A numerical algorithm for constructing a mathematical model of intracellular dynamics for individual HIV treatment

(1)

model of intraellular dynamis for individual HIV

treatment

S.I.Kabanikhin, O.I.Krivorotko, D.V. Yermolenko,H.T. Banks

Institute of ComputationalMathematis and Mathematial Geophysis SB RAS,

Novosibirsk State University

and

Center for Researh inSienti Computation

North CarolinaState University

Raleigh,NC

E-mails: kabanikhinss.ru, olga.krivorotkoss.ru, ermolenko.dashamail.ru,

htbanksnsu.edu

Abstrat. In this paper, a problem of speifying HIV-infetion parameters and

immune response using additional measurements of the onentrations of the

Ò-lymphoytes, thefreevirus, and theimmuneeetors atxed timesfora

mathemat-ial model of HIV dynamis is investigated numerially. The problem of speifying

theparametersofthemathematialmodel(aninverseproblem) isredued toa

prob-lem of minimizingan objetive funtion desribing the deviation of the simulation

resultsfromthe experimentaldata. Ageneti algorithmfor solvingthe least squares

funtionminimizationproblemmethodisimplementedandinvestigated. Theresults

of anumerial solutionof the inverse problem are analyzed.

Keywords: mathematialmodelofHIVdynamis,parameterspeiationproblem,

inverse problem,optimizationapproah, genetialgorithm,ondene intervals.

Introdution

The human immunodeieny virus (HIV) was disovered independently in 1983 in Frenh

and USA laboratories. This disovery initiated numerous studies of the ation of this virus in

humans(inludingthoseofA.S.Perelsonetal.[1℄,B.M.Adamsetal.[2℄,H.T.Banksetal.[3℄,G. Boharov etal.[4℄). Thevirusstruture israther simple: the envelopeonsists ofadoublelayer of lipids, with glyoprotein mushrooms in it; it has two RNA hains inside ontaining a virus

genetiprogram and proteins (reverse transriptase, integrase,and protease). The HIVattaks

the organism at ertain blood ells (Ò-lymphoytes and marophages) with CD4 moleules on

the surfae. Meeting a ell, the glyoprotein mushrooms of the virus stik to these moleules.

The virus envelope and the ell merge, and the geneti material of the virus enters the ell.

Thenthe infeted elldevelops and grows with the help ofthe enzymesof reverse transriptase,

integrase,and protease. Theseenzymesplay animportantrole, and mostantiviraldrugsinhibit

theproessesthattakeplaeowingtotheseenzymes. Theinfetiondevelopmentintheorganism

an be deelerated(inhibited) by slowing theseproesses.

Mathematial simulation is one of the methods used for identifying the degree of aetion

of the immune system by speifying the parameters of the disease and immune response. The

purposeofthispaperistonumeriallyspeifysomeparametersharaterizingthepeuliaritiesof

(2)

Banks [2,3℄ and G. Boharov [4℄.

Thepaperisorganizedasfollows. Setion1desribestheformulationofthediretproblemfor

the mathematial modelof the dynamis of HIV. Setion2 desribes the problem of speifying

the parameters (an inverse problem) for the mathematial model of HIV dynamis by using

weighted leastsquares. The stabilityof theinverse problemis studiedinSetion3. InSetion4,

agenetialgorithmforsolvingtheproblemofminimizingafuntionalisinvestigated. Condene

intervalsfor parameters are analyzed in Setion 5 and numerialresults of the inverse problem

solutionare presented. Conlusions are given inSetion6.

1 Mathematial model of HIV dynamis

The mathematial model of HIV dynamis we employ is desribed by the following system of

nonlinear dierentialequations (1.1) as given in [2℄:











˙

T

1

=

λ

1

−

d

1

T

1

−

k

1

V T

1

,

˙

T

2

=

λ

2

−

d

2

T

2

−

k

2

V T

2

,

˙

T

∗

1

=

k

1

V T

1

−

δT

1

∗

−

m

1

ET

1

∗

,

˙

T

∗

2

=

k

2

V T

2

−

δT

2

∗

−

m

2

ET

2

∗

,

t

∈

(0

, T

)

,

˙

V

=

NT

δ

(

T

∗

1

+

T

∗

2

)

−

cV

−

[

ρ

1

k

1

T

1

+

ρ

2

k

2

T

2

]

V,

˙

E

=

λE

+

b

E

(

T

1

∗

+

T

2

∗

)

(

T

∗

1

+

T

2

∗

)+

K

b

E

+

d

E

(

T

1

∗

+

T

2

∗

)

(

T

∗

1

+

T

2

∗

)+

K

d

E

−

δEE.

(1.1)

Here

(

T

1

, T

∗

1

)

are theuninfeted and infetedT-lymphoytes,respetively,

(

T

2

, T

∗

2

)

are the

unin-feted and infeted marophages,

V

isthe free virus (that is, the virus that has not penetrated the ell), and

E

representthe immune eetors. The Ò-lymphoytes and immune eetors play an important role in the immune response of the organism. When the virus penetrates the

or-ganism ells, the T-lymphoytes send a danger signal to the immune eetors. Upon reeiving

this signal the immune eetors begin a long term eort to eradiate the virus. The following

vetor is taken asinitialdata for the mathematialmodel(1.1):

T

1

(0) = 500000

, T

2

(0) = 4800

, T

∗

1

(0) = 5000

, T

∗

2

(0) = 10

, V

(0) = 10000

, E

(0) = 15

.

(1.2)

These initialonditions haraterize a state of the patient that is intermediate between that of

anuninfeted patient with healthy T-ellsand that at whih the healthy ells of the target are

pratially depleted and the organism has a high viral load. As a rule, the patient rst seeks

medialassistane when inthe state (1.2).

The mathematialmodel(1.1)has

19

parametersharaterizingthe diseaseand the immune system of the patient:

λ

1

,

λ

2

,

d

1

,

d

2

,

k

1

,

k

2

,

δ

,

m

1

,

m

2

,

NT

,

c

,

ρ

1

,

ρ

2

,

λE

,

bE

,

dE

,

Kb

,

Kd

,

δE

(see Table 1). A.S. Perelson et al. [5℄ argue that most of these

19

parameters an be rather aurately approximated using results from the available statistial information, and only four

of them, namely,

k

1

,

k

2

,

λ

1

,

λ

2

, need to be speied for eah individualpatient. Here

k

1

and

k

2

are the infetion rates of Ò-lymphoytes and marophages, respetively, and

λ

1

and

λ

2

are the T-lymphoytes and marophages prodution rates. These four parameters are patient-spei

(3)

λ

1

10000

ml

cells

·

day

Target elltype 1 prodution(soure)rate

d

1

0

.

