Citation: Tyson Loudon, Stephen Pankavich. Mathematical analysis and dynamic active subspaces for a long term model of HIV[J]. Mathematical Biosciences and Engineering, 2017, 14(3): 709-733. doi: 10.3934/mbe.2017040
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The Human Immunodeficiency Virus (HIV) disables many components of the body's immune system and, without antiretroviral treatment, leads to the onset of Acquired Immune Deficiency Syndrome (AIDS). Despite the vast amount of study devoted to understanding viral pathogenesis and developing new therapeutics, no procedure or medication currently exists to reliably eliminate the virus from a host. However, new advances in long-term treatment strategies and insight into disease dynamics have stemmed from mathematical and computational modeling approaches, in addition to clinical experimentation.
A variety of mathematical models have been proposed to describe HIV infection and disease dynamics [1,12,13,14,15,16,17,19,23,24,26]. Unfortunately, using a model to capture the entire time course of infection within the body can be troublesome as many oversimplify the biological dynamics of the disease in an effort to gain mathematical and rudimentary biological insight, and fail to capture all stages of infection. The majority of models accurately capture only the first stage(s) of infection [1,12,13,16,17,18,21] with the T-cell count and viral load asymptotically approaching a nonzero limit -the latter referred to as the viral set point. One recent description has been able to provide a holistic understanding of disease dynamics by accurately capturing all three stages of infection. This model, proposed in [8], is comprised of a system of seven nonlinear autonomous differential equations that are fully coupled and augmented by twenty-seven distinct parameters. In the current study we investigate the dynamical properties of the model and establish a result concerning its large-time asymptotic behavior. Further, we analyze this model, utilizing mathematical and statistical methods to elucidate the contribution of the parameters to an infected individual's T-cell count. In particular, we conduct a global sensitivity analysis in order to determine which of the twenty-seven parameters possess the greatest influence over the time course of the T-cell population. With this information, a reduced model (in the parameter space) can be constructed and solutions can be approximated as a function of time.
The paper proceeds as follows. Within the next section, all infection-free steady states of the model are determined and the local asymptotic stability properties of the biologically-relevant equilibrium are studied. The associated theorems have interesting implications for the model's predictive nature, especially upon the introduction of antiretroviral therapy. In Section
http://inside.mines.edu/~pankavic/activeHIV
while an interactive iPython notebook is freely accessible at
http://nbviewer.jupyter.org/github/paulcon/as-data-sets/blob/master/HIV/HIV.ipynb.
Additional examples of biological and physical models for which active subspace methods have been implemented are also accessible at this website.
To begin, we consider the following long term model of HIV disease dynamics within a host, as recently formulated in [8]:
dTdt=s1+p1C1+VTV−δ1T−(K1V+K2MI)TdTIdt=ψ(K1V+K2MI)T+α1TL−δ2TI−K3TICTLdTLdt=(1−ψ)(K1V+K2MI)T−α1TL−δ3TLdMdt=s2+K4MV−K5MV−δ4MdMIdt=K5MV−δ5MI−K6MICTLdCTLdt=s3+(K7TI+K8MI)CTL−δ6CTLdVdt=K9TI+K10MI−K11TV−(K12+K13)MV−δ7V.} | (1) |
The population of CD4
The second equation describes changes within the actively-infected T-cell population (
Similar to T-cells, the macrophage population (
The main defender of the body against infected cells is the cytotoxic lymphocyte, the population of which is denoted by
Lastly, the growth of the virion population
Note that all parameter values in (1) are positive. Typical values and ranges for the parameters taken from [8] can be found within Table 1. The time course of infection predicted by this model with the established parameter values was shown to agree well with clinical data from [5,6,22], and representative simulations of (1) with realistic parameter values are displayed in Figure 1.
