the stochastic stability of internal hiv models with...

Research ArticleThe Stochastic Stability of Internal HIV Models withGaussian White Noise and Gaussian Colored Noise

XiyingWang 1 Yuanxiao Li1 and Xiaomei Wang2

1College of Science Henan University of Technology Zhengzhou 450001 China2School of Mathematics Science University of Electronic Science and Technology Chengdu 610054 China

Correspondence should be addressed to Xiying Wang wangxiying668163com

Received 20 January 2019 Accepted 5 March 2019 Published 19 March 2019

Academic Editor Vicenc Mendez

Copyright copy 2019 Xiying Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the stochastic stability of internal HIVmodels driven by Gaussian white noise and Gaussian colored noise is analyzedFirst the stability of deterministic models is investigated By analyzing the characteristic values of endemic equilibrium we couldobtain that internal HIVmodels reach a steady state under the influence of RTI and PI drugsThenwe discuss the stochastic stabilityof internal HIV models driven by Gaussian white noise and Gaussian colored noise based on probability density functions Thefunctionalmethods are carried out to derive the approximate Fokker-Planck equation of stochastic internalHIV systems and furtherobtain the marginal probability density functions Finally numerical results show that the noise intensities have a great influenceon uninfected cell infected cell and virus particles for predicting the stability of stochastic dynamic systems subjected to Gaussianwhite noise and Gaussian colored noise

1 Introduction

Governments and scientists all over the world have beenconcerning about the epidemic of HIV with its high speed ofspread around the world As we all know that HIV has causedmillions of deaths and there are some millions of peopleliving with HIV alone [1] In June 2001 at a special sessionof the General Assembly on AIDS world leaders made acommitment to ensure that resources for the global responseto HIVAIDS are substantial sustained and geared towardsachieving results As yet there is no cure for HIVAIDSTogether with the people of all sections of society themedical world is working at utmost strain go study on thepathogenesis and properties of epidemic diseases [2 3]

Mathematical models play a very important role indescribing the Immunological response to infection withHIV and making predictions about their behavior Earlymodels of HIV infection [4ndash6] were studied analyticallyand numerically by defining ordinary differential equationswhich are deterministic models There are many authors toinvestigate how to control and predict HIV virus based ondeterministic HIV models [7ndash10]

In recent years some authors [11ndash13] have added stochas-tic terms to incorporate variability introduced by a fluctuatingenvironment or others And Renshaw pointed out that themost natural phenomena do not follow strictly deterministiclaws but rather oscillate randomly about some averagesso that the deterministic equilibrium is not an absolutelyfixed state [14] In fact stochasticity plays a vital role inthe structure and function of biological systems Nowadaysstochastic internal HIV systems have been concerned withthe study of Gaussian white noise [15ndash17] However there arefew studies on internal HIV systems subjected to combinedGaussian white noise and Gaussian colored noise What ismore Gaussian distributions are not appropriate in somepractical cases Many experimental evidences particularlyin biological virus systems indicate that most of the noisesare not only Gaussian white noise and there may beGaussian colored noise or Non-Gaussian noise or others[18] In this paper we mainly discuss internal HIV systemssubjected to Gaussian white noise and Gaussian colorednoise

The discussions of stochastic systems play a key roleespecially for those analyses on the basis of characteristics of

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 6951389 8 pageshttpsdoiorg10115520196951389

2 Discrete Dynamics in Nature and Society

Lyapunov exponents stationary densities and characteristicfunction equations [19 20] Up to now the authors [15ndash17] have proposed many theories and methods to studystochastic HIV systems and the most important one is aboutstationary densities which have become an important wayto examine basic statistical properties of stochastic systemssuch as the stability chaos and bifurcation of stochasticsystems FPK method is an effect means to obtain stationarydensities of stochastic dynamic systems and is often usedin prediction of response process For instance Cetto et al[21] showed that a closed formula for the effective diffusioncoefficient might be used to derive Fokker-Planck equationsof the different approximate expressions Ditlevsen et al[22] raised doubt about the validity of the spectral Fokker-Planck equation in its standard formulation and solvedthe equation with respect to stationary solutions in theparticular case where the noise was Cauchy noise and thedrift function was polynomial Until now stochastic stabilityof stochasticHIVmodels excited byGaussianwhite noise andGaussian colored noise based on FPK equation has not beenconsidered

In this paper the characteristic values of epidemicequilibrium are calculated to consider the stability of HIVdeterministic systemsAndwemainly derive the approximateFokker-Planck equation of internal HIV stochastic dynamicsystems and get the general expression of their stationarydensities Considering variety of stochastic noise intensitieswe analyze the changes of uninfected cell infected celland virus particles make predictions about the stability ofstochastic dynamic systems subjected toGaussianwhite noiseand Gaussian colored noise and discuss the fact that whennoises are both Gaussian noises these two cases agree verywell [23]

The paper is organized as follows in Section 2 we brieflyreview some basic facts about the stability HIV deterministicmodels which have involved the concentration of uninfectedtarget cell infected cell and virus particles with the help ofthe characteristic values of epidemic equilibrium Section 3is devoted to derive the approximate Fokker-Planck equationof the HIV system with Gaussian white noise and Gaussiancolored noise and further obtain the general expression oftheir stationary densities in Section 4 numerical simulationsresults for the different stochastic noise intensities are carriedout to predict about the behavior of the uninfected target cellinfected cell and virus particles in Section 5 we will presentthe conclusions and future directions to close this paper

