An overview and comparison of a selective pinning control method with a random pinning control method. Networks in which HIV is spread are used. Presentation was for the 19th IFAC World Congress in Cape Town, 2014.

1. 1
Pinning Control
of Disease Networks
Departement Elektriese, Elektroniese & Rekenaar-Ingenieurswese
Department of Electrical, Electronic & Computer Engineering
Kgoro ya Merero ya Mohlagase, Elektroniki & Bointšinere bja
Khomphutha
IFAC 2014
Eben du Toit
and
Ian K. Craig
University of Pretoria
*

2. 2
Background (3 min.)
HIV network: Pinning Strategies (7 min.)
Control for incidence steady-state (5 min.)

3. 3
Background

4. 4
Complex
Networks
Epidemiology
Control
Systems
Pinning Control of
Disease Networks
A synergy of fieldsBackground
*

5. 5
How many?
Budget Target
Research QuestionBackground
Which?
Nodes
Pin / Control
(Medicate)
Incidence Target

6. 6
How many?
Budget Target
Research QuestionBackground
Which?
Nodes
Pin / Control
(Medicate)
Incidence Target
IFAC 2014 paper

7. 7
How many?
Budget Target
Research QuestionBackground
Which?
Nodes
Pin / Control
(Medicate)
Incidence Target
IFAC 2014 paper
Also Presented today

8. 8
How many?
Budget Target
Research QuestionBackground
Which?
Nodes
Pin / Control
(Medicate)
Incidence Target
IFAC 2014 paper
Also Presented today
Near future

9. 9
Overview : Sexual contact network
Background
*
* – Virus
ζi
Person
ζi – Transmission function:
Probability of node “i” becoming
infected by its neighbours
Ui – Pinning Control (RTI)
Ui
Ui

10. 10
Overview : Sexual contact network
à Nodes represent people
à Vertices represent sexual contacts
Example network:
• Nodes: 59
• Alpha: -2.40 [1]
• Type: Scale-free
• Avg Degree: 2.95
Assume: Static network
[1] F Liljeros, CR Edling, and LAN Amaral. The web of
human sexual contacts. Nature , 411:907–908, 2001.
Background

11. 11
Complex Networks : Scale-Free networks
Avg. degree distribution P(k) follows a power law with
Typically: 2 < α < 3
or, simpler:
A network with hubs
[1] F Liljeros, CR Edling, and LAN Amaral. The web of
human sexual contacts. Nature , 411:907–908, 2001.
Lilieros [1] suggests sexual contact network with α ~ 2.4
Background

12. 12
Sexual Contact Network with HIV transmission (1)
1. “3D” HIV Immune Response Model
à T-cells
à Infected T-cells
à virus
Background

13. 13
Sexual Contact Network with HIV transmission (2)
2. HIV Immune Response Network Model
Viral load of node “i”
Transmission function
Network topology
Coupling-strength
(Set equal to 1)
Neighbour’s
viral load
Background

14. 14
Sexual Contact Network with HIV transmission (3)
3. Transmission function (probability of transmission)
Actual amount of virus transferred
(Unknown, 1 virion assumed)
[1] James P Hughes, Determinants of per-coital-act
HIV-1 infectivity among African HIV-1-
serodiscordant couples. The Journal of Infectious
Diseases, 205(3):358–65, February 2012.
[1]
Increased
Risk of
Transmission
[2] [2] Paul Arora, Nico J D Nagelkerke, and Prabhat Jha.
A systematic review and meta-analysis of risk
factors for sexual transmission of HIV in India. PloS
One, 7(8): e44094, January 2012.
Background
j’th neighbour’s
viral load
Probability of
transmission
based on
viral load of
neighbours

15. 15
Sexual Contact Network with HIV transmission (4)
3. Transmission function (visualisation)
Background

16. 16
Sexual Contact Network with HIV transmission (5)
4. Pinning control (intervention)
[1] Alan S Perelson and Ruy M Ribeiro. Modeling the
within- host dynamics of HIV infection. BMC biology,
11(1):96, August 2013
Effectiveness of
Reverse Transcriptase
Inhibitor ~ <= 80% [2]
[2] S. Duwal, C. Schütte, and M. von Kleist,
“Pharmacokinetics and pharmacodynamics of the
reverse transcriptase inhibitor tenofovir and
prophylactic efficacy against HIV-1 infection.,” PLoS
One, vol. 7, no. 7, Jan. 2012.
Control
Term [1]
Background

