Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Math Biol. Author manuscript; available in PMC May 1, 2011.
Published in final edited form as:
PMCID: PMC2841722

An Accurate Two-Phase Approximate Solution to an Acute Viral Infection Model


During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate its accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the fall in virus concentration from its peak was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of model parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and in investigating host and virus heterogeneities.

Keywords: Acute Virus Infection, Influenza, Virus Dynamics Model, Approximation of Nonlinear Differential Equations

1 Introduction

Mathematical models have been used to study the infection kinetics of viruses, such as HIV, hepatitis B and C, and influenza, and have resulted in estimation of such quantities as in vivo viral replication rates, virus half-lives and infected cell life-spans (Baccam et al., 2006; Ho et al., 1995; Lewin et al., 2001; Neumann et al., 1998; Nowak et al., 1996; Perelson et al., 1996, 1997). These models are frequently nonlinear and analysis is usually limited to parameter estimation and numerical simulation. Approximate analytically tractable solutions could provide easier interpretation of virus dynamics and identify the extent to which each parameter influences infection kinetics including viral rise, peak and decay and the associated time scales.

The basic model structure (Equations (1)(3)), summarized in Nowak and May (2001) and Perelson (2002), describes a viral infection using three state variables: target cells (T), infected cells (I), and free virus (V).


In Equations (1)(3), uninfected cells are supplied at constant rate s, die at rate d and become infected at rate βV. Infected cells are lost, either to apoptosis or removal by immune cells, at rate δ. Virion production occurs at rate p per cell and virions are cleared at rate c per day.

Virus initially grows exponentially while the target cell population remains relatively constant (Fig. 1). Nowak et al. (1997) found if T is constant, then viral growth occurs according to er0t, where r0 is the leading eigenvalue which solves r02+(c+δ)r0cδ(R01)=0. Here, R0 = βps/(cδd) is the basic reproductive number, which describes how many infected cells are produced per infected cell at the initiation of infection.

Fig. 1Fig. 1
Numerical solution of Equations (4)(7) using Patient 4’s parameters (Table 1) and data (boxes). (a) Free virus (V), given in log10 TCID50/mL of nasal wash. (b) Cell populations: target cells (T, dotted line), infected cells not yet producing ...

Following the initial exponential growth phase, virus levels reach a peak and then virus begins to slowly decrease. Target cells decline sharply, and the infected cell population peaks (Fig. 1). If target cell regeneration (s) is not considered, then following the peak virus asymptotically decays as eδt and the infection is cleared. On the other hand, if s is large, then following the peak a short period of viral decay takes place before a steady state is reached (Bonhoeffer et al., 1997; Stafford et al., 2000).

Viral dynamics models have largely been applied to persistent infections; however, the basic model has also been used to study an influenza A virus (IAV) infection. Baccam et al. (2006) and Handel et al. (2007) applied simple target cell limited models to human nasal wash data collected daily from six individuals infected with IAV. Influenza is an acute infection usually lasting 7–10 days; therefore, some modifications of the basic model were made in their analyses. They first generalized the basic model to a model that included an eclipse phase to describe the time it takes for an infected cell to produce virions after it is initially infected. As previously done in HIV models (Perelson et al., 1993), infected cells were split into two class: I1, representing infected cells that have not yet begun producing virions and I2, representing infected cells that are producing virus. Transition from I1 to I2 is assumed to occur at rate k with no death occurring before virus production begins. Productively infected cells are lost at rate δ. Lastly, the models did not consider processes which have longer time scales, such as natural cell death (d = 0). The source of new target cells, s, was also set to zero as little epithelial cell regeneration occurs before virus is cleared, and estimates of s based on data fitting were indistinguishable from zero (Baccam et al., 2006). This target cell limited model (with s = 0, d = 0) is described by Equations (4)(7) and is the focus of our work.