2 The Stability Analysis of HIVDeterministic Models

In the early stage of HIV infection after reverse transcriptaseinhibitor (RTI) and protease inhibitor (PI) drugs are givenvirus particles are classified as either infections not influ-enced or as non-infection On the basis of standard internalviral dynamics models [15 16 24ndash26] we will considerthe following three-dimensional deterministic models whichhave involved the concentration of uninfected target cell1198851(119905) infected cell 1198852(119905) and virus particles 1198853(119905)

1198851 (119905) = 120582 minus 1205751198851 (119905) minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905)1198852 (119905) = (1 minus 120574) 1205731198851 (119905) 1198853 (119905) minus 1198861198852 (119905)1198853 (119905) = (1 minus 120578)1198731198861198852 (119905) minus 1205831198853 (119905)

minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905) (1)

The initial conditions are 1198851(0) = 11988510 1198852(0) =11988520 1198853(0) = 11988530 Here 1198851(119905) 1198852(119905) and 1198853(119905) isin 119877+ andall parameters are in 119877+ (1 minus 120574) ((0 lt 120574 lt 1)) presentsthe reverse transcriptase inhibitor drug effect and (1 minus 120578)((0 lt 120578 lt 1)) is the protease inhibitor drug effect Theconstant 120582 is the total rate of production of healthy cells perunit time 120575 is the per capita death rate of healthy cells 120573is the transmission coefficient between uninfected cells andinfective virus particles 119886 is the per capital death rate ofinfected cells 119873 is the average number of infective virusparticles produced by an infected cell in the absence ofHAART during its entire infectious lifetime and 120583 presentsthe per capita death rate of infective virus particles

The Jacobian matrix for model system (1) is given as

119869

= [[[[

minus120575 minus (1 minus 120574) 1205731198853 (119905) 0 minus (1 minus 120574) 1205731198851 (119905)(1 minus 120574) 1205731198853 (119905) minus119886 (1 minus 120574) 1205731198851 (119905)minus (1 minus 120574) 1205731198853 (119905) (1 minus 120578)119873119886 minus120583 minus (1 minus 120574) 1205731198851 (119905)]]]] (2)

The deterministic modes have been analyzed by Tuckwelet al [25] They show that if 1198770 = (1 minus 120574)120573119873(1 minus 120578)(120575120583 +120573120582(1minus120574)) le 1 then the disease free equilibrium is the uniqueequilibriumAnd if1198770 = (1minus120574)120573119873(1minus120578)(120575120583+120573120582(1minus120574)) gt 1as well as the disease free equilibrium then there is a uniqueequilibrium 1198750 given by

1198750 = (119885lowast1 119885lowast2 119885lowast3 ) (3)

in which

119885lowast1 = 120583120573 (1 minus 120574) [119873 (1 minus 120578) minus 1]119885lowast2 = 120573120582 (1 minus 120574)119873 (1 minus 120578) minus 120573120582 (1 minus 120574) minus 120575120583120572120573 (1 minus 120574) [119873 (1 minus 120578) minus 1]119885lowast3 = 120573120582 (1 minus 120574)119873 (1 minus 120578) minus 120573120582 (1 minus 120574) minus 120575120583120573120583 (1 minus 120574)

(4)

Take parameters 120575 = 01119889119886119910minus1 119886 = 05119889119886119910minus1 120583 =5119889119886119910minus1 120574 = 05 120578 = 05 1205901 = 05 and 1205902 = 05 120573 =1 times 10minus8119889119886119910minus11198891198983 120582 = 107119889119886119910minus11198891198983 119873 = 100 per cellThe characteristic values of Jacobian matrix for (1) which isin equilibrium 1198750 are 1205821 = minus03589+04210119894 1205822 = minus03589minus04210119894 1205823 = minus56355 From that we can see that one ofthe characteristic values is real number and less than zeroand others are conjugate complex whose real parts are lessthan zeroTherefore based on Lyapunov stabilityrsquos law inter-nal viral dynamics models are asymptotically stable which

Discrete Dynamics in Nature and Society 3

implies that the system trajectories are ultimately confided toa fixed point In other words the HIV deterministic systemsreach a steady state under the influence of RTI and PI drugs

When there is randomness in parameters such as thedisease death rate it is a standard technique to introduceenvironmental noise into the parameters in this way [14]Stochastic effects are considered by Gaussian white noisewhich is only ideal noise and may not exist in the realword Therefore both Gaussian white noise and Gaussiancolored noise are investigated to reach on upon uninfectedand infected CD4 cells even CD4 cells and virus particles

3 Stationary Probability Densities of InternalHIV Models with Gaussian White Noise andGaussian Colored Noise

The Fokker-Planck equations have played an important rolein the investigation of unusual statistic properties of dynamicsystems such as biology systems In this paper their statisticcharacteristics are predicted by deriving the approximateexpressions of stationary probability densities for uninfectedtarget cell infected cell and virus particles

The models with Gaussian white noise and Gaussiancolored noise in the paper seek to describe the dynamics ofHIV-1 rival load during primary infection

1198851 (119905) = 120582 minus 1205751198851 (119905) minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905)minus 12059011198851 (119905) 1205851 (119905)

