Mathematical models have proven valuable in understanding the dynamics of viral infections in vivo within host cells and were originally devised to examine HIV infection (reviewed by Perelson and Ribeiro, 2013). For interactions of that sort, a basic three-component dynamical systems model consisting of an uninfected target-cell population, an infected cell population, and the free virus population was proposed (see Figure 1). This model implied that the propagation of the virus was limited by the availability of susceptible target-cells and hence is now characterized as target-cell-limited (Phillips, 1996). Assuming a rapid enough time-scale for the free virus dynamics so that a quasi-steady-state approximation could be employed, Tuckwell and Wan (2004) formally reduced this basic target-cell-limited viral model system to a two-component one consisting of the uninfected and infected target-cells. They then showed that there were no periodic solutions for the two-component model and that the trajectories of both systems remained quite close. DeLeenheer and Smith (2003) and Prüss et al. (2008) studied the global stability of the biologically relevant equilibrium points for this basic target-cell-limited viral model system and found that its behavior depended upon the size of a particular non-dimensional parameter R₀, the basic reproductive number, to be defined in the next section. If R₀ < 1, they demonstrated that the virus-free equilibrium point was globally asymptotically stable, while if R₀ > 1, this property shifted to the disease-persistence equilibrium point.

The results cited above use either standard techniques of dynamical systems theory or numerical simulations. Defining α as the ratio of the death rates of the infected to the uninfected cells, Burg et al. (2009) classified such viral interactions to be either cytopathic or non-cytopathic depending upon whether α > 1 or α = 1, respectively. During cytopathic viral interactions the infected cells are killed by the virus during the course of infection. Some viruses are intrinsically non-cytopathic because they replicate in a relatively benign manner while others actively maintain such a state by shutting down all destructive processes, activating non-destructive mechanisms, or inducing alternate non-damaging replication programs (Plesa et al., 2006).

In what follows, we shall consider non-cytopathic retroviral interactions; that is, interactions that satisfy α = 1, which is believed to be the case for the equine infectious anemia virus (EIAV) (Schwartz et al., 2018). EIAV shows many characteristics similar to other retroviruses, including a very rapid replication rate and high levels of antigenic variation. It, however, is unusual among retroviruses in that most infected animals, after a few episodes of fever and high viral load, progress to a stage with low viral load and an absence of clinical disease symptoms. The horses effectively control viral replication through adaptive immune mechanisms. Given that this differs from the retroviruses human immunodeficiency virus (HIV) and simian immunodeficiency virus (SIV), in which the infected develop immunodeficiency and disease, EIAV is especially interesting to study in clinical research as well as by using mathematical models. When adopting the mathematical model depicted in Figure 1, the viral clearance rate γ captures these adaptive immune system response mechanisms. In section 2, we shall employ a systematic two-time method (Matkowsky, 1970) to deduce a quasi-steady-state asymptotic closed-form analytic solution of that basic target-cell-limited viral dynamics model.

Although such non-linear problems can be solved numerically the computation must be performed sequentially for each different set of parameter values. The advantage of this asymptotic approach is that it yields an analytic representation, involving the parameters as well as time, required for least-squares parameter-identification curve-fitting procedures to experimental data. We conclude by applying this approach to an experimental data set on EIAV infection.

The basic model for viral dynamics (see Anderson and May, 1992; Tuckwell and Wan, 2004; Burg et al., 2009; Stancevic et al., 2013) that describes the interactions of a virus with target-cells is given by

where T represents the uninfected target-cell population, I is the population of infected cells, and V is quantity of free virus while t, as usual, represents time. It is assumed that the target-cells are produced at a constant rate λ and die at a rate ρT. Free virus infects target-cells at a rate βTV and infected cells die at a rate δI. New virus particles are produced at a rate bI and are cleared at a rate γV. For the model under consideration, we assume that the viral interaction is non-cytopathic and therefore take ρ = δ in the analysis which follows.

