The long time behavior of DI SIR epidemic model with stochastic perturbation. (English) Zbl 1194.92053

Summary: We present a differential infectivity (DI) SIR epidemic model with two categories of stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of the deterministic system under different conditions.


92C60 Medical epidemiology
34F05 Ordinary differential equations and systems with randomness
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92D30 Epidemiology
Full Text: DOI


[1] Anderson, R.; May, R.; Medley, G.; Johnson, A., A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. Appl. Med. Biol., 3, 229-263 (1986) · Zbl 0609.92025
[2] Stochastic, L. Arnold, Differential Equations: Theory and Applications (1972), Wiley: Wiley New York
[3] Beddington, J.; May, R., Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465 (1977)
[4] Beretta, E.; Kolmanowskii, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation, 45, 269-277 (1998) · Zbl 1017.92504
[5] Blythe, S.; Anderson, R., Distributed incubation and infections periods in models of transmission dynamics of human immunodeficiency virus (HIV), IMA J. Math. Appl. Med. Biol., 5, 1-19 (1988) · Zbl 0686.92015
[6] Blythe, S.; Anderson, R., Variable infectiousness in HIV transmission models, IMA J. Math. Appl. Med. Biol., 5, 181-200 (1988) · Zbl 0655.92021
[7] Blythe, S.; Anderson, R., Heterogeneous sexual activity models of HIV transmission in male homosexual populations, IMA J. Math. Appl. Med. Biol., 5, 237-260 (1988) · Zbl 0688.92010
[8] Carletti, M., On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175, 117-131 (2002) · Zbl 0987.92027
[9] Carletti, M., Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection, Math. Med. Biol., 23, 297-310 (2006) · Zbl 1117.92032
[10] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325, 36-53 (2007) · Zbl 1101.92037
[11] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341, 1084-1101 (2008) · Zbl 1132.92015
[12] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007
[13] Hyman, J.; Stanley, E., Using mathematical models to understand the AIDS epidemic, Math. Biosci., 90, 415-473 (1988) · Zbl 0727.92025
[14] Hyman, J.; Li, J.; Stanley, E., Threshold conditions for the spread of the HIV infection in age-structured populations of homosexual men, J. Theoret. Biol., 166, 9-31 (1994)
[15] Hyman, J.; Li, J., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 77-109 (1999) · Zbl 0942.92030
[16] Hyman, J.; Jia, L.; Stanley, E., The initialization and sensitivity of multigroup models for the transmission of HIV, J. Theoret. Biol., 208, 227-249 (2001)
[17] Hyman, J.; Jia, L., The reproductive number for a HIV model with differential infectivity and staged progression, Linear Algebra Appl., 398, 101-116 (2005) · Zbl 1062.92060
[18] Ida, A.; Oharu, S.; Oharu, Y., A mathematical approach to HIV infection dynamics, J. Comput. Appl. Math., 204, 172-186 (2007) · Zbl 1112.37328
[19] Imhof, L.; Walcher, S., Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217, 26-53 (2005) · Zbl 1089.34041
[20] Isham, V., Mathematical modelling of the transmission dynamics of HIV infection and AIDS: a review with discussion, J. Roy. Statist. Soc. Ser. A, 151, 5-30 (1988), 44-49 · Zbl 1001.92550
[21] Jacquez, J.; Simon, C.; Koopman, J.; Sattenspiel, L.; Perry, T., Modelling and analyzing HIV transmission: the effect of contact patterns, Math. Biosci., 92, 119-199 (1988) · Zbl 0686.92016
[22] Kwon, H., Optimal treatment strategies derived from a HIV model with drug-resistant mutants, Appl. Math. Comput., 188, 1193-1204 (2007) · Zbl 1113.92035
[23] Ma, Z.; Liu, J.; Li, J., Stability analysis for differential infectivity epidemic models, Nonlinear Anal. Real World Appl., 4, 841-856 (2003) · Zbl 1025.92012
[24] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050
[25] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 95-110 (2002) · Zbl 1058.60046
[26] Mukandavire, Z.; Garira, W.; Chiyaka, C., Asymptotic properties of an HIV/AIDS model with a time delay, J. Math. Anal. Appl., 330, 916-933 (2007) · Zbl 1110.92043
[27] Tan, W.; Zhu, X., A stochastic model for the HIV epidemic in homosexual populations involving age and race, Math. Comput. Modelling, 24, 67-105 (1996) · Zbl 0884.92027
[28] Tan, W.; Zhu, X., A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes: I. The probabilities of HIV transmission and pair formation, Math. Comput. Modelling, 24, 47-107 (1996) · Zbl 0885.92032
[29] Tan, W.; Zhu, X., A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes: II. The chain multinomial model of the HIV epidemic, Math. Comput. Modelling, 26, 17-92 (1997) · Zbl 1185.92086
[30] Tan, W.; Xiang, Z., A state space model for the HIV epidemic in homosexual populations and some applications, Math. Biosci., 152, 29-61 (1998) · Zbl 0941.92027
[31] Yu, J.; Jiang, D.; Shi, N., Global stability of two-group SIR model with random perturbation, J. Math. Anal. Appl., 360, 235-244 (2009) · Zbl 1184.34064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.