Optimal Estimation of Parameters for an HIV Model

doi:10.4236/eng.2013.510B084

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Engineering, 2013, 5, 413-415

http://dx.doi.org/10.4236/eng.2013.510B084 Published Online October 2013 (http://www.scirp.org/journal/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: [email protected], *[email protected]

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. ([1]), 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

dx s x xy

dy xy y

dv cy v

µβ

βα

−+ +=



− +=





−+ =





(1)

with initial values

(0) 0x>

(0) 0y≥

and

(0) 0v>

Here s,

, c and

are model parameters

which interpreted as follows:

s: is the rate of pro duc tion of CD4 + T-cells

: is their per capita death rate

: is the rate of infection of CD4 + T-cells by virus

: is the per capita rate of disappearance of infected

cells

c: is the rate of production of virus by infected cells

: 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) 200

(0) 0y=

and

(0) 1v=

, and typical

parameters values

0.272s=

0.00136

, β =

0.00027,

0.33

50c=

and

. 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.

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 22

12 3

1(()()())

To oo

Jw xxwyywvvdt=−+ −+−

∫

(2)

Where

and

present observation of x, y and

v, respectively.

and 3

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:

()

LJXLxYLyVLv dt=+ ++

∫

(3)

where X, Y and V are adjoint variables.

, and

are the left hand part of Equations (1). It is easy to

get the following adjoint equations:

()()0

( )0

() ()0

dX XvXYw xx

dY YcVw yy

dV VxXYw vv

µβ

γβ

− ++−+− =



−+−+− =





−+ +−+−=



(4)

Now the parameters

and

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,()0

Xt YTVT=== (5)

The gradients can be easily obtained through Equation

(3), which are

JxXdt

JyYdt

JvVdt

∂

=

∂



∂

=

∂



∂

=

∂





∫

(6)

For the sake of convenience, we denotes

(,,)P

µαγ

(7)

(, ,)

JJJJ

µαγ

∂∂∂

′

∇=

∂∂∂

(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 pagd

⇐ −×

(9 )

where

is the optimal step.

D. N. SUN ET AL.

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

0ww= =

. 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)”.

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