Robust Model-Free Software Sensors for the HIV/AIDS Infection Process

This paper considers the problem of the HIV/AIDS Infection Process filtering characterized by three compounds, namely, the number of healthy T-cells, the number of infected T-cells and free virus particles. Only the first and third of them can be measurable during the medical treatment process. Moreover, the exact parameter values are admitted to be also unknown. So, here we deal with an uncertain dynamic model that excludes the application of classical filtering theory and requires the application of robust filters successfully working in the absence of a complete mathematical model of the considered process. The problem is to estimate the number of infected T-cells based on the available information. Here we admit the presence of stochastic “white noise” in current observations. To do that we apply the Luenberger-like filter (software sensor) with a matrix gain, which should be adjusted at the beginning of the process in such a way that the filtering error would be as less as possible using the Attractive Ellipsoid Method (AEM). It is shown that the corresponding trajectories of the filtering error converge to an ellipsoidal set of a prespecified form in mean-square sense. To generate the experimental data sequences in the test-simulation example, we have used the well-known simplified HIV/ AIDS model. The obtained results confirm the effectiveness of the suggested approach.

systems that affect us. The process of using partial observations and a stochastic model to make inferences about an evolving system is known as stochastic state estimation (or filtering). In this paper, we consider the problem of the HIV/ AIDS Infection Process filtering characterized by three compounds: the number of healthy T-cells, the number of infected T-cells and free virus particles. Only the first and third of them can be measurable during the medical treatment process. The problem is to estimate the number of infected T-cells (to create a software sensor) based on the available information. Here we admit the presence of stochastic "white noise" in current observations as well as in the dynamics of other components.

HIV/AIDS Infection Process
Human Immunodeficiency Virus ( The dynamic HIV/AIDS have been studied by many researchers (see, for example, [1] [2]). Experimental data show that the treated disease progression rate is significantly varied between individuals: from two weeks up to 20 years. Figure 1 shows the natural history of HIV infection dynamics. One can see that if an individual has been received HIV virus as primary infection, a number of HIV virus will dramatically increase in the first 30 days (resulting CD4 + T-cell reduction). Then after the primary infection period, a body builds HIV antibodies for agent virus so that, the infection still stabilizes an approximate steady state. In fact, up to now no effective cure for HIV exists, but nevertheless by some proper treatment and medical care, HIV can be controlled. To realize these treatments on-line information data are extremely required. The special physic equipment for these on-line measurements is very expensive and unavailable in many health clinics. So, the problem of designing the cheap and easy realizable that cells increase at a constant rate σ and die in proportion to their size with death rate µ , so the average number of cells that die at time t is given by T µ . Similarly, the specialized CD4 + T-cells are activated at a mean dTV , a term that describes the encounter of the T cells with infectious viral particles. It is assumed that specialized CD4 + T-cells also die at an average a T µ . If the population of Tes cells infected by the virus with a probability β , the principle of mass action allows to establish that the average number of cells that are infected at a time t is TV β , corresponding to a classical consideration in this type of models. In the same way CD4 + T-cells Specialized a T are susceptible to infection and are infected at an average of a T ψ at a time t.
Based on the results of ([1] [2]) we may conclude that the current of an AIDS person state at time 0 t ≥ can be characterized by the following three compounds:  : , , x t x t x t x t =  is not available now. Some simplified mathematical model, given in [1], we will use here to generate ( ) x t during the test-numerical simulation but not for the filter designing.