Parameter | Value | Range | Value taken from: | Units |
10 | 5 -36 | [13] | mm |
|
0.15 | 0.03 -0.15 | [13] | mm |
|
5 | - | [8] | mm |
|
0.2 | 0.01 -0.5 | [8] | d |
|
55.6 | 1 -188 | [8] | mm |
|
3.87 x |
10 |
[8] | mm |
|
[13] | mm |
|||
4.5 x 10 |
10 |
[8] | mm |
|
7.45 x 10 |
- | [8] | mm |
|
5.22 x 10 |
4.7 x 10 |
[8] | mm |
|
3 x 10 |
- | [8] | mm |
|
3.3 x 10 |
10 |
[8] | mm |
|
6 x 10 |
- | [8] | mm |
|
0.537 | 0.24 -500 | [8] | d |
|
0.285 | 0.005 -300 | [8] | d |
|
7.79 x 10 |
- | [8] | mm |
|
10 |
- | [8] | mm |
|
4 x 10 |
- | [8] | mm |
|
0.01 | 0.01 -0.02 | [8] | d |
|
0.28 | 0.24 -0.7 | [8] | d |
|
0.05 | 0.02 -0.069 | [8] | d |
|
0.005 | 0.005 | [13] | d |
|
0.005 | 0.005 | [13] | d |
|
0.015 | 0.015 -0.05 | [27] | d |
|
2.39 | 2.39 -13 | [13] | d |
|
3 x 10 |
- | [8] | d |
|
0.97 | 0.93 -0.98 | [8] | - |
Though the system (1) possesses a large number of steady states -the authors have discovered at least ten using standard parameter values and a computational root finder -one is often most interested in understanding the dynamical properties of the disease-free equilibrium. In this section, we identify such equilibria and investigate the stability of the biologically-relevant state. Our first result demonstrates that only one such equilibrium state exists when all populations of (1) are positive.
Theorem 2.1. The model (1) possesses exactly two virus-free (i.e.
E:=(s1δ1−ωK2K9,ωK10,ωK10ξK6(α1+δ3ψ),s2δ4,−ωK9,−δ5K6,0) |
achieves negative values, where
ω=s3K6+δ5δ6δ5(K7K10−K8K9)andξ=(1−ψ)(δ2K6−δ5K3). |
The only nonnegative (i.e. biologically relevant) steady state of (1) satisfying
ENI:=(s1δ1,0,0,s2δ4,0,s3δ6,0). |
Hence, the only guarantee of viral clearance as
T=1010.39mm−3,TI=54.57mm−3TL=−15.77mm−3,M=30mm−3MI=−102.83mm−3,CTL=−1666.67mm−3,V=0mm−3 |
Since the parameter values in (1) are positive, the steady state
Next, we provide necessary and sufficient conditions which guarantee the local asymptotic stability of the disease-free equilibrium
Theorem 2.2. The equilibrium state
R0=max{R1,R2,R3} |
and
R1=K1K9δ2K11,R2=K5K10(K12+K13)δ5 |
R3=K2K5K9s1s2δ1δ2δ4δ5δ7+δ4δ5K1K9s1+δ1δ2K5K10s2. |
The proof of Theorem 2.2 is also contained within Appendix A. Computing the basic reproduction number of Theorem 2.2 by using the standard parameter values given in Table 1, we find that
Introducing antiretroviral therapy, or ART, into the system provides additional insight into this result. Two specific classes of ART drugs, namely Reverse Transcriptase Inhibitors (RTIs) and Protease Inhibitors (PIs), serve to reduce the amount of new virus produced by either reducing the ability of virions to replicate through reverse transcription or disabling the capability of newly-produced virions to mature, thereby rendering them uninfective. The efficacies of these classes of drug, denoted
K1→K1(1−ϵRTI), K5→K5(1−ϵRTI), K9→K9(1−ϵPI), K10→K10(1−ϵPI). |
As these constants appear within the stability result and decrease each of the ratios
1−ϵ:=(1−ϵRTI)(1−ϵPI), |
we find
Notice that the asymptotic stability result in Theorem 2.2 depends only upon a relatively few number (
In this section we will use active subspace methods to approximate the T-cell count at a specific time given the parameter values in (1). An active subspace is a low-dimensional linear subspace of the set of parameters, in which input perturbations along these directions alter the model's predictions more, on average, than perturbations which are orthogonal to the subspace. These subspaces allow for a global measurement of sensitivity of output variables with respect to parameters, and often the construction of reduced-order models that greatly decrease the dimension of the parameter space.
