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.
Many aspects of phenomena critical to our lives cannot be measured directly. Fortunately, models of these phenomena, together with more limited obser- vations frequently allow us to make reasonable inferences about the state of the 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.
Human Immunodeficiency Virus (HIV) stands for human immunodeficiency virus. If left untreated, HIV can lead to the disease AIDS (acquired immuno- deficiency syndrome). Unlike some other viruses, the human body can’t get rid of HIV completely. So once you have HIV, you have it for life. That’s why the problems of HIV/AIDS are very important from medical and human points of view. HIV attacks the body’s immune system, specifically the CD4 cells (T-cells), which help the immune system fight off infections. If left untreated, HIV reduces the number of CD4 cells (T-cells) in the body (directly and indirectly destroys CD4 + T-cells), making the person more likely to get infections or infection- related cancers. Over time, HIV can destroy so many of these cells that the body can’t fight off infections and disease. These opportunistic infections or cancers take advantage of a very weak immune system and signal that the person has AIDS, the last state of HIV infection. The medicine used to treat HIV is called antiretroviral therapy or ART. If taken the right way, every day, this medicine can dramatically prolong the lives of many people with HIV, keep them healthy, and greatly lower their chance of transmitting the virus to others. Today, a person who is diagnosed with HIV, treated before the disease is far advanced, and stays in treatment can live a nearly as long as someone who does not have HIV.
The dynamic HIV/AIDS have been studied by many researchers (see, for example, [
on-line software sensors seems to be very actual.
The model is proposed to include the activation process of CD4 + T-cells and their intervention in the HIV infection dynamics. Let T be the average number of CD4 + T-cells at time t,
Based on the results of ( [
・
・
・
The states
Problem. Based on the available measurements
Remark 1. Notice that any real mathematical model describing the exact behavior of the state vector
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 [
The Kalman-Busy-Stratonovich (time-domain) filtering. The next major deve- lopment 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 [
The extended Kalman filter (EKF). Following the success of the linear filter, scientists started to explore different avenues. Firstly they extended the appli- cation 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 [
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 [
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 pre- dicting dynamic performance requires multiple models that match field pro- duction 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 [
Advanced developments in the time domain filtering. In the mid-1960s in [
In this paper we follows the approach suggested in [
・ 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 Elli- psoid Method [
The HIV dynamics (human immunodeficiency virus (HIV) causes the acquired immune deficiency syndrome, knows as AIDS) were analyzed in [
where
the vector functions
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.
Here we suppose that the uncertain vector functions
The physical sense of the “linear represents” A and C of the classes
To clarify the parameters of the Quasi-Lipschitz mapping class:
- The parameters A,
-A characterizes the gradient of a linear mapping
as an approximation of
-
-
The following properties will be required hereafter:
The pair of matrices
The stochastic system (1) is quadratically stable, that is,
The state vector
where
Now our problem can be formulated in the following manner: find the ob- server-gain matrix L which provides the closeness of
The next definition will be in use hereafter.
Definition 1. The ellipsoid
(with the center point in 0 and the ellipsoidal matrix
1) In mean-square sense, if
2) With probability one (or almost sure), if for any time-subsequences
finite number of times, that is,
where
Let us introduce the “measure of closeness”
where the weighting matrix should be done as much as possible satisfying
(the biggest eigenvalues of
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,
then the ellipsoid
with the center in the origin
is attractive in mean-square sense, that is,
where
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
where
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 [
1) First, we fix the scalar parameters
2) Second, for the found matrix variables
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
of the considered optimization problem (15).
For the numerical implementation, then we consider the approximation of the Gaussian noise signal
The Mathematical models of HIV dynamics (human immunodeficiency virus (HIV) causes the acquired immune deficiency syndrome, knows as AIDS) were derived several years ago. In this study the third-order model of HIV dynamics is considered. It captures the time rate of healthy and infected white blood cells (T-cells) and the number of HIV viruses.
There are more complex models of HIV dynamics that can be found in the literature (see, for example, and. The methodology presented in these papers can be applied with minor modifications to the other models of HIV dynamics. However, as indicated by an anonymous reviewer of the manuscript, it should be emphasized that 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."
Consider now the simplified nonlinear HIV-dynamics model (see [
The constant parameters in (16) are as follows:
-
-
-
-
-
-
Here C corresponds to the real situation when only the first and the third states, corrupted by the random noises, are available in time. The random va- riables
so that
1) In (18) select
and
Here
The matrix parameters P and L, obtained by the application of the suggested approach and realizing the robust output linear controller, are as follows:
The Figures 2-4 show the state trajectories
The corresponding zoom-images are given in
mation errors to the attractive ellipsoid
We also have
2) To demonstrate that the suggested approach is sufficiently robust with re- spect to selection of matrix A, let us repeat the numerical example with another filter corresponding the following linear represents with
One can see that
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 de- signed, is suggested to be applied to the estimation process and the Itô calculus
should be used to derive the corresponding attractive ellipsoids where almost all trajectories of the state estimation errors converge.
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.
Alazki, H. and Poznyak, A. (2017) Robust Model-Free Software Sensors for the HIV/AIDS Infection Process. International Journal of Modern Nonlinear Theory and Application, 6, 39-58. http://dx.doi.org/10.4236/ijmnta.2017.62004
Consider a filtered probability space
We are interested in the following nonlinear stochastic differential equations
Here
for any integrable function
First, let us represent the system (18) in the, so-called, quasi-linear format:
For the estimation error
For the storage function
using the Itô formula (see, for example, [?]) we obtain
In the integral form this relation can be expressed as
Using the property of the Itô integral
applying the operator
we get
Since
we obtain
where
and
Then, finally we get
and if
Then from (21) for the function
it follows:
Taking
implying
In view of
we conclude the proof.
The “best” gain matrix L of the filter is a solution of the following optimization problem:
This problem is equivalent to the following one
Notice that
where the matrices H satisfies
or, equivalently, by the Shour’s lemma
Finally, in new variables
our problem (with the supporting functional (
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