Baccam et al. (2006) fit Equations (4)(7) to estimate parameter values for all six patients. The dynamics are shown in Fig. 1, which plots free virus (panel (a)) and each cell population (panel (b)) against time. Values for parameters β, p, k, c, δ, the initial viral titer, V0, and the basic reproductive ratio, R0, estimated by Baccam et al. (2006) are provided in Table 1. For this model,

Table 1
Estimates of parameter values for the target cell limited model with an eclipse phase, Equations (4)(7), using nasal wash data of six patients infected with IAV taken from Baccam et al. (2006).


where T (0) = T0 is the initial number of target cells, which was fixed at T0 = 4 × 108 based on an estimate of the number of epithelial cells in the upper respiratory tract.

For each patient, the viral dynamics break into two distinct phases that are linear on a log scale. As illustrated by Fig. 2, an initial period of exponential growth is followed by a phase of exponential decay, a behavior shown by the data and numerical solutions of the model. Understanding this characteristic behavior through analytical approximations is the central goal of this paper.

Fig. 2
Log-linear fits to data (Patient 4 from Baccam et al. (2006)) separating the initial virus rise (days 1–3) and subsequent virus decay (days 4–7). The number of data points included in each phase was found by finding which two lines produced ...

While Equations (4)(7) are nonlinear and cannot be solved analytically, approximate solutions in each of the log-linear phases can be obtained. In this paper, we derive these approximations for the initial viral growth phase and the subsequent viral decay phase. Using the data and parameters (Table 1) presented in Baccam et al. (2006), we verify the accuracy of the approximations. Finally, we use the approximate solutions to compare slopes and durations of growth and decay among the six patients in order to develop hypotheses about the host heterogeneity causing differences in dynamics.

2 Phase I Approximate Solution: Initial Viral Growth

Phase I of acute viral infection kinetics is characterized by an initial dip in virus levels followed by exponential growth, as shown in Fig. 1. The dip occurs because virus is assumed to be continually lost, but there is a delay before virus production occurs. The exponential growth of virus is expected as long as the target cell level remains relatively constant because the system is linear when T (t) is constant. The total number of cells, Z(t)=T (t)+I1(t)+I2(t), during Phase I is also approximately constant (say, ZT0) as infected cells are at low levels during this phase, Z [dbl greater-than sign] I1 + I2. Throughout Phase I, target cells decline as they become infected eventually rendering this approximation inaccurate. A reduced system (Equations (9)(11)) is obtained by using this approximation in Equations (4)(7).


Equations (9)(11) are linear and can be solved. The characteristic equation is given by λ3 + 2 + + D = 0 which has a discriminant, Δ = B2C2 − 4B3D − 4C3 + 18BCD − 27D2, that is less than zero since B > 0, C > 0 and D < 0. Therefore, we find one real (λ1) and two complex (λ2,3) eigenvalues with associated eigenvectors, ξ1,2,3. Values of λ1,2,3 for each patient are provided in Table 2.

Table 2
Values for all six patients of the eigenvalues, λ1,2,3 (day−1), in the Phase I solution, the final time (t1, days) the Phase I solution is valid, and the start time (t2 or t2, days) of the Phase II solution.








Using initial conditions I1(0) = I2(0) = 0 and V (0) = V0, we obtain the Phase I solution,





with constants


The initial dip in virus levels can be attributed to λ2 and λ3 having negative real parts with magnitudes greater than λ1 (Table 2). Following this, however, eλ2t and eλ3t decay leaving eλ1t, which has positive real part, driving the exponential growth.

Ideally, we could identify a small number of parameters or parameter combinations that determine the slope of viral growth. However, the expression for λ1 is complicated, and we cannot find any single parameter that has the most influence on this early phase. Using linear regressions, as in Fig. 2, can provide a rough estimate of λ1. Although specific parameters cannot be estimated by either the approximate solution or linear analysis, numerical exploration of each parameter indicates that all play large roles except for the initial virus concentration, V0.

