The standard model that has been used and modified for studying hepatitis C viral dynamics is the Neumann et al. model [30]. The three variables this model keeps track of are the target cells T, in Eq. (1a), the infected cells I in Eq. (1b) and the free virus V in Eq. (1c). The target cells T are produced at constant rate s, die at per capita rate d and are infected by virus V at constant rate β. The infected cells I increase with the new infections at rate βV (t)T (t) and die at constant rate δ. The virus V is produced at rate p by each infected cells and is cleared at constant rate c. The ε term denotes the effectiveness of the anti-viral treatment that decreases the production from p to (1 − ε)p. Formally the ensemble of ODEs for this model is: (1a)dT(t)dt=s−βV(t)T(t)−dT(t) (1b)dI(t)dt=βV(t)T(t)−δT(t) (1c)dV(t)dt=(1−ε)pI(t)−cV(t).

From the mathematical perspective, the standard model is realtively much simpler than the multiscale model. Although it is nonlinear, it can be solved analytically when assuming that T is constant.

The multiscale model of hepatitis C virus (HCV) dynamics has been formulated in recent years [18, 36, 4] as a more complex system in order to study HCV dynamics during therapy. It keeps track as in the standard viral dynamics model of uninfected target cells in equation (2a), productively infected cells in equation (2b), and free virus in equation (2c) along with considering the intracellular viral RNA dynamics in an infected cell by adding equation (2d).

The equations were formulated as follows: (2a)dT(t)dt=s−βV(t)T(t)−dT(t) (2b)∂I(a,t)∂t+∂I(a,t)∂a=−δ(a)I(a,t) (2c)dV(t)dt=(1−εs)∫0∞ρ(a)R(a,t)I(a,t)da−cV(t) (2d)∂R(a,t)∂t+∂R(a,t)∂a=(1−εα)α(a)−[(1−εs)ρ(a)+κμ(a)]R(a,t), subject to the initial and boundary conditions V(0)=V¯, T(0)=T¯, I(0, t) = βV (t)T (t), I(a,0)=I¯(a), R(0, t) = 1, and R(a,0)=R¯(a).

Model parameters of T, Eq.(2a), and I, Eq. (2b), are similar to the standard model where a is the age of infection and t is the time duration from therapy initiation. The quantity of intracellular viral RNA R, in Eq. (2d), depends on its production α its degradation μ and expulsion from the cell ρ. The quantity of free viruses V shown in Eq. (2c) depends on the number of assembled and released virions and their clearance rate c. While those parameters should depend on the cell age a, in practice they are considered constant.

An important consideration in this model is that the treatment starts after the infection has reached its steady state. The steady states of the different functions are R¯(a), I¯(a), V¯ and T¯. Given N, the total number of virions produced by a cell in its life-span, it can be shown that those values are: (3)N=ρ(α+δ)δ(ρ+μ+δ)R¯(a)=αρ+μ+(1−αρ+μ)e−(ρ+μ)aI¯(a)=βV¯T¯e−δaT¯=c/(βN)V¯=(βNs−dc)/(βc).

Unlike the standard model, three different antiviral effects of therapy can be distinguished in the multiscale model. The decrease in viral RNA synthesis is represented by ε_α, the reduction in secretion by ε_s and the increase in viral degradation by κ ≥ 1.

Through the method of characteristics, as was derived in [18] and explained in more detail herein in Appendix A, an analytical solution can be found the variable R(a, t). The same method can be applied to derive I(a, t) solution. The ensemble of equations (2) represent the full model. (4)R(a,t)={ a<tαρ+μ+(1−αρ+μ)e−[ρ+μ]a  a≥tα(ρ+μ)+(R¯(a−t)−α(ρ+μ))e−(ρ+μ)t  (5)I(a,t)={βV(t−a)T(t−a)e−δaa<tI¯(a−t)e−δta≥t

From equations (2) it can be noticed that computing V (t) necessitates an integral. If a < t, in other words the cell age is younger than the time of treatment, i.e., infection occurs after initiation of treatment, the term I(a, t) of the integral in Eq. (5) depends itself on V (t) and T (t). As it will be shown, this makes the analytical solution approximative and impedes the use of canned numerical solvers for this system of equations.

To derive an analytical solution for V (t) the dependancies of I(a, t) makes it difficult unless some additional assumptions are taken into account. The solution will be divided into two cases, the short term approximation and long term approximation [18, 36], which mainly differ on how I(a, t) is treated.