1198852 (119905) = (1 minus 120574) 1205731198851 (119905) 1198853 (119905) minus 1198861198852 (119905)minus 12059011198852 (119905) 1205851 (119905)

1198853 (119905) = (1 minus 120578)1198731198861198852 (119905) minus 1205831198853 (119905)minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905) minus 12059021198853 (119905) 1205852 (119905)

(5)

Where 1205851(119905) and 1205852(119905) are dependent Gaussian whitenoise andGaussian colored noises with the intensity of noises1205901 and 1205902 respectively the following statistical properties

⟨1205851 (119905)⟩ = ⟨1205852 (119905)⟩ = 0⟨1205851 (119905) 1205851 (1199051015840)⟩ = 12059011205911 exp[minus

10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205911 ]

⟨1205852 (119905) 1205852 (1199051015840)⟩ = 12059021205912 exp[minus10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205912 ]

⟨1205851 (119905) 1205852 (1199051015840)⟩ = ⟨1205852 (1199051015840) 1205851 (119905)⟩ = 0

(6)

in which 1205911 and 1205912 are the self-correlation time of the noises

Let 119896119894119895 be 119894119895119905ℎ order intensity coefficient then

11989611 = 12059011205911 11989622 = 12059021205912 11989612 = 11989621 = 0

(7)

In order to derive the approximate Fokker-Planck equa-tion of the HIV driven by Gaussian white noise and Gaussiancolored noise some signs are defined

119885 (119905) = (1198851 (119905) 1198852 (119905) 1198853 (119905)) 119865 (119885 (119905) 119905) = (1198651 (119885 (119905) 119905) 1198652 (119885 (119905) 119905) 1198653 (119885 (119905) 119905))119882 (119905) = (1205851 (119905) 1205851 (119905) 1205852 (119905))119879119866 (119885 (119905) 119905)

= [[[1198661 (119885 (119905) 119905) 0 0

0 1198662 (119885 (119905) 119905) 00 0 1198663 (119885 (119905) 119905)

]]]

(8)

in which

1198651 (119885 (119905) 119905) = 120582 minus 1205751198851 (119905) minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905)1198652 (119885 (119905) 119905) = (1 minus 120574) 1205731198851 (119905) 1198853 (119905) minus 1198861198852 (119905)1198653 (119885 (119905) 119905) = (1 minus 120578)1198731198861198852 (119905) minus 1205831198853 (119905)

minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905) 1198661 (119885 (119905) 119905) = minus12059011198851 (119905)1198662 (119885 (119905) 119905) = minus12059011198852 (119905)1198663 (119885 (119905) 119905) = minus12059021198853 (119905)

(9)

Hence we can write (5) as

(119905) = 119865 (119885 (119905) 119905) + 119866 (119885 (119905) 119905)119882 (119905) (119905 isin 119879 119885 (1199050) = 1198850) (10)

Based on above definitions time-dependent joint proba-bility density function of 119885(119905) satisfies the FPK equation

120597 [119875 (119885 119905 | 1198850 1199050)]120597119905 = 119871119885 [119875 (119885 119905 | 1198850 1199050)] (11)


where 119871119885[119906] is a partial differential operator defined as

119871119885 [119906] = minus 3sum119894=1

120597 [119891119894 (119885 119905) 119906]120597119885119894 + 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119906]120597119885119894120597119885119895 (12)

Consider Wong-Zakairsquos modification terms drift coeffi-cient and diffusion coefficient are provided as

119891119894 (119885 119905) = 119865119894 (119885 119905)+ 120587 3sum119895=1

3sum119897=1

3sum119904=1

119896119897119904119866119895119897 (119885 119905) 120597120597119885119895119866119894119904 (119885 119905)

119887119894119895 (119885 119905) = 2120587 3sum119897=1

3sum119904=1

119896119897119904119866119894119897 (119885 119905) 119866119895119904 (119885 119905) (13)

where

1198911 (119885 119905) = 1198651 (119885 119905) minus 12058712059011198961111986611 (119885 119905)1198912 (119885 119905) = 1198652 (119885 119905) minus 12058712059011198962211986622 (119885 119905)1198913 (119885 119905) = 1198653 (119885 119905) minus 12058712059021198963311986633 (119885 119905) 11988711 (119885 119905) = 21205871198961111986611 (119885 119905) 11986611 (119885 119905)11988712 (119885 119905) = 21205871198961211986611 (119885 119905) 11986622 (119885 119905)11988713 (119885 119905) = 21205871198961311986611 (119885 119905) 11986633 (119885 119905) 11988721 (119885 119905) = 21205871198962111986622 (119885 119905) 11986611 (119885 119905)11988722 (119885 119905) = 21205871198962211986622 (119885 119905) 11986622 (119885 119905)11988723 (119885 119905) = 21205871198962311986622 (119885 119905) 11986633 (119885 119905) 11988731 (119885 119905) = 21205871198963111986622 (119885 119905) 11986611 (119885 119905)11988732 (119885 119905) = 21205871198963211986633 (119885 119905) 11986622 (119885 119905)11988733 (119885 119905) = 21205871198963311986633 (119885 119905) 11986633 (119885 119905)

(14)

