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FIGURE 5. An example of a switch in stability resulting from a stepwise reduction in r over the course of 5,000 days (13.7 years).

Figure 5 An example of a switch in stability resulting from a stepwise reduction in r over the course of 5,000 days (13.7 years).

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Like Gourley et al. [4] we employ a standard incidence function to describe virion infec- tion of healthy cells and explicitly consider a delay between viral infection and production. These considerations produce the following model:
Like Gourley et al. [4] we employ a standard incidence function to describe virion infec- tion of healthy cells and explicitly consider a delay between viral infection and production. These considerations produce the following model:
3.4. Stability. Here we focus on the stability of the biologically relevant equilibria anc their bifurcation behavior. In particular, we show that the disease-free state, Ey, is asymp totically stable when Ro < 1, as required by biological intuition. The equilibrium repre: senting chronic infection, E*, is treated computationally. This analysis leads to the mos important biological result of the research, viz. that E* undergoes a clinically relevant bi furcation. That is, for clinically relevant parameter values this fixed point experiences « stability switch as certain parameters change. This bifurcation should be clinically eviden in certain cases as the sudden onset of oscillations in viral load and liver damage. 3.3. Steady States. There exist three steady states: Ey = (0,0,0,0), Ey = (K,0,0,0), anc E* = (x*, p*,y*,v*). Ep is biologically meaningless except, perhaps, in the context of developing liver, and Ey represents a healthy, mature liver. E* is a state of persistent chronic HBV infection. Explicitly, E* requires
3.4. Stability. Here we focus on the stability of the biologically relevant equilibria anc their bifurcation behavior. In particular, we show that the disease-free state, Ey, is asymp totically stable when Ro < 1, as required by biological intuition. The equilibrium repre: senting chronic infection, E*, is treated computationally. This analysis leads to the mos important biological result of the research, viz. that E* undergoes a clinically relevant bi furcation. That is, for clinically relevant parameter values this fixed point experiences « stability switch as certain parameters change. This bifurcation should be clinically eviden in certain cases as the sudden onset of oscillations in viral load and liver damage. 3.3. Steady States. There exist three steady states: Ey = (0,0,0,0), Ey = (K,0,0,0), anc E* = (x*, p*,y*,v*). Ep is biologically meaningless except, perhaps, in the context of developing liver, and Ey represents a healthy, mature liver. E* is a state of persistent chronic HBV infection. Explicitly, E* requires
FIGURE |. Bifurcation diagrams for r, k, B, and a. Fixed parameters are d = 0.0039, pw = 0.693, T= 1, and K =2 x 10!!. The other parameters used are (a) a= 0.0693, 6B =0.0014, and k = 200, (b) r=0.7, a= 0.0693 , and k = 300, (c) r= 1.0, a = 0.0693, B = 0.0012, (d) r = 0.675, B = 0.0014, and k = 200.
FIGURE |. Bifurcation diagrams for r, k, B, and a. Fixed parameters are d = 0.0039, pw = 0.693, T= 1, and K =2 x 10!!. The other parameters used are (a) a= 0.0693, 6B =0.0014, and k = 200, (b) r=0.7, a= 0.0693 , and k = 300, (c) r= 1.0, a = 0.0693, B = 0.0012, (d) r = 0.675, B = 0.0014, and k = 200.
FIGURE 2. Bifurcation diagram for t. Note that the chronic steady state is very “close” to unstable, as determined by the other parameter values, which are fixed at r= 0.55, d = 0.0039, u = 0.693, T= 1, K =2 x 10!!, a = 0.0693, B = 0.0014, and k = 200.
FIGURE 2. Bifurcation diagram for t. Note that the chronic steady state is very “close” to unstable, as determined by the other parameter values, which are fixed at r= 0.55, d = 0.0039, u = 0.693, T= 1, K =2 x 10!!, a = 0.0693, B = 0.0014, and k = 200.
FIGURE 3. Simulations of two new infections leading to chronic HBV under different values of r. Other parameter values are r = 1, a = .0693, B = .0018, d = .0039, k = 200, wp = .693, t= 1, and K =2x 10!!. Lau et al. [23] present data from a clinical study of patients suffering chronic HBV infection but undergoing a course of lamivudine treatment, and Min et al. [5] fit a basic virus infection model to this data. In Lau’s study, treatment lasted 48 weeks, and the patients were monitored for an additional 24. Viral loads were reduced by 4 to 5 orders of magnitude during the course of treatment, but quickly rebounded following cessation of therapy. As lamivudine interferes only with viral replication within the cell it is not expected to affect viral infection of healthy cells. Therefore, within our model, we assume
FIGURE 3. Simulations of two new infections leading to chronic HBV under different values of r. Other parameter values are r = 1, a = .0693, B = .0018, d = .0039, k = 200, wp = .693, t= 1, and K =2x 10!!. Lau et al. [23] present data from a clinical study of patients suffering chronic HBV infection but undergoing a course of lamivudine treatment, and Min et al. [5] fit a basic virus infection model to this data. In Lau’s study, treatment lasted 48 weeks, and the patients were monitored for an additional 24. Viral loads were reduced by 4 to 5 orders of magnitude during the course of treatment, but quickly rebounded following cessation of therapy. As lamivudine interferes only with viral replication within the cell it is not expected to affect viral infection of healthy cells. Therefore, within our model, we assume
FIGURE 4. Comparison of model results to clinical results for a course of lamivudine therapy.
FIGURE 4. Comparison of model results to clinical results for a course of lamivudine therapy.
that lamivudine treatment only reduces virion production by infected cells. The modified model follows: Since patients had chronic hepatitis B, we assume that they were at the chronic disease steady state, E*, before the initiation of treatment. For at least one set of parameter values within the ranges presented in Table 1, namely y = .9999, a = .011, d = .0039, B = 4.8 x 10->, k = 200, K =2x 10!!, r=1, t=1, and Lt = .693, we find excellent qualitative agreement between the model and actual data (Figure 4).
that lamivudine treatment only reduces virion production by infected cells. The modified model follows: Since patients had chronic hepatitis B, we assume that they were at the chronic disease steady state, E*, before the initiation of treatment. For at least one set of parameter values within the ranges presented in Table 1, namely y = .9999, a = .011, d = .0039, B = 4.8 x 10->, k = 200, K =2x 10!!, r=1, t=1, and Lt = .693, we find excellent qualitative agreement between the model and actual data (Figure 4).
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