×

zbMATH — the first resource for mathematics

The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. (English) Zbl 1231.92058
Summary: We include stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number \(R_{0}\). The long time behavior of the two stochastic systems is studied. Mainly, we utilize stochastic Lyapunov functions to show under some conditions that the solution has the ergodic property as \(R_{0}\)>1, and exponential stability as \(R_{0}\leq 1\). At last, we make simulations to conform our analytical results.

MSC:
92D30 Epidemiology
34F05 Ordinary differential equations and systems with randomness
34D10 Perturbations of ordinary differential equations
34A99 General theory for ordinary differential equations
37N25 Dynamical systems in biology
PDF BibTeX XML Cite
Full Text: DOI
References:
[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] Anderson, R.M.; May, R.M., Infectious diseases in humans: dynamics and control, (1991), Oxford University Press Oxford
[3] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[4] Gopal, K.B.; Rabi, N.B., Stability in distribution for a class of singular diffusions, Ann. probab., 20, 312-321, (1992) · Zbl 0749.60073
[5] Beddington, J.; May, R., Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465, (1977)
[6] Berettaa, E.; Kolmanovskiib, V.; Shaikhetc, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Math. comput. simulation, 45, 269-277, (1998)
[7] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal., 47, 4107-4115, (2001) · Zbl 1042.34585
[8] Brown, G.C.; Hasibuan, R., Conidial discharge and transmission efficiency of neozygites floridana, an entomopathogenic fungus infecting two-spotted spider mites under laboratory conditions, J. invertebr. pathol., 65, 10-16, (1995)
[9] Capasso, V.; Serio, G., A generalisation of the kermack-mckendrick deterministic epidemic model, Math. biosci., 42, 43-61, (1978) · Zbl 0398.92026
[10] Caraballo, T.; Kloeden, P.E., The persistence of synchronization under environmental noise, Proc. R. soc. lond. ser. A math. phys. eng. sci., 461, 2257-2267, (2005) · Zbl 1206.60054
[11] 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
[12] Carletti, M., Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection, Math. med. biol., 23, 297-310, (2006) · Zbl 1117.92032
[13] Cooke, K.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. math. biol., 35, 240-260, (1996) · Zbl 0865.92019
[14] 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
[15] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model for internal HIV dynamics, J. math. anal. appl., 341, 1084-1101, (2008) · Zbl 1132.92015
[16] Gakkhar, S.; Negi, K., Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate, Chaos solitons fractals, 35, 626-638, (2008) · Zbl 1131.92052
[17] Gao, S.; Chen, L.; Teng, Z., Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear anal. real world appl., 9, 599-607, (2008) · Zbl 1144.34390
[18] Gao, S.; Liu, Y.; Nieto, J.J.; Andrade, H., Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. comput. simulation, 81, 1855-1868, (2011) · Zbl 1217.92066
[19] Hasminskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoff & Noordhoff Alphen aan den Rijn, The Netherlands · Zbl 0419.62037
[20] Hethcote, H.W.; van den Driessche, P., An SIS epidemic model with variable population size and a delay, J. math. biol., 34, 177-194, (1995) · Zbl 0836.92022
[21] Hethcote, H.W.; van den Driessche, P., Two SIS epidemiologic models with delays, J. math. biol., 40, 3-26, (2000) · Zbl 0959.92025
[22] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033
[23] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007
[24] Hu, Y.; Wu, F.; Huang, C., Stochastic Lotka-Volterra models with multiple delays, J. math. anal. appl., 375, 42-57, (2011) · Zbl 1245.92063
[25] Imhof, L.; Walcher, S., Exclusion and persistence in deterministic and stochastic chemostat models, J. differential equations, 217, 26-53, (2005) · Zbl 1089.34041
[26] Ji, C.; Jiang, D., Dynamics of a stochastic density dependent predator-prey system with beddington-deangelis functional response, J. math. anal. appl., 381, 441-453, (2011) · Zbl 1232.34072
[27] Ji, C.; Jiang, D.; Shi, N., Multigroup SIR epidemic model with stochastic perturbation, Phys. A, 390, 1747-1762, (2011)
[28] Jiang, D.; Ji, C.; Shi, N.; Yu, J., The long time behavior of DI SIR epidemic model with stochastic perturbation, J. math. anal. appl., 372, 162-180, (2010) · Zbl 1194.92053
[29] Jiang, D.; Yu, J.; Ji, C.; Shi, N., Asymptotic behavior of global positive solution to a stochastic SIR model, Math. comput. modelling, 54, 221-232, (2011) · Zbl 1225.60114
[30] Jin, Y.; Wang, W.; Xiao, S., An SIRS model with a nonlinear incidence rate, Chaos solitons fractals, 34, 1482-1497, (2007) · Zbl 1152.34339
[31] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bull. math. biol., 69, 1871-1886, (2007) · Zbl 1298.92101
[32] Kutoyants, Yury A., Statistical inference for ergodic diffusion processes, (2003), Springer London · Zbl 1038.62073
[33] Li, G.; Jin, Z., Global stability of an SEIR epidemic model with infectious force in latent, infected and immune period, Chaos solitons fractals, 25, 1177-1184, (2005) · Zbl 1065.92046
[34] Liu, J.; Zhou, Y., Global stability of an SIRS epidemic model with transport-related infection, Chaos solitons fractals, 40, 145-158, (2009) · Zbl 1197.34098
[35] Lou, J.; Ruggeri, T., The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network, J. math. anal. appl., 365, 210-219, (2010) · Zbl 1181.92074
[36] Lucas, A.R., A one-parameter family of stationary solutions in the susceptible-infected-susceptible epidemic model, J. math. anal. appl., 374, 258-271, (2011) · Zbl 1202.92078
[37] Meng, X., Stability of a novel stochastic epidemic model with double epidemic hypothesis, Appl. math. comput., 217, 506-515, (2010) · Zbl 1198.92041
[38] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050
[39] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[40] Ponciano, J.M.; Capistrán, M.A., First principles modeling of nonlinear incidence rates in seasonal epidemics, Plos comput. biol., 7, (2011), art. No. e1001079
[41] Strang, G., Linear algebra and its applications, (1988), Thomson Learning, Inc. United States
[42] 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
[43] 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
[44] 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
[45] 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
[46] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036
[47] Wanduku, D.; Ladde, G.S., Global stability of two-scale network human epidemic dynamic model, Neural parallel sci. comput., 19, 65-90, (2011) · Zbl 1228.92069
[48] Wanduku, D.; Ladde, G.S., Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear anal. real world appl., 13, 794-816, (2012) · Zbl 1238.60046
[49] Wen, L.; Zhong, J., Global asymptotic stability and a property of the SIS model on bipartite networks, Nonlinear anal. real world appl., 13, 967-976, (2012) · Zbl 1238.34110
[50] Yang, Q.; Jiang, D., A note on asymptotic behaviors of stochastic population model with allee effect, Appl. math. model., 35, 4611-4619, (2011) · Zbl 1225.34058
[51] 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
[52] Yuan, Z.; Zou, X., Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population, Nonlinear anal. real world appl., 11, 3479-3490, (2010) · Zbl 1208.34134
[53] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. control optim., 46, 4, 1155-1179, (2007) · Zbl 1140.93045
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.