Furthermore initial condition boundary condition andnormalization condition of 119875(119885 119905|1198850 1199050) can be expressed asfollows respectively

lim119905997888rarr1199050

119875 (119885 119905 | 1198850 1199050) = 120575 (119885 minus 1198850)119875 (119885 119905 | 1198850 1199050)119885119894997888rarrplusmninfin = 0 (119894 = 1 2 3)int+infinminusinfin119875 (119885 119905 | 1198850 1199050) 119889119885 = 1

(15)

For convenience we define

119875 (119885 119905) = intΩ0

119875 (119885 119905 | 1198850 1199050) 119875 (1198850 1199050) 1198891198850 (16)

where Ω0 is a defined domain determined by initial vector1198850

Deterministic solutionStochastic solution

10 20 30 40 50 60 70 80 90 1000Time (days)

0

2

4

6

8

10

12

14

1

x 105

Figure 1 The approximate stationary solutions of 1198851 obtained forequations (1) and (5) with 1205901 = 05 and 1205902 = 05

When 119905 997888rarr infin 119875(119885 119905) is independent of time andsatisfies reduced FPK equation

119871119885 [119875 (119885 119905)] = 0 997904rArrminus 3sum119894=1

120597 [119891119894 (119885 119905) 119875 (119885 119905)]120597119885119894+ 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119875 (119885 119905)]120597119885119894120597119885119895 = 0(17)

By utilizing (17) the marginal probability density func-tion of 119885119894(119894 = 1 2 3) can be determined as

119875 (119885119894 119905) = int+infinminusinfin119875 (119885 119905) 119889119885119894 (18)

where 119885119894 = [1198851 119885119894minus1 119885119894+1 119885119899] 119899 = 34 Numerical Simulations

In order to illustrate some of the effects of Gaussian whitenoise and Gaussian colored noise we numerically solve thedeterministic system of differential equations and stochasticsystem of stochastic differential equations The parametervalues for Figures 1ndash3 are 120575 = 01119889119886119910minus1 119886 = 05119889119886119910minus1120583 = 5119889119886119910minus1 120574 = 05 120578 = 05 1205901 = 05 and 1205902 = 05120573 = 1 times 10minus8119889119886119910minus11198891198983 120582 = 106119889119886119910minus11198891198983 119873 = 100 percell When 1205851(119905) and 1205852(119905) are both Gaussian white noises theresults are consistent with findings in [23]

Figures 1ndash3 demonstrate the influence of Gaussian whitenoise and Gaussian colored noise onto the approximatestationary solutions It is seen from Figure 1 that when thetime increases the number of uninfected target cells1198851(119905) fornoiseless conditions increases rapidly and reaches a set point



1 2 3 4 5 6 7 8 9 100Time (days)

0

2000

4000

6000

8000

10000

12000

2

Figure 2 The approximate stationary solutions of 1198852 obtained for equations (1) and (5) with 1205901 = 05 and 1205902 = 05


x 104

0

05

1

15

2

25

3

35

4

3

1 2 3 4 5 6 7 8 9 100Time (days)


level However for uninfected target cells 1198851(119905) with noisethe number firstly increases rapidly and then changes in anappropriate range And compared with the numerical resultswe can find that the range is bigger and noise influenced ismore intense Figures 2-3 show that noises have great effecton infected cell 1198852(119905) and virus particles 1198853(119905) respectivelyInfected cells and virus particles with noises in number arefewer than ones without noise which means that Gaussianwhite noise andGaussian colored noise are helpful to improvehumanrsquos immune system

When the parameters 120575 = 01119889119886119910minus1 120573 = 1 times10minus8119889119886119910minus11198891198983 120582 = 106119889119886119910minus11198891198983 119873 = 100 per cell 119886 =05119889119886119910minus1 120583 = 5119889119886119910minus1 120574 = 05 120578 = 05 are fixed the systemfactors are considered one is the intensity of Gaussian white

noise 1205901 and the other is the intensity of Gaussian colorednoise 1205902

Figures 4ndash6 show that when 1205901 and 1205902 take differentvalues respectively noises have a great influence on thestationary marginal probability density functions It is shownfrom Figure 5 that the peak of the stationary marginalprobability density 119875(1198851) appears between 095 times 106 and105 times 106 and remains unchanged with 1205901 and 1205902 changedwhich means that the noise-induced phase transitions donot occur It is also clear that the decreases in stochasticnoise intensity 1205901 can lead to higher peaks of the stationaryprobability density 119875(1198851) That is litter intensity values 1205901lead to larger probability that system will stay close to theequilibrium state It can be seen from Figures 5-6 that


2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)

Figure 4 The stationary marginal probability density of uninfected target cells 1198851(119905) (a) 1205901 = 01 is fixed with 1205902 = 01 and 1205902 = 06 (b)1205902 = 01 is fixed with 1205901 = 01 and 1205901 = 015

2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)

Figure 5 The stationary marginal probability density of infected target cells 1198852(119905) (a) 1205901 = 008 is fixed with 1205902 = 05 and 1205902 = 08 (b)1205902 = 05 is fixed with 1205901 = 01 and 1205901 = 001

the probabilities of infected cells and virus particles aredecreasing rapidly and reach level no matter whether theparameters 1205901 and 1205902 are changed which means that theinfected cells and virus particles are approximate steady state

