Abstract
In this paper, the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise is analyzed. First, the stability of deterministic models is investigated. By analyzing the characteristic values of endemic equilibrium, we could obtain that internal HIV models reach a steady state under the influence of RTI and PI drugs. Then we discuss the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise, based on probability density functions. The functional methods are carried out to derive the approximate Fokker-Planck equation of stochastic internal HIV systems and further obtain the marginal probability density functions. Finally, numerical results show that the noise intensities have a great influence on uninfected cell, infected cell, and virus particles, for predicting the stability of stochastic dynamic systems subjected to Gaussian white noise and Gaussian colored noise.
1. Introduction
Governments and scientists all over the world have been concerning about the epidemic of HIV, with its high speed of spread around the world. As we all know that HIV has caused millions of deaths, and there are some millions of people living with HIV alone [1]. In June 2001, at a special session of the General Assembly on AIDS, world leaders made a commitment to ensure that resources for the global response to HIV/AIDS are substantial, sustained, and geared towards achieving results. As yet, there is no cure for HIV/AIDS. Together with the people of all sections of society, the medical world is working at utmost strain go study on the pathogenesis and properties of epidemic diseases [2, 3].
Mathematical models play a very important role in describing the Immunological response to infection with HIV, and making predictions about their behavior. Early models of HIV infection [4–6] were studied analytically and numerically by defining ordinary differential equations which are deterministic models. There are many authors to investigate how to control and predict HIV virus, based on deterministic HIV models [7–10].
In recent years, some authors [11–13] have added stochastic terms to incorporate variability introduced by a fluctuating environment or others. And Renshaw pointed out that the most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some averages so that the deterministic equilibrium is not an absolutely fixed state [14]. In fact, stochasticity plays a vital role in the structure and function of biological systems. Nowadays, stochastic internal HIV systems have been concerned with the study of Gaussian white noise [15–17]. However, there are few studies on internal HIV systems subjected to combined Gaussian white noise and Gaussian colored noise. What is more, Gaussian distributions are not appropriate in some practical cases. Many experimental evidences, particularly in biological virus systems, indicate that most of the noises are not only Gaussian white noise, and there may be Gaussian colored noise or Non-Gaussian noise or others [18]. In this paper, we mainly discuss internal HIV systems subjected to Gaussian white noise and Gaussian colored noise.
The discussions of stochastic systems play a key role especially for those analyses on the basis of characteristics of Lyapunov exponents, stationary densities, and characteristic function equations [19, 20]. Up to now, the authors [15–17] have proposed many theories and methods to study stochastic HIV systems, and the most important one is about stationary densities which have become an important way to examine basic statistical properties of stochastic systems, such as the stability, chaos, and bifurcation of stochastic systems. FPK method is an effect means to obtain stationary densities of stochastic dynamic systems and is often used in prediction of response process. For instance, Cetto, et al. [21] showed that a closed formula for the effective diffusion coefficient might be used to derive Fokker-Planck equations of the different approximate expressions. Ditlevsen, et al. [22] raised doubt about the validity of the spectral Fokker-Planck equation in its standard formulation and solved the equation with respect to stationary solutions in the particular case where the noise was Cauchy noise and the drift function was polynomial. Until now, stochastic stability of stochastic HIV models excited by Gaussian white noise and Gaussian colored noise based on FPK equation has not been considered.
In this paper, the characteristic values of epidemic equilibrium are calculated to consider the stability of HIV deterministic systems. And we mainly derive the approximate Fokker-Planck equation of internal HIV stochastic dynamic systems and get the general expression of their stationary densities. Considering variety of stochastic noise intensities, we analyze the changes of uninfected cell, infected cell, and virus particles, make predictions about the stability of stochastic dynamic systems subjected to Gaussian white noise and Gaussian colored noise, and discuss the fact that when noises are both Gaussian noises, these two cases agree very well [23].
The paper is organized as follows: in Section 2, we briefly review some basic facts about the stability HIV deterministic models which have involved the concentration of uninfected target cell, infected cell, and virus particles, with the help of the characteristic values of epidemic equilibrium; Section 3 is devoted to derive the approximate Fokker-Planck equation of the HIV system with Gaussian white noise and Gaussian colored noise and further obtain the general expression of their stationary densities; in Section 4, numerical simulations results for the different stochastic noise intensities are carried out to predict about the behavior of the uninfected target cell, infected cell, and virus particles; in Section 5, we will present the conclusions and future directions to close this paper.
2. The Stability Analysis of HIV Deterministic Models
In the early stage of HIV infection, after reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) drugs are given, virus particles are classified as either infections, not influenced, or as non-infection. On the basis of standard internal viral dynamics models [15, 16, 24–26], we will consider the following three-dimensional deterministic models which have involved the concentration of uninfected target cell , infected cell , and virus particles .
The initial conditions are . Here and and all parameters are in . () presents the reverse transcriptase inhibitor drug effect and () is the protease inhibitor drug effect. The constant is the total rate of production of healthy cells per unit time, is the per capita death rate of healthy cells, is the transmission coefficient between uninfected cells and infective virus particles, is the per capital death rate of infected cells, is the average number of infective virus particles produced by an infected cell in the absence of HAART during its entire infectious lifetime, and presents the per capita death rate of infective virus particles.
