In this section, the use of the Jacobian prior for parameter estimation in a nonlinear system of differential equations will be studied. The system that will be used to conduct this study is a model for influenza A virus infection from [6]:
V9 “ rI ´ cV H9 “ ´βHV I9 “ βHV ´ δI
(4.8)
where V is the concentration of infectious viral particles with units TCID50{ml, H is the number of uninfected target cells, and I is the number of productively infected cells. By shedding viral titers, infected cells increase the concentration of viral particles at a rate of r per cell, and free viral particles are cleared at a rate of c per day. Uninfected cells interact with virus particles and become infected at a rate βHV . The infected cells die at a rate of δ per cell (where 1{δ is the average life span of an infected cell). The initial number of infected cells is taken to be zero, and the initial viral concentrations, V p0q “ V0, and initial number of target cells, Hp0q “ H0, are considered as parameters. Then, the vector of model parameters is a “ pV0, H0, β, r, c, δq P R6 “ A, with units pTCID50{ml, cells, pTCID50{mlq´1day´1, pTCID50{mlqday´1,day´1,day´1q, respectively.
Approach 2 again provides a systematic way to study which prior makes the posterior a better approximation of the parameter density. Figure 25gives a roadmap for the approach and the details for this specific setting are discussed here. The parameter density ρpaq is defined to be a normal multivariate density with mean parameter vector ¯a and covariance matrix Σ “ diag pp¯a1σ1q2, p¯a2σ2q2, . . . , p¯a6σ6q2q. In the examples below, ρpaq is constructed by prescribing ¯a and the vector σ “ pσ1, σ2, . . . , σ6q. As before, ρpaq is sampled to obtain A “ ta1, a2, . . . aNu, aj P A. The number of observation times for the data will be taken
to be m “ 2 (so that p “ mn “ 6 and DaF paq is a square matrix). With this setup, for a fixed a, the solution map gives, F paq “ `
pV pt1q, Hpt1q, Ipt1qqJ, pV pt2q, Hpt2q, Ipt2qqJ˘
“ y.
The forward solution, F pajq “ yj, is computed for all j, yielding the collection of data Y “ ty1, y2, . . . yNu, yj P Y . As in the previous treatment, multivariate kernel density estimation is used to represent the sample Y with a density function ˜ηpyq, and the Metropolis-Hastings algorithm is implemented using various priors, to obtain Mπ. The posteriors, Mπ, will be compared to the A, the sample of the parameter density, and to each other. A series of four examples using this approach is now presented.
Example 6. In this example, ρpaq is constructed by selecting ¯a “ p0.093, 4 ˆ 108, 2.7 ˆ 10´5, 0.012, 3.0, 4.0q and σ “ p0.01, 0.05, 0.01, 0.05, 0.02, 0.01q. Here, the mean parameter values are chosen to be the average of the best-fit parameter values from [6] and σ was prescribed to contain small values. The observation times are taken to be tt1, t2u “ t1, 2u, and the number of samples for A and Y is N “ 1000. In Figure 32 (a), the solution to the system for the mean parameter ¯a is given by the blue curves, and the box plots represent the sample of data Y , graphed at the associated time points for the corresponding variables.
The figure shows that there is a large spread in the data for this selection of σ. Part (b) of Figure32shows marginal histograms of Y , and marginals of the kernel density estimate are given by the red curves. Finally, in part (c) of the figure, A is compared to the posterior samples MJ ef f, MU nif, and MJ ac obtained from the Metropolis-Hastings algorithm. We can see that MJ ac most accurately approximates A for the parameters H0, β, r, and δ. None of the posteriors approximate the marginals for V0 or c well. The parameter c is related to the virus clearance; we can see from Figure 32, that for this choice of t1 and t2, the data do not capture any information about the decay of the virus. As before, a two sample Kolmogorov-Smirnov test was conducted, however the p-values were too small in each of the cases to provide any useful insight in comparing MJ ef f, MU nif, and MJ ac; thus we will rely on the marginal depictions for comparison. This will also be the case in the subsequent examples.
Example 7. In this example, everything is the same as Example 6, but the value of σ is increased to σ “ p0.05, 0.05, 0.05, 0.1, 0.04, 0.02q for the construction of ρpaq. Figure 33 (a) again shows the solution for the mean parameter and the box plot representations of Y . As expected, the broadness in the spread of the data is increased. In Figure33(b), we compare
0 1 2 3 4
Figure 32: Example 6: (a) Solutions curves from the mean parameter value and box plot representations of Y . The box plots at t “ 1 for V and I are difficult to see because they are tight relative to the figure scale. (b) marginal histograms of Y and the kernel density estimate, (c) curves representing marginalized histograms for A (red), MJ ef f (blue), MU nif (black), and MJ ac (green).
A, MJ ef f, MU nif, and MJ ac and see that again, MJ ac most accurately approximates A for the parameters H0, β, r, and δ. In this example, there is a more pronounced discrepancy between MU nif and MJ ef f and A than in Example 6. The observation times remained the same; still no information is gathered about the decay rate of the virus, and thus the marginal for c is not approximated well by any of the posteriors.
Example 8. In this example, different time points are chosen in an effort to obtain more information about c. Let ¯a and N be the same as the previous two examples, and choose
0 1 2 3 4
Figure 33: Example 7: (a) Solutions curves from the mean parameter value and box plot representations of Y . The box plots at t “ 1 for V and I are difficult to see because they are tight relative to the figure scale. (b) curves representing marginalized histograms for A (red), MJ ef f (blue), MU nif (black), and MJ ac (green).
σ “ p0.01, 0.02, 0.01, 0.01, 0.02, 0.01q and tt1, t2u “ t2, 3u. Figure 34 (a) shows the solution for the mean parameter and the box plot representations of Y . The new observation times now capture data in both the increasing and decreasing portions of the V solution. Figure 34 (b) depicts marginals of A, MJ ef f, MU nif, and MJ ac. There is a vast improvement in the estimation of c. We find that all of the priors produce very similar posteriors, and all of the marginals approximate A well, except for V0. In this example, the accuracy of the approximation of the parameter density between the different priors is indistinguishable.
Example 9. For the final example, the mean parameter used to define ρpaq is changed to
¯a “ p0.25, 4ˆ108, 1.4ˆ10´2, 2.7ˆ10´5, 3.2, 3.2q, from [77]. We choose σ “ p0.01, 0.0001, 0.01, 0.05, 0.02, 0.01q, N “ 1000, and tt1, t2u “ t1, 2u. As before, Figure 35(a) shows the solution for the mean parameter ,¯a, and the box plot representations of Y , and (b) depicts marginals of A, MJ ef f, MU nif, and MJ ac. In a similar manner as Example 8, the observation times capture data in both the increasing and decreasing portions of the V solution, so the posteriors for
0 1 2 3 4
Figure 34: Example 8: (a) Solutions curves from the mean parameter value and box plot representations of Y . The box plot at t “ 3 for H is difficult to see because it is tight relative to the figure scale. (b) curves representing marginalized histograms for A (red), MJ ef f (blue), MU nif (black), and MJ ac (green).
c are meaningful. In the marginals of the posteriors, we see that both initial condition parameters are not approximated well with any of the three priors. For the remaining parameters, the marginals of MJ ef f, MU nif, and MJ ac are all quite similar to each other.
A is matched well for the parameters β, c, δ, but less accurately for r. Again, we cannot distinguish between the performance of the three priors.