Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i...
Transcript of Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i...
![Page 1: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/1.jpg)
Probability Modeling for HIV Viral Blips
Megan Osborne1 and Tamantha Pizarro2
The University of Scranton1
Iona College2
University of Michigan-Dearborn REU
Mentor: Dr. Hyejin Kim
February 2, 2020
![Page 2: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/2.jpg)
Overview
1. Motivation
2. ODE Model
3. SDE Model
4. Random Activation Function
6. Future Work
Megan Osborne, Tamantha Pizarro February 2, 2020 1 / 22
![Page 3: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/3.jpg)
Motivation
L. Rong and A. Perelson, Mathematical Biosciences 2009
Megan Osborne, Tamantha Pizarro February 2, 2020 2 / 22
![Page 4: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/4.jpg)
HIV ODE ModelL. Rong and A. Perelson Mathematical Biosciences 2009
T L
V I
λ
dT
c Nδ
ak
αL
1− αL
dL
δ
dTdt
= λ− dTT − (1− ε)kV T
dLdt
= αL(1− ε)kV T − dLL− aL
dIdt
= (1− αL)(1− ε)kV T − δI + aL
dVdt
= NδI − cV
Megan Osborne, Tamantha Pizarro February 2, 2020 3 / 22
![Page 5: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/5.jpg)
Parameter Values
L. Rong and A. Perelson, Mathematical Biosciences 2009Megan Osborne, Tamantha Pizarro February 2, 2020 4 / 22
![Page 6: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/6.jpg)
ODE ModelL. Rong and A. Perelson Mathematical Biosciences 2009
Figure: ODE Model: T, Latent, Infected, Virus
Megan Osborne, Tamantha Pizarro February 2, 2020 5 / 22
![Page 7: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/7.jpg)
Stochastic Model
Megan Osborne, Tamantha Pizarro February 2, 2020 6 / 22
![Page 8: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/8.jpg)
Diffusion Process:1.
lim∆t→0
E(|∆X(t)|δ|X(t) = x)
∆t= 0 for δ > 2
2.lim
∆t→0
E(|∆X(t)||X(t) = x)
∆t= b(x)
3.
lim∆t→0
E(|∆X(t)|2|X(t) = x)
∆t= a(x),
where ∆X(t) = X(t+ ∆t)−X(t). Here b(x) denotes the drift term and a(x)denotes the diffusion term.
Stochastic Differential Equations
dX(t) = b(X(t))dt+ σ(X(t))dWt,
where Wt is a Wiener process and a(x) = σ(x) ∗ σ(x).
Megan Osborne, Tamantha Pizarro February 2, 2020 7 / 22
![Page 9: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/9.jpg)
Diffusion Coefficients: ~X = [T, L, I, V ]
i (∆ ~X) pi∆t
1[1 0 0 0
]T λ∆t
2[−1 0 0 0
]T dTT∆t
3[−1 1 0 0
]T αL(1− ε)kTV∆t
4[−1 0 1 0
]T (1− αL)(1− ε)kTV∆t
5[0 −1 0 0
]T δLL∆t
6[0 −1 1 0
]T aL∆t
7[0 0 −1 N
]T δI∆t
8[0 0 0 −1
]T cV∆t
9[0 0 0 0
]T 1−
∑8i=1 Pi∆t
Megan Osborne, Tamantha Pizarro February 2, 2020 8 / 22
![Page 10: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/10.jpg)
HIV Model
Covariance Matrixλ + dT T + (1 − ε)kTV -αL(1 − ε)kTV -(1-αL)(1 − ε)kTV 0
-αL(1 − ε)kTV αL(1 − ε)kTV + (δL + a)L -aL 0-(1-αL)(1 − ε)kTV -aL (1-αL)(1 − ε)kTV + aL + δI -NδI
0 0 -NδI N2δI + cV
Diffusion Matrix√λ+ dTT -
√αL(1− ε)kTV −
√(1− αL)(1− ε)kTV 0 0 0 0
0√αL(1− ε)kTV 0
√δLL -
√aL 0 0
0 0√
(1− αL)(1− ε)kTV 0√aL -
√δI 0
0 0 0 0 0 N√δI -
√cV
Megan Osborne, Tamantha Pizarro February 2, 2020 9 / 22
![Page 11: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/11.jpg)
SDE HIV Model
dT = [λ− dTT − (1− ε)kTV ]dt
+κ(√λ+ dTTdW1 −
√αL(1− ε)kTV dW2 −
√(1− αL)(1− ε)kTV dW3)
dL = [αL(1− ε)kTV − (δL + a)L]dt
+κ(√αL(1− ε)kTV dW2 +
√δLLdW4 −
√aLdW5)
dI = [(1− αL)(1− ε)kTV + aL− δI]dt+κ(
√(1− αL)(1− ε)kTV dW3 +
√aLdW5 −
√δIdW6)
dV = [NδI − cV ]dt
+κ(N√δIdW6 −
√cV dW7),
where Wi are independent Wiener processes.
Megan Osborne, Tamantha Pizarro February 2, 2020 10 / 22
![Page 12: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/12.jpg)
SDE Model
Megan Osborne, Tamantha Pizarro February 2, 2020 11 / 22
![Page 13: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/13.jpg)
SDE Virus
Figure: SDE Model Virus with Detection Limit
Megan Osborne, Tamantha Pizarro February 2, 2020 12 / 22
![Page 14: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/14.jpg)
Interarrival Time
)
( > c >A
inter arrival time
~
exp CA )
.