We begin by introducing the dimensionless quantities

which upon substitution in Equations (1) yields the dimensionless system

In this section we examine the qualitative behavior of the quasi-steady-state approximation given by Equations (18) and (19). We then compare the quasi-steady-state approximation with a numerical simulation of Equations (2) using equine infectious anemia virus (EIAV) data (Schwartz et al., 2018).

From the form of x₀(τ), it is readily seen that when R₀ = 1, x₀(τ) = f(τ) → 1 as τ → ∞. If R₀ < 1 then x₀(τ) = g(τ) → 1 while if R₀ > 1, x₀(τ) = g(τ) → 1/R₀ as τ → ∞, where x₀(τ) is expressed as a percent of its initial population. This is consistent with the global stability results mentioned in section 1.

Figure 2 is a plot of the three uniformly valid composite functions xu(0)(τ), yu(0)(τ), and vu(0)(τ). Parameter values used are median values reported in Schwartz et al. (2018) for the equine infectious anemia virus. Specifically, we take

Given that a dimensionless time unit (τ = 1) corresponds to 21 days, we see that the uninfected cell population remains relatively constant for approximately 7 days (τ = 0.33). This is followed by a period of eight to ten days of rapid infection of the uninfected cell population at the end of which approximately 95% of the population has been infected by the EIAV.

Figure 3A provides a comparison of the one-term asymptotic representation of the cell population (solid black curve) given by Equation (16) with a numerical simulation (dashed curve) of Equation (2) using the parameter values given above. Figure 3B provides a comparison of the one-term asymptotic representation of the free virus population (solid black curve) with its numerical simulation (dashed curve). The initial virus population was taken to be 450 × β/ρ ≈ 0.00307 viral RNA copies/ml. We note the excellent agreement between the analytic asymptotic representation and numerical simulations.

Researchers that employ the basic viral dynamics model now have an analytic representation involving the parameters that provides a vehicle for least-squares parameter-identification curve-fitting procedures to experimental data. In particular, given a time series population data set {(tn,Tn)}n=1N and our analytic solution for uninfected target-cells in dimensional variables denoted by T(t; λ, ρ, β, b, γ), a parameter identification residual least squares fit to that data is determined by defining (Torres-Cerna et al., 2016)

and minimizing this function by solving for λ_c, ρ_c, β_c, b_c, γ_c such that

employing the appropriate algorithm. This procedure can be accomplished much more efficiently if one has a closed form representation for T(t; λ, ρ, β, b, γ) as in our case.

We note that for the basic target-cell-limited viral dynamics model, the deduction of an analytic solution for the quasi-steady-state approximation is crucially dependent on the non-cytopathic condition α = δ/ρ = 1 and we have selected parameter values relevant to this scenario for EIAV. If this were the only non-cytopathic virus, our development restricted to the spread of infection in horse populations might not be representative enough to enlist general interest from virologists. Besides EIAV, however, it has been shown that this non-cytopathic assumption is reasonable for a fairly wide class of important viral interactions in human and other animal populations as well, for example, Hepatitis B and C viruses (Wieland and Chisari, 2005). In addition, non-cytopathic enteroviruses such as the coxsackie virus B, one of the agents suspected to be responsible for chronic fatigue syndrome (Landay et al., 1991), cause persistent infections in their host's cells. Another non-cytopathic virus infecting human populations is the Newcastle disease virus (Carver et al., 1967). Finally, Table II in Marcus and Carver (1967) lists a collection of similar non-cytopathic viruses inducing intrinsic interference, among which is the hemadsorption simian virus.

We have been investigating the non-cytopathic interaction of EIAV infection. While similar to human immunodeficiency virus (HIV), EIAV differs from the latter in that it is not fatal, partially because the horses' immune systems help to effectively control the virus. Thus, studies of EIAV infection are of importance since they serve as useful prototypes of viral dynamics and immune control, which may have implications in the development of vaccines for HIV and other retroviral infections.

RAC led the project, performed model analysis, ran numerical simulations, and wrote the paper. EJS initiated the project, gathered data for application of the model, and assisted with the interpretation of results and writing the paper. DJW introduced the non-cytopathic assumption, performed model analysis, and assisted with writing the paper.