A Viral Infection Model with a Nonlinear Infection Rate

Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 958016, 19 pages doi:10.1155/2009/958016 Research Article A Viral Infection Model with a Nonlinear Infection Rate Yumei Yu,1Juan J. Nieto,2Angela Torres,3and Kaifa Wang4 1School of Science, Dalian Jiaotong University, Dalian 116028, China 2Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santioga de compostela, Spain 3Departamento de Psiquiatr´ ıa, Radiolog´ ıaySaludP´ ublica, Facultad de Medicina, Universidad de Santiago de Compostela, 15782 Santioga de compostela, Spain 4Department of Computers Science, Third Military Medical University, Chongqing 400038, China Correspondence should be addressed to Kaifa Wang, [email protected] Received 28 February 2009; Revised 23 April 2009; Accepted 27 May 2009 Recommended by Donal O’Regan A viral infection model with a nonlinear infection rate is constructed based on empirical evidences. Qualitative analysis 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 behaviors indicate that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus. Copyright q2009 Yumei Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Mathematical models can provide insights into the dynamics of viral load in vivo. A basic viral infection model 1has been widely used for studying the dynamics of infectious agents such as hepatitis B virus HBV, hepatitis C virus HCV, and human immunodeficiency virus HIV, which has the following forms: dx dtλ−dx −βxv, dy dtβxv −ay, dv dtky −uv, 1.1 2 Boundary Value Problems where susceptible cells xt are produced at a constant rate λ, die at a density-dependent rate dx, and become infected with a rate βuv; infected cells yt are produced at rate βuv and die at a density-dependent rate ay; free virus particles vt are released from infected cells at the rate ky and die at a rate uv. Recently, there have been many papers on virus dynamics within-host in different aspects based on the 1.1. For example, the influences of spatial structures on virus dynamics have been considered, and the existence of traveling waves is established via the geometric singular perturbation method 2. For more literature, we list 3,4and references cited therein. Usually, there is a plausible assumption that the amount of free virus is simply proportional to the number of infected cells because the dynamics of the virus is substantially faster than that of the infected cells, ua, k λ. Thus, the number of infected cells yt can also be considered as a measure of virus load vte.g., see 5–7. As a result, the model 1.1is reduced to dx dtλ−dx −βxy, dy dtβxy −ay. 1.2 As for this model, it is easy to see that the basic reproduction number of virus is given by R0βλ/ad, which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. Furthermore, we know that the infection-free equilibrium E0λ/d, 0is globally asymptotically stable if R0<1, and so is the infection equilibrium E1a/β, βλ −ad/aβif R0>1. Note that both infection terms in 1.1and 1.2are based on the mass-action principle Perelson and Nelson 8; that is, the infection rate per susceptible cell and per virus is a constant β. However, infection experiments of Ebert et al. 9and McLean and Bostock 10 suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose and is usually sigmoidal in shape. Thus, as Regoes et al. 11, we take the nonlinear infection rate into account by relaxing the mass-action assumption that is made in 1.2and obtain dx dtλ−dx −βyx, dy dtβyx−ay, 1.3 where the infection rate per susceptible cell, βy, is a sigmoidal function of the virus parasiteconcentration because the number of infected cells ytcan also be considered as a measure of virus load e.g., see 5–7, which is represented in the following form: βyy/ID50κ 1y/ID50κ,κ>1.1.4 Here, ID50 denotes the infectious dose at which 50% of the susceptible cells are infected, κ measures the slope of the sigmoidal curve at ID50 and approximates the average number Boundary Value Problems 3 of virus that enters a single host cell at the begin stage of invasion, y/ID50κmeasures the infection force of the virus, and 1/1y/ID50κmeasures the inhibition effect from the behavioral change of the susceptible cells when their number