5 Conclusions

The internal HIV models subjected to Gaussian white noiseand Gaussian colored noise are mainly studied in this

article It is found that the intensities of noises influencegreatly not only uninfected target cells infected cell andvirus particles but also their stationary marginal proba-bility density functions And compared with the internalHIV models with Gaussian white noises systems drivenby Gaussian white noise and Gaussian colored noise aremore stable and more conform to reality In our futurework we will need deep-going study on how to controlinfected cells and virus particles in short time to HIV


2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)

Figure 6 The stationary marginal probability density of virus particles 1198853(119905) (a) 1205901 = 01 is fixed with 1205902 = 01 and 1205902 = 001 (b) 1205902 = 001is fixed with 1205901 = 01 and 1205901 = 05

systems driven byGaussian white noise andGaussian colorednoise

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 11847081) the Key ScientificResearch Projects of Henan Province (Grant nos 17A110039and 19A110007) the Fundamental Research Funds for theHenan Provincial Colleges andUniversities inHenanUniver-sity of Technology (Grant no 2017QNJH18) and High-LevelPersonal Foundation of Henan University of Technology (no2017BS009)

References

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[2] R A Littlewood and P A Vanable ldquoA global perspective oncomplementary and alternative medicine use among peopleliving with HIVAIDS in the era of antiretroviral treatmentrdquoCurrent HIVAIDS Reports vol 8 no 4 pp 257ndash268 2011

[3] A Hillmann M Crane and H J Ruskin ldquoHIV models fortreatment interruption adaptation and comparisonrdquo Physica A

Statistical Mechanics and its Applications vol 483 pp 44ndash562017

[4] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[5] PW Nelson and A S Perelson ldquoMathematical analysis of delaydifferential equation models of HIV-1 infectionrdquoMathematicalBiosciences vol 179 no 1 pp 73ndash94 2002

[6] N M Dixit and A S Perelson ldquoHIV dynamics with multipleinfections of target cellsrdquo Proceedings of the National Acadamyof Sciences of the United States of America vol 102 no 23 pp8198ndash8203 2005

[7] B G Williams S Gupta M Wollmers and R GranichldquoProgress and prospects for the control of HIV and tuberculosisin SouthAfrica a dynamicalmodelling studyrdquoTheLancet PublicHealth vol 2 no 5 pp e223ndashe230 2017

[8] X Liu H Wang Z Hu andW Ma ldquoGlobal stability of an HIVpathogenesis model with cure raterdquo Nonlinear Analysis RealWorld Applications vol 12 no 6 pp 2947ndash2961 2011

[9] XWangWXu I Y Cui andX IWang ldquoMathematical analysisof hivmodels with switching nonlinear incidence functions andpulse controlrdquo Abstract and Applied Analysis vol 2014 ArticleID 853960 8 pages 2014

[10] R F Stengel ldquoMutation and control of the human immunode-ficiency virusrdquoMathematical Biosciences vol 213 no 2 pp 93ndash102 2008

[11] D Jasmina J S Cristiana and F M T Delfim ldquoA stochasticSICA epidemic model for HIV transmissionrdquo Applied Mathe-matics Letters vol 84 pp 168ndash175 2018

[12] A J McKane and T J Newman ldquoStochastic models in popu-lation biology and their deterministic analogsrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 70 no 4p 041902 2004

[13] X Wang X Liu W Xu and K Zhang ldquoStochastic dynamicsof HIV models with switching parameters and pulse controlrdquoJournal of The Franklin Institute vol 352 no 7 pp 2765ndash27822015


[14] G P Samanta ldquoA stochastic two species competition modelnonequilibrium fluctuation and stabilityrdquo International Journalof Stochastic Analysis vol 2011 Article ID 489386 7 pages 2011

[15] Z Huang Q Yang and J Cao ldquoComplex dynamics in astochastic internal HIV modelrdquo Chaos Solitons amp Fractals vol44 no 11 pp 954ndash963 2011

[16] H C Tuckwell and E Le Corfec ldquoA stochastic model for earlyHIV-1 population dynamicsrdquo Journal ofTheoretical Biology vol195 no 4 pp 451ndash463 1998

[17] N Dalal D Greenhalgh and X R Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[18] H G Solari and M A Natiello ldquoStochastic population dynam-ics the poisson approximationrdquo Physical Review E StatisticalNonlinear and SoftMatter Physics vol 67 no 3 p 031918 2003

[19] Y Xu X Wang H Zhang and W Xu ldquoStochastic stability fornonlinear systems driven by Levy noiserdquo Nonlinear Dynamicsvol 68 no 1-2 pp 7ndash15 2012

[20] D Schertzer M Larcheveque J Duan V V Yanovsky andS Lovejoy ldquoFractional Fokker-Planck equation for nonlinearstochastic differential equations driven by non-Gaussian Levystable noisesrdquo Journal of Mathematical Physics vol 42 no 1 pp200ndash212 2001

[21] A M Cetto L De La Pena and R M Velasco ldquoApproximateFokker-Planck equation with colored Gaussian noiserdquo PhysicalReview A Atomic Molecular and Optical Physics vol 39 no 5pp 2747-2748 1989

[22] O Ditlevsen ldquoInvalidity of the spectral Fokker-Planck equationfor Cauchy noise driven Langevin equationrdquo Probabilistic Engi-neering Mechanics vol 19 no 4 pp 385ndash392 2004

