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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
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
Motivation
L. Rong and A. Perelson, Mathematical Biosciences 2009
Megan Osborne, Tamantha Pizarro February 2, 2020 2 / 22
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
Parameter Values
L. Rong and A. Perelson, Mathematical Biosciences 2009Megan Osborne, Tamantha Pizarro February 2, 2020 4 / 22
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
Stochastic Model
Megan Osborne, Tamantha Pizarro February 2, 2020 6 / 22
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
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
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
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
SDE Model
Megan Osborne, Tamantha Pizarro February 2, 2020 11 / 22
SDE Virus
Figure: SDE Model Virus with Detection Limit
Megan Osborne, Tamantha Pizarro February 2, 2020 12 / 22
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
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
ODE with Random Activation Function
Figure: ODE Model with Poisson Process: T, Latent, Infected, Virus
Megan Osborne, Tamantha Pizarro February 2, 2020 15 / 22
ODE with Random Activation Function: Virus
Figure: ODE Model with Poisson Process for Virus Cell
Megan Osborne, Tamantha Pizarro February 2, 2020 16 / 22
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
SDE with Random Activation Function
Figure: SDE Model with Poisson Process: T, Latent, Infected, Virus
Megan Osborne, Tamantha Pizarro February 2, 2020 18 / 22
Detection Limit
Figure: Virus Model with Detection Limit
Megan Osborne, Tamantha Pizarro February 2, 2020 19 / 22
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
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
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
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