increases or from the production of immune response which depends on the infected cells. In fact, many investigators have introduced different functional responses into related equations for epidemiological modeling, of which we list 12–17and references cited therein. However, a few studies have considered the influences of nonlinear infection rate on virus dynamics. When the parameter κ1, 18,19considered a viral mathematical model with the nonlinear infection rate and time delay. Furthermore, some different types of nonlinear functional responses, in particular of the form βxqyor Holling-type functional response, were investigated in 20–23. Note that κ>1in1.4. To simplify the study, we fix the slope κ2 in the present paper, and system 1.3becomes dx dtλ−dx −y2 ID2 50 y2x, dy dty2 ID2 50 y2x−ay. 1.5 To be concise in notations, rescale 1.5by Xx/ID50,Y y/ID50. For simplicity, we still use variables x, y instead of X, Y and obtain dx dtm−dx −y2 1y2x, dy dty2 1y2x−ay, 1.6 where mλ/ID50.Notethat1/d is the average life time of susceptible cells and 1/a is the average life-time of infected cells. Thus, a≥dis always valid by means of biological detection. If ad, the virus does not kill infected cells. Therefore, the virus is non cytopathic in vivo. However, when a>d, which means that the virus kills infected cells before its average life time, the virus is cytopathic in vivo. The main purpose of this paper is to study the effect of the nonlinear infection rate on the dynamics of 1.6. We will perform a qualitative analysis and derive the Allee-type dynamics which result from the appearance of bistable states or saddle-node state in 1.6. The bifurcation analysis indicates that 1.6undergoes a Bogdanov-Takens bifurcation at the degenerate singular infection equilibrium which includes a saddle-node bifurcation, a Hopf bifurcation, and a homoclinic bifurcation. Thus, the nonlinear infection rate can induce the complex dynamic behaviors in the viral infection model. The organization of the paper is as follows. In Section 2, the qualitative analysis of system 1.6is performed, and the stability of the equilibria is obtained. The results indicate that 1.6can display an Allee effect. Section 3 gives the bifurcation analysis, which indicates that the dynamics of 1.6is more complex than that of 1.1and 1.2. Finally, a brief discussion on the direct biological implications of the results is given in Section 4. 4 Boundary Value Problems 2. Qualitative Analysis Since we are interested in virus pathogenesis and not initial processes of infection, we assume that the initial data for the system 1.6are such that x0>0,y 0>0.2.1 The objective of this section is to perform a qualitative analysis of system 1.6and derive the Allee-type dynamics. Clearly, the solutions of system 1.6with positive initial values are positive and bounded. Let gyy/1y2, and note that 1.6has one and only one infection-free equilibrium E0m/d, 0. Then by using the formula of a basic reproduction number for the compartmental models in van den Driessche and Watmough 24,weknow that the basic reproduction number of virus of 1.6is R01 a·m d·g00,2.2 which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process as zero. Although it is zero, we will show that the virus can still persist in host. We start by studying the equilibria of 1.6. Obviously, the infection-free equilibrium E0m/d, 0always exists and is a stable hyperbolic node because the corresponding characteristic equation is ωdωa0. In order to find the positive infectionequilibria, set m−dx −y2 1y2x0, y 1y2x−a0, 2.3 then we have the equation a1dy2−my ad 0.2.4 Based on 2.4, we can obtain that ithere is no infection equilibria if m2<4a2d1d; iithere is a unique infection equilibrium E1x∗,y∗if m24a2d1d; iiithere are two infection equilibria E11 x1, y1and E12 x2, y2if m2>4a2d1d. Boundary Value Problems 5 Here, y∗m 2a1d,x ∗a1y∗2 y∗, y1m−m2−4a2d1d 2a1d,x1 a1y2 1 y1 , y2mm2−4a2d1d 2a1d,x2 a1y2 2 y2 . 