We consider a coe cient inverse problems for one model of HIV infection. The problem is formulated as an minimization problem of a quadratic residual functional. The last one is turned to the xed point problem. We study two approaches to obtaining approximations such as the adaptive Wegstein method to solutions of the xed point problem. One way is the vector approach, this method uses the ratio of the norms of residuals for nding parameters of the method. The other way is the componentwise approach that shift tracking of approximations in each coordinate more subtle. Numerical results are presented.
There are an impressive number of methods for the approximate solutions of nonlinear scalar equations and systems. The search for new methods of solving such equations and ways to improve the e ciency of the wellknown and practically proven methods remains important. Newton's method and simple iterations method, being the simplest to build and structure and most capable of an exhaustive study, often serve as the starting point for this.
One way to improve the convergence and extend the scope of applicability of linearly convergent iterative
processes is to build processes of the Fejer type [1] which are called Mann iterations (see [2, 3, etc.]). Suppose
that the problem of a xed point is de ned by the equality
Here
: Rn ! Rn is a given continuous nonlinear mapping. These iterations are based on the formula
+ (
Here is a scalar parameter or a sequence of parameters = (k). Di erent approaches to the method of
xing this parameter lead to di erent speci c processes of the family (
The resulting iterative method is identical to the Aitken 2 method, in which the formula for acceleration is
applied every other step of the method of simple iterations. The Aitken method in the form (
When solving scalar equations, the parameters can be calculated using the formula which allows extending this method to the case of systems of nonlinear equations.
The method (
Let the given nonlinear system (
to the value x~(k+1) = x~(k) i x~(k+1) i
is carried out based on formulas x(k+1) = 'i x~(1k); x~(2k); : : : ; x~(k) ;
i n x~(k+1) = i
(k) 1
1 + (k) x~i(k) + 1 + (k) xi(k+1)
(k = 0; 1; 2; : : : );
The matrix-vector form of the method (
; i = 1; : : : ; n: (k); if sgn x~(k)
i (k); if sgn x~(k) i x(k+1) i x(k+1) i = sgn 'i(x(k+1)) = sgn x(k+1) i x(k+1) ;
i 'i(x(k+1)) ; where (k) =
(x(k+1)) x(k+1) x~(k) : Theorem 1. Let a vector-valued function be continuously di erentiable in some domain M contains its xed point : Let there exist nonnegative constants ak; bk; k = 0; 1; : : : ; such that If there exist q; that the following inequality holds miax 1 +1 i(k) 6 ak; max i (11) and all members of the sequence x~(k) and x(k) ; de ned by (10), do not leave the M; then the sequence x~(k) converges to provided that the initial point x~(0) 2 M: P r o o f. Regardless of the way in which parameters i(k); i = 1; : : : ; n is chosen, we deduce from equality (10) that
@x1 @x2 @xn i(k); i = 1; : : : ; n: If the following inequality holds 6 x~(k) ; equality (12) where k =
1(k); : : : ; n(k) calculated with x = implies the inequality Successive iterations of this inequality yield x~(k+1)
6 6 akbk x~(k) k + 6 q (k)
x~(k) x~(k) : x~(k+1) We have (11). It is now obvious that the above estimate for the error of the approximation x~(k+1) implies the convergence of the sequence x~(k) to for each x~(0) 2 M: Theorem 2. Let a vector-valued function be continuously di erentiable in some domain M contains its xed point : Let there exist nonnegative constants a; b c; such that max i
1 1 + i(k) 6 a; max i (k) i 6 b;
n max X i j=1 If the following inequality holds and all members of the sequence x~(k) and x(k) ; de ned by (10), do not leave the M; then the sequence x~(k) converges to provided that the initial point x~(0) 2 M:
To illustrate good properties of the elaborated method we apply it for solving one inverse coe cient problem for a model of HIV dynamics. An observer measures the concentration of viral particles in blood as well as a serum level of immunocompetent cells. These data are used to estimate coe cients of the model. This is a typical inverse dynamic problem and to deal with it a quadratic residual functional is introduced and the identi cation problem is formulated as a minimization problem. As a necessary condition for extremum dictates the last problem is reduced to a nonlinear system to be solved numerically. To accelerate a convergence of iterative process we use method (10). The implementation of these ideas in a formal and precise way is given in a paragraph 3 while a concise description of the HIV model is given in a paragraph 2. 2
Mathematical models provide a means to understand the human immunode ciency virus (HIV)-infected immune system as a dynamic process. Models formulated as di erential equations for the dynamic interactions of CD4+ lymphocytes and virus populations are useful in identifying essential characteristics of HIV pathogenesis and chemotherapy [7, 8]. The equations for the model with treatment are as follows: 8 dT (t) >>>> dt >> dTS (t) > > >>>> dt <> dTr(t) > dt >> dVS (t) > > >>>> dt >> dVr(t) > > : dt = S1
S2VS (t)
BS + VS (t) = kS VS (t)T (t)
Ti TS (t) = krVr(t)T (t)
Ti Tr(t) = (1 =
3Tr(t) Ci + VS (t) q) Ci + V3S (t) TS (t)V (t)
3TS (t) Ci + VS (t)
1 T (t)V (t) C + VS (t)
2 Ci + VS (t) TS (t)V (t);
2 Ci + VS (t) Tr(t)V (t); kV T (t)VS (t) +
GS VS B + V (t)
; V (t) + q
V (t)
kV T (t)Vr(t) + Gr (V (t)) T T (t) + (kS VS (t) + krVr(t)) T (t);
Vr(t) B + V (t) ; (13) where V (t) = VS (t) + Vr(t); Gr = (GS e(V V0)p)=(1 + e(V V0)p); the initial conditions are T (0) = T0; TS (0) = 0; Tr(0) = 0; V (0) = V0; the dependent variables are T is uninfected CD4+ T-cell population, Ts is CD4+ T-cell population infected by virus sensitive to the treatment, Tr is CD4+ T-cell population infected by virus restrictive to the treatment, Vs is infectious HIV population sensitive to the treatment, Vr is infectious HIV population restrictive to the treatment.