Briefly on the Most Popular Filtering Methods
All classical filtering methods require the exact knowledge of the dynamic model which components are intended to be estimated. The main of them are as follows.
The Wiener (frequency domain) filtering. The origins of the filtering problem in discrete time can be traced back to the works [3] and [4]. In the continuous time case the first analysis of the optimal state estimation of dynamic systems in the presence of noise was given in [5]. The results were included in a classified National Defense Research Council report issued in January/February 1942.
Subsequently declassified, it appeared as a book in 1949.
The Kalman-Busy-Stratonovich (time-domain) filtering. The next major development in stochastic filtering was the introduction of the linear filter. In this case, the signal satisfies a stochastic differential equation with linear coefficients and Gaussian initial condition. The linear filter can be solved explicitly in a finite-dimensional format: the distribution of the state estimate is shown to be Gaussian, and hence completely determined by its mean and its covariance matrix. These were the reasons for the linear filter's widespread success in the 1960s. Bucy and Kalman were the pioneers in this field. Kalman was the first to publish in a wide circulation journal. In [6], he solved the discrete time version of the linear filter. Bucy obtained similar results independently [7]. Stratonovich suggested the same filter, but for another type if stochastic integrals [8].
The extended Kalman filter (EKF). Following the success of the linear filter, scientists started to explore different avenues. Firstly they extended the application of the Kalman filter beyond the linear/Gaussian framework. The basis of this extension is the fact that, locally, all systems behave linearly. So, at least locally, one can apply the Kalman filter equation. This gave rise to a class of algorithm called the extended Kalman filter [9]. At the time of writing these algorithms, most of which are empirical and without theoretical foundation, are still widely used in a variety of applications.
The ensemble Kalman filter (EnKF). The progress in data assimilation is related with both increased computational power and the introduction of techniques that are capable of handling large amounts of data and more severe nonlinearities. The EnKF has been introduced to petroleum science recently [10] and, in particular, has attracted attention as a promising method for solving the history matching problem.
The singular evaluative interpolated Kalman filter (SEIKF). Inherent data and model uncertainties render the history-matching inverse problem extremely non-unique. Therefore, a reliable uncertainty quantification framework for predicting dynamic performance requires multiple models that match field production data. An efficient variant of the ensemble Kalman filter, namely, Singular Evaluative Interpolated Kalman Filter (SEIKF) is applied to the multi-model history-matching problem [11].
Advanced developments in the time domain filtering. In the mid-1960s in [12], [13] there were derived and analyzed the stochastic models using the Itô (and not Stratonovich) calculus. In [14] there was provided the first rigorous derivation in the case of a general observation process where the signal and observation noises may be correlated. In 1968, in [15] there was introduced the innovation approach to linear filtering. This new method for deducing the filtering equations was extended in the early 1970s in [16] and [17]. Similarly, the other type of the filtering equation was introduced in the same period in [18] [19] [20] [21], is consequently referred to as the Zakai or the Duncan-Mortensen-Zakai equation. The stochastic partial differential equations (SPDEs) associated with the filtering equations were rigorously analyzed and extended in the late 1970s in [

Main Contribution of the Paper
In this paper we follows the approach suggested in [29] [30].
• This paper considers the problem of designing a robust model-free filter for a respectively wide class of uncertain nonlinear stochastic system where classical filtering theory can not be applied.
• The behavior of these systems is given in Itô form and contains both a regular part, which assumed to be the Quasi-Lipschitz type but unknown exactly, as well as a stochastic part generated by a standard vector Wiener process.
• Filtering itself is suggested to be realized by a Luenberger-like filter with a matrix gain which should be adjusted in the beginning of the process in such a way that the filtering error would be as less as possible.
• It is shown that the corresponding trajectories of the filtering error converge (in the mean-square sense) to an ellipsoidal set of a prespecified form.
• We show that the HIV/AIDS infection process can be effectively realized by the suggested model-free technique based on the, so-called, Attractive Ellipsoid Method [31].

Class of Possible Dynamics for the Main State Variables of HIV Dynamics
The HIV dynamics (human immunodeficiency virus (HIV) causes the acquired immune deficiency syndrome, knows as AIDS) were analyzed in [32] [33] [34].
It captures the time rate of healthy, infected white blood cells (T-cells) and the number of HIV viruses. The presented analysis is limited in the sense that it "cannot take into the account patient factors as physiological/genetic level, physicochemical factors at cell-protein-viral interactions level, and viral factors that relate to the various HIV strains and clades". Based on the corresponding numerical data it is possible to conclude that the dynamics of the HIV/AIDS Infection Process can be completely (without consideration the limiting factors mentioned above) described by the following systems of stochastic differential equations: is the state vector of the system in time 0 t ≥ which may contain unmeasurable components, Notice that in our case of the HIV/AIDS Infection Process we have Remark 2. The system (1) is written in the, so-called, engineering format.
The rigorous mathematical description (which we are working with) is given in Appendix. It is represented by the system of stochastic differential equations of the Itô type.