The general structure of an active subspace decomposition begins by letting
∫ρ(x) dx=1. |
Here, the space
C=∫(∇xf)(∇xf)Tρdx. | (2) |
For any smooth
Considering each entry of the matrix
Cij=∫∂f∂xi∂f∂xjρ(x)dx |
we note that
C=WΛWT,whereΛ=diag(λ1,…,λm),λ1≥…≥λm≥0. | (3) |
Here,
From (3) we can further solve for the eigenvalues of
λi=∫((∇xf)Twi)2ρ(x)dx,i=1,…,m. | (4) |
From (4) we see that the eigenvalues of the
Once the eigendecomposition (3) has been determined, the eigenvalues and eigenvectors can be separated in the following way:
Λ=[Λ100Λ2],W=[W1W2]. | (5) |
where
With the decomposition (5), we can represent any element
x=WWT⏟Ix=W1WT1x⏟y+W2WT2x⏟z=W1y+W2z. | (6) |
Thus, evaluating the quantity of interest at
f(x)=f(W1y+W2z). |
By the definition of
In general, the eigenvalues and eigenvectors of
1. Draw
2. For each parameter sample
∂xif(xj)≈f(xj+hi)−f(xj)|hi| |
where
(hi)k={δif i=k0if i≠k. |
represents a vector perturbation from the sampled parameter values and
3. Approximate the matrix
C≈ˆC=1NN∑j=1(∇xfj)(∇xfj)T |
4. Compute the eigendecompositions
We note that the last step is equivalent to computing the singular value decomposition of the matrix
1√N[∇xf1…∇xfN]=ˆW√ˆΛˆV, | (7) |
where it can be shown that the singular values are the square roots of the eigenvalues of
Next, we will demonstrate the active subspace method at one specific point in time by applying the aforementioned algorithm to the HIV model (1). A suitable quantity of interest would likely involve the T-cell or virus population evaluated at a fixed time. For this example we select
In order to compute
p=12(diag(xu−xl)xj+(xu+xl)), | (8) |
for each of the random samples
ˆλk=λk−λk+1λ1,k=1,2,…,26. | (9) |
Then, the dimension of the active subspace, i.e. the number of eigenvalues stored within
dim=argmaxk=1,…,26ˆλk. | (10) |
While the index of the largest value of
Clearly, with this measure of separation, the optimal choice for the dimension of the active subspace is merely one. Consequently, we store
Plotting
T(1700;y)=−79.2532−492.5680 tan−1(0.8933y−1.9069), | (11) |
and the resulting curve can be found in the plot on the right side of Figure 5. In order to test the accuracy of the nonlinear approximation,
Of course, not every eigenvalue decomposition will necessarily result in a one-dimensional active subspace of the model parameters. For instance, should one wish to approximate the T-cell count
The two weight vectors corresponding to these dominant eigenvalues are then computed and sufficient summary plots are obtained. Figure 8 shows the one and two dimensional sufficient summary plots, respectively, for the approximation of
Since this algorithm can be used to approximate the T-cell count at any fixed time, we can further compute active subspaces to reduce complexity in the parameter space and obtain a simple time course for
Now that the approximation for
Once the approximations have been computed at each time step, the final task is to dynamically assemble them using linear basis functions, namely
T(t;x)≈86∑i=1Ti(x⋅w(t))ϕi(t) | (12) |
where
ϕi(t)={t−ti−1ti−ti−1,ti−1≤t≤titi+1−tti+1−ti,ti≤t≤ti+10,t∉[ti−1,ti+1]. |
Recall that the original parameter values
The results of computing the active subspace and sufficient summary plot for
The time-course of HIV infection is characterized by three distinct stages: acute (or primary) infection, chronic infection (also referred to as clinical latency), and the transition to AIDS. The first of these phases takes place within
From the sufficient summary plots in Figure 10 and Appendix B, the three distinct stages of the disease become clear. The three functional forms arising from sufficient summary plots precisely separate the three distinct stages of HIV disease progression within an infected individual -the initial arctangent function representing the acute phase, followed by a slow linear decline that denotes the asymptomatic or chronic phase, and finally the heaviside arctangent function detailing the decline of the T-cell count as a patient develops AIDS. This description of the T-cell population provides a detailed visual account of the three stages and the transitions between them. For clarity, a representative graph of each trend in the data is provided in Figure 10. Therefore, the abrupt changes in the T-cell count arising within the one-dimensional active subspace of parameters completely categorize the stages of disease pathogenesis.