01

∗∗

day

1

Target elltype 1 deathrate

ε

∈

[0

,

1]

Population

1

treatment eay

k

1

8

.

0

×

10

−

7

ml

virions

·

day

Population

1

infetion

λ

2

31

.

98

ml

cells

·

day

Target elltype 2 prodution(soure)rate

d

2

0

.

01

∗∗

day

1

Target elltype 2 deathrate

f

0

.

34(

∈

[0

,

1])

Treatment eay redution inpopulation2

k

2

1

×

10

−

4

ml

virions

·

day

Population

2

infetion rate

δ

0

.

7

∗

1

day

Infeted ell deathrate

m

1

.

0

×

10

−

5

cells

ml

·

day

Immune-indued learane rate for population1

m

2

1

.

0

×

10

−

5

cells

ml

·

day

Immune-indued learane rate for population2

NT

100

∗

virions

cell

Virions produed perinfeted rate

c

13

∗

1

day

Virus naturaldeath rate

ρ

1

virions

cell

Average numberof virions infetingatype1 ell

ρ

2

1

virions

cell

Average numberof virions infetingatype2 ell

λE

1

cells

ml

·

day

Immune eetor prodution (soure) rate

b

E

0

.

3

day

1

Maximum birth rate forimmune eetors

Kb

100

cells

ml

Saturation onstant for immuneeetor birth

dE

0

.

25

day

1

Maximum death rate for immune eetors

Kd

500

cells

ml

Saturation onstant for immuneeetor death

δE

0

.

1

∗

1

day

Natural death rate for immuneeetors

Table 1: Parameters used in model(1.1). Those inthe top setionof the table are taken diretlyfrom Callaway and Perelson. Parameters inthe bottom setion of the table are taken

fromBonhoeer etal., with

Kb

and

Kd

saled toreet the volumetriunits used inour model and alsoadjusted. The supersripts

∗

denoteparameters estimated from human data and

∗∗

denote thoseestimated frommaaque data.

2 Speifying the immune response parameters by using

weighted least squares

When an infeted patient seeks medial assistane, the dotor rst determines the patients'

individual harateristis, and then presribes mediation. Here we investigate the problem

(4)

model(1.1)with the initialonditions (1.2) onsists of speifying the vetor of parameters

q

=

(

k

1

, k

2

, λ

1

, λ

2

)

T

from additional information about the onentrations of T-lymphoytes in the

patient

T

1

+

T

∗

1

, the free virus

V

,and the immune eetors

E

at xed times

tk, k

= 1

, . . . , K

:

Y

1

(

t

k

) = Φ

1

(

t

k

;

q

ex

) +

ε

1

k

,

Y

2

(

tk

) = Φ

2

(

tk

;

qex

) +

ε

2

k

,

k

= 1

, . . . , K.

Y

3

(

tk

) = Φ

3

(

tk

;

qex

) +

ε

3

k

.

(2.1)

Here

Y

= (

Y

1

, Y

2

, Y

3

)

T

isavetorof noisydata ofuninfeted plus infetedT-lymphoytes,free

virus, and immune eetors,

Φ = (Φ

1

,

Φ

2

,

Φ

3

)

T

= (

T

1

+

T

1

∗

, V, E

)

T

, and

qex

is a vetor of exat

knownvaluesof

q

,the parameters

εk

= (

ε

1

k

, ε

2

k

, ε

3

k

)

T

, k

= 1

, . . . , K

,representavetorof Gaussian

noises with zero meanand ovarianematrix given by

V

0

=

V ar

(

εk

) =

diag

(

σ

2

0

,

1

, σ

0

2

,

2

, σ

0

2

,

3

)

.

(2.2)

for

k

= 1

, . . . , K

.

The inverse problemonsists of nding the minimizer

qOLS

=

arg

min

q

J

(

q

) =

arg

min

q

K

P

k

=1

[

Yk

−

Φ(

tk

;

q

)]

T

V

0

−

1

[

Yk

−

Φ(

tk

;

q

)]

,

(2.3)

where

qOLS

isa randomvetor (sine

εk

=

Yk

−

Φ(

tk

;

q

)

isa randomvetor). Hene if

{yk}

K

k

=1

is

aolletionof realizations of the random vetors

{Y

k

}

K

k

=1

, then solving

ˆ

qOLS

=

arg

min

q

J

(

q

) =

arg

min

q

K

P

k

=1

[

yk

−

Φ(

tk

;

q

)]

T

V

0

−

1

[

yk

−

Φ(

tk

;

q

)]

,

(2.4)

provides a realization

q

ˆ

= ˆ

qOLS

for

qOLS

. Bythe denition of the ovariane matrix we have

V

0

=

diag

E

1

K

P

k

=1

[

Yk

−

Φ(

tk

;

qex

)][

Yk

−

Φ(

tk

;

qex

)]

T

ii

(2.5)

Thus an unbiased approximationfor

V

0

is given by

ˆ

V

=

diag

1

K

−

4

K

P

k

=1

[

yk

−

Φ(

tk

;

qex

)][

yk

−

Φ(

tk

;

qex

)]

T

ii

(2.6)

However, theestimate

q

ˆ

of (2.4)requiresthe (generallyunknown) matrix

V

0

,and

V

0

requires the unknown vetor

qex

, sowe willinstead use the followingexpressions toalulate

q

ˆ

and

V

ˆ

qOLS

=

arg

min

q

K

P

k

=1

[

yk

−

Φ(

tk

;

q

)]

T

V

ˆ

−

1

[

yk

−

Φ(

tk

;

q

)]

(2.7)

ˆ

V

=

diag

1

K

−

4

K

P

k

=1

[

y

k

−

Φ(

t

k

; ˆ

q

)][

y

k

−

Φ(

t

k

; ˆ

q

)]

T

ii

.

(2.8)

We do this in a generalized least squares or more preisely, an iteratively reweighted weighted

(5)

The numerialalgorithmof nding

qOLS

ˆ

onsists of the following steps [8,9℄: 1. Set

V

ˆ

= ˆ

V

(0) =

I

3

and solve forthe initialestimate

q

ˆ

(0)

using (2.7). Set

l

= 0

.

2. Use

q

ˆ

(

l

)

to alulate

V

ˆ

(

l

+1)

using (2.8).

3. Re-estimate

q

ˆ

by solving (2.7) with

V

ˆ

= ˆ

V

(

l

+1)

toobtain

q

ˆ

(

l

+1)

.

4. Set

l

=

l

+ 1

and return to step 2. Terminate the proess and set

qIRW LS

ˆ

= ˆ

q

(

l

+1)

when

two suessive estimates for

q

ˆ

are suiently lose to one another.

