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)

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


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


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


Stochastic Model


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).


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


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


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.


SDE Model


SDE Virus

Figure: SDE Model Virus with Detection Limit


Interarrival Time

)

( > c >A

inter arrival time

~

exp CA )

.

⇒ I = 0.01

I



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



ODE with Random Activation Function

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


ODE with Random Activation Function: Virus

Figure: ODE Model with Poisson Process for Virus Cell


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.


SDE with Random Activation Function

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


Detection Limit

Figure: Virus Model with Detection Limit


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.


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.


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.


Documents

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)