2.5 Thus, the surface SN m, d, a:m24a2d1d2.6 is a Saddle-Node bifurcation surface, that is, on one side of the surface SN system 1.6has not any positive equilibria; on the surface SN system 1.6has only one positive equilibrium; on the other side of the surface SN system 1.6has two positive equilibria. The detailed results will follow. Next, we determine the stability of E11 and E12. The Jacobian matrix at E11 is JE11  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −d−y2 1 1y2 1 −2x1y1 1y2 12 y2 1 1y2 1 −a2x1y1 1y2 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .2.7 After some calculations, we have detJE11 − a1d4a2d1dmm2−4a2d1d−m 2a21dmm−m2−4a2d1d.2.8 Since m2>4a2d1din this case, 4a2d1dmm2−4a2d1d−m >0 is valid. Thus, detJE11 <0 and the equilibrium E11 is a saddle. The Jacobian matrix at E12 is JE12  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −d−y2 2 1y2 2 −2x2y2 1y2 22 y2 2 1y2 2 −a2x2y2 1y2 22 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .2.9 6 Boundary Value Problems By a similar argument as above, we can obtain that detJE12 >0. Thus, the equilibrium E12 is a node, or a focus, or a center. For the sake of simplicity, we denote mε2ad1d, m0a212d a−d1ad,if a>2d1d. 2.10 We have the following results on the stability of E12. Theorem 2.1. Suppose that equilibrium E12 exists; that is, m>m ε.ThenE12 is always stable if d≤a≤2d1d.Whena>2d1d, we have iE12 is stable if m>m 0; iiE12 is unstable if m<m 0; iiiE12 is a linear center if mm0. Proof. After some calculations, the matrix trace of JE12 is trJE12  2a31d12d−m1admm2−4a2d1d 2a21dmmm2−4a2d1d,2.11 and its sign is determined by Fm2a31d12d−m1admm2−4a2d1d.2.12 Note that Fm−1ad2mm2−4a2d1dm2 m2−4a2d1d<0,2.13 which means that Fmis a monotone decreasing function of variable m. Clearly, Fmε2a21da−2d1d⎧ ⎨ ⎩ >0,if a>2d1d, ≤0,if a≤2d1d. 2.14 Note that Fm0 implies that 2a31d12d m1ad−mm2−4a2d1d.2.15 Boundary Value Problems 7 Squaring 2.15we find that 4a61d212d2 m21ad2−4a31d12d 1adm2m2−4a2d1d.2.16 Thus, a41d12d2 m21ad2a12d 1ad−da−d1d 1ad, ma212d a−d1ad. 2.17 This means that Fm00. Thus, under the condition of m>m εand the sign of Fm, tr JE12 <0 is always valid if a≤2d1d. When a>2d1d,trJE12 <0ifm>m 0, trJE12 >0ifm<m 0,andtrJE12 0ifmm0. For 1.6, its asymptotic behavior is determined by the stability of E12 if it does not have a limit cycle. Next, we begin to consider the nonexistence of limit cycle in 1.6. Note that E11 is a saddle and E12 is a node, a focus, or a center. A limit cycle of 1.6 must include E12 and does not include E11. Since the flow of 1.6moves toward down on the line where yy1and x<x1and moves towards up on the line where yy1and x>x1, it is easy to see that any potential limit cycle of 1.6must lie in the region where y>y1. Take a Dulac function D1y2/y2, and denote the right-hand sides of 1.6by P1and P2, respectively. We have ∂DP1 ∂x ∂DP2 ∂y −1ady2−a−d y2,2.18 which is negative if y2>a−d/1ad. Hence , we can obtain the following result. Theorem 2.2. There is no limit cycle in 1.6if y2 1>a−d 1ad.2.19 Note that y1>0 as long as it exists. Thus, inequality 2.19is always valid if a d. When a>d, using the expression of y1in 2.5, we have that inequality 2.19that is equivalent to 2a31d12d 1ad<m 2<a412d2 a−d1ad.2.20 8 Boundary Value Problems Indeed, since y2 1m2 2a21d2−d 1d−mm2−4a2d1d 2a21d2, m2 2a21d2−d 1d−a−d 1adm2 2a21d2−a12d 1d1ad, 2.21 we have 2.19that is equivalent to m2 2a21d2−a12d 1d1ad>mm2−4a2d1d 2a21d2,2.22 that is, m2−2a31d212d 1d1ad>m m2−4a2d1d.2.23 Thus, m2>2a31d212d 1d1ad.2.24 On the other hand, squaring 2.23we find that m4−4a31d212d 1d1adm24a61d412d2 1d21ad2>m 4−4a2d1dm2,2.25 which is equivalent to m2<a412d2 a−d1ad.2.26 The combination of 2.24and 2.26yields 2.20. Furthermore, 4a2d1d<a412d2 a−d1ad2.27 Boundary Value Problems 9 is equivalent to a/ 2d1d,both 2a31d12d 1ad<a412d2 a−d1ad, 2a31d12d 1ad<4a2d1d 2.28 are equivalent to a<2d1d. Consequently, we have the following. Corollary 2.3. There is no limit cycle in 1.6if either of the following conditions hold: iadand m2>4a2d1d; iid<a<2d1dand 4a2d1d<m 2<a 412d2/a−d1ad. When m24a2d1d,system1.6has a unique infection equilibrium E1.The Jacobian matrix at E1is JE1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −d−y∗2 1y∗2−2x∗y∗ 1y∗22 y∗2 1y∗2−a2x∗y∗ 1y∗22 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .2.29 The determinant of JE1is detJE1−a1d4a2d1d−m2 m24a21d20,2.30 and the trace of JE1is trJE14a21da−2d1d m24a21d2.2.31 Thus, E1is a degenerate singular point. Since its singularity, complex dynamic behaviors may occur, which will be studied in the next section. 