We specifically looked at the sensitivity of λ1 to the eclipse phase parameter, k. Baccam et al. (2006) found that including an eclipse phase gave more realistic parameter estimates than the simpler system without k, though the added complexity may not be statistically justifiable. The target cell limited model without k is equivalent to the basic model with s = 0 and d = 0, and the leading eigenvalue, r0, satisfies r02+(c+δ)r0cδ(R01)=0, as discussed by Nowak et al. (1997) in the context of an HIV infection and Lee et al. (2009) in the context of an influenza infection. The resulting solution is r0=(c+δ)/2±(δc)2+4βpT0/2. To compare the model with and without an eclipse phase, we plot λ1 and r0 versus k using the parameters of Patient 4 (Fig. 3). As expected, the calculated λ1 from our analysis converges to r0 as k → ∞ (Fig. 3 inset). We also find that for k in biologically feasible ranges, e.g. k [set membership] [2, 6], small variations lead to significant changes in the slope of viral growth.

Fig. 3
The Phase I leading eigenvalue, λ1, over a range of values of the eclipse phase parameter, k. Also shown is the leading eigenvalue, r0, of the basic model with s = 0 and d = 0 (dashed line). As k → ∞, λ1 converges to r ...

The Phase I solution is valid from t = 0 until some time t = t1, which is defined to be the time for which T0 is no longer a good approximation for Z(t). We set this to occur when T (t) has been depleted by ~ 10% of its initial value. This value is somewhat arbitrary; however, smaller values (5%) were not sufficient and larger values (15%) began to result in large deviations of the approximations from the numerical solution. Therefore, t1 solves T (t1) = 0.9T0. Terms involving exponentials of λ2 and λ3 can be ignored in the calculation of t1 since eλ1t is the dominating term. Solving 0.9T0 = T0κ1(1 + ξ12)eλ1t1, we obtain the expression,


Using the parameters in Table 1, we calculated the value of t1 for each patient (Table 2). Figure 4 illustrates the Phase I solution, numerical simulation of the target cell limited model, and t1 for each of the six patients.

Fig. 4
Phase I and II virus (V) approximate solutions (solid lines, respectively) plotted against experimental data (squares) and fits (circles) of Equations (4)(7) for all six patients. Virus titers are given in TCID50/mL of nasal wash. The final time, ...

3 Phase II Approximate Solution: Virus Decay

Following the initial exponential rise in Phase I, viral growth begins to slow as target cells sharply decline. At this point, most remaining cells are infected since target cells are nearly depleted. This behavior characterizes Phase II of the infection kinetics; therefore, during this phase, Z [dbl greater-than sign] T implying that I1ZI2. Using this approximation in Equations (4)(7) results in the reduced system


We define t2 to be the time at which the Phase II solution begins. Solving Equations (20)(23) yields the expressions,







This approximation is a combination of exponentials involving the parameters k, c and δ. This solution takes on a different form if k = c, k = δ and/or c = δ, which can be obtained by appropriate use of l’Hôpital’s rule.

During Phase II, growth of free virus has slowed significantly and a peak is reached. From here, virus decline begins and becomes exponential. In the region around the peak, all three exponential terms are involved; however, the rate of asymptotic decay depends on the values of k, c and δ. If all three are similar in value, each contributes throughout much of the infection resolution with the smallest dominating towards the end. On the other hand, if one is much smaller than the other two, it will dictate the slope of viral decay. If we have an a priori hypothesis about which is smallest, the slope obtained from doing a linear fit (Fig. 2) provides an estimate of this parameter.

The Phase II solution is valid from t = t2, the time at which T (t) becomes sufficiently small, until infection resolution. We assumed that T (t) ≈ 0 to find the approximate solution given by Equations (24)(27). However, T (t) is not identically equal to zero in the initial portion of this phase and is necessary to define t2. To derive an approximate solution for T (t), we expand each state variable as a power series in ε and write, for example, V (t) = V (0) + εV (1) + O(ε2). Because T (t) is small in this phase, we assume T (t) = εT (1) + O(ε2), i.e., T (0) = 0. The solution in Equations (24)(27) is equivalent to substituting these expansions into Equations (4)(7) and solving the zeroth order (O(ε0)) subsystem. In the order ε (O(ε1)) subsystem, we find that a solution for T (1) can be obtained by substituting Equation (27) into dT/dt = −βT V and integrating, yielding


The remaining order ε differential equations are a set of linear, non-autonomous equations that cannot be directly solved. However, numerically solving this subsystem, we find that the zeroth order approximation (Equations (24)(27)) is nearly indistinguishable from the order ε solution for the other state variables, particularly V (t).