In addition to separating these stages of the disease, the active subspace method allows for the creation of three different types of approximate models. First, a visual representation of the quantity of interest -in this case, the T-cell count -as a function of the most pertinent linear combination of parameters within the original model is provided. Additionally, the method allows for the construction of an explicit, analytic model by combining nonlinear function approximations such as (11) over an interval of time. In this direction a second approximation method is available instead of utilizing basis functions as in (12). Namely, one may prescribe the functional form during a particular stage of the disease and fit time-dependent coefficients to the transitions within sufficient summary plots. For instance, one may express the arctangent approximation over the timespan
T(t;y)=a(t)+b(t) tan−1(c(t)y−d(t)) |
for well-behaved functions
T(t;y)=m(t)y+b(t) |
and the slope and
Lastly, the nonlinear fits arising from the sufficient summary data can be easily stored and supply the basis for a low-cost computational approximation without the need to simulate the original model over long time periods. In contrast to simulating the full system of ODEs for each new set of parameter values, this computational model need only be precomputed once and can easily describe which of the parameters are most important and during which stages they significantly alter the biological quantities of interest. As parameters within the original HIV model (1) vary amongst patients, they must be recomputed for each individual, and the computational savings provided by the reduced model is vital.
Utilizing the dynamical algorithm represented by (12), this computational model can be constructed and compared with a representative simulation of the full dynamical system. Figure 12 displays both the simulated T-cell count and its global-in-time approximation using dynamic active subspaces. We note that because the active subspace method is global with respect to the parameter space, the precise values of parameters in Table 1 do not necessarily influence the structure of the active subspace model, as long as a feasible range of parameter values is available.
In order to test the accuracy of the active subspace approximation to solutions of (1),
As determined in the previous section, not all parameters are required to describe the behavior of solutions within the model and those that are needed may not be important during each stage. Therefore, it makes sense to investigate the construction of reduced models, namely those which eliminate the contributions of certain parameters. Though the active subspace method has reduced the dimension of the parameter space upon which the T-cell count depends, all of the parameter values are still needed in order to compute the approximate solution, as the reduced parameter space has been expressed merely as a linear combination of the original parameter values, i.e.
While it would be most beneficial to determine those weight vectors that remain small throughout the entire course of infection and remove the corresponding parameters, this is an unlikely scenario as different parameters will typically influence different stages in some substantial manner. For this reason, a true global-in-time parameter reduction is highly unlikely. Indeed, the computed weight vectors for this study show that at most
As an illustrative example, we consider the weight vectors arising within the first
dTdt=s1+p1C1+VTV−δ1T−K1TVdTIdt=K1TV−δ2TIdVdt=K9TI−δ7V.} | (13) |
This model is an augmented form of the well-known three-component model (see [11]) with an additional Michaelis-Menten term within the T-cell population, which accounts for the homeostatic proliferation of such cells upon depletion of this compartment due to interactions with virions. This model was recently analyzed in detail in [20] and found to possess many desirable properties, including bistable equilibria which explain the dependence of infection dynamics on the initial T-cell count and viral load, as well as the existence of a Hopf bifurcation which describe oscillations within the system. Using the fitted parameter values given in [20] for the model (13) and plotting against the full model (1) with the standard parameter values given in Table 1 results in Figure 14.
