastic epidemic model incorporating media coverage. Commun Math Sci (in press)
6. Cai Y, Kang Y, Banerjee M, Wang W (2015b) A stochastic SIRS epidemic model with infectious force under intervention strategies. J Differ Equ. doi:10.1016/j.jde.2015.08.024
7. Capasso V (1993) Mathematical structures of epidemic systems. In: Levin SA (ed) Lecture notes in biomathematics, vol 97. Springer-Verlag, Berlin, Heidelberg
8. Capasso V, Serio G (1978) A generalization of the Kermack–McKendrick deterministic epidemic model. Math Biosci 42:43
9. Casagrandi R, Bolzoni L, Levin SA, Andreasen V (2006) The SIRC model and influenza A. Math Biosci 200:152
10. Chowell G, Nishiura H, Bettencourt LMA (2007) Comparative estimation of the reproduction number for pandemic influenza from daily case notification data. J R Soc Interface 4:155
11. Cui J, Sun Y, Zhu H (2008) The impact of media on the control of infectious diseases. J Dyn Differ Equ 20:31
12. Cui J, Tao X, Zhu H (2008) An SIS infection model incorporating media coverage. Rocky Mt J Math 38:1323
13. Douglas R (1975) Influenza in man. In: Kilbourne E (ed) The influenza viruses and influenza. Academic Press, New York, p 395
14. Dushoff J, Plotkin JB, Levin SA, Earn DJD (2004) Dynamical resonance can account for seasonality of influenza epidemics. Proc Natl Acad Sci USA 101:16915
15. Enatsu Y, Nakata Y, Muroya Y (2012) Global stability of SIRS epidemic models with a class of nonlinear incidence rates and distributed delays. Acta Math Sci 32:851
16. Feng Z, Thieme HR (2000) Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model. SIAM J Appl Math 61:803
17. Frank A, Taber L, Wells C, Wells J, Glezen W, Parades A (1981) Patterns of shedding of myxoviruses and paramyxoviruses in children. J Infect Dis 144:433
18. Germann TC, Kadau K, Longini IM, Macken CA (2006) Mitigation strategies for pandemic influenza in the United States. Proc Natl Acad Sci USA 103:5935
19. Gog JR, Grenfell BT (2002) Dynamics and selection of many-strain pathogens. Proc Natl Acad Sci USA 99:17209
20. Gray A, Greenhalgh D, Hu L, Mao X, Pan J (2011) A stochastic differential equation SIS epidemic model. SIAM J Appl Math 71:876
21. Gu J, Gao Z, Li W (2011) Modeling of epidemic spreading with white Gaussian noise. Chin Sci Bull 56:3683
22. Haque M (2011) A detailed study of the Beddington–DeAngelis predator–prey model. Math Biosci 234:1
23. Has’minskii RZ (1980) Stochastic stability of differential equations. Sijthoof & Noordhoof, Alphen aan den Rijn, The Netherlands
24. Hay AJ, Gregory V, Douglas AR, Lin YP (2001) The evolution of human influenza viruses. Proc R Soc Lond B 356:1861
25. Hethcote HW (1976) Qualitative analyses of communicable disease models. Math Biosci 28:335
26. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599
27. Hethcote HW, van den Driessche P (1991) Some epidemiological models with nonlinear incidence. J Math Biol 29:271
28. Hooten MB, Anderson J, Waller LA (2010) Assessing North American influenza dynamics with a statistical SIRS model. Spat Spat Temporal Epidemiol 1:177
29. Ji C, Jiang D, Shi N (2011) Multigroup SIR epidemic model with stochastic perturbation. Phys A 390:1747
30. Keeling MJ, Rohani P (2008) Modeling infectious diseases in human and animals. Princeton University Press, New Jersey
31. Khasminskii R (2012) Stochastic stability of differential equations, 2nd edn. Spring, Berlin
32. Kloeden PE, Platen E, Schurz H (1992) Numerical solution of stochastic differential equations. Spring, Berlin
33. Koelle K, Cobey S, Grenfell B, Pascual M (2006) Epochal evolution shapes the phylodynamics of influenza A (H3N2) in humans. Science 314:1898
34. Korobeinikov A (2007) Global properties of infectious disease models with nonlinear incidence. Bull Math Biol 69:1871
35. Kushner HJ (1967) Stochastic stability and control. Academic Press, New York
36. Lahodny GE Jr, Allen LJS (2013) Probability of a disease outbreak in stochastic multipatch epidemic models. Bull Math Biol 75:1157
37. Lahrouz A, Omari L (2013) Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Stat Probab Lett 83:960
38. Lahrouz A, Settati A (2014) Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. Appl Math Comput 233:10
39. Levin SA, Hallam TG, Gross LJ (1989) Applied mathematical ecology. Springer, New York
40. Li D, Cui J, Song G (2015) Permanence and extinction for a single-species system with jump-diffusion. J Math Anal Appl 430:438
41. Liu WM, Levin SA, Iwasa Y (1986) Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models. J Math Biol 23:187
42. Liu M, Wang K, Wu Q (2011) Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull Math Biol 73:1969