We then set the time the Phase II solution begins to be when T (t) is at 10% of its initial value (T0 = 4×108 cells). Again, we found that smaller values are not sufficient while larger values produced inaccuracy. We are not able to derive an expression for t2 as we did for t1 because initial values of the state variables (e.g. Z(t2)) are unknown. There is a short time period where neither the Phase I nor the Phase II approximation is valid; therefore, we cannot use the values of state variables at the conclusion of Phase I for the initial value of Phase II. Because of this, a numerical method was used to find the value of t2 that satisfies T (t2) = 0.1T0 = 4 × 107 cells and the values of state variables at this time. Table 2 contains the calculated t2 for each patient. These times are plotted in Fig. 4 along with the Phase II solution for all six patients.

3.1 An Alternate Phase II Approximate Solution

The Phase II solution (Equations (24)(28)) is accurate only if the number of target cells is small at some arbitrary time t2 and continues to approach zero as t → ∞. In the initial stages of this phase, free virus peaks and infected cells make up the majority of the total cell population (i.e. T [double less-than sign] I1 +I2Z). This approximation continues throughout infection resolution so long as target cells continue to become infected. However, should free virus and/or infected cells be cleared too quickly for further infection of target cells to take place, then the total cell population will eventually consist mostly of susceptible cells (i.e. ZT [dbl greater-than sign] I1 + I2).

Using the numerical solution, we found the final number of target cells as t → ∞, denoted by T. With the estimated parameters, the value of T is zero cells for Patient’s 1, 2 and 3, two cells for Patient 4, and 17 cells for Patient 6. However, Patient 5 retains a significant number of cells, 1.37 × 107 cells, and therefore, the approximation of Phase II is initially valid but eventually fails (Fig. 4).

A dependence of the final number of target cells on R0 was shown for the classic epidemic model by Kermack and McKendrick (1927) and later by Diekmann and Heesterbeek (2000). For the epidemic model, target cell final size was determined to satisfy ln(T/T0) = R0(T/T0 − 1). The solution was found to be TT0eR0 for R0 [dbl greater-than sign] 1. Because of the similarities between Equations (4)(7) and the Kermack-McKendrick model, we find this equation to be a good approximation of T for the target cell limited model. Estimates of the target cell final size for each patient can then be easily calculated for a given value of R0, e.g. T ≈ 4 × 108e−19.1 = 2 cells for Patient 4 and T ≈ 4 × 108e−3.5 = 1.21 × 107 cells for Patient 5.

We further examined R0 (Equation (8)) to determine when the Phase II solution becomes invalid. A plot of target cell final sizes, given different parameter combinations, and associated R0 values revealed that low basic reproductive numbers result in significant levels of susceptible cells late in the infection (Fig. 5). The number of surviving target cells increases significantly for R0 < 7 leading to failure of the approximation. For Patient 5, R0 = 3.5 is significantly lower than that of the other five patients; therefore, an alternative approximation is needed.

Fig. 5
Target cell final size, T, with the corresponding value of the basic reproductive number, R0, for various parameter combinations.

In Phase I, we used the fact that the total number of cells (Z) remained relatively constant and that Z [dbl greater-than sign] I1 + I2 to derive an approximation. We find a similar behavior in Phase II as Z reaches a non-zero constant state when target cells are not fully depleted. We can then take advantage of the approximation used in Phase I to derive an alternate approximation for Phase II. Here, we have TT and define t2 as the time that the alternate Phase II solution begins. This approximation results in the alternate solution being defined by Z(t) = T8 + I1 + I2 and Equations (16)(18) but with T in place of T0 and with constants,


The approximation is valid for a shorter time period ( t2>t2) than the previous Phase II solution. This will always be the case since the previous Phase II solution only requires target cells to be at low levels but not necessarily constant. In that instance, target cells are still changing with respect to time before they reach a constant steady state. On the other hand, the alternate solution does not begin until target cells have reached this constant state.