3 The stability of the inverse problem

In this Setion the stability of the inverse problemsolution is analyzedusing the singular value

deomposition for linearized matrix of disrete inverse problem. It means that the stability of

theparameteridentiationproblemisinvestigated onthebasis ofthe analysisofthemagnitude

of the ondition number for the linearized matrix of the inverse problem (in the ases when we

dene19, 4and 2 parameters).

3.1 Linearization algorithm for the inverse problem

The diretproblem (1.1)-(1.2) an be writtenin vetor form:

(

˙

X

(

t

) =

P

(

X

(

t

)

, q

)

, t

∈

(0

, T

)

X

(0) =

X

0

,

(3.1)

Here

X

= (

X

1

, X

2

, X

3

, X

4

, X

5

, X

6

)

T

= (

T

1

, T

2

, T

∗

1

, T

∗

2

, V, E

)

T

is the vetor of system (1.1) vari-ables,

q

∈

R

N

is the vetor of the parameters of the system (1.1), haraterizing the features of thepatientimmunityand disease,

X

0

is thevetor ofinitialonditions,

P

isthe righthand side vetor.

We onsider the ase when additional information about only three funtions

X

1

(

t

) +

X

3

(

t

)

, X

5

(

t

)

, X

6

(

t

)

at xed times

tk, k

= 1

, . . . , K

is known (see 2.1).

Let us present the vetor

q

in the form

q

=

q

0

+

δq

, where

q

0

∈

R

N

is the vetor of initial

approximations,

δq

∈

R

N

is the vetor of unknown inrementsof parameters.

Solving the Cauhy problem(3.1) for the set of parameters

q

0

by the Runge-Kutta method of the fourth approximation order [10℄, we obtain the values

Xi

(

tk, q

0

) =

X

e

k

i

, i

= 1

, . . . ,

6

in

the time points

tk, k

= 1

, . . . , K

of the time interval

(0

, T

)

. Denote

X

(

t, q

0

) =

X

e

,

X

(

t, q

) =

X

(

t, q

0

+

δq

) =

X

. WeapplyTaylor'sformulatothefuntion

Z

=

X

−

X

e

=

X

(

t, q

0

+

δq

)

−X

(

t, q

0

)

atthe point

(

X, q

e

0

)

:

˙

Z

=

P

(

X, q

0

+

δq

)

−

P

(

X, q

e

0

) =

P

(

X, q

e

0

) +

P

′

e

X

(

X, q

e

0

)

·

Z

+

P

q

′

0

(

X, q

e

0

)

·

δq

−

P

(

X, q

e

0

) +

o

(

q

e

X

2

+

q

0

2

)

.

(6)

writtenas follows:











˙

Z

(

t

) =

P

′

e

X

(

X, q

e

0

)

·

Z

+

P

′

q

0

(

X, q

e

0

)

·

δq, t

∈

(0

, T

)

Z

(0) = 0

,

Z

1

(

tk

) +

Z

3

(

tk

) =

Y

1

(

tk

)

−

(

X

e

1

k

+

X

e

3

k

)

, k

= 1

, . . . , K,

Z

5

(

tk

) =

Y

2

(

tk

)

−

X

e

5

k

, k

= 1

, . . . , K,

Z

6

(

tk

) =

Y

3

(

tk

)

−

X

e

6

k

, k

= 1

, . . . , K.

(3.3)

Here

Y

(

tk

)

is the vetor of known measurement data,

X

e

k

=

X

e

(

tk, q

0

)

is the diret problem

solution(3.1) for the set of parameters

q

0

inxed times

tk, k

= 1

, . . . , K

.

3.2 Algorithm of disretization

Usinganexpliitdierenesheme ofthe rst orderof approximation,we onstrut thedisrete

matrix of the linearized inverse problem(3.3):

Z

j

+1

−

Z

j

h

=

P

′

e

X

(

X

e

j

, q

0

)

·

Z

j

+

P

′

q

0

(

X

e

j

, q

0

)

·

δq,

Z

j

=

Z

(

jh

)

, h

=

T /Nt, j

= 1

, . . . , Nt.

Z

j

+1

= (

I

+

P

X

′

e

(

X

e

j

, q

0

))

·

Z

j

+

hP

′

q

0

(

X

e

j

, q

0

)

·

δq.

We obtain equationsfor eah

Z

j

:

Z

0

= 0

,

Z

1

=

hP

q

′

0

(

X

e

0

, q

0

)

·

δq

=

B

0

·

δq,

Z

2

= (

I

+

P

′

e

X

(

X

e

1

, q

0

))

·

Z

1

+

hP

′

q

0

(

X

e

1

, q

0

)

·

δq

=

M

1

·

B

0

·

δq

+

B

1

·

δq

= (

M

1

·

B

0

+

B

1

)

·

δq,

.

Z

j

= (

j

−

1

Y

i

=1

M

i

·

B

0

+

j

−

1

Y

i

=2

M

i

·

B

1

+

. . .

+

M

j

−

1

·

B

j

−

2

+

B

j

−

1

)

·

δq,

where

Mj

=

I

+

hP

′

e

X

(

X

e

j

, q

0

)

, Bj

=

hP

′

q

0

(

X

e

j

, q

0

)

, j

= 1

, . . . , Nt.

We apply linear interpolation onthe grid nodes

ω

=

{Z

j

=

Z

(

jh

)

, h

=

T /Nt, j

= 1

, . . . , Nt}

to the data of the inverse problem (2.1). Then the required matrix of the linearized inverse problem(3.3) has a blok form:

(

A

)

(3

·

N

t

)

×

N

= (

A

1

|

. . .

|A

N

t

)

T

,

(

A

k

)

1

j

=

Q

k

−

1

i

=1

Mi

·

B

0

+

Q

k

−

1

i

=2

Mi

·

B

1

+

. . .

+

Mk

−

1

·

Bk

−

2

+

Bk

−

1

j

+

Q

k

−

1

i

=1

Mi

·

B

0

+

Q

k

−

1

i

=2

Mi

·

B

1

+

. . .

+

Mk

−

1

·

Bk

−

2

+

Bk

−

1

(7)

(

A

k

)

2

j

=

Q

k

−

1

i

=1

Mi

·

B

0

+

Q

k

−

1

i

=2

Mi

·

B

1

+

. . .

+

Mk

−

1

·

Bk

−

2

+

Bk

−

1

5

j

,

(

A

k

)

3

j

=

Q

k

−

1

i

=1

Mi

·

B

0

+

Q

k

−

1

i

=2

Mi

·

B

1

+

. . .

+

Mk

−

1

·

Bk

−

2

+

Bk

−

1

6

j

k

= 1

, . . . , Nt, j

= 1

, . . . , N.

A

=







A

1

A

2

. . .