43. Liu M, Bai C (2015) Optimal harvesting of a stochastic logistic model with time delay. J Nonlinear Sci 25:277
44. Liu Q, Chen Q (2015) Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys A 428:140
45. Lowen AC, Mubareka S, Steel J, Palese P (2007) Influenza virus transmission is dependent on relative humidity and temperature. PLOS Pathog. doi:10.1371/journal.ppat.0030151
46. Lowen AC, Steel j (2014) Roles of humidity and temperature in shaping influenza seasonality. J Virol 88:7692
47. Mao X (2002) A note on the LaSalle-type theorems for stochastic differential delay equations. J Math Anal Appl 268:125
48. Mao X, Marion G, Renshaw E (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stoch Process Appl 97:95
49. Mode CJ, Jacobson ME (1987) On estimating critical population size for an endangered species in the presence of environmental stochasticity. Math Biosci 85:185
50. Mukhopadhyay B, Bhattacharyya R (2012) Effects of deterministic and random refuge in a prey-predator model with parasite infection. Math Biosci 239:124
51. Nuño M, Feng Z, Martcheva M, Chavez CC (2005) Dynamics of two-strain influenza with isolation and cross-protection. SIAM J Appl Math 65:964
52. Øksendal B (2005) Stochastic differential equations: an introduction with applications, 6th edn. Springer, New York
53. Palese P, Young J (1982) Variation of influenza A, B, and C viruses. Science 215:1468
54. Pease CM (1987) An evolutionary epidemiological mechanism, with applications to type A influenza. Theor Popul Biol 31:422
55. Pica N, Bouvier NM (2012) Environmental factors affecting the transmission of respiratory viruses. Curr Opin Virol 2:90
56. Plotkin JB, Dushoff J, Levin SA (2002) Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus. Proc Natl Acad Sci USA 99:6263
57. Ripa J, Lundberg P (1996) Noise color and the risk of population extinctions. Proc R Soc Lond Ser B 263:1751
58. Ruan S, Wang W (2003) Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Differ Equ 188:135
59. Shaman J, Pitzer VE, Viboud C, Grenfell BT, Lipsitch M (2010) Absolute humidity and the seasonal onset of influenza in the continental United States. PLoS Biol 8:e1000316
60. Shaman J, Kohn M (2009) Absolute humidity modulates influenza survival, transmission, and seasonality. Proc Natl Acad Sci USA 106:3243
61. Simonsen L, Clarke M, Schonberg L, Arden N, Cox N, Fukuda K (1998) Pandemic versus epidemic influenza mortality: a pattern of changing age distribution. J Infect Dis 178:53
62. Simonsen L (1999) The global impact of infuenza on morbidity and mortality. Vaccine 17:S3
63. Steele JH (1985) A comparison of terrestrial and marine ecological systems. Nature 313:355
64. Turelli M (1977) Random environments and stochastic calculus. Theor Popul Biol 12:140
65. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29
66. van den Driessche P, Watmough J (2008) Further notes on the basic reproduction number. In: Brauer F, van den Driessche P, Wu J (eds) Mathematical epidemiology. Springer-Verlag, Berlin, Heidelberg, p 159
67. Vasseur DA, Yodzis P (2004) The color of environmental noise. Ecology 85:1146
68. Wang K (2010) Random mathematical biology model. Science Press, Beijing
69. Wang X, Liu S, Wang L, Zhang W (2015) An epidemic patchy model with entry-exit screening. Bull Math Biol 77:1237
70. Webster R, Laver W, Air G, Schild G (1982) Molecular mechanisms of variation in influenza viruses. Nature 296:115
71. Wissel C, Stöcker S (1991) Extinction by random influences. Theor Popul Biol 39:315
72. Xia Y, Gog JR, Grenfell BT (2005) Semiparametric estimation of the duration of immunity from infectious disease time series: influenza as a case-study. Appl Stat 54:659
73. Yang F, Yuan L, Tan X, Huang C, Feng J (2013) Bayesian estimation of the effective reproduction number for pandemic influenza A H1N1 in Guangdong Province, China. Ann Epidemiol 23:301
74. Yang Q, Mao X (2013) Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Anal RWA 14:1434
75. Yang Q, Mao X (2014) Stochastic dynamics of SIRS epidemic models with random perturbation. Math Biosci Eng 11:1003
76. Yuan Y, Allen LJS (2011) Stochastic models for virus and immune system dynamics. Math Biosci 234:84
77. Yuan H, Koelle K (2012) The evolutionary dynamics of receptor binding avidity in influenza A: a mathematical model for a new antigenic drift hypothesis. Philos Trans R Soc B 368:20120204
78. Zhao Y, Yuan S, Ma J (2015) Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull Math Biol 77:1285
79. Zhao Y, Jiang D, Mao X, Gray A (2015) The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete Contin Dyn Syst Ser B 20:1277
80. Zhu C, Yin G (2007) Asymptotic properties of hybrid diffusion systems. SIAM J Control Optim 46:1155