Probability Modeling for HIV Viral Blips Megan Osborne 1 and Tamantha - - PowerPoint PPT Presentation

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 Model

• L. Rong and A. Perelson Mathematical Biosciences 2009

T L V I

λ dT c Nδ a k αL 1 − αL dL δ

dT dt = λ − dT T − (1 − ǫ)kV T dL dt = αL(1 − ǫ)kV T − dLL − aL dI dt = (1 − αL)(1 − ǫ)kV T − δI + aL dV dt = NδI − cV

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Parameter Values

• L. Rong and A. Perelson, Mathematical Biosciences 2009

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ODE Model

• L. Rong and A. Perelson Mathematical Biosciences 2009

Figure: ODE Model: T, Latent, Infected, Virus

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Stochastic Model

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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).

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Diffusion Coefficients: X = [T, L, I, V ]

i (∆ X) pi∆t 1 1 0T λ∆t 2 −1 0T dT T ∆t 3 −1 1 0T αL(1 − ǫ)kT V ∆t 4 −1 1 0T (1 − αL)(1 − ǫ)kT V ∆t 5 −1 0T δLL∆t 6 −1 1 0T aL∆t 7 −1 NT δI∆t 8 −1T cV ∆t 9 0T 1 − 8

i=1 Pi∆t Megan Osborne, Tamantha Pizarro February 2, 2020 8 / 22

HIV Model

Covariance Matrix

    λ + dT T + (1 − ǫ)kT V

• αL(1 − ǫ)kT V
• (1-αL)(1 − ǫ)kT V
• αL(1 − ǫ)kT V

αL(1 − ǫ)kT V + (δL + a)L

• aL
• (1-αL)(1 − ǫ)kT V
• aL

(1-αL)(1 − ǫ)kT V + aL + δI

• NδI
• NδI

N2δI + cV    

Diffusion Matrix

    √λ + dT T

• αL(1 − ε)kT V

−

• (1 − αL)(1 − ε)kT V
• αL(1 − ε)kT V

√δLL

• √

aL

• (1 − αL)(1 − ε)kT V

√ aL

• √

δI N √ δI

• √

cV    

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SDE HIV Model

dT = [λ − dT T − (1 − ǫ)kTV ]dt +κ(√λ + dT TdW1 −

• αL(1 − ǫ)kTV dW2 −
• (1 − αL)(1 − ǫ)kTV dW3)

dL = [αL(1 − ǫ)kTV − (δL + a)L]dt +κ(

• αL(1 − ǫ)kTV dW2 + √δLLdW4 −

√ aLdW5) dI = [(1 − αL)(1 − ǫ)kTV + aL − δI]dt +κ(

• (1 − αL)(1 − ǫ)kTV dW3 +

√ aLdW5 − √ δIdW6) dV = [NδI − cV ]dt +κ(N √ δIdW6 − √ cV dW7), where Wi are independent Wiener processes.

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SDE Model

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SDE Virus

Figure: SDE Model Virus with Detection Limit

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Interarrival Time

( > c > A inter arrival

time

~

exp

C A )

⇒

I

0.01

• L. Rong and A. Perelson, Mathematical Biosciences 2009

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Random Activation Function

dLt dt = αL(1 − ε)kV T − dLL − f(t)(2pL − 1)aL dIt dt = (1 − αL)(1 − ε)kV T − δI + f(t)(2 − 2pL)aL

Figure: Antigen Stimulation

• L. Rong and A. Perelson, Mathematical Biosciences 2009

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ODE with Random Activation Function

Figure: ODE Model with Poisson Process: T, Latent, Infected, Virus

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ODE with Random Activation Function: Virus

Figure: ODE Model with Poisson Process for Virus Cell

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SDE HIV Model with Random Activation Function

dT = [λ − dT T − (1 − ǫ)kTV ]dt +κ(√λ + dT TdW1 −

• αL(1 − ǫ)kTV dW2 −
• (1 − αL)(1 − ǫ)kTV dW3)

dL = [αL(1 − ǫ)kTV − dLL − f(t)(2pL − 1)aL]dt +κ(

• αL(1 − ǫ)kTV dW2 + √δLLdW4 −

√ aLdW5) dI = [(1 − αL)(1 − ǫ)kTV − δI + f(t)(2 − 2pL)aL]dt +κ(

• (1 − αL)(1 − ǫ)kTV dW3 +

√ aLdW5 − √ δIdW6) dV = [NδI − cV ]dt +κ(N √ δIdW6 − √ cV dW7), where Wi are independent Wiener processes. The random activation function is defined by f(t) = χ{N(t)−N(t−∆t)=0} where N(t) is a Poisson process with λ = 0.01.

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SDE with Random Activation Function

Figure: SDE Model with Poisson Process: T, Latent, Infected, Virus

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Detection Limit

Figure: Virus Model with Detection Limit

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Future Work

Within the time limit of about 300 days, the model can approximate viral blips. However, after that time, the blips become too small to be

• detectable. This is not accurate to life, so extending these detectable

blips out further is a goal. Further simulations in order to compare the probability of viral blips

• ccurring to experimental data are desirable in order to compare more

accurate data.

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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 for Longevity 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 of the HIV-1 Infection by Stimulation of Latently Infected Cells in Patients under Highly Active Anti-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 Viremia Can 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- and Postexposure 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) in Mathematical Analysis and Applications at the University of Michigan-Dearborn. We would like to thank the National Science Foundation, National Security Agency, University of Michigan-Dearborn (SURE 2019), and the University of Michigan-Ann Arbor for their support.

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