To calculate t2, we find the time at which T becomes a good approximation for T(t). As with t2, we used numerical methods to determine the time when T(t2)T=25% T = 1.71 × 107 cells. We tried larger values than 25% but found they were inadequate.

Using the parameters for Patient 5 (Table 1) and values of the state variables at t2, we find one real (λ1 = −4.42) and two complex (λ2,3 = − 14.29 ± 3.71i) eigenvalues. Plugging T in for T0 in Equation (12) results in a negative value of λ1 in the second phase since R0=βpT/cδ<1. While this solution does not capture the viral peak, information concerning the slope can still be obtained. As before, λ1 dominates and defines the rate of viral decay. The alternate Phase II solution is plotted with t1, t2 and the numerical results in Fig. 6.

Fig. 6
Phase I and alternate Phase II virus (V) approximate solutions plotted against experimental data (squares) and fits (circles) of Equations (4)(7) for Patient 5. Virus titers are given in TCID50/mL of nasal wash. The final time, t1, of the Phase ...

4 Discussion

We studied an acute virus infection model, specifically the target cell limited influenza model (Equations (4)(7)), to find a two-phase approximate analytical solution. We focused on investigating the log-linear property of viral load data using human nasal wash data from six patients and their associated parameter values (Table 1), estimated in Baccam et al. (2006). Although the published estimates do not include confidence intervals on the parameters for individual patients, we use the estimated values to illustrate the accuracy of the approximations and to investigate the slopes and timing of virus growth and decay.

The initial phase, Phase I, is defined by target cells remaining relatively constant and lasts until approximately 10% of the target cells have been infected. During this phase, we found exponential viral growth to be driven by the leading eigenvalue, λ1, for a time t1 (Table 2, Fig. 4). While we cannot extract one or two parameters controlling the value of this complicated expression, we can use the equation to evaluate differences among hosts or among virus strains.

We examined how changes in k, the eclipse phase parameter, affect the value of λ1 because including an eclipse phase complicates the basic viral infection model but does not appear to change its characteristic behavior and may not be justified statistically (Baccam et al., 2006). We verified that λ1 converges to the leading eigenvalue, r0, of the basic model as k → 8 (Fig. 3). However, for k in biologically realistic ranges, λ1 is significantly lower than r0, given the same parameter values. This suggests that parameters will estimate to significantly different values with each of these models, a finding consistent with the conclusions made by Baccam et al. (2006). We also find that λ1 is quite sensitive to k such that small changes in the eclipse phase length could lead to large changes in the rate of viral growth and, thus, produce distinct growth phases.

When parameter values have been estimated, we find the general trend that fast viral growth (large λ1) results in rapid target cell depletion and an early peak (small t1). Patient 2 had the largest λ1 and, hence, a short Phase I lasting for only about 30 hours. In contrast, the smallest growth parameters belonged to Patient 5 whose exponential growth phase was twice as long.

This variability highlights the challenge in effectively prescribing influenza antivirals. Currently, the antiviral Tamiflu is recommended during the first 12–24 hours to ensure maximum efficacy (Aoki et al., 2003). In our analysis, the length of Phase I (t1) may be an indication of the available window to treat a patient effectively. The opportunity for antiviral administration may last longer in some patients (e.g. Patient 5) than in others (e.g. Patient 2).

We have not fully addressed the gap between Phases I and II. This middle phase is truly nonlinear, and we were unable to find a simple approximation. Although this intermediate phase is not necessary to understand the exponential nature of acute viral infections, it does remain a focus of future work.

In the final phase, Phase II, infection resolution takes place. In this phase, target cells have markedly declined and both infected cells and free virus have a peak followed by exponential decay. The rate of virus decay depends on the smallest of k, c and δ while the viral peak depends on all three of these parameters. This result is similar to that found for the basic model where virus was determined to asymptotically decline according to eδt when c [dbl greater-than sign] δ (Bonhoeffer et al., 1997).