A

N

t







=







a

1

11

a

1

12

· · ·

a

1

N

a

1

21

a

1

22

· · ·

a

1

2

N

a

1

31

a

1

32

· · ·

a

1

3

N

. . . . . . . . . . . . . . . . . . . . . . . .

a

N

t

11

a

N

12

t

· · ·

a

N

1

N

t

a

N

t

21

a

N

22

t

· · ·

a

N

2

N

t

a

N

t

31

a

N

32

t

· · ·

a

N

3

N

t







·

Hene, the inverse problem(3.3)isredued tothe system oflinear equations

A

·

δq

=

f

with the vetor

f

∈

R

3

·

N

t

of the right-hand side of the form:

f

= (

Z

1

+

Z

3

1

, Z

5

1

, Z

6

1

,

| · · · |Z

1

N

t

+

Z

N

t

3

, Z

5

N

t

, Z

6

N

t

)

T

.

3.3 The stability investigation on the basis of the analysis of the

on-dition number of the linearized inverse problem matrix

By the singular value deomposition theorem [6℄ for the

3

·

Nt

×

N

-matrix

A

, we an nd the orthogonal

3

·

Nt

×

3

·

Nt

-matrix

U

and

N

×

N

-matrix

V

and alsothediagonal

3

·

Nt

×

N

-matrix

Σ =

diag

(

σ

1

, σ

2

, . . . , σN

)

,

suh that

A

=

U

Σ

V

T

,

(3.4)

0

≤

σN

≤

σN

−

1

≤ · · · ≤

σ

2

≤

σ

1

.

Thenumbers

σi

=

σi

(

A

)

, i

= 1

, . . . , N

,areuniquelydeterminedandarealledsingularvalues of the matrix

A

.

Our linearized inverse probleman bewritten as:

A

·

δq

=

f,

(3.5)

where

A

is the linearized matrix of an inverse problem of the order

3

·

Nt

×

N

;

δq

∈

R

N

is a

vetor of unknown parameters,

f

∈

R

3

·

N

t

is the vetor of the right-handside.

In these ases we have the estimate of the relativeauray error of the solution [6℄:

kδq

−

δqexk

kδq

ex

k

≤

Cond

(

A

)

kεk

kf

−

εk

.

(3.6)

Here

δq

ex

=

q

ex

−

q

0

is the vetor of exat parameters,

ε

isthe vetor of noise in data.

Thus,theerrorofthesolutionisdeterminedbytheonstant

Cond

(

A

) =

σmax/σmin

=

σ

1

/σN

, whih is alled the ondition number of the matrix. Systems with a relatively large ondition

number are alled ill-onditioned. Systems with ill-onditioned matries an be onsidered

(8)

Consider a time interval

T

= 10

days and onstrut a partition of the domain

(0;

T

)

:

ω

e

=

{tj

:

tj

=

jht, ht

=

T /Nt, j

= 1

, . . . , Nt, Nt

= 1000

}

. Using the linearization and disretization

algorithm,we obtain the matrix

A

of the linearized inverse problem(3.3).

3.4.1 The stability investigation in the ase of 19 dened parameters

To begin with, we investigate the stability of the inverse problem (1.1)-(2.1) in the ase

q

=

(

λ

1

, λ

2

, d

1

, d

2

, k

1

, k

2

, δ, m

1

, m

2

, NT

, c, ρ

1

, ρ

2

, λE

, bE, dE, Kb, Kd, δE

)

T

∈

R

19

. Inthis asethe matrix

A

of thelinearized inverse problem(3.3)has order

(3

·

Nt

)

×

19

. The vetorshosen asthe exat

parameters

qex

and initialparameters

q

0

are given in the table 2.

Exat parameters

qex

Initial parameters

q

0

Units

λ

1

10000

20000

ml

cells

·

day

λ

2

31

.

98

40

ml

cells

·

day

d

1

0

.

01

0

.

02

day

1

d

2

0

.

001

0

.

002

day

1

k

1

8

.

0

×

10

−

7

.

0

×

10

−

7

virions

ml

·

day

k

2

1

.

0

×

10

−

7

2

.

0

×

10

−

7

virions

ml

·

day

δ

0

.

7

0

.

8

day

1

m

1

.

0

×

10

−

5

2

.

0

×

10

−

5

ml

cells

·

day

m

2

1

.

0

×

10

−

5

2

.

0

×

10

−

5

ml

cells

·

day

NT

100

200

virions

cell

c

13

30

day

1

ρ

1

2

virions

cell

ρ

2

1

2

virions

cell

λ

E

1

2

ml

cells

·

day

bE

0

.

3

0

.

4

1

day

dE

0

.

25

0

.

35

day

1

Kb

100

200

cells

ml

Kd

500

600

cells

ml

δE

0

.

1

0

.

2

day

1

Table 2: Seleted values of exat parameters

qex

∈

R

19

q

0

∈

R

19

.

Using the singular expansion (3.4), we obtain the singular values for the matrix

A

of the linearized inverse problems (3.3) (see Table 3 and Figure 1).

(9)

prob-values of the

matrix

A

values of the

matrix

A

σ

1

.

9938

×

10

5

σ

11

3

.

1954

×

10

0

σ

2

.

7625

×

10

4

σ

12

9

.

7718

×

10

−

2

σ

3

7

.

5309

×

10

3

σ

13

8

.

7167

×

10

−

2

σ

4

.

9575

×

10

3

σ

14

4

.

4632

×

10

−

2

σ

5

3

.

3978

×

10

3

σ

15

1

.

3928

×

10

−

2

σ

6

0

.

9306

×

10

3

σ

16

9

.

4850

×

10

−

3

σ

7

2

.

6617

×

10

2

σ

17

8

.

421

×

10

−

5

σ

8

5

.

8252

×

10

1

σ

18

2

.

4

×

10

−

7

σ

9

7

.

356

×

10

0

σ

19

0

σ

10

6

.

6257

×

10

0

Table 3: Singularvalues for the matries

A

of linearizedinverse problem(3.3) (inthe ase

q

∈

R

19

).