[23] N Dalal D Greenhalgh and X Mao ldquoA stochastic model forinternal HIV dynamicsrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 1084ndash1101 2008

[24] D Li and W Ma ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007

[25] F Neri J Toivanen L Cascella and Y Ong ldquoAn adaptivemultimeme algorithm for designing HIV multidrug therapiesrdquoIEEE Transactions on Computational Biology and Bioinformat-ics 2009

[26] M S Ciupe B L Bivort D M Bortz et al ldquoEstimatingkinetic parameters from HIV primary infection data throughthe eyes of three different mathematical modelsrdquoMathematicalBiosciences vol 200 no 1 pp 1ndash27 2006

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Lyapunov exponents stationary densities and characteristicfunction equations [19 20] Up to now the authors [15ndash17] have proposed many theories and methods to studystochastic HIV systems and the most important one is aboutstationary densities which have become an important wayto examine basic statistical properties of stochastic systemssuch as the stability chaos and bifurcation of stochasticsystems FPK method is an effect means to obtain stationarydensities of stochastic dynamic systems and is often usedin prediction of response process For instance Cetto et al[21] showed that a closed formula for the effective diffusioncoefficient might be used to derive Fokker-Planck equationsof the different approximate expressions Ditlevsen et al[22] raised doubt about the validity of the spectral Fokker-Planck equation in its standard formulation and solvedthe equation with respect to stationary solutions in theparticular case where the noise was Cauchy noise and thedrift function was polynomial Until now stochastic stabilityof stochasticHIVmodels excited byGaussianwhite noise andGaussian colored noise based on FPK equation has not beenconsidered

In this paper the characteristic values of epidemicequilibrium are calculated to consider the stability of HIVdeterministic systemsAndwemainly derive the approximateFokker-Planck equation of internal HIV stochastic dynamicsystems and get the general expression of their stationarydensities Considering variety of stochastic noise intensitieswe analyze the changes of uninfected cell infected celland virus particles make predictions about the stability ofstochastic dynamic systems subjected toGaussianwhite noiseand Gaussian colored noise and discuss the fact that whennoises are both Gaussian noises these two cases agree verywell [23]

The paper is organized as follows in Section 2 we brieflyreview some basic facts about the stability HIV deterministicmodels which have involved the concentration of uninfectedtarget cell infected cell and virus particles with the help ofthe characteristic values of epidemic equilibrium Section 3is devoted to derive the approximate Fokker-Planck equationof the HIV system with Gaussian white noise and Gaussiancolored noise and further obtain the general expression oftheir stationary densities in Section 4 numerical simulationsresults for the different stochastic noise intensities are carriedout to predict about the behavior of the uninfected target cellinfected cell and virus particles in Section 5 we will presentthe conclusions and future directions to close this paper

2 The Stability Analysis of HIVDeterministic Models

In the early stage of HIV infection after reverse transcriptaseinhibitor (RTI) and protease inhibitor (PI) drugs are givenvirus particles are classified as either infections not influ-enced or as non-infection On the basis of standard internalviral dynamics models [15 16 24ndash26] we will considerthe following three-dimensional deterministic models whichhave involved the concentration of uninfected target cell1198851(119905) infected cell 1198852(119905) and virus particles 1198853(119905)

1198851 (119905) = 120582 minus 1205751198851 (119905) minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905)1198852 (119905) = (1 minus 120574) 1205731198851 (119905) 1198853 (119905) minus 1198861198852 (119905)1198853 (119905) = (1 minus 120578)1198731198861198852 (119905) minus 1205831198853 (119905)

minus (1 minus 120574) 1205731198851 (119905) 1198853 (119905) (1)

The initial conditions are 1198851(0) = 11988510 1198852(0) =11988520 1198853(0) = 11988530 Here 1198851(119905) 1198852(119905) and 1198853(119905) isin 119877+ andall parameters are in 119877+ (1 minus 120574) ((0 lt 120574 lt 1)) presentsthe reverse transcriptase inhibitor drug effect and (1 minus 120578)((0 lt 120578 lt 1)) is the protease inhibitor drug effect Theconstant 120582 is the total rate of production of healthy cells perunit time 120575 is the per capita death rate of healthy cells 120573is the transmission coefficient between uninfected cells andinfective virus particles 119886 is the per capital death rate ofinfected cells 119873 is the average number of infective virusparticles produced by an infected cell in the absence ofHAART during its entire infectious lifetime and 120583 presentsthe per capita death rate of infective virus particles

The Jacobian matrix for model system (1) is given as

119869

= [[[[

minus120575 minus (1 minus 120574) 1205731198853 (119905) 0 minus (1 minus 120574) 1205731198851 (119905)(1 minus 120574) 1205731198853 (119905) minus119886 (1 minus 120574) 1205731198851 (119905)minus (1 minus 120574) 1205731198853 (119905) (1 minus 120578)119873119886 minus120583 minus (1 minus 120574) 1205731198851 (119905)]]]] (2)

The deterministic modes have been analyzed by Tuckwelet al [25] They show that if 1198770 = (1 minus 120574)120573119873(1 minus 120578)(120575120583 +120573120582(1minus120574)) le 1 then the disease free equilibrium is the uniqueequilibriumAnd if1198770 = (1minus120574)120573119873(1minus120578)(120575120583+120573120582(1minus120574)) gt 1as well as the disease free equilibrium then there is a uniqueequilibrium 1198750 given by