We implemented a variety of numerical methods from the simplest to the more advanced ones that are tailored to our needs, starting from first order methods and describing them in detail in order for our derivations to remain self-contained. We then build from bottom up towards higher order methods, adaptive stepsize methods and implicit schemes in order to better characterize the equations of our model using advanced numerical schemes [43, 44].

The general scheme of a numerical method in our context requires to solve a function f(t, x) which depends on time and a variable x. In this section we explore the solutions of the functions T and V and—except in subsection 2.4.4 about the Rosenbrock method—f(t, x) will be always defined as follows. As in Eq. (2) in the case of V we have f(t,x)=(1−εs)∫0∞ρ(a)R(a,t)I(a,t)da−c∗x and for T, f(t, x) = s – d * x – βV(t) * x.

As mentioned in Sec. 2.4.1, the structure of the equations is problematic when trying to use some general fitting methods as canned solvers (e.g., calling the Levenberg-Marquardt [43] as a canned method). In passing, we briefly show here how the multiscale model equations can be adapted for the fitting of parameter s when combined with a full implementation of the Levenberg–Marquardt method, without the use of canned solvers. Such a scheme can be generalized for other variables besides s and is left for future work that is beyond the scope of this contribution.

Briefly stated, the main idea for the development of a new parameter fitting scheme that should be more flexible than what was used in [18], which utilized Levenberg-Marquardt as a canned method, is how the the derivatives of T and V with respect to s can be evaluated. As schemes to solve dT(t)dt and dV(t)dt being already implemented, we can notice that d∂T(t)∂sdt and d∂T(t)∂sdt can be determined and thereafter solved with the already implemented schemes. Those values are exactly: (23a)d∂T(t)∂sdt=1−β(∂V(t)∂sT(t)+V(t)∂V(t)∂s)−d∂V(t)∂s (23b)d∂T(t)∂sdt=(1−εs)∫0∞ρ(a)R(a,t)∂I(a,t)∂sda−c∂V(t)∂s where: (24)∂I(a,t)∂s{β(∂V(t−a)∂sT(t−a)+V(t−a)∂T(t−a)∂s)eδaa<te−δaa≥t.Let us note that the partial derivative of R(a, t) is 0 and thus ignored.

These equations can be directly integrated in the previously discussed schemes since they are working on a set of ODEs in parallel. Practically, we need to increase the dimensions of the vector f, defined in subsection 2.4.4, to include those new equations. We now need to solve for f(yn+x):=[dT(t)dt,dV(t)dt,d∂T(t)∂sdt,d∂V(t)∂sdt]. The only additional variable to be inserted is a tolerance on the error of the fitting. In section 3.6, we report on a successful proof of concept for the new parameter fitting scheme that was briefly outlined here and will be fully dealt with in future work.

The short-term approximation is shown in Fig. 1a in blue. The results are shown for only two days since the behaviour is smooth and conserved on longer time scales in agreement with previous results (Fig. 2A in [18]). It is clear that after twelve hours the value converges far from the PDE model solution (the PDE model solution is computed by the canned solver Default implementation as described at the beginning of the section). This is expected since the effect of the treatment on the infection rate is not taken into account. The analysis will therefore focus on the long-term approximation.

In [36] it is hypothesized that “The long-term approximation is an underestimate of the PDE model solution since some infection events are being ignored. However, with realistic parameters characteristic of potent therapy, the difference between them is very small.” Fig. 1b depicts the result of our scheme, measuring the difference between the long-term approximation and the solution of the multiscale PDE model (Figure 2B in [18]). We show that the long-term approximation is converging to a number with the numerical solution and we are indeed obtaining that the long-term approximation is an underestimate of the PDE model solution (as the difference between the long term approximation and the solution of the PDE model is negative). The difference of less than 0.01 that we are obtaining is very small compared to the y-axis units shown in Figure 1 of [36]. Therefore we are actually viewing in Fig. 1b this small effect that was inferred in [36] regarding the long-term approximation. This limitation of the long-term approximation is worthwhile mentioning although it is less severe than the limitation of the short-term approximation. Therefore in practice, the long-term approximation is favored over the short-term approximation, although the numerical solution offers an attractive alternative to analytical approximations and could be easier to adjust when introducing changes to the model.

A significant observation that can be inferred already when starting to experiment with the multiscale model equations by numerical solutions is their stiffness, or how they can be numerically unstable even with a small stepsize. The most straightforward numerical method to implement is forward Euler, described in Sec. 2.4.2, followed by RK4 discussed in Sec. 2.4.3. An implicit version of RK4 in the same section is the Gauss-Legendre method. With small time-steps, the quality of the results is similar between all methods. But as we increase the size of the steps, the instability of the equations can be observed as show in Fig. 2 over 20 days. Both forward Euler and RK4 quickly diverge from the long-term approximation, which is close to the numerical solution as shown previously. On the other hand the implicit method of second order Gauss-Legendre that belongs to the Runge-Kutta family tends to the correct solution.