10

-8

10

-6

10

-4

10

-2

10

0

10

2

10

4

10

6

0

2

4

6

8

10

12

14

16

18

Figure 1: Singularvalues

σi

(

A

)

in the logarithmi sale with known additionalinformation about all funtions

T

1

, T

2

, T

∗

1

, T

∗

2

, V, E

(ñirles) and singularvalues

σi

(

A

e

)

in the logarithmi

sale with known additionalinformationabout three funtions

T

1

+

T

∗

1

, V, E

(inthe ase of 19

parameters).

lems (3.3) are unstable. Consequently, the problems of determining the parameter vetor

q

= (

λ

1

, λ

2

, d

1

, d

2

, k

1

, k

2

, δ, m

1

, m

2

, NT

, c, ρ

1

, ρ

2

, λE, bE

, dE

, Kb, Kd, δE

)

T

∈

R

19

of the

mathemati-almodel(1.1) for the additionalinformation about three funtions

T

1

(

t

) +

T

∗

1

(

t

)

, V

(

t

)

, E

(

t

)

is

(10)

Now we investigate the stability of the inverse problem (1.1)-(2.1) in the ase

q

=

(

λ

1

, λ

2

, k

1

, k

2

,

)

T

∈

R

4

. In this ase the matrix

A

of the linearized inverse problem (3.3) has

order

(3

·

N

t

)

×

4

. The vetors hosen as the exat parameters

q

ex

and the initialparameters

q

0

are given inthe Table 4(the 1st Case).

Exat parameters

qex

Initial parameters

q

0

(the 1st Case)

Initial parameters

q

0

(the 2nd Case)

Units

λ

1

10000

20000

40000

ml

cells

·

day

λ

2

31

.

98

32

40

ml

cells

·

day

k

1

8

.

0

×

10

−

7

9

.

0

×

10

−

7

6

.

0

×

10

−

7

ml

virions

·

day

k

2

1

.

0

×

10

−

7

1

.

1

×

10

−

7

3

.

0

×

10

−

7

virions

ml

·

day

qex

∈

R

4

q

0

∈

R

4

.

The Table 5(the1stCase) and Figure2listthe obtainedsingularvaluesfor the matrix

A

of the linearized inverse problem

(

3

.

3

)

. In this ase, the ondition number for the matrix

A

takes a ratherlarge value:

Cond

(

A

) =

σ

1

(

A

)

/σ

4

(

A

) = 2

.

4076

×

10

3

. Aording to the estimate (3.6), small hanges in the right-hand sides of the inverse problem (3.3) an lead to signiant large hanges inthe solutions.

The singular values of the matrix

A

(the 1st Case)

A

(the 2nd Case)

σ

1

2

.

7680

×

10

4

≈

9

.

9

×

10

48

σ

2

1

.

9166

×

10

4

≈

5

.

1

×

10

48

σ

3

1

.

9637

×

10

3

≈

3

.

3

×

10

48

σ

4

1

.

1497

×

10

1

≈

7

.

5

×

10

46

Table 5: Singular values for the matrix

A

of linearized inverse problem (3.3)(inthe ase

q

∈

R

4

).

Now onsider another set of the initialparameters

q

0

(see Table 4(the 2nd Case)). Byusing thesingulardeomposition(3.4),weobtainthesingularvalues forthematrix

A

ofthe linearized inverse problem (3.3) (see Table 5 (the 2nd Case) and Figure 3). In this ase, the ondition number for the matrix

A

take the following values:

Cond

(

A

) =

σ

1

(

A

)

/σ

4

(

A

) = 1

.

32

×

10

2

.

This value is smaller than the values in the previous ase. But in spite of this, aording to

(11)

0

5000

10000

15000

20000

25000

30000

0

1

2

3

Figure2: Singularvalues

σi

(

A

)

with known additionalinformationabout three funtions

T

1

+

T

1

∗

, V, E

(inthe ase of 4parameters from

the Table 4 (the 1st Case)).

0

1x10

48

2x10

48

3x10

48

4x10

48

5x10

48

6x10

48

7x10

48

8x10

48

9x10

48

1x10

49

0

1

2

3

Figure3: Singularvalues

σi

(

A

)

with known additionalinformation about three funtions

T

1

+

T

1

∗

, V, E

(inthe ase of 4 parametersfrom

(12)

Nextweinvestigatethestabilityof theinverseproblem(1.1)-(2.1)inthease

q

= (

λ

1

, k

1

)

T

∈

R

2

.

In this ase, the matrix

A

of the linearized inverse problem(3.3) has order

(3

·

Nt

)

×

2

. In the Table 6 we an see the vetors hosen as the exat parameters

q

ex

and the initial parameter estimates

q

0

.

Exat parameters

q

ex

Initial parameters

q

0

Units

λ

1

10000

40000

ml

cells

·

day

k

1

8

.

0

×

10

−

7

5

.

0

×

10

−

7

virions

ml

·

day

qex

∈

R

2

q

0

∈

R

2

.

In the Table 7 the singular values of the inverse problem (3.3) matrix

A

,obtained by means of a singularexpansion (3.4)are presented. In this ase, the ondition number of the matrix

A

takes a small value:

Cond

(

A

) =

σ

1

(

A

)

/σ

2

(

A

)

≈

2

.

65

. Therefore in this situation, the inverse problem(3.3)an be onsidered stable, sine, aording to the estimate(3.6), smallhanges in the right-hand sides of the inverse problem (3.3) result in minor hanges in the solution of the problem.

A

σ

1

6

.

6737

×

10

4

σ

2

.

5158

×

10

4

Table 7: Singular values for the matrix

A

of linearized inverse problem (3.3).

3.4.4 Conlusions of stability investigations

In the Setion 3.4 the stability of the inverse problem solution was analyzed using the singular

value deomposition for linearized matrix of disrete inverse problem. This means that the

stability of the parameter identiation problem was investigated on the basis of the analysis

of the magnitude of the ondition number for the linearized matrix of the inverse problem (in

ases when we onsider 19, 4 and 2 parameters to be estimated). In the ase of determining

4 parameters, it was shown that the form of the matrix of the linearized inverse problem and,

onsequently,the onditionalnumberofthismatrix depends onthe hoseninitialapproximation

of the parameters (see Table 5). In the ase of determining 19 parameters it was demonstrated thattheonditionalnumberofthelinearizedinverseproblem3.3)isofhugevalue. Therefore, we an say that the nonlinearinverse problem (1.1)-(2.1)of determining19 parametersis unstable. Intheasesofdetermining4parametersand2parameterstheonditionnumbersofthelinearized

(13)

Theproblemofminimizingafuntionalanbesolved by linearprogrammingmethods, gradient

methods of zero [11℄, rst [12℄ and higher orders, and by other methods. A general drawbak of the deterministi methods is that the initial approximation must be hosen suiently lose

to the exat solution. This is often a diult task. We here use instead a stohasti method

(a geneti algorithm) [13℄ for solving the inverse problem desribed above. This method is as follows:

1. Choosing an initial population: hoose

N

arbitrary vetors of parameters

q

i

=

(

k

i

1

, k

i

2

, λ

i

1

, λ

i

2

)

T

,

i

= 1

, . . . , N

, whose elements belong to the admissible intervals. For

eah

q

i

, alulatethe objetive funtional

J

(

q

i

)

given informula(2.3).