The Jacobian matrix for model system (1) is given as
The deterministic modes have been analyzed by Tuckwel et al. [25]. They show that if , then the disease free equilibrium is the unique equilibrium. And if , as well as the disease free equilibrium, then there is a unique equilibrium given byin which
Take parameters , , , , and , , , per cell. The characteristic values of Jacobian matrix for (1) which is in equilibrium are . From that, we can see that one of the characteristic values is real number and less than zero, and others are conjugate complex whose real parts are less than zero. Therefore, based on Lyapunov stability’s law, internal viral dynamics models are asymptotically stable, which implies that the system trajectories are ultimately confided to a fixed point. In other words, the HIV deterministic systems reach a steady state under the influence of RTI and PI drugs.
When there is randomness in parameters such as the disease death rate, it is a standard technique to introduce environmental noise into the parameters in this way [14]. Stochastic effects are considered by Gaussian white noise, which is only ideal noise and may not exist in the real word. Therefore, both Gaussian white noise and Gaussian colored noise are investigated to reach on upon uninfected and infected CD4 cells, even CD4 cells and virus particles.
3. Stationary Probability Densities of Internal HIV Models with Gaussian White Noise and Gaussian Colored Noise
The Fokker-Planck equations have played an important role in the investigation of unusual statistic properties of dynamic systems, such as biology systems. In this paper, their statistic characteristics are predicted, by deriving the approximate expressions of stationary probability densities for uninfected target cell, infected cell, and virus particles.
The models with Gaussian white noise and Gaussian colored noise in the paper seek to describe the dynamics of HIV-1 rival load during primary infection.
Where and are dependent, Gaussian white noise and Gaussian colored noises with the intensity of noises and , respectively, the following statistical properties:in which and are the self-correlation time of the noises.
Let be order intensity coefficient; then
In order to derive the approximate Fokker-Planck equation of the HIV driven by Gaussian white noise and Gaussian colored noise, some signs are defined: in which
Hence we can write (5) as
Based on above definitions, time-dependent joint probability density function of satisfies the FPK equation.where is a partial differential operator, defined as
Consider Wong-Zakai’s modification terms, drift coefficient and diffusion coefficient are provided as where
Furthermore, initial condition, boundary condition, and normalization condition of can be expressed as follows, respectively.
For convenience, we definewhere is a defined domain, determined by initial vector .
When , is independent of time and satisfies reduced FPK equation.
By utilizing (17), the marginal probability density function of can be determined aswhere , .
4. Numerical Simulations
In order to illustrate some of the effects of Gaussian white noise and Gaussian colored noise, we numerically solve the deterministic system of differential equations and stochastic system of stochastic differential equations. The parameter values for Figures 1–3 are , , , and , , , per cell. When and are both Gaussian white noises, the results are consistent with findings in [23].
Figures 1–3 demonstrate the influence of Gaussian white noise and Gaussian colored noise onto the approximate stationary solutions. It is seen from Figure 1 that when the time increases, the number of uninfected target cells for noiseless conditions increases rapidly and reaches a set point level. However, for uninfected target cells with noise, the number firstly increases rapidly and then changes in an appropriate range. And compared with the numerical results, we can find that the range is bigger and noise influenced is more intense. Figures 2-3 show that noises have great effect on infected cell and virus particles , respectively. Infected cells and virus particles with noises in number are fewer than ones without noise, which means that Gaussian white noise and Gaussian colored noise are helpful to improve human’s immune system.
When the parameters , , per cell, are fixed, the system factors are considered: one is the intensity of Gaussian white noise and the other is the intensity of Gaussian colored noise .
Figures 4–6 show that when and take different values, respectively, noises have a great influence on the stationary marginal probability density functions. It is shown from Figure 5 that the peak of the stationary marginal probability density appears between and and remains unchanged with and changed, which means that the noise-induced phase transitions do not occur. It is also clear that the decreases in stochastic noise intensity can lead to higher peaks of the stationary probability density . That is, litter intensity values lead to larger probability that system will stay close to the equilibrium state. It can be seen from Figures 5-6 that the probabilities of infected cells and virus particles are decreasing rapidly and reach level no matter whether the parameters and are changed, which means that the infected cells and virus particles are approximate steady state.
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5. Conclusions
The internal HIV models subjected to Gaussian white noise and Gaussian colored noise are mainly studied in this article. It is found that the intensities of noises influence greatly not only uninfected target cells, infected cell, and virus particles, but also their stationary marginal probability density functions. And compared with the internal HIV models with Gaussian white noises, systems driven by Gaussian white noise and Gaussian colored noise are more stable and more conform to reality. In our future work, we will need deep-going study on how to control infected cells and virus particles in short time to HIV systems driven by Gaussian white noise and Gaussian colored noise.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11847081), the Key Scientific Research Projects of Henan Province (Grant nos. 17A110039 and 19A110007), the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (Grant no. 2017QNJH18), and High-Level Personal Foundation of Henan University of Technology (no. 2017BS009).