D. N. SUN ET AL.

Copyright © 2013 SciRes. ENG

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3. Numerical Tests

In real applications, the model must be calibrated against

experimental data. In this numerical study, however, a

twin experiment is carried out: a reference solution is

generated with the model itself using the parameters the

same as in Section I.

In this part, we design 3 numerical tests, 2 tests with

random error in the synthetic data, which denoted TX1

with no errors, TX2 with 1% errors, TX3 with 3% errors,

respectively. Only the infected CD4 + T-cells observed

data are valid, that is

. The first guess of

them are 1.0E−3, 0.2 and 1.5, respectively. The observed

dada are plotted in Figure 4. Th e numerical resu lts listed

in Table 1. The cost function descent curve showed in

Figure 5.

4. Summary and Discussion

Adjoint data assimilation method is employed to estimate

the parameters of a kind of HIV model. The parameters

estimated are μ, α and γ. The method is based on an op-

timal control approach where by a cost function measur-

ing the discrepancies between numerical computed and

measured is minimized, subject to constrains consisting

of equations of the model. The numerical scheme used is

the forth order Runge-Kutta method. In order to test the

validity of this method, we have designed several expe-

riments. The results showed the method is valid. The

research results are useful to understand the HIV model

and helpful to establish a robust and effective HIV con-

trol and prediction.

Figure 4. The observed data.

Table 1. Estimated parameters values.

TX1 TX2 TX3

μ 1.371E−3 1.412E−3 1.501E−3

α 0.328 0.343 0.352

γ 1.996 2.051 2.132

Figure 5. The cost f unction desc ent curve.

5. Acknowledgements

This work is supported by “A Project of Shandong Prov-

ince Higher Educational Science and Technology Pro-

gram (J09LA12)” and “Shandong Provincial Natural Sci-

ence Foundation (ZR2009AL012)”.

REFERENCES

[1] H. C. Tuckwell and F. Y. M. Wan, “On the Behavior of

Solutions in Viral Dynamical Models,” BioSystems, Vol.

73, 2004, pp. 157-161.

http://dx.doi.org/10.1016/j.biosystems.2003.11.004

[2] J. F. Zhang and X. H. Xia, “Identifiability Problems of

Time-Delay HIV Models,” Proceedings of the 17th

World Congress, The International Federation of Auto-

matic Control, Seoul, 2008, pp. 283-288.

[3] R. V. Culshaw, S. G. Ruan and R. J. Spiteri, “Optimal

HIV Treatment by Maximising Immune Response,” Ma-

thematical Biology, Vol. 48, 2004, pp. 545-562.

http://dx.doi.org/10.1007/s00285-003-0245-3

[4] C. Myburgh and K. H. Wong, “Computational Control of

an HIV Model,” Annals of Operations Research, Vol. 133,

2005, pp. 277-283.

http://dx.doi.org/10.1007/s10479-004-5038-6

[5] J. M. Hy ma n and J. Li, “The Reproductive Number for an

HIV Model with Differential Infectivity and Staged Pro-

gression,” Linear Algebra and Its Applications, Vol. 398,

2005, pp. 101-116.

http://dx.doi.org/10.1016/j.laa.2004.07.017

[6] H.-D. Kwon, “Optimal Treatment Strategies Derived from

a HI V Model with Drug-Resistant Mutants,” Applie d Ma-

thematics and Computation, Vol. 188, 2007, pp. 1193-

1204. http://dx.doi.org/10.1016/j.amc.2006.10.071