2. Seleting: hoose

N

pairs of parents. The probability that a member of the population gets into a pair is the greater the smaller is the value of its funtional. Calulate the

probabilitythat an

i

thindividual gets intoa pair by the formula

P

i

=

J

(

q

i

)

N

P

i

=1

J

(

q

i

)

.

3. Crossing: rossing eah pair

(

q

i

, q

j

)

,

i, j

= 1

, . . . , N

, by rossing-over, we get

N

de-sendants. For this purpose, we hoose two random numbers: one is a random integer

Q

∈

[1

, N

−

1]

, and the other is a random integer

R

whih an be either

1

or

2

. The

number

Q

haraterizes the dividing line of the parents, the number

R

shows whih part (leftor right) fromthe dividinglinethe desendant inheritsfrom the mother and father.

•

If

Q

=

s

,

R

= 1

,

s

∈

[1

, N

−

1]

mother:

(

q

i

1

, . . . , q

s

i

,

|q

s

i

+1

, . . . , q

M

i

)

T

, father:

(

q

j

1

, . . . , q

s

j

,

|q

j

s

+1

, . . . , q

j

M

)

T

−→

desen-dant:

(

q

i

1

, . . . , q

s

i

,

|q

j

s

+1

, . . . , q

j

M

)

T

.

•

If

Q

=

s

,

R

= 2

,

mother:

(

q

i

1

, . . . , q

s

i

,

|q

s

i

+1

, . . . , q

M

i

)

T

, father:

(

q

j

1

, . . . , q

s

j

,

|q

j

s

+1

, . . . , q

j

M

)

T

−→

desen-dant:

(

q

j

1

, . . . , q

s

j

,

|q

s

i

+1

, . . . , q

M

i

)

T

.

4. Mutating: makerandom hanges inthe desendants, i.e.,

•

hoose a random number of desendants

A

whih an mutate. Here

A

is a random

integer from

1

to

N

;

•

then hoose

Bi, i

= 1

, . . . , A

random integers from

1

to

N

whih haraterize the

numbers of mutating desendants;

•

for eah mutating desendant

Bi

, hoose a random number of mutating elements

CB

i

, i

= 1

, . . . , A

. Here

CB

i

is arandom integer from

1

to

M

;

•

then hoose

Dk, k

= 1

, . . . , CB

i

, i

= 1

, . . . , A

: random integers from

1

to

M

whih haraterize the numbers of mutating elements, and replae eah mutating element

by anew random value fromthe admissibleperiod.

5. Forming a new generation: hoose the ttestindividualsfromthe parents and

desen-dants, that is, those having the lowest value of the funtional

J

(

q

i

)

. Also, hoose a few

luky individuals that badly minimizethe funtionalbut may bringdiversity.

(14)

•

J

(

q

i

)

<

∆

, where

J

(

q

i

)

is the minimum value of the funtional over the population

and

∆

isa given number. In the paper, we take

∆ = 0

.

0001

.

•

The minimum value of the funtional overthe population

J

(

q

i

)

hanges by less than

10

−

8

inmore than 500 onseutive iterations.

Ifat least one of the onditions is satised, the resultingpopulationisthe sought-for one.

Choose from the population a vetor with the minimum value of the funtional. If the

onditionsare not satised, return tostep 2.

If the geneti algorithm is stuk in a loal minimum, the step of mutation will help to get

out of it. Experiene has shown that the global minimum an be found by using the geneti

algorithmforthe optimizationproblem. This is very importantwhen working with real data.

5 Condene intervals

We an determine the asymptoti properties of the IRWLS estimator (2.3) obtained using the implementation of Se. 2.1. As

K

→ ∞

,

q

IRW LS

has the following asymptoti properties [8,9,

14,15℄:

q

IRW LS

∼

N

(

q

ex

,

Σ

K

0

)

,

(5.1)

where

Σ

K

0

≈

K

P

k

=1

D

T

k

(

qex

)

V

0

−

1

Dk

(

qex

)

!

−

1

,

(5.2)

and the

3

×

4

matrix

D

k

(

q

ex

)

isgiven by







∂

Φ

1

k

(

q

ex

)

∂q

1

∂

Φ

1

k

(

q

ex

)

∂q

2

∂

Φ

1

k

(

q

ex

)

∂q

3

∂

Φ

1

k

(

q

ex

)

∂q

4

∂

Φ

2

k

(

q

ex

)

∂q

1

∂

Φ

2

k

(

q

ex

)

∂q

2

∂

Φ

2

k

(

q

ex

)

∂q

3

∂

Φ

2

k

(

q

ex

)

∂q

4

∂

Φ

3

k

(

q

ex

)

∂q

1

∂

Φ

3

k

(

q

ex

)

∂q

2

∂

Φ

3

k

(

q

ex

)

∂q

3

∂

Φ

3

k

(

q

ex

)

∂q

4







=







∂

Φ

1

k

(

q

ex

)

∂λ

1

∂

Φ

1

k

(

q

ex

)

∂λ

2

∂

Φ

1

k

(

q

ex

)

∂k

1

∂

Φ

1

k

(

q

ex

)

∂k

2

∂

Φ

2

k

(

q

ex

)

∂λ

1

∂

Φ

2

k

(

q

ex

)

∂λ

2

∂

Φ

2

k

(

q

ex

)

∂k

1

∂

Φ

2

k

(

q

ex

)

∂k

2

∂

Φ

3

k

(

q

ex

)

∂λ

1

∂

Φ

3

k

(

q

ex

)

∂λ

2

∂

Φ

3

k

(

q

ex

)

∂k

1

∂

Φ

3

k

(

q

ex

)

∂k

2







.

(5.3)

Sine the true values ofthe parameters

qex

and

V

0

are unknown, their estimates

q

ˆ

and

V

ˆ

are used toapproximate the asymptotiproperties of the IRWLS estimator

qIRW LS

:

qIRW LS

∼

N

(

qex,

Σ

K

0

)

≈

N

(ˆ

q,

Σ

ˆ

K

)

,

(5.4)

where

Σ

K

0

≈

Σ

ˆ

K

=

K

P

k

=1

D

T

k

(ˆ

q

) ˆ

V

−

1

Dk

(ˆ

q

)

!

−

1

.