Hence, for the acute stage we have reduced (1) with
The current study concerns an analysis of the system (1), which is one of the only mathematical models to accurately represent all three stages of HIV infection within a host. A unique, biologically-relevant virus-free equilibrium was shown to exist, and conditions were determined that guarantee the local asymptotic stability of this state. Then, using dynamic active subspaces, the system of seven ODEs with
As with any study, a number of future questions arise from our investigation. First, as many parameter values within (1) are not specifically known and vary greatly in the literature, one would like to quantify the uncertainty inherent in choosing a particular value for each. While this can be investigated using the approximations established herein, it was not the focus of the study and more complex mathematical tools are needed to do so. Another direction to consider is the explicit quantification of error within the lower-dimensional approximations provided by active subspaces. Some preliminary results appear in [2], but exact bounds and convergence theorems for dynamic, rather than time-independent, active subspaces are currently unavailable within the literature. Finally, we note that these methods can be used for other in-host models of HIV (or other physical and biological models [4]) that possess high-dimensional parameter spaces. For instance, a recent refinement of (1) was proposed in [9], and appears to display less sensitivity to variations in parameter values. Hence, a similar technique would likely be useful to conduct a parameter study or construct a reduced-order model for this system.
In the first appendix, we outline the proofs of the theorems stated in Section 2.
Proof of Theorem 2.1. Beginning with (1), we search for steady states by assuming that all time derivatives are zero within the equations, and attempt to solve for the constant states
Consider the latter case first. Multiplying the
0=(α1+ψδ3)TL+((1−ψ)K3δ5K6−(1−ψ)δ2)TI. | (14) |
Creating a linear system with (14), the
TI=K10ω,TL=K10ξωK6(α1+δ3ψ),MI=−K9ω |
where
ω=s3K6+δ5δ6δ5(K7K10−K8K9)andξ=(1−ψ)(δ2K6−δ5K3). |
Lastly, inserting the value of
T=s1δ1−ωK2K9 |
Finally, consider the former case. Then, it follows from the equation for
ENI:=(s1δ1,0,0,s2δ4,0,s3δ6,0). |
We note that assuming all parameter values are strictly positive implies that the only non-infective steady state of biological significance is
Finally, we sketch the proof of the asymptotic stability result, which utilizes standard methods from the theory of dynamical systems (i.e. the Hartman-Grobman and Routh-Hurtwitz theorems) to determine the qualitative behavior of the
Proof of Theorem 2.2. As the model is not positivity-preserving a simple technique like the next-generation method does not apply. Hence, we will utilize the Hartman-Grobman Theorem to arrive at the stated result and omit some of the more technical details. We first compute the Jacobian of (1) evaluated at the steady states
J(ENI)=(−δ1000−K2s1δ10(p1−c1K1)s1c1δ10−δ2δ6+K3s3δ6a10pK2s1δ10pK1s1δ100−a1−δ30−(p−1)K2s1δ10−(p−1)K1s1δ1000−δ400(K4−K5)s2δ40000−δ5δ6+K6s3δ60K5s2δ40K7s3δ600K8s3δ6−δ600K900K100−δ7−K11s1δ1−(K12+K13)s2δ4). |
From this, we can see that three eigenvalues are certainly real and negative
λ1=−δ1,λ2=−δ4,λ3=−δ6. |
The remaining four eigenvalues are more difficult to identify as they are determined by the quartic equation
a4λ4+a3λ3+a2λ2+a1λ+a0=0 |
where
a4=δ1δ4δ26>0 |
a3=α1δ1δ4δ26+δ4δ26K11s1+δ1δ26K12s2+δ1δ26K13s2+δ1δ4δ6K3s3+δ1δ4δ6K6s3+δ1δ2δ4δ26+δ1δ3δ4δ26+δ1δ4δ5δ26+δ1δ4δ7δ26>0 |
and
Instead, we must impose conditions on each term to guarantee positivity, which is needed for the roots of the quartic to possess negative real part by the Routh-Hurwitz criteria. In particular, the two negative terms in
K1K9≤δ2K11 |
K5K10≤(K12+K13)δ5. |
The same conditions imply the positivity of
K2K5K9s1s2≤δ1δ2δ4δ5δ7+δ4δ5K1K4s1+δ1δ2K5K10s2. |
The final inequalities of the Routh-Hurwitz criteria are also implied by these conditions. Hence, defining
R1=K1K9δ2K11,R2=K5K10(K12+K13)δ5,R3=K2K5K9s1s2δ1δ2δ4δ5δ7+δ4δ5K1K9s1+δ1δ2K5K10s2, |
and
R0=max{R1,R2,R3} |
we see that the equilibrium is locally asymptotically stable if and only if all three conditions are satisfied, and thus
In Figures 15, 16, and 17, we provide a more temporally refined representation of the three stages of infection displayed by the model.