We believe that δ, the infected cell death rate, will often determine the rate of virus decay. This is not true for all infections since the eclipse phase length can vary from 4–6 hours for influenza to a day or more for human immunodeficiency virus (HIV) (Dixit et al., 2004; Nelson et al., 2001). However, for many acute infections, infected cells are reasonably long lived compared to both free virus and the eclipse phase length. For several patients, the parameter values predicted by Baccam et al. (2006) do not reflect this observation. One possible explanation is that parameter estimation schemes find it difficult to distinguish between k, c and δ as each contributes to the viral decay in the latter phase, leading them to estimate two or all three parameters to similar value. To remedy this, one could fit parameters constraining k and c to be larger than δ.

With k [dbl greater-than sign] δ and c [dbl greater-than sign] δ, parameter estimation is not necessary to understand viral decay, and δ can be approximated by a linear fit such as that in Fig. 2. Additionally, the up slope obtained from fitting data in Phase I could be used to estimate λ1 and to compare hosts and/or virus strains. This presents an opportunity for biologists to utilize the tools developed here in a simple manner.

There are other ways to compare virus strains without access to specific parameter values as well. For example, genetically engineered influenza strains which differ only in their hemagglutinin (the surface protein used in cell entry) may have similar parameter values except for different infectivity rates, β. Using reasonable choices for the other parameters, the behavior of each strain (i.e. how quickly and for how long virus levels rise) could still be tested. We can also determine the extent of damage to the epithelium and virus transmissibility without the need for parameter estimation. These quantities, in terms of the parameters, can be investigated by taking the integrals of I2(t) and V (t) in each phase.

As illustrated in Fig. 4, the Phase II solution, given by Equations (24)(28), does not agree with numerical results for Patient 5. There was not an obvious indication in the data or linear fit that this patient was significantly different in any way. However, plotting the change in target cells over time (not shown) revealed that a large number of these cells remained late in infection. A reasonable estimate of this value can be easily found by calculating TT0eR0. The estimated parameters for this patient indicate a much smaller value of the basic reproductive number, R0, than the other five patients and, therefore, a larger target cell final size. When parameters fall in ranges that produce an R0 [less, similar] 7 (Fig. 5), as with Patient 5, the Phase II solution fails and an alternate approximation is required. We use a similar approach to that in Phase I to derive a solution which does allow for small R0’s. This approximate solution (Fig. 6) begins at a later time than the previous Phase II solution and does not capture the viral peak. However, if one is only interested in this late phase of decay, this approximation can be used for all patients where information on the rate of viral decay lies once again in λ1.

In summary, we have derived approximate solutions to the two exponential phases of acute virus infection dynamics. The approximations and the techniques we used are not restricted to an influenza A virus infection and could be easily extended to other virus models. The tools we describe are intended to complement parameter estimation and numerical simulation typically used for analysis of virus dynamics; however, several aspects are also amenable to use by experimentalists. This simplified analysis presents an opportunity to uncover new virus properties, investigate the implications of various drug therapies and aid in the development of therapies customized to individual patients based on the initial course of an infection.


This material is based upon work supported by the National Science Foundation under grant DMS-0354259 (AMS), the Modeling the Dynamics of Life Fund at the University of Utah and the 21st Century Science Initiative Grant from the James S. McDonnell Foundation (FRA). Portions were done under the auspices of the U. S. Department of Energy under contract DE-AC52-06NA25396 and supported in part by NIH contract N01-AI-50020 and grants RR06555-17 and AI28433-18 (ASP).

Contributor Information

Amber M. Smith, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA, Tel.: +801-581-6851, Fax: +801-581-4148.

Frederick R. Adler, Departments of Mathematics and Biology, University of Utah, Salt Lake City, UT 84112, USA.

Alan S. Perelson, Theoretical Biology and Biophysics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.


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