(5.5)

The standard errors

SEi

(ˆ

qIRW LS

)

an then be alulated forthe

i

thelement of

qIRW LS

ˆ

by

SEi

(ˆ

qIRW LS

)

≈

q

ˆ

Σ

K

(15)

Toompute the ondeneintervals(at the

100(1

−

α

)

%level)forthe estimatedparameters inourmodel,wedenetheondeneintervalsassoiatedwiththe estimatedparameterssothat

P rob{qi

−

t

1

−

α/

2

SEi

(ˆ

q

)

< q

exi

< qi

+

t

1

−

α/

2

SEi

(ˆ

q

)

}

= 1

−

α

where

α

∈

[0

,

1]

and

t

1

−

α/

2

∈

R

+

. For a realization

y

and estimates

q

ˆ

, the orresponding ondeneintervals are given by

[

qi

−

t

1

−

α/

2

SEi

(ˆ

q

);

qi

+

t

1

−

α/

2

SEi

(ˆ

q

)]

.

(5.7)

Given asmall

α

value,the ritialvalue

t

1

−

α/

2

is omputedfromthe Student'sdistribution

t

K

−

4

with

K

−

4

degrees of freedom.

5.1 Algorithm for onstruting ondene intervals

1. Modeling noisy data. We add Gaussian noise of about 10 % in the data in the following

form:

Y

k

i

= Φ

i

(

tk

;

qex

)(1 + 0

.

1

·

ε

ˆ

i

k

)

, i

= 1

,

2

,

3

, k

= 1

, . . . , K,

where

ˆ

ε

i

k

is Gaussianrandomvariable:

ε

i

k

= 1

/

3

q

−

2 ln

α

ki

1

·

sin (2

πα

ki

2

)

,

and here

(

α

ki

1

, α

ki

2

)

isa pair of independentstandard values.

2. Compute the matrix

Σ

K

0

≈

Σ

ˆ

K

=

K

P

k

=1

D

T

k

(ˆ

q

) ˆ

V

−

1

Dk

(ˆ

q

)

!

−

1

(see (5.5)). To do this, we

perform the followingsteps:

(a) Ñalulate

V

ˆ

by the formula (2.8). Here

q

ˆ

is a parameter estimate, obtained by the geneti algorithm.

(b) For eah

k

= 1

, . . . , K

alulate matries

Dk, D

T

k

(see (5.3)). To do this, we need to arry out the followingsteps:

i. Let's onsider the individual (traditional) sensitivity funtions (TSF) dened by

the derivatives:

sq

i

(

t

) =

∂X

∂q

i

(

t

)

, i

= 1

, . . . ,

4

.

(5.8)

Here

X

(

t

) = (

T

1

(

t

)

, T

2

(

t

)

, T

∗

1

(

t

)

, T

∗

2

(

t

)

, V

(

t

)

, E

(

t

))

T

,

q

= (

q

1

, q

2

, q

3

, q

4

)

T

=

(

λ

1

, λ

2

, k

1

, k

2

)

T

.

Eah

sq

i

satises the following equation:

(

˙

s

q

i

(

t

) =

∂P

∂X

(

t, X

(

t

;

q

)

, q

)

s

q

i

(

t

) +

∂P

∂q

i

(

t, X

(

t

;

q

)

, q

)

.

s

q

i

(

t

0

) = 0

,

(5.9)

Here

P

= (

P

1

, P

2

, P

3

, P

4

, P

5

, P

6

)

T

isavetorof righthandside ofthe model(1.1). ii. Bysolvingthediretproblem(5.9),weobtainthevaluesof

∂X

j

∂q

i

,

i

= 1

, . . . ,

4

, j

=

(16)

iii. Calulate the values

∂

Φ

l

∂q

i

, l

= 1

,

2

,

3

, i

= 1

, . . . ,

4

by using following formulae:

∂

Φ

1

∂q

i

=

∂X

1

+

∂X

3

∂q

i

,

∂

Φ

2

∂q

i

=

∂X

5

∂q

i

,

∂

Φ

3

∂q

i

=

∂X

6

∂q

i

.

() By using formula(5.3) we alulatematries

Dk

(ˆ

q

)

,

D

T

k

(ˆ

q

)

.

(d) By using formula(5.5) we alulatematrix

Σ

0

(ˆ

q

)

.

3. We nd standard errors

SEi

(ˆ

q

)

for parametersby using formula(5.6).

4. We hoose

α

= 0

.

05

(thus we onsider ondene intervalsof 95 %).

5. Compute the value

t

1

−

α/

2

from the Student's distribution

t

K

−

4

with

K

−

4

degrees of freedom.

6. Calulateondene intervals forthe parameters by using formulae (5.7)

5.2 Numerial results of onstruting ondene intervals and solving

inverse problems

Inthepresentpaper,modeldataareusedforsolvingtheinverseproblem. Consideratimeinterval

T

= 100

days and onstrut apartitionof the domain

(0

, T

)

:

ω

1

=

{tj

:

tj

=

jht, ht

=

T /Nt, j

=

1

, . . . , Nt, Nt

= 10000

}

. To determine the vetor of parameters

q

= (˜

k

1

,

k

˜

2

,

λ

˜

1

,

˜

λ

2

)

T

= (

k

1

·

10

6

, k

2

·

10

3

, λ

1

·

10

−

5

, λ

2

·

10

−

2

)

T

fortheproblem(1.1)-(2.1),weonstrutthesynthetidata(2.1) as follows. For this we hoose, as the vetor of exat parameters,

qex

= (˜

k

ex

1

,

k

˜

2

ex

,

˜

λ

ex

1

,

˜

λ

ex

2

)

T

=

(

k

ex

1

·

10

6

, k

2

ex

·

10

3

, λ

ex

1

·

10

−

5

, λ

ex

2

·

10

−

2

)

T

= (0

.

1

,

₀

.

3198

,

₀

.

8

,

₀

.

1)

T

fromTable 1haraterizingthe typialHIV-infeted patient [2℄.

Exat

param-eters

qex

Estimates of

parameters

q

ˆ

Standard

er-rors

SEi

(ˆ

q

)

Condene intervals

e

λ

1

0

.

1

0

.

10060888

0

.

06521128

[

−

0

.

04469091; 0

.

24590866]

e

λ

2

0

.

3198

0

.

39198582

0

.

00543713

[0

.

37987113; 0

.

40410051]

e

k

1

0

.

8

0

.

80331192

0

.

00000567

[0

.

80329930; 0

.

80332455]

e

k

2

0

.

1

0

.

01818300

0

.

00000567

[0

.

01032578; 0

.

09135427]

Table 8: Numerialresults of onstruting ondene intervals.

The numerialresults onstruting ondene intervals are presented inthe Table 8. In the Figure 4 the ondene intervals for the parameters are presented in the logarithmsale. It is worthobservingthatinthisasetheondeneintervalfor

k

1

isquitesmallrelativetothatof

λ

1

whilethe reonstrution (estimation)of

λ

1

itselfisquitegood. Anexplanationof this liesinthe fat that when multiple parameters are being estimated, there may not be a diret orrelation

between size of ondene intervals and ability to estimate well a given parameter in a set of

parameters. We have seen this in other appliations. In partiular we refer to Se. 3.8 of [8℄ where it is found that

k

N

on

is not important to be estimated vs.

k

N

(17)

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

1

2

3

4

Figure4: Condeneintervals of the parameters

λ

1

, λ

2

, k

1

, k

2

in the logarithmsale.