In equation 1 the term S1 S2VS (t)=(BS + VS (t)) represents the external input of uninfected CD4+ T-cells from the thymus, bone marrow, or other sources. It is assumed that there is a deterioration of this source as the viral level increases during the course of HIV infection, T is the death rate of uninfected CD4+ T-cells whose average lifespan is 1= T . The term 1T (t)V (t)=(C + VS (t)) represents CD4+ T-cell proliferation in the plasma due to an immune response that incorporates both direct and indirect e ects of antigen stimulation, where C is a saturation constant. The form assumed here idealizes the growth mechanisms of CD4+ T-cells, since subpopulations of antigen speci c CD4+ T-cells are not modeled. In equation 1 kS is the infection rate of CD4+ T cells by virus (it is assumed that the rate of infection is governed by the mass action term kS VS (t)T (t)). In the absence of virus the CD4+ T-cell population converges to a steady state of S1= T .
In equation 2 there is a gain term kS VS (t)T (t) of CD4+ T-cells infected by drug-sensitive virus, a loss term Ti TS (t) due to the death of these cells independent of the virus population, and a loss term 2TS (t)VS (t)=(Ci + VS (t)) dependent on the virus population due to bursting or other causes. The dependence of the loss term 2TS (t)VS (t)=(Ci +VS (t)) allows for an increased rate of bursting of infected cells as the immune system collapses and fewer of these cells are removed by CD8+ T-cells. The structure of equation 3 is the same.
In equation 4 the virus population is increased by the term 3TS (t)V (t)=(Ci +VS (t)). This term corresponds to the internal production of virus in the blood. The dependence of this term on TS (t) allows for a decreased rate of viral production in the plasma when the infected CD4+ T-cell population in the plasma collapses. Since most of the plasma virus is contributed by the external lymph source, the plasma virus population still increases steeply at the end stage of the disease. In equation 4 the virus population is decreased by the loss term kV T (t)VS (t), which represents viral clearance. In equation 4 there is a source of virus from the external lymphoid compartment, which is represented by the term GS VS =(B + V (t)) (B is a saturation constant). This term accounts for most of the virus present in the blood. the structure of equation 5 is the same.
A complete list of parameters and their estimated values for model (13) is given in [8]. Figure 1 shows a schematic diagram of the entire immune response process. A minimization problem for a functional (14) is turned to solving of nonlinear equation
= 0; it is easy to check that @ (a) = 2 P5 Pm 1 (xi; k(a) yi; k) @xi; k(a) ; l = 1; : : : ; p:
@al i=1 k=1 i @al
Method (10) is applied for solving the problem (15). We use a priory information to set the initial approximation of the vector a0: Calculation of the function in (15) implies a numerical solving of the di erential equations (13). Because of sti ness we use implicit fourth order Runge-Kutta method each step of which implies solving of an appropriate nonlinear system, and adaptive method (10) is used for this too.
To illustrate good properties of the proposed method (10) and e ciency of the approach described above we performed several numerical experiments. Let B and BS are unknown parameters to be estimated, so a = (B; BS ): To simulate the system dynamics and obtain the observations a was de ned according to [8] as a = (2; 13:8): The initial approximation was taken as a0 = (3; 18):
The system dynamics for true parameters value is depicted by solid lines, for initial value a0 by green and blue
The advantage of the proposed method (10) for solving nonlinear systems is that only one function calculation is required on each iterative step, while the calculation of the derivatives and matrix inversion is not required at all. In one-dimensional case the proposed method has the quadratic convergence; in general case according to the results of numerical estimations method (10) converges not slower than Newton's method.
This research is supported by the program 02.A03.21.0006 on 27.08.2013, Russian Science Foundation (RSF) 14-35-00005, RFBR 14-01-00065. 2 Process. Compu