Without taking a detailed examiniation of the equations, a straight-forward strategy that is simple to implement and pursue is the utilization of a canned solver (an ODE solver called from by a command within a program that is written in a higher level language such as Matlab and Mathematica, or Python). Those canned solvers solve the equations automatically, determining the time steps given a tolerance provided by the user (only initial and extremal values for the time step can be given). However, a restriction of such a strategy is that the equations need to be self contained in a set of ODEs without additional integrals that may complicate the matter, as discussed in Sec. 2.4.1. The integral term (1−εs)∫0∞ρ(a)R(a,t)I(a,t) da cannot be solved analytically and thus a numerical approach must be used. Doing so requires to know for which times t the values of V and T are known, and what they are. Such a scheme is incompatible with the canned solver strategy. A workaround to not fully implement a solver within a program is therefore to use a canned solver for small increments of times h and after each iteration, compute the integral again. We illustrate this approach in Fig. 3 and it is the strategy used to evaluate the system with what we call a Default method. This is not sustainable for long time intervals as the number of iterations grows quickly and defeats the purpose of the implementation of advanced ODE solvers that utilize multistep methods or adaptive timestep methods. In this strategy, there is no advantage in using in the canned solver a sophisticated multistep method such as the Backward Differentiation Formula (BDF) or the Rosenbrock method that are especially suited for stiff equations because of the small time increments that do not provide a chance to exploit the advanced method used in the canned solver.

In contrast, a full implementation of the Rosen-brock method gives full access to the error term. With that information, a more subtle integration of the problematic integral is performed as shown in Fig. 4. There are two main differences. First, we need to keep track of the times at which the ODE system is solved in the array ℐ, which is required to compute the integral since those are not fixed anymore. Second, after each iteration we use the known error to modify if possible the time step h to potentially decrease the number of required iterations. This is shown to be a successful strategy as presented in Table 6.

As previously explained, the integral term (1 – ε_s) ∫0∞ρ(a)R(a,t)I(a,t) da impedes the use of adaptive stepsize in the canned solvers. Such methods are essential to allow fast computations of complex models like ours. We present in Table. 6 the number of iterations computed using the Rosenbrock and Dormand-Prince methods, shown in Sec. 2.4.4, in comparison with the default canned method for three different values of h and h_a. With the smallest step-size, the Rosenbrock method is more than nine times as efficient as the default canned method, giving results of similar accuracy as shown in Fig. 5 and with simulations extended to 14 days. This is crucial since the methods with fixed stepsize can take up to tens of seconds per day of simulation with h = 0.001 and h_a = 0.01. Under those conditions a nine-fold increase in speed, through the decrease in the number of iterations, provides a much needed advantage when testing large number of parameters over long time periods. This is the main inference from Table. 6, showing the advantage of adaptive stepsize methods. Interestingly, with the highest value of h, the Rosenbrock method takes proportionately additional steps. This is due to the greediness of the stepsize adjustment, which when increased too quickly (thereby inducing a large error) backtracks and starts again with a smaller value. The Dormand-Prince adaptive stepsize method performs somewhat better than the Rosenbrock method in terms of number of iterations but is vulnerable to the stiffness of the multiscale model equations, as elaborated below.

Simpler methods than Rosenbrock exist with adaptive stepsize. A well used scheme is the Dormand-Prince method that was described in Sec. 2.4.4. The latter is an explicit method, while Rosenbrock is implicit, and therefore Dormand-Prince is vulnerable to the stiffness of the equations. We show in Fig. 5 that with a modest increase in the stepsize h the Dormand-Prince method already converges to the wrong value compared to the default method and the Rosenbrock method. We note that Rosenbrock needs almost a third of the number of iterations as the Default method in that case with similar results.

In sub-section 2.5, we briefly sketched an idea for a new strategy that performs flexible parameter fitting and is in line with our overall strategy to tackle the problematic issues with the integral mentioned in sub-section 2.4.1. As a proof of concept, to evaluate our method, we generated an ensemble of data points over one day with our implementation of the Rosenbrock method and the parameters in Table 5.

We then applied our implementation of the Levenberg–Marquardt method with different starting points for the parameter s, which can not be recovered when using the long-term approximation. Using 20 values of s equally distributed in [0, 2s] we observed that in all cases the method converged to the correct value of s used to generate the data points, inside the provided tolerance on the error of 0.001.