(tothe sizes of the parameters being estimated) of the ondene intervals are omparable. We

believe this has to do with the information ontent (see [8℄) in the data set relative to a given parameteror pair of parameters.

In Figure5, the diret problemsolutions(1.1)(solid lines) for the thus-obtained parameters andnoisymeasurements (2.1)ofonentrationsatxedtimes(points)arepresented. Theurves showthat theommonrelativeaurayerrorofthe fourparametersidentiationissuiently

small for a good mathematial model that has a solution quite lose to the additional noisy

measurementsof CD4 T-lymphoytes (

T

1

+

T

∗

1

), immune eetors

E

,and freeviruses

V

.

15

20

25

30

35

40

0

30

60

90

E(t)

0⋅10

0

1⋅10

5

2⋅10

5

3⋅10

5

4⋅10

5

5⋅10

5

6⋅10

5

7

⋅

10

5

8

⋅

10

5

9

⋅

10

5

0

30

60

90

V(t)

5⋅10

4

1⋅10

5

2⋅10

5

2⋅10

5

2

⋅

10

5

3⋅10

5

4⋅10

5

4⋅10

5

5⋅10

5

5⋅10

5

6

⋅

10

5

0

30

60

90

T

1

(t) + T

1

*

(t)

Figure5: Conentrations of immuneeetors

E

(

t

)

(left),free virus

V

(

t

)

(enter), and T-lymphoytes

T

1

(

t

) +

T

∗

1

(

t

)

(right). Points onthe urves are noisy measurements (2.1) of onentrationsat xed times.

It shouldbenoted that the programrun time ona PC with proessor Intel(R) Core(TM) i3

(18)

Inthis paper, the problemofestimatingthe HIV-infetionparameters andthe immuneresponse

using additionalmeasurements of the onentrations of Ò-lymphoytes, free virus, and immune

eetorsatxed timesfor themathematialmodelofHIVdynamis (1.1)has been investigated numerially. The problem of speifying the parameters of the mathematial model (i.e., an

inverse problem) was redued tothe problemof minimizinganobjetive funtiondesribing the

deviationof the simulation results fromthe experimentaldata.

The linearized matrix ofthe inverse problemwasobtained by using methods oflinearization

anddisretization. The stabilityofthe inverse problemsolutionwasanalyzedusing the singular

valuedeompositionforlinearizedmatrixofdisreteinverseproblem. Bythismeansthestability

of the parameter identiation problem was investigated on the basis of the analysis of the

magnitude of the ondition number for the linearized matrix of the inverse problem (in ases

wherewe attempted toestimate 19,4 or2 parameters).

A geneti algorithm for solving a least squares minimization problemwas implemented and

investigated. Formeasurements performed one a week, individual parameters of patients have

been obtained using the geneti algorithm. In the paper [2℄by B.M.Adams et al.,the problem of parameter speiation for the mathematial model (1.1) was also onsidered. To solve this problem,the authorsusedageneralizedleastsquares method. Paper[2℄presentsthe results(for two parameters

k

1

and

k

2

) obtained by this method. In the present paper, we have determined fourparameters,namely

λ

1

, λ

2

, k

1

, k

2

,and have shown that the relativeerrorindetermining the modelparameters issuiently smallso that the modelagrees wellwith the measurements.

Aknowledgements

ThisworkwassupportedinpartbyPresidentGrantofRussianFederation(No.MK-1214.2017.1)

and in part by the U.S. Air Fore Oe of Sienti Researh under grant number AFOSR

FA9550-15-1-0298.

Referenes

1. A.S.Perelson, P.W.Nelson, Mathematialanalysis of HIV-I:Dynamis invivo,SIAM Rev.,

41 (1999),344.

2. B.M.Adams,H.T.Banks,M.Davidian,Hee-DaeKwon, H.T.Tran,S.N.Wynne,E.S.

Rosen-berg, HIVdynamis: Modeling, data analysis, and optimaltreatment protools, Journal of

Computational and Applied Mathematis,184 (2005), 1049.

3. H.T. Banks, MarieDavidian, ShuhuaHu,Grae M.Kepler,E. S.Rosenberg, ModelingHIV

immune response and validation with linial data, CRSC-TR07-09, Center for Researh

in Sienti Computation, N. C. State University, Raleigh, NC, Marh, 2007; J. Biologial

Dynamis,2 (2008), 357385.

4. G.Boharov,A.Kim,A.Krasovskii,V.Chereshnev,V.Glushenkova,A.Ivanov,Anextremal

shiftmethodforontrolofHIVinfetiondynamis,Russ. J.Numer.Anal.Math. Modelling,

(19)

Biol., 64 (2001),2964.

6. S.I.Kabanikhin, Inverse and Ill-Posed Problems,De Gruyter, 2011.

7. H.W. Engl, C. Flamm, P. Kugler, J. Lu, S. Muller and P. Shuster, Inverse problems in

systems biology,Inverse Problems, 25 (2009),123014, 51pp.

8. H.T. Banks, M.L. Joyner, Information ontent in data sets: A review of methodsfor

inter-rogationand model omparison, CRSC-TR17-15, Center for Researh in Sienti

Compu-tation, N. C. State University, Raleigh, NC, June, 2017; J. Inverse Ill-Posed Probl.,(2018),

toappear.

9. H.T. Banks, Shuhua Hu,and W.ClaytonThompson, Modelingand InverseProblems in the

Presene of Unertainty, Boa Raton, CRCPress, 2014.

10. B.P. Demidovih, I.A. Maron, Numerial Methods of Analysis, 3rd ed., Moskva: Nauka,

1967.

11. J. Nelder, R. Mead, A simplex method for funtion minimization, Computer Journal, 7

(1965),308313.

12. S.I.Kabanikhin, O.I.Krivorotko,Identiationof biologialmodels desribed by systems of

nonlinear dierential equations,J. Inverse Ill-Posed Probl., 23(2015), 519527.

13. S. Qiu, A. Msweeny, S. Choulet, A. Saha-Mandal, L. Fedorova, A. Fedorov, Genome

evo-lution by matrix algorithms(GEMA): Cellular automata approah to populationgenetis,

Genome Biol Evol., 6 (2014), 988 999; PubMed PMID:24723728.

14. M.Davidian,D.Giltinan,NonlinearModelsforRepeatedMeasurementData,London:

Chap-man &Hall, 1998.