3. Bifurcation Analysis In this section, the Bogdanov-Takens bifurcation for short, BT bifurcationof system 1.6is studied when there is a unique degenerate infection equilibrium E1. 16 Boundary Value Problems HIII SN II SN 0 I II III IV HL H SN− μ1 μ2 0IV HL ISN− Figure 1: The bifurcation set and the corresponding phase portraits of system 3.26at origin. Theorem 3.2. Let the assumptions (H1) and (H2) hold. Then 1.6admits the following bifurcation behaviors: ithere is a saddle-node bifurcation curve SN±{λ1,λ 2:μ10,μ 2>0or μ2<0}; iithere is a Hopf bifurcation curve H{λ1,λ 2:μ1−μ2 2oλ2,q 1<0}; iiithere is a homoclinic-loop bifurcation curve HL{λ1,λ 2:μ1−49/25μ2 2oλ2}. Concretely, as the statement in 28, Chapter 3, when μ1,μ 2∈Δ, the orbital topical structure of the system 3.26at origin corresponding system 1.6at E1is shown in Figure 1. 4. Discussion Note that most infection experiments suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose, usually sigmoidal in shape. In this paper, we study a viral infection model with a type of nonlinear infection rate, which was introduced by Regoes et al. 11. Qualitative analysis Theorem 2.1implies that infection equilibrium E12 is always stable if the virus is noncytopathic, ad, or cytopathic in vivo but its cytopathic effect is less than or equal to an appropriate value, a≤2d1d. When the cytopathic effect of virus is greater than the threshold value, a>2d1d, the stability of the infection equilibrium E12 depends on the value of parameter m, which is proportional to the birth rate of susceptible cells λand is in inverse proportion to the infectious dose ID50. The infection equilibrium is stable if m>m 0and becomes unstable if m<m 0. When mgets to the critical value, mm0, the infection equilibrium is a linear center, so the oscillation behaviors may occur. If our model 1.6does not have a limit cycle see Theorem 2.2 and Corollary 2.3, its asymptotic behavior is determined by the stability of E12. When E12 is stable, there is a region outside which positive semiorbits tend to E0as ttends to infinity and inside Boundary Value Problems 17 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 y 0 5 10 15 20 x E0 ExtinctionExtinction UM SM E11 UM SM E12 Persistence Figure 2: Illustrations of the Allee effect for 1.5. Here, λ17.06,d1.0,a3.0,ID50 2. E017.06,0 is stable, E11 13.2311,1.2763is a saddle point, E12 12.3589,1.567is stable. Note that SM is the stable manifolds of E11 solid line, UM is the unstable manifolds of E11 dash line, and the phase portrait of 1.6is divided into two domains of extinction and persistence of the virus by SM. which positive semi-orbits tend to E12 as ttends to infinity; that is, the virus will persist if the initial position lies in the region and disappear if the initial position lies outside this region. Thus, besides the value of parameters, the initial concentration of the virus can also affect the result of invasion. An invasion threshold may exist in these cases, which is typical for the so-called Allee effect that occurs when the abundance or frequency of a species is positively correlated with its growth rate see 11. Consequently, the unrescaled model 1.5can display an Allee effect see Figure 2, which is an infrequent phenomenon in current viral infection models though it is reasonable and important in viral infection process. Furthermore, when infection equilibrium becomes a degenerate singular point, we have shown that the dynamics of this model are very rich inside this region see Theorems 3.1 and 3.2 and Figure 1. Static and dynamical bifurcations, including saddle-node bifurcation, Hopf bifurcation, homoclinic bifurcation, and bifurcation of cusp-type with codimension two i.e., Bogdanov-Takens bifurcation, have been exhibited. Thus, besides the Allee effect, our model 1.6shows that the viral oscillation behaviors can occur in the host based on the appropriate conditions, which was observed in chronic HBV or HCV carriers see 29– 31. These results inform that the viral infection is very complex in the development of a better understanding of diseases. 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