Table 3: Estimated constant parameters with PSO compared with the results of the multistage smoothing-based (MSSB)
approach and the spline-enhanced nonlinear least squares (SNLS) approach, see (Liang et al., 2010).
Patient I Patient II
PSO MSSB SNLS PSO MSSB SNLS
s 368.94 254.49 397.09 238.40 18.38 45.45
d 0.46 0.34 0.49 0.58 0.04 0.10
ξ 2.16 1.09 1.09 2.48 0.43 0.43
k 1317.35 1284.27 288.57 1283.17 1190.81 479.18
c 3.60 2.46 2.46 2.50 3.78 3.78
rameters, the time-varying parameter shows similar
behavior to those discovered with SNLS: After an ini-
tial perturbation they remain mostly constant. Patient
I data shows much more significant overshoot in β
than patient II, this can be due the different treatment
effects, but it is also possible that this is an artifact
of the method (e.g. numerical instability). Unlike
SNLS algorithm we avoided the usage of splines for
parameter estimation, that would address the design
questions: what kind of spline and of which order to
choose, moreover it would introduce new parameters.
5 CONCLUSIONS AND FUTURE
WORK
Here we introduced a novel strategy for the estimation
of constant and time-varying parameters of the long-
term HIV-1 dynamic time-series. As this methodol-
ogy depends on the noise, first the noise model was
estimated from the data. The noise was found to be
multiplicative zero mean Gaussian. Constant param-
eters were then estimated with PSO, while the time-
varying parameter was further refined using an EKF
algorithm adopted to continuous models. In com-
parison with existing methods here we found much
smaller deviations between the estimated parameters,
on the two patients. No approximation with smooth-
ing or splines were used, instead, with EKF we ap-
plied the noise model directly to estimate the time-
varying parameter in a Bayesian manner. Thus we
believe this method offers a simpler and more flexible
framework for estimating the time-varying parameter.
Our future work aims at better characterization of
the noise and the time-varying parameter involving
more patients’ data and simulated experiments.
ACKNOWLEDGEMENTS
This work was supported by FCT (Portugal) un-
der project PTDC/EEA-CRO/100128, HIVControl -
Control based on dynamic modeling of HIV-1 infec-
tion for therapy design, and INESC-ID multi-annual
funding through the PIDDAC program funds. Andr´as
Hartmann received PhD fellowship from FCT - Fun-
dao para a Ciˆencia e a Tecnologia under the reference
SFRH/BD/69336/2010.
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