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A viral infection model with a nonlinear infection rate. (English) Zbl 1187.34062

A viral infection model with a nonlinear infection rate is constructed based on empirical evidences. This leads to the study of the dynamics of the following two-dimensional autonomous system
\[ \begin{aligned} \frac{dx}{dt}&=m-dx-\frac{y^2}{1+y^2}x,\\ \frac{dy}{dt}&=\frac{y^2}{1+y^2}x-ay. \end{aligned} \tag{1} \]
The main purpose of the paper is to study the effect of the nonlinear infection rate on the dynamics of (1). Qualitative analysis of system (1) shows that there is a degenerate singular infection equilibrium. Furthermore, bifurcation of cusp-type with codimension two (i.e., Bogdanov-Takens bifurcation) is confirmed under appropriate conditions. As a result, the rich dynamical behavior indicates that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus. Thus, the nonlinear infection rate can induce complex dynamic behavior in the viral infection model. A brief discussion on the direct biological implications of the results is given.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
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[1] Nowak MA, May RM: Virus Dynamics. Oxford University Press, Oxford, UK; 2000:xii+237. · Zbl 1101.92028
[2] Wang, K; Wang, W, Propagation of HBV with spatial dependence, Mathematical Biosciences, 210, 78-95, (2007) · Zbl 1129.92052
[3] Campos, D; MĂ©ndez, V; Fedotov, S, The effects of distributed life cycles on the dynamics of viral infections, Journal of Theoretical Biology, 254, 430-438, (2008) · Zbl 1400.92467
[4] Srivastava PKr, Chandra P:Modeling the dynamics of HIV and [InlineEquation not available: see fulltext.] T cells during primary infection. Nonlinear Analysis: Real World Applications. In press
[5] Bartholdy, C; Christensen, JP; Wodarz, D; Thomsen, AR, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with lymphocytic choriomeningitis virus, Journal of Virology, 74, 10304-10311, (2000)
[6] Bonhoeffer, S; Coffin, JM; Nowak, MA, Human immunodeficiency virus drug therapy and virus load, Journal of Virology, 71, 3275-3278, (1997)
[7] Wodarz, D; Christensen, JP; Thomsen, AR, The importance of lytic and nonlytic immune responses in viral infections, Trends in Immunology, 23, 194-200, (2002)
[8] Perelson, AS; Nelson, PW, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41, 3-44, (1999) · Zbl 1078.92502
[9] Ebert, D; Zschokke-Rohringer, CD; Carius, HJ, Dose effects and density-dependent regulation of two microparasites of daphnia magna, Oecologia, 122, 200-209, (2000)
[10] McLean, AR; Bostock, CJ, Scrapie infections initiated at varying doses: an analysis of 117 titration experiments, Philosophical Transactions of the Royal Society B, 355, 1043-1050, (2000)
[11] Regoes, RR; Ebert, D; Bonhoeffer, S, Dose-dependent infection rates of parasites produce the allee effect in epidemiology, Proceedings of the Royal Society B, 269, 271-279, (2002)
[12] Gao, S; Chen, L; Nieto, JJ; Torres, A, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045, (2006)
[13] Ruan, S; Wang, W, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188, 135-163, (2003) · Zbl 1028.34046
[14] Ruan, S; Xiao, D, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61, 1445-1472, (2001) · Zbl 0986.34045
[15] Sharomi, O; Gumel, AB, Re-infection-induced backward bifurcation in the transmission dynamics of chlamydia trachomatis, Journal of Mathematical Analysis and Applications, 356, 96-118, (2009) · Zbl 1162.92024
[16] Wang, W, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3, 267-279, (2006) · Zbl 1089.92052
[17] Zhang, H; Chen, L; Nieto, JJ, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Analysis: Real World Applications, 9, 1714-1726, (2008) · Zbl 1154.34394
[18] Li, D; Ma, W, Asymptotic properties of a HIV-1 infection model with time delay, Journal of Mathematical Analysis and Applications, 335, 683-691, (2007) · Zbl 1130.34052
[19] Song, X; Neumann, AU, Global stability and periodic solution of the viral dynamics, Journal of Mathematical Analysis and Applications, 329, 281-297, (2007) · Zbl 1105.92011
[20] Cai, L; Wu, J, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons & Fractals, 41, 175-182, (2009) · Zbl 1198.34076
[21] Wang, W; Shen, J; Nieto, JJ, Permanence and periodic solution of predator-prey system with Holling type functional response and impulses, No. 2007, 15, (2007) · Zbl 1146.37370
[22] Wang, X; Song, X, Global stability and periodic solution of a model for HIV infection of [inlineequation not available: see fulltext.] T cells, Applied Mathematics and Computation, 189, 1331-1340, (2007) · Zbl 1117.92040
[23] Yang, J, Dynamics behaviors of a discrete ratio-dependent predator-prey system with Holling type III functional response and feedback controls, No. 2008, 19, (2008) · Zbl 1161.39020
[24] van den Driessche, P; Watmough, J, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180, 29-48, (2002) · Zbl 1015.92036
[25] Perko L: Differential Equations and Dynamical Systems, Texts in Applied Mathematics. Volume 7. 2nd edition. Springer, New York, NY, USA; 1996:xiv+519.
[26] Bogdanov, R, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Mathematica Sovietica, 1, 373-388, (1981) · Zbl 0518.58029
[27] Bogdanov, R, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Mathematica Sovietica, 1, 389-421, (1981)
[28] Zhang Z, Li C, Zheng Z, Li W: The Base of Bifurcation Theory about Vector Fields. Higher Education Press, Beijing, China; 1997.
[29] Chun, YK; Kim, JY; Woo, HJ; etal., No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection, Hepatology, 32, 1154-1162, (2000)
[30] Deng, G-H; Wang, Z-L; Wang, Y-M; Wang, K-F; Fan, Y, Dynamic determination and analysis of serum virus load in patients with chronic HBV infection, World Chinese Journal of Digestology, 12, 862-865, (2004)
[31] Pontisso, P; Bellati, G; Brunetto, M; etal., Hepatitis C virus RNA profiles in chronically infected individuals: do they relate to disease activity?, Hepatology, 29, 585-589, (1999)
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