The authors would like to thank Prof. Paul Constantine for helpful discussions and advice, and Ryan Howard for constructing an openly-available iPython notebook that details the work in our computational study.
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1. | Hyunjung Lee, Elaine T. Spiller, Susan E. Minkoff, 2019, Dimension reduction and global sensitivity metrics using active subspaces for coupled flow and deformation modeling, 3240, 10.1190/segam2019-3215234.1 | |
2. | C. A. Pena Fernandez, 2020, Fractional Complex-order Hopfield Neural Networks to Analyze the Effect of Drug-resistance in the HIV Infection, 978-1-7281-4248-7, 8, 10.1109/ICACI49185.2020.9177513 | |
3. | Jeffrey M. Hokanson, Paul G. Constantine, Data-Driven Polynomial Ridge Approximation Using Variable Projection, 2018, 40, 1064-8275, A1566, 10.1137/17M1117690 | |
4. | Tongqian Zhang, Junling Wang, Yi Song, Zhichao Jiang, Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells, 2020, 2020, 1076-2787, 1, 10.1155/2020/2313102 | |
5. | James C. Gross, Pranay Seshadri, Geoff Parks, 2020, Optimisation with Intrinsic Dimension Reduction: A Ridge Informed Trust-Region Method, 978-1-62410-595-1, 10.2514/6.2020-0157 | |
6. | Charlotte Lew, 2022, Neural Network Modeling of HIV Acute and Chronic Phases With and Without Antiretroviral Intervention, 978-1-6654-7184-8, 47, 10.1109/TransAI54797.2022.00014 | |
7. | Wouter Edeling, On the Deep Active-Subspace Method, 2023, 11, 2166-2525, 62, 10.1137/21M1463240 | |
8. | Zixiao Ma, Bai Cui, Zhaoyu Wang, Dongbo Zhao, Parameter Reduction of Composite Load Model Using Active Subspace Method, 2021, 36, 0885-8950, 5441, 10.1109/TPWRS.2021.3078671 | |
9. | Qiang-hui Xu, Ji-cai Huang, Yue-ping Dong, Yasuhiro Takeuchi, A Delayed HIV Infection Model with the Homeostatic Proliferation of CD4+ T Cells, 2022, 38, 0168-9673, 441, 10.1007/s10255-022-1088-2 | |
10. | Xifu Sun, Barry Croke, Anthony Jakeman, Stephen Roberts, Benchmarking Active Subspace methods of global sensitivity analysis against variance-based Sobol' and Morris methods with established test functions, 2022, 149, 13648152, 105310, 10.1016/j.envsoft.2022.105310 | |
11. | Cameron Clarke, Stephen Pankavich, Three-stage modeling of HIV infection and implications for antiretroviral therapy, 2024, 88, 0303-6812, 10.1007/s00285-024-02056-1 | |
12. | Soraya Terrab, Stephen Pankavich, Global sensitivity analysis of plasma instabilities via active subspaces, 2024, 134, 10075704, 107994, 10.1016/j.cnsns.2024.107994 | |
13. | Wouter Edeling, Maxime Vassaux, Yiming Yang, Shunzhou Wan, Serge Guillas, Peter V. Coveney, Global ranking of the sensitivity of interaction potential contributions within classical molecular dynamics force fields, 2024, 10, 2057-3960, 10.1038/s41524-024-01272-z | |