1198750 = (119885lowast1 119885lowast2 119885lowast3 ) (3)

in which

119885lowast1 = 120583120573 (1 minus 120574) [119873 (1 minus 120578) minus 1]119885lowast2 = 120573120582 (1 minus 120574)119873 (1 minus 120578) minus 120573120582 (1 minus 120574) minus 120575120583120572120573 (1 minus 120574) [119873 (1 minus 120578) minus 1]119885lowast3 = 120573120582 (1 minus 120574)119873 (1 minus 120578) minus 120573120582 (1 minus 120574) minus 120575120583120573120583 (1 minus 120574)

(4)

Take parameters 120575 = 01119889119886119910minus1 119886 = 05119889119886119910minus1 120583 =5119889119886119910minus1 120574 = 05 120578 = 05 1205901 = 05 and 1205902 = 05 120573 =1 times 10minus8119889119886119910minus11198891198983 120582 = 107119889119886119910minus11198891198983 119873 = 100 per cellThe characteristic values of Jacobian matrix for (1) which isin equilibrium 1198750 are 1205821 = minus03589+04210119894 1205822 = minus03589minus04210119894 1205823 = minus56355 From that we can see that one ofthe characteristic values is real number and less than zeroand others are conjugate complex whose real parts are lessthan zeroTherefore based on Lyapunov stabilityrsquos law inter-nal viral dynamics models are asymptotically stable which










(5)


⟨1205851 (119905)⟩ = ⟨1205852 (119905)⟩ = 0⟨1205851 (119905) 1205851 (1199051015840)⟩ = 12059011205911 exp[minus

10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205911 ]

⟨1205852 (119905) 1205852 (1199051015840)⟩ = 12059021205912 exp[minus10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205912 ]

⟨1205851 (119905) 1205852 (1199051015840)⟩ = ⟨1205852 (1199051015840) 1205851 (119905)⟩ = 0

(6)



11989611 = 12059011205911 11989622 = 12059021205912 11989612 = 11989621 = 0

(7)


119885 (119905) = (1198851 (119905) 1198852 (119905) 1198853 (119905)) 119865 (119885 (119905) 119905) = (1198651 (119885 (119905) 119905) 1198652 (119885 (119905) 119905) 1198653 (119885 (119905) 119905))119882 (119905) = (1205851 (119905) 1205851 (119905) 1205852 (119905))119879119866 (119885 (119905) 119905)

= [[[1198661 (119885 (119905) 119905) 0 0

0 1198662 (119885 (119905) 119905) 00 0 1198663 (119885 (119905) 119905)

]]]

(8)

in which



(9)


(119905) = 119865 (119885 (119905) 119905) + 119866 (119885 (119905) 119905)119882 (119905) (119905 isin 119879 119885 (1199050) = 1198850) (10)


120597 [119875 (119885 119905 | 1198850 1199050)]120597119905 = 119871119885 [119875 (119885 119905 | 1198850 1199050)] (11)



119871119885 [119906] = minus 3sum119894=1

120597 [119891119894 (119885 119905) 119906]120597119885119894 + 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119906]120597119885119894120597119885119895 (12)


119891119894 (119885 119905) = 119865119894 (119885 119905)+ 120587 3sum119895=1

3sum119897=1

3sum119904=1

119896119897119904119866119895119897 (119885 119905) 120597120597119885119895119866119894119904 (119885 119905)

119887119894119895 (119885 119905) = 2120587 3sum119897=1

3sum119904=1

119896119897119904119866119894119897 (119885 119905) 119866119895119904 (119885 119905) (13)

where


(14)


lim119905997888rarr1199050


(15)


119875 (119885 119905) = intΩ0

119875 (119885 119905 | 1198850 1199050) 119875 (1198850 1199050) 1198891198850 (16)



10 20 30 40 50 60 70 80 90 1000Time (days)

0

2

4

6

8

10

12

14

1

x 105



119871119885 [119875 (119885 119905)] = 0 997904rArrminus 3sum119894=1

120597 [119891119894 (119885 119905) 119875 (119885 119905)]120597119885119894+ 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119875 (119885 119905)]120597119885119894120597119885119895 = 0(17)








1 2 3 4 5 6 7 8 9 100Time (days)

0

2000

4000

6000

8000

10000

12000

2



x 104

0

05

1

15

2

25

3

35

4

3

1 2 3 4 5 6 7 8 9 100Time (days)







2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)


2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)



5 Conclusions




2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018

















(5)


⟨1205851 (119905)⟩ = ⟨1205852 (119905)⟩ = 0⟨1205851 (119905) 1205851 (1199051015840)⟩ = 12059011205911 exp[minus

10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205911 ]

⟨1205852 (119905) 1205852 (1199051015840)⟩ = 12059021205912 exp[minus10038161003816100381610038161003816119905 minus 1199051015840100381610038161003816100381610038161205912 ]

⟨1205851 (119905) 1205852 (1199051015840)⟩ = ⟨1205852 (1199051015840) 1205851 (119905)⟩ = 0

(6)



11989611 = 12059011205911 11989622 = 12059021205912 11989612 = 11989621 = 0

(7)


