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Get checked online

Mathematical models on Increasing HIV testing in Vancouver

Work in progress, June 9th 2015

Bernhard Konrad, Dan Coombs (UBC)

Warren Michelow, Mark Gilbert (BCCDC)

Get checked online

Mathematical models on Increasing HIV testing in Vancouver

This talk may contain explicit language - please attend at your own discretion.

HIV in Canada

Canada

British Columbia

Vancouver

71,000 HIV+ (2011)
3,200 new diagnosis every year
one new infection every 2.5h

300 new diagnosis every year
3,500 unaware of infection (25%!)

6,000 HIV+ (1.2%)
175 new infections every year
150 new diagnosis every year
~50% MSM, ~25% IDU, ~25%

Understand! Non-improvement for MSM

GetCheckedOnline (GCO)

BCCDC survey data

Sex Now (Canada-wide, ongoing since 2000)
ManCount (Vancouver, 2008)

=> MSM use the internet (you don't say...) to meet sexual partners and to look for health information.

=> 23% of young men under age of 30 never tested for HIV

GCO

Print lab form, go to LifeLab, get results online/phone
Check for Chlamydia, Gonorrhea, HIV, Syphilis, HCV
Goal: Start treatment (as prevention), change risk behaviour

Mathematical models - starting simple

Find total number of new infections as function of testing rate

\beta = {\color{blue}p}{\color{red}r}S\frac{I_u + {\color{red}\varepsilon}I_a}{{\color{blue}N}}

β = p r S \frac{​ I ​ u ​ ​ + ε I ​ a ​ ​ ​ ​}{​ N ​}

\color{red}\varepsilon\!:

ε :

Change in risk behaviour if aware of infection

\color{red}r\!:

r :

Rate of risky events

Sensitivity Analysis

Data: ~175 annual new infections

Find total number of new infections as function of testing rate

The Direct Differential Method

\frac{d\vec{y}}{dt} = f(\vec{y}, \vec{p}, t),\, \vec{y}(0) = \vec y_0

\frac{​ d ​ y ​ ⃗ ​ ​ ​ ​}{​ d t ​} = f (​ y ​ ⃗ ​ ​, ​ p ​ ⃗ ​ ​, t), ​ y ​ ⃗ ​ ​ (0) = ​ y ​ ⃗ ​ ​ ​ 0 ​ ​

Let

ODE with parameters

\vec p

​ p ​ ⃗ ​ ​

We want

\vec s_j := \frac{d \vec y}{dp_j}

​ s ​ ⃗ ​ ​ ​ j ​ ​ : = \frac{​ d ​ y ​ ⃗ ​ ​ ​ ​}{​ d p ​ j ​ ​ ​}

\frac{d}{dt}\vec s_j = \frac{d}{dt} \frac{d\vec y}{dp_j}

\frac{​ d ​ ​}{​ d t ​} ​ s ​ ⃗ ​ ​ ​ j ​ ​ = \frac{​ d ​ ​}{​ d t ​} \frac{​ d ​ y ​ ⃗ ​ ​ ​ ​}{​ d p ​ j ​ ​ ​}

= \frac{d}{dp_j}\frac{d\vec y}{dt}

= \frac{​ d ​ ​}{​ d p ​ j ​ ​ ​} \frac{​ d ​ y ​ ⃗ ​ ​ ​ ​}{​ d t ​}

= \frac{d}{dp_j}f(\vec y, \vec p, t)

= \frac{​ d ​ ​}{​ d p ​ j ​ ​ ​} f (​ y ​ ⃗ ​ ​, ​ p ​ ⃗ ​ ​, t)

= \frac{\partial f}{\partial \vec y}\cdot \frac{d\vec y}{dp_j} + \frac{\partial f}{\partial p_j}

= \frac{​ \partial f ​ ​}{​ \partial ​ y ​ ⃗ ​ ​ ​} \cdot \frac{​ d ​ y ​ ⃗ ​ ​ ​ ​}{​ d p ​ j ​ ​ ​} + \frac{​ \partial f ​ ​}{​ \partial p ​ j ​ ​ ​}

= J\vec s_j + F_j

= J ​ s ​ ⃗ ​ ​ ​ j ​ ​ + F ​ j ​ ​

where J is the Jacobian and

(F_j)_k = \frac{\partial f_k}{\partial p_j}

(F ​ j ​ ​) ​ k ​ ​ = \frac{​ \partial f ​ k ​ ​ ​ ​}{​ \partial p ​ j ​ ​ ​}

can be found analytically.

The Direct Differential Method

s_{ji} \cdot \frac{p}{y_i} = 1.16

s ​ j i ​ ​ \cdot \frac{​ p ​ ​}{​ y ​ i ​ ​ ​} = 1.16

It is more typical to consider the relative change, e.g. in the simple model:

That is, doubling per-exposure risk leads to a relative increase of total new infections of a factor of 1.16.

Effect of change in testing rate:

Susceptibles: 0.01

Infected unaware: -1.03

Infected aware: 0.15

new infections: -0.19

new dections: 0.34

Extracting parameters from surveys

\beta = {\color{blue}p}{\color{red}r}S\frac{I_u + {\color{red}\varepsilon}I_a}{{\color{blue}N}}

β = p r S \frac{​ I ​ u ​ ​ + ε I ​ a ​ ​ ​ ​}{​ N ​}

\color{red}\varepsilon\!:

ε :

Change in risk behaviour if aware of infection

\color{red}r\!:

r :

Rate of risky events

Sex Now (Canada-wide, ongoing since 2000)
ManCount (Vancouver, 2008)
BCCDC Acute study (HIV- cohort)

Survey data

BCCDC Acute study (HIV-)

166 MSM recruited at HIV testing clinics in Vancouver between June 2011 - March 2012
Four questionnaires, four network grids on days 7, 30, 180, 360

Q: Testing behaviour, general risk behaviour, HIV knowledge

Grids: Detailed interview on partnerships and risk events

BCCDC Acute study (HIV-)

"What best describes your usual testing pattern."

Questionnaires: Testing behaviour

BCCDC Acute study (HIV-)

Questionnaires: general risk behaviour

Number of male partners

BCCDC Acute study (HIV-)

Grid: Relationships and risk events

Length and number of recent relationships

"During the past year, how often did you have sex with this partner."

The role of GCO in a diverse population

Identify/Classify subgroup boundaries (hand-picked or clustering)
Average relevant parameters to obtain average subgroup risk, testing
Estimate mixing rates of subgroups
Report resulting sensitivity of GCO effect

WIP