Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response. (English) Zbl 1239.92071

Summary: A viral infection model with nonlinear incidence rate and delayed immune response is investigated. It is shown that if the basic reproduction ratio of the virus is less than unity, and the infection-free equilibrium is globally asymptotically stable. By analyzing the characteristic equation, the local stability of the chronic infection equilibrium of the system is discussed. Furthermore, the existence of Hopf bifurcations at the chronic infection equilibrium is also studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the chronic infection equilibrium. Numerical simulations are carried out to illustrate the main results.


92C60 Medical epidemiology
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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[1] Burić, N.; Mudrinic, M.; Vasović, N., Time delay in a basic model of the immune response, Chaos solitons fract, 12, 483-489, (2001) · Zbl 1026.92015
[2] Canabarro, A.A.; Gléria, I.M.; Lyra, M.L., Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342, 234-241, (2004)
[3] Culshaw, R.V.; Ruan, S.; Webb, G., A mathematical model of cell-to-cell HIV-1 that include a time delay, J math biol, 46, 425-444, (2003) · Zbl 1023.92011
[4] Gumel, A.B.; Moghadas, S.M., HIV control in vivo: dynamical analysis, Commun nonlinear sci numer simul, 9, 561-568, (2004) · Zbl 1121.92300
[5] Hale, J.K., Theory of functional differential equations, (1997), Springer New York
[6] Ji, Y.; Min, L.Q.; Zheng, Y.; Su, Y.M., A viral infection model with periodic immune response and nonlinear CTL response, Math comput simulat, 80, 2309-2316, (2010) · Zbl 1195.92037
[7] Leenheer, P.D.; Smith, H.L., Virus dynamic: a global analysis, SIAM J appl math, 63, 1313-1327, (2003) · Zbl 1035.34045
[8] Nelson, P.W.; Perelson, A.S., Mathematical analysis of a delay differential equation models of HIV-1 infection, Math biosci, 179, 73-94, (2002) · Zbl 0992.92035
[9] Nowak, M.A.; Bangham, C.R.M., Population dynamics of immune responses to persistent viruses, Science, 272, 74-79, (1996)
[10] Nowak, M.A.; May, R.M., Virus dynamics: mathematical principles of immunology and virology, (2000), Oxford University Press Oxford · Zbl 1101.92028
[11] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev, 41, 3-44, (1999) · Zbl 1078.92502
[12] Revilla, T.; Garci-Ramos, G., Fighting a virus with a virus: a dynamic model for HIV-1 therapy, Math biosci, 185, 191-203, (2003) · Zbl 1021.92015
[13] Song, X.Y.; Wang, S.L.; Dong, J., Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J math anal appl, 373, 345-355, (2011) · Zbl 1208.34128
[14] Wang, K.F.; Wang, W.D.; Pang, H.Y.; Liu, X.N., Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226, 197-208, (2007) · Zbl 1117.34081
[15] Wodarz, D., Hepatitis C virus dynamics and pathology: the role of CTL and antibody response, J gen virol, 84, 1743-1750, (2003)
[16] Wodarz, D.; Christensen, J.P.; Thomsen, A.R., The importance of lytic and nonlytic immune responses in viral infections, Trends immunol, 23, 194-200, (2002)
[17] Wodarz, D.; Krakauer, D.C., Defining CTL-induced pathology: implications for HIV, Virology, 274, 94-104, (2000)
[18] Xu, R.; Ma, Z.E., Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure, Chaos solitons fract, 38, 669-684, (2008) · Zbl 1146.34323
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