17. 17
Sexual Contact Network with HIV transmission (6)
5. Controlled Network
Background

18. 18
Assumptions for now
• Homogenous network (nodes the same)
• The same HIV immune response model per node
• Continuous-time sexual contact network. Links of the
network represent sexual contact with other nodes.
• Discrete nature of transmission is captured with
a stochastic function.
• Re-infection occurs after the first infection: virus transferred
• Reverse-transcriptase inhibitors maximum effectiveness
set at 80%. [1].
• 1 virions transferred during transmission
• HIV sexual transmission network is stable.
[1] S. Duwal, C. Schütte, and M. von Kleist, “Pharmacokinetics
and pharmacodynamics of the reverse transcriptase
inhibitor tenofovir and prophylactic efficacy against HIV-1
infection.,” PLoS One, vol. 7, no. 7, Jan. 2012.
Background

19. 19
HIV network: Pinning Strategy

20. 20
Three main pinning strategies
à Give the medicine to everyone
• Not economically viable, although ideal
à Give the medicine to a random selection of individuals
• In reality, this is the norm
à Give the medicine to the highest connected individuals
• Ethics in question of course!
• With limited resources, this is a consideration
• Explored with this research
HIV network: Pinning Strategy
Note: Pinned proportion always an increasing
function, to reflect public health ethics!

21. 21
Random pinning vs. Selective pinning (1)
HIV network: Pinning Strategy

22. 22
Random pinning vs. Selective pinning (2)
HIV network: Pinning Strategy

23. 23
Control for Incidence Steady-State

24. 24
Sexual contact network (revisited as a control system)
Control for Incidence Steady-State
* Person
ζavg – Avg Transmissibility,
across the whole network
r – Target incidence %
Medicate (Ui) highest connected
nodes first (Selective pinning)
y – Actual incidence %
c – Nodes to pin %
ζ1
ζ2
ζ3
ζ4
ζ5
ζ6

25. 25
Control for Incidence Steady-State
Two methods tested
• Proportional control
• Steady-state observer using bond-percolation

26. 26
Proportional control
• 0 <= y(x) <= 1
• 0 <= r <= 1
Control for Incidence Steady-State

27. 27
Steady-State Observer using Bond Percolation
• Bond Percolation : Complex Network technique used in
this work to estimate the size of an epidemic
Bond Percolation
Model
Average
Transmissibility
Node
Degree
Distribution
(intrinsic network
structure)
Final Epidemic
Size Estimate
Control for Incidence Steady-State
[1] M. Newman, “Spread of epidemic disease on
networks,” Phys. Rev. E, vol. 66, no. 1, p. 016218: 1–
11, Jul. 2002.
[1]
ζavg

28. 28
Steady-State Observer using Bond Percolation
Control for Incidence Steady-State
Bond Percolation
Model Observed
Steady-State

29. 29
Bond Percolation: estimates final infected proportion
Bond percolation
estimate
Actual network
Control for Incidence Steady-State
ζavg

30. 30
Results – Proportional Control
Control for Incidence Steady-State
r=0.1
c(x)
• Note: Gain adjusted manually after each
reference, thus not a useful strategy.
y(x)

31. 31
Results – Steady-State Observer Control
Control for Incidence Steady-State
r=0.5
c(x)
• Now: Loop gain tuned only the first time to
reduce steady-state error.
y(x)

32. 32
Results – Impact of control action on
Control for Incidence Steady-State
c(x)
y(x)
ζavg
ζavg

33. 33
Papers:
1. Pinning Strategy à Accepted for IFAC2014 : ✔
2. Predicting Disease Steady-State à çSubmission to BSPC
3. Optimal Gain Scheduling Control à ç Near future
4. Intervention Budgets à ç Near future
E. F. Du Toit and I. K. Craig, “Quantifying the Impact of
Two Pinning Control Strategies on HIV Incidence.”
E. F. Du Toit and I. K. Craig, “Pinning Control of
an HIV Sexual Contact Network.”
E. F. Du Toit and I. K. Craig, “Estimating optimal
interventions given budget adherence for disease networks”
E. F. Du Toit and I. K. Craig, “Gain Scheduled Pinning
Control of HIV Network Incidence Steady State.”

34. Thank you!
Eben du Toit
Student - Ph.D (Electronic Engineering)
Dept.: Electrical, Electronic & Computer Engineering
University of Pretoria
Pretoria 0002
SOUTH AFRICA
Tel: +27-(82)-318-7773
E-mail: ebendutoit@tuks.co.za
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