119885 (119905) = (1198851 (119905) 1198852 (119905) 1198853 (119905)) 119865 (119885 (119905) 119905) = (1198651 (119885 (119905) 119905) 1198652 (119885 (119905) 119905) 1198653 (119885 (119905) 119905))119882 (119905) = (1205851 (119905) 1205851 (119905) 1205852 (119905))119879119866 (119885 (119905) 119905)

= [[[1198661 (119885 (119905) 119905) 0 0

0 1198662 (119885 (119905) 119905) 00 0 1198663 (119885 (119905) 119905)

]]]

(8)

in which



(9)


(119905) = 119865 (119885 (119905) 119905) + 119866 (119885 (119905) 119905)119882 (119905) (119905 isin 119879 119885 (1199050) = 1198850) (10)


120597 [119875 (119885 119905 | 1198850 1199050)]120597119905 = 119871119885 [119875 (119885 119905 | 1198850 1199050)] (11)



119871119885 [119906] = minus 3sum119894=1

120597 [119891119894 (119885 119905) 119906]120597119885119894 + 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119906]120597119885119894120597119885119895 (12)


119891119894 (119885 119905) = 119865119894 (119885 119905)+ 120587 3sum119895=1

3sum119897=1

3sum119904=1

119896119897119904119866119895119897 (119885 119905) 120597120597119885119895119866119894119904 (119885 119905)

119887119894119895 (119885 119905) = 2120587 3sum119897=1

3sum119904=1

119896119897119904119866119894119897 (119885 119905) 119866119895119904 (119885 119905) (13)

where


(14)


lim119905997888rarr1199050


(15)


119875 (119885 119905) = intΩ0

119875 (119885 119905 | 1198850 1199050) 119875 (1198850 1199050) 1198891198850 (16)



10 20 30 40 50 60 70 80 90 1000Time (days)

0

2

4

6

8

10

12

14

1

x 105



119871119885 [119875 (119885 119905)] = 0 997904rArrminus 3sum119894=1

120597 [119891119894 (119885 119905) 119875 (119885 119905)]120597119885119894+ 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119875 (119885 119905)]120597119885119894120597119885119895 = 0(17)








1 2 3 4 5 6 7 8 9 100Time (days)

0

2000

4000

6000

8000

10000

12000

2



x 104

0

05

1

15

2

25

3

35

4

3

1 2 3 4 5 6 7 8 9 100Time (days)







2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)


2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)



5 Conclusions




2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018










119871119885 [119906] = minus 3sum119894=1

120597 [119891119894 (119885 119905) 119906]120597119885119894 + 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119906]120597119885119894120597119885119895 (12)


119891119894 (119885 119905) = 119865119894 (119885 119905)+ 120587 3sum119895=1

3sum119897=1

3sum119904=1

119896119897119904119866119895119897 (119885 119905) 120597120597119885119895119866119894119904 (119885 119905)

119887119894119895 (119885 119905) = 2120587 3sum119897=1

3sum119904=1

119896119897119904119866119894119897 (119885 119905) 119866119895119904 (119885 119905) (13)

where


(14)


lim119905997888rarr1199050


(15)


119875 (119885 119905) = intΩ0

119875 (119885 119905 | 1198850 1199050) 119875 (1198850 1199050) 1198891198850 (16)



10 20 30 40 50 60 70 80 90 1000Time (days)

0

2

4

6

8

10

12

14

1

x 105



119871119885 [119875 (119885 119905)] = 0 997904rArrminus 3sum119894=1

120597 [119891119894 (119885 119905) 119875 (119885 119905)]120597119885119894+ 123sum119894=1

3sum119895=1

1205972 [119887119894119895 (119885 119905) 119875 (119885 119905)]120597119885119894120597119885119895 = 0(17)








1 2 3 4 5 6 7 8 9 100Time (days)

0

2000

4000

6000

8000

10000

12000

2



x 104

0

05

1

15

2

25

3

35

4

3

1 2 3 4 5 6 7 8 9 100Time (days)







2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)


2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)



5 Conclusions




2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018










1 2 3 4 5 6 7 8 9 100Time (days)

0

2000

4000

6000

8000

10000

12000

2



x 104

0

05

1

15

2

25

3

35

4

3

1 2 3 4 5 6 7 8 9 100Time (days)







2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)


2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)



5 Conclusions




2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018









2=06

2=01

x 10minus6

x 106

0

1

2

3

4

5

6P

(1)

085 09 095 1 105 11 115 12081

(a)

1=01

1=015

x 106

x 10minus6

085 09 095 1 105 11 115 12081

0

1

2

3

4

5

6

P (

1)

(b)


2=05

2=08

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 4500 500002

0

05

1

15

2

25

P (

2)

(a)

x 10minus3

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000002

0

05

1

15

2

25

3

P (

2)

1=01

1=001

(b)



5 Conclusions




2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018









2=01

2=001

x 10minus3

500 1000 1500 2000 250003

0

05

1

15

2

25

3

35

4P

(3)

(a)

1=01

1=05

x 10minus3

500 1000 1500 2000 2500 300003

0

1

2

3

4

5

6

7

P (

3)

(b)



Data Availability




Acknowledgments


References




































Journal of












Journal of







Volume 2018






Volume 2018





























Journal of












Journal of







Volume 2018






Volume 2018








the stochastic stability of internal hiv models with...

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