 Engineering, 2013, 5, 413-415 http://dx.doi.org/10.4236/eng.2013.510B084 Published Online October 2013 (http://www.scirp.org/journal/eng) Copyright © 2013 SciRes. ENG Optimal Estimation of Parameters for an HIV Model Danna Sun, Zhaoying Jiang, Ziku Wu* College of Science and Information, Qingdao Agricultural University, Qingdao, China Email: msj1958@126.com, *zkwu1968@126.com Received 2013 ABSTRACT An HIV model was considered. The parameters of the model are estimated by adjoint dada assimilation method. The results showed the method is valid. This method has potential application to a wide variety of models in biomathemat-ics. Keywords: HIV Model; Optimal Estimation; Adjoint Data Assimilation 1. Introduction One of the worst diseases in the world is AIDS (Ac-quired Immunity Deficiency Syndrome). It is caused by the human immunodeficiency virus (HIV). There has been much interest recently in mathematical models of HIV. During the last decade, many scholars have done a great of work about HIV model ([1-6]). We consider a model which presented by Henry et al. (), which de- scribes the interaction of HIV and the immune system of the body. In this model, the variables are uninfected CD4 + T-cells, infected such cells and free virus, whose densi- ties at time t are denoted respectively by ()xt, ()yt and ()vt. These quantity satisfy 000dx s x xydtdy xy ydtdv cy vdtµββαγ−+ +=− +=−+ = (1) with initial values (0) 0x>, (0) 0y≥ and (0) 0v>. Here s, µ, β, α, c and γ are model parameters which interpreted as follows: s: is the rate of pro duc tion of CD4 + T-cells µ: is their per capita death rate β: is the rate of infection of CD4 + T-cells by virus α: is the per capita rate of disappearance of infected cells c: is the rate of production of virus by infected cells γ: is the death rate of virus particles. It’s impossible to get the above model’s analytic solu-tions due to its strong nonlinearity. Thus we computed its solutions using Runge-Kutta methods of order 4. This was done with initial Thu s we computed its solutions us- ing Runge-Kutta methods of order 4. This was done with initial (0) 200x=, (0) 0y= and (0) 1v=, and typical parameters values 0.272s=, 0.00136µ=, β = 0.00027, 0.33α=, 50c= and 2γ=. The time series of uninfected CD4+T-cells and infected CD4 + T-cells are listed in Figures 1 and 2, res pectively. The peak val-ues exist round 20 days. The phase portraits are shown in Figure 3 for a time period of 200 d ays and it can be seen that the orbits are quite close. These model parameters are chosen to be consistent with those in the model, whose values may not be very well known. There has recently been considerable inter-est in the inverse problem of determining such values by incorporating measured data into the numerical model. The adjoint assimilation method involves minimizing a certain cost function which consists basically of a norm of the difference between the computed and the observed values of the measured variables. The purpose of the Figure 1. The time series of uninfected CD4 + T-cell. *Corresponding a uthor. D. N. SUN ET AL. Copyright © 2013 SciRes. ENG 414 Figure 2. The time series of infected CD4 + T-cell. Figure 3. The phase portraits. present paper is to estimate the model parameters (μ, α, λ) by constructing the adjoint model. The basis of the me-thod is to minimize the cost function which is equal to a norm of difference between the computed and the ob-served data. An algorithm is obtained, via so-called ad-joint equation, for construction of the gradient of the function with respect to the parameters. In Section 2 we describe the adjoint assimilation me-thod. A detailed discussion of the numerical tests results is given in Section 3. Section 4 summarizes the results and conclusions. 2. The Adjoint Numerical Model For parameter estimation and data assimilation, differ-ences between predicted and measured values of these variables must be quantified by a single misfit number, the objective function. A lot of options are available for choosing the misfit function. In this study, we choose the classical least-squares approach. However, the results from above governing equations are not good as with the observed values. The error between observed and model calculated can be defined as 2 2212 301(()()())2To ooJw xxwyywvvdt=−+ −+−∫ (2) Where ox, oy and ov present observation of x, y and v, respectively. 1w, 2w and 3w are weight coeffi-cients and [0, ]T stands for assimilation window. In order to get the adjoint Equations of (1), we intro-duce the adjoint control variables and Lagrange function: 0()TLJXLxYLyVLv dt=+ ++∫ (3) where X, Y and V are adjoint variables. Lx, Ly, and Lv are the left hand part of Equations (1). It is easy to get the following adjoint equations: 123()()0( )0() ()0ooodX XvXYw xxdtdY YcVw yydtdV VxXYw vvdtµβαγβ− ++−+− =−+−+− =−+ +−+−= (4) Now the parameters µ, α and γ are unknown, which need to estimate by the adjoint assimilation me-thod. The numerical scheme of the adjoint Equation (4) are the same as Equation (1), however it needs to inte-grate backward, and its initial conditions are set to zeros, that is ()0,()0,()0Xt YTVT=== (5) The gradients can be easily obtained through Equation (3), which are 000TTTJxXdtJyYdtJvVdtµαγ∂=∂∂=∂∂=∂∫∫∫ (6) For the sake of convenience, we denotes '(,,)Pµαγ= (7) (, ,)JJJJPµαγ∂∂∂′∇=∂∂∂ (8) Having determined the gradients which respect to the unknown parameters, we can perform the minimization by the descend method, the modified parameters are pa pagdλ⇐ −× (9 ) where λ is the optimal step. D. N. SUN ET AL. Copyright © 2013 SciRes. ENG 415 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 130ww= =. 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  H. C. Tuckwell and F. Y. M. 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