⇒ I = 0.01
I
L. Rong and A. Perelson, Mathematical Biosciences 2009
Megan Osborne, Tamantha Pizarro February 2, 2020 13 / 22
![Page 15: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/15.jpg)
Random Activation Function
dLtdt = αL(1− ε)kV T − dLL− f(t)(2pL − 1)aL
dItdt = (1− αL)(1− ε)kV T − δI + f(t)(2− 2pL)aL
Figure: Antigen Stimulation
L. Rong and A. Perelson, Mathematical Biosciences 2009
Megan Osborne, Tamantha Pizarro February 2, 2020 14 / 22
![Page 16: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/16.jpg)
ODE with Random Activation Function
Figure: ODE Model with Poisson Process: T, Latent, Infected, Virus
Megan Osborne, Tamantha Pizarro February 2, 2020 15 / 22
![Page 17: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/17.jpg)
ODE with Random Activation Function: Virus
Figure: ODE Model with Poisson Process for Virus Cell
Megan Osborne, Tamantha Pizarro February 2, 2020 16 / 22
![Page 18: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/18.jpg)
SDE HIV Model with Random Activation Function
dT = [λ− dTT − (1− ε)kTV ]dt
+κ(√λ+ dTTdW1 −
√αL(1− ε)kTV dW2 −
√(1− αL)(1− ε)kTV dW3)
dL = [αL(1− ε)kTV − dLL− f(t)(2pL − 1)aL]dt
+κ(√αL(1− ε)kTV dW2 +
√δLLdW4 −
√aLdW5)
dI = [(1− αL)(1− ε)kTV − δI + f(t)(2− 2pL)aL]dt
+κ(√
(1− αL)(1− ε)kTV dW3 +√aLdW5 −
√δIdW6)
dV = [NδI − cV ]dt
+κ(N√δIdW6 −
√cV dW7),
where Wi are independent Wiener processes. The random activation function isdefined by f(t) = χ{N(t)−N(t−∆t)6=0} where N(t) is a Poisson process with λ = 0.01.
Megan Osborne, Tamantha Pizarro February 2, 2020 17 / 22
![Page 19: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/19.jpg)
SDE with Random Activation Function
Figure: SDE Model with Poisson Process: T, Latent, Infected, Virus
Megan Osborne, Tamantha Pizarro February 2, 2020 18 / 22
![Page 20: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/20.jpg)
Detection Limit
Figure: Virus Model with Detection Limit
Megan Osborne, Tamantha Pizarro February 2, 2020 19 / 22
![Page 21: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/21.jpg)
Future Work
Within the time limit of about 300 days, the model can approximateviral blips. However, after that time, the blips become too small to bedetectable. This is not accurate to life, so extending these detectableblips out further is a goal.
Further simulations in order to compare the probability of viral blipsoccurring to experimental data are desirable in order to compare moreaccurate data.
Megan Osborne, Tamantha Pizarro February 2, 2020 20 / 22
![Page 22: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/22.jpg)
References
[1] Jessica M. Conway and Alan S. Perelson, Post-Treatment Control of HIV Infection PNAS, vol. 112, no. 17,2015, pp. 5467-5472.
[2] Yen Ting Lin, Hyejin Kim, and Charles R. Doering, Features of Fast Living: On the Weak Selection forLongevity in Degenerate Birth-Death Processes, Journal of Statistical Physics, 2012.
[3] Libin Rong and Alan S. Perelson, Modeling HIV Persistence, the Latent Reservoir, and Viral Blips,Journal of Theoretical Biology, 2009, pp. 308-331.
[4] Sukhitha W. Vidurupola and Linda J. S. Allen, Basic Stochastic Models for Viral Infection within a Host,Mathematical Biosciences and Engineering, vol. 9, no. 4, 2012, pp. 915-935.
[5] Daniel Sánchez-Taltavull, Arturo Vieiro, and Tómas Alarcón, Stochastic Modelling of the Eradication ofthe HIV-1 Infection by Stimulation of Latently Infected Cells in Patients under Highly ActiveAnti-Retroviral Therapy, Journal of Mathematical Biology, 2016.
[6] Wenwen Huang et al, Exactly Solvable Dynamics of Forced Polymer Loops, New Journal of Physics, 2018,pp. 1-18.
[7] Wenjing Zhang, Lindi M. Wahl, and Pei Yu, Viral Blips May Not Need a Trigger: How Transient ViremiaCan Arise in Deterministic In-Host Models, SIAM Review, vol. 56, no. 1, 2014, pp. 127-155.
[8] Jessica M. Conway, Bernhard P. Konrad, and Daniel Coombs, Stochastic Analysis of Pre- andPostexposure Prophylaxis Against HIV Infection, SIAM Journal on Applied Mathematics, vol. 73, no. 2,2013, pp. 904-928.
Megan Osborne, Tamantha Pizarro February 2, 2020 21 / 22
![Page 23: Probability Modeling for HIV Viral Blips · dV = [N I cV]dt + (N p IdW 6 p cVdW 7); whereW i areindependentWienerprocesses. Therandomactivationfunctionis definedbyf(t) = ˜ fN(t)](https://reader035.fdocuments.in/reader035/viewer/2022071217/604b2d4d76e7c651521d3ab4/html5/thumbnails/23.jpg)
Big Thanks!
This research was conducted at the NSF REU Site (DMS-1659203) inMathematical Analysis and Applications at the University ofMichigan-Dearborn. We would like to thank the National ScienceFoundation, National Security Agency, University of Michigan-Dearborn(SURE 2019), and the University of Michigan-Ann Arbor for their support.
Megan Osborne, Tamantha Pizarro February 2, 2020 22 / 22