The Class of Uncertainties
Here we suppose that the uncertain vector functions The physical sense of the "linear represents" A and C of the classes ( ) , , C c c  is discussed in [30] and [31]. The parameters of these are c characterizes the growth rates of ( ) f x as x → ∞ which is not faster than linear function of x .
The following properties will be required hereafter: The pair of matrices ( ) , C A is observable. The stochastic system (1) is quadratically stable, that is,

The Structure of the Filter
The state vector t x will be estimated by the Luenberger-type filter where ˆt x is referred to as the "state estimate" of t x and n p L × ∈  is called the observer-gain matrix to be designed. Here the initial condition of the dynamic filter 0 x  (6) may differ from the initial conditions 0 x of the system (1). Now our problem can be formulated in the following manner: find the observer-gain matrix L which provides the closeness of ˆt x to t x in some probabilistic sense. Below we will give the formal formulation.

The Best Selection of the Observer-Gain Matrix
The next definition will be in use hereafter. where the weighting matrix should be done as much as possible satisfying 0 filt filt (the biggest eigenvalues of filt P correspond the smallest attractive ellipsoid semi-axis). So, we intend to obtain the gain-matrix L which asymptotically guarantee the boundedness (in mean-square sense) of all trajectories in an attractive region containing the origin, and, moreover, which minimize the "size" (in this case, the trace of the inverse ellipsoidal matrix) of the attractive ellipsoid, containing this bounded region. Below we formulate the 1-st main result.
Theorem 1 (on the mean-square attractive ellipsoid). If the assumptions H1-H2 are fulfilled and additionally, the following matrix inequality holds for some L, is attractive in mean-square sense, that is, The proofs of this and the next theorem are in Appendix. The 2-nd main result is as follows: Theorem 2 (on the best observer-gain matrix) The best observer-gain ma matrix L * is given by (14) where X * and Y * are the solution of the following constraint optimization problem

On the Numerical Solution of the Constraint Optimization Problem
The solution of the suboptimal constraint optimization problem (15) can be obtained by the SEDUMI and YALMIP toolboxes of MATLAB which effectively use the Interior Point Method (see, for example, the details in [31]). These toolboxes are used at each step of the following iterative procedure: 1) First, we fix the scalar parameters , α ε and solve the problem (15) varying only the matrix variables , X H and Y using the toolboxes mentioned above; 2) Second, for the found matrix variables , X H and Y we solve the problem (15) with respect to the scalar parameters α and ε ; 3) Then the process iterated up to the iteration which has no solution (the toolboxes mentioned above provide this information); 4) Returning one-step back we declare this iterative approximation as a solution , , , and X H Y α ε * * * * * of the considered optimization problem (15).
For the numerical implementation, then we consider the approximation of the Gaussian noise signal t W by the noisy signal in Simulink is given in terms of a Gaussian noise signal generator t W , using the following approximation: , 0 1

Filtering of HIV/AIDS Infection Model Mathematical Model of HIV/AIDS
The The constant parameters in (16) are as follows: The Figures 2-4 show the state trajectories 1 2 3 , , x x x and their estimates 1 2, The corresponding zoom-images are given in Figure 5 and Figure 6. It can be seen from these figures that the filtering process has a good performance maintaining the filter estimates close to their real states.   x 2) To demonstrate that the suggested approach is sufficiently robust with respect to selection of matrix A, let us repeat the numerical example with another filter corresponding the following linear represents with The results are as follows: x Figure 6. Zoom-image of 3 x and its estimate 3 .

Conclusions
In this paper, the proposed Attractive Ellipsoid Method can be successfully applied to the filtering process of nonlinear uncertain stochastic models given in Itô form, where the Luenberger-like filter, whose gain matrix should be designed, is suggested to be applied to the estimation process and the Itô calculus  To minimize the size of this ellipsoid the standard technique, under the LMI constraints, may be applied, also the suggested method is respectively robust with respect to the selection of the linear represents participating in the filter structure. Finally, the well-working of the suggested method is illustrated by the application to the filtering of the HIV/AIDS infection model.

The Mathematical Description of the Considered System: The Itô-Type Model
In the integral form this relation can be expressed as Using the property of the Itô integral and if 0 W ≤ , from (21) we may conclude that , 0 Taking β µ α = from the last differential inequality we obtain

Proof of Theorem 2
The "best" gain matrix L of the filter is a solution of the following optimization problem:  S P H α ε )) has the form (15). Theorem is proven.
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