14. | Valentin Breaz, Richard Wilkinson, 2024, Randomized Maximum Likelihood Via High-Dimensional Bayesian Optimization, 979-8-3503-4485-1, 5300, 10.1109/ICASSP48485.2024.10446755 | |
15. | Sanket Jantre, Nathan M. Urban, Xiaoning Qian, Byung-Jun Yoon, 2024, Learning Active Subspaces for Effective and Scalable Uncertainty Quantification in Deep Neural Networks, 979-8-3503-4485-1, 5330, 10.1109/ICASSP48485.2024.10448265 | |
16. | Hui Duan, Giray Okten, DERIVATIVE-BASED SHAPLEY VALUE FOR GLOBAL SENSITIVITY ANALYSIS AND MACHINE LEARNING EXPLAINABILITY , 2025, 15, 2152-5080, 1, 10.1615/Int.J.UncertaintyQuantification.2024051548 | |
17. | Yueping Dong, Jicai Huang, Yasuhiro Takeuchi, Qianghui Xu, 2024, Chapter 2, 978-981-97-7849-2, 11, 10.1007/978-981-97-7850-8_2 | |
18. | Xia Wang, Yue Wang, Yueping Dong, Libin Rong, Dynamics of a delayed HIV infection model with cell-to-cell transmission and homeostatic proliferation, 2024, 139, 2190-5444, 10.1140/epjp/s13360-024-05845-1 | |
19. | Jing Cai, Jun Zhang, Kai Wang, Zhixiang Dai, Zhiliang Hu, Yueping Dong, Zhihang Peng, Evaluating the long-term effects of combination antiretroviral therapy of HIV infection: a modeling study, 2025, 90, 0303-6812, 10.1007/s00285-025-02196-y |
Parameter | Value | Range | Value taken from: | Units |
10 | 5 -36 | [13] | mm |
|
0.15 | 0.03 -0.15 | [13] | mm |
|
5 | - | [8] | mm |
|
0.2 | 0.01 -0.5 | [8] | d |
|
55.6 | 1 -188 | [8] | mm |
|
3.87 x |
10 |
[8] | mm |
|
[13] | mm |
|||
4.5 x 10 |
10 |
[8] | mm |
|
7.45 x 10 |
- | [8] | mm |
|
5.22 x 10 |
4.7 x 10 |
[8] | mm |
|
3 x 10 |
- | [8] | mm |
|
3.3 x 10 |
10 |
[8] | mm |
|
6 x 10 |
- | [8] | mm |
|
0.537 | 0.24 -500 | [8] | d |
|
0.285 | 0.005 -300 | [8] | d |
|
7.79 x 10 |
- | [8] | mm |
|
10 |
- | [8] | mm |
|
4 x 10 |
- | [8] | mm |
|
0.01 | 0.01 -0.02 | [8] | d |
|
0.28 | 0.24 -0.7 | [8] | d |
|
0.05 | 0.02 -0.069 | [8] | d |
|
0.005 | 0.005 | [13] | d |
|
0.005 | 0.005 | [13] | d |
|
0.015 | 0.015 -0.05 | [27] | d |
|
2.39 | 2.39 -13 | [13] | d |
|
3 x 10 |
- | [8] | d |
|
0.97 | 0.93 -0.98 | [8] | - |
Parameter | Value | Range | Value taken from: | Units |
10 | 5 -36 | [13] | mm |
|
0.15 | 0.03 -0.15 | [13] | mm |
|
5 | - | [8] | mm |
|
0.2 | 0.01 -0.5 | [8] | d |
|
55.6 | 1 -188 | [8] | mm |
|
3.87 x |
10 |
[8] | mm |
|
[13] | mm |
|||
4.5 x 10 |
10 |
[8] | mm |
|
7.45 x 10 |
- | [8] | mm |
|
5.22 x 10 |
4.7 x 10 |
[8] | mm |
|
3 x 10 |
- | [8] | mm |
|
3.3 x 10 |
10 |
[8] | mm |
|
6 x 10 |
- | [8] | mm |
|
0.537 | 0.24 -500 | [8] | d |
|
0.285 | 0.005 -300 | [8] | d |
|
7.79 x 10 |
- | [8] | mm |
|
10 |
- | [8] | mm |
|
4 x 10 |
- | [8] | mm |
|
0.01 | 0.01 -0.02 | [8] | d |
|
0.28 | 0.24 -0.7 | [8] | d |
|
0.05 | 0.02 -0.069 | [8] | d |
|
0.005 | 0.005 | [13] | d |
|
0.005 | 0.005 | [13] | d |
|
0.015 | 0.015 -0.05 | [27] | d |
|
2.39 | 2.39 -13 | [13] | d |
|
3 x 10 |
- | [8] | d |
|
0.97 | 0.93 -0.98 | [8] | - |