Probability Modeling for HIV Viral Blips Megan Osborne 1 and Tamantha Pizarro 2 The University of Scranton 1 Iona College 2 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 λ d L T L d T α L k a 1 − α L δ V I c Nδ dT dt = λ − d T T − (1 − ǫ ) kV T dL dt = α L (1 − ǫ ) kV T − d L L − aL dI dt = (1 − α L )(1 − ǫ ) kV T − δI + aL dV dt = NδI − cV Megan Osborne, Tamantha Pizarro February 2, 2020 3 / 22
Parameter Values L. Rong and A. Perelson, Mathematical Biosciences 2009 Megan Osborne, Tamantha Pizarro February 2, 2020 4 / 22
ODE Model L. 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. E ( | ∆ X ( t ) | δ | X ( t ) = x ) lim = 0 for δ > 2 ∆ t ∆ t → 0 2. E ( | ∆ X ( t ) || X ( t ) = x ) lim = b ( x ) ∆ t ∆ t → 0 3. E ( | ∆ X ( t ) | 2 | X ( t ) = x ) lim = a ( x ) , ∆ t ∆ t → 0 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 )) dW t , where W t 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 ] (∆ � X ) p i ∆ t i � 1 0 � T 1 0 0 λ ∆ t � − 1 0 � T 0 0 d T T ∆ t 2 � − 1 0 � T 1 0 α L (1 − ǫ ) kT V ∆ t 3 0 � T � − 1 0 1 (1 − α L )(1 − ǫ ) kT V ∆ t 4 0 � T � 0 − 1 0 δ L L ∆ t 5 0 � T � 0 − 1 1 6 aL ∆ t N � T � 0 7 0 − 1 δI ∆ t − 1 � T � 0 8 0 0 cV ∆ t 1 − � 8 � 0 0 � T 9 0 0 i =1 P i ∆ 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 0 - αL (1 − ǫ ) kT V αL (1 − ǫ ) kT V + ( δL + a ) L -aL 0 -(1- αL )(1 − ǫ ) kT V -aL (1- αL )(1 − ǫ ) kT V + aL + δI -N δI N 2 δI + cV 0 0 -N δI Diffusion Matrix √ λ + d T T � � α L (1 − ε ) kT V (1 − α L )(1 − ε ) kT V 0 0 0 0 - − √ δ L L √ � 0 α L (1 − ε ) kT V 0 aL 0 0 - √ √ � 0 0 (1 − α L )(1 − ε ) kT V 0 aL δI 0 - √ √ 0 0 0 0 0 δI cV N - Megan Osborne, Tamantha Pizarro February 2, 2020 9 / 22
SDE HIV Model dT = [ λ − d T T − (1 − ǫ ) kTV ] dt + κ ( √ λ + d T TdW 1 − � � α L (1 − ǫ ) kTV dW 2 − (1 − α L )(1 − ǫ ) kTV dW 3 ) dL = [ α L (1 − ǫ ) kTV − ( δ L + a ) L ] dt √ α L (1 − ǫ ) kTV dW 2 + √ δ L LdW 4 − � + κ ( aLdW 5 ) = [(1 − α L )(1 − ǫ ) kTV + aL − δI ] dt dI √ √ � + κ ( (1 − α L )(1 − ǫ ) kTV dW 3 + aLdW 5 − δIdW 6 ) dV = [ NδI − cV ] dt √ √ + κ ( N δIdW 6 − cV dW 7 ) , where W i 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 I C A ) ⇒ 0.01 ~ exp = . I L. Rong and A. Perelson, Mathematical Biosciences 2009 Megan Osborne, Tamantha Pizarro February 2, 2020 13 / 22
Random Activation Function dL t dt = α L (1 − ε ) kV T − d L L − f ( t )(2 p L − 1) aL dI t dt = (1 − α L )(1 − ε ) kV T − δI + f ( t )(2 − 2 p L ) 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 = [ λ − d T T − (1 − ǫ ) kTV ] dt dT + κ ( √ λ + d T TdW 1 − � � α L (1 − ǫ ) kTV dW 2 − (1 − α L )(1 − ǫ ) kTV dW 3 ) dL = [ α L (1 − ǫ ) kTV − d L L − f ( t )(2 p L − 1) aL ] dt √ α L (1 − ǫ ) kTV dW 2 + √ δ L LdW 4 − � + κ ( aLdW 5 ) = [(1 − α L )(1 − ǫ ) kTV − δI + f ( t )(2 − 2 p L ) aL ] dt dI √ √ � + κ ( (1 − α L )(1 − ǫ ) kTV dW 3 + δIdW 6 ) aLdW 5 − dV = [ NδI − cV ] dt √ √ + κ ( N δIdW 6 − cV dW 7 ) , where W i 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 . 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 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 occurring to experimental data are desirable in order to compare more accurate 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 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. Megan Osborne, Tamantha Pizarro February 2, 2020 22 / 22
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