Download Mathematical models for the efficacy of Gardasil on instances of

How Effective is the HPV
Vaccine?
Aaron Steinberg
10 April 2008
Math Comps
Overview
 HPV, the genital HPV infection, and
Gardasil
 Previous models
 Markov Models
 My previous work
 New Markov model
 Results and discussion of new model
What is HPV?
 HPV (Human papillomavirus) is the name of a group of
viruses, 30 of those types are sexually transmitted
 The “high-risk” types of HPV are responsible for 90
percent of genital warts and 70 percent of cervical
cancers
 The genital HPV infection is thought to be the most
common sexually transmitted disease in the world
 To combat this infection a vaccine called Gardasil was
created by Merck Pharmaceuticals
Previous Models
 Models by R.V. Barnabus et al. and E.J.
Dasbach et al. explore the affects of Gardasil
on HPV
Markov Models
 A Markov model is a discrete-time stochastic
process
 In such a process, future states are
independent of past states
 At each time step the model may transition into
another state or remain in the same state
according to a given probability distribution and
transition diagram
 Transition probabilities pij= probability of
moving from state i to state j
My Previous Work
 A simple Markov
model with three
states
 Concerned with HPV
and Cervical Cancer
 Population is 101
million women
Transition Matrices for Previous Model
0% Vaccinated
Susceptible
HPV Infected
Cervical Cancer
Susceptible
HPV Infected
Cervical Cancer
0.97
0
0.6
0.03
0.996
0
0
0.004
0.4
50 % Vaccinated
Susceptible
HPV Infected
Cervical Cancer
Susceptible
HPV Infected
Cervical Cancer
0.985
0
0.6
0.015
0.996
0
0
0.004
0.4
100 % Vaccinated
Susceptible
HPV Infected
Cervical Cancer
Susceptible
HPV Infected
Cervical Cancer
0.999
0
0.6
0.001
0.996
0
0
0.004
0.4
Graphed Results of Previous Model
HPV
Cervical
Cancer
100
0.6
80
0.5
Percentage of 0.4
60
Percentage
Population of 0.3
40
Population
20
0.2
0.10
0
0% Vaccinated
Vaccinated
0%
50% Vaccinated
Vaccinated
50%
100% Vaccinated
Vaccinated
100%
Issues Leading to New Model
 People naturally cure HPV infection so
proportion ending in HPV state is too
large
 Three dose structure of HPV vaccine is
not taken into account in previous models
New Markov Model
New Model Assumptions
 Since humans naturally cure the HPV infection
it is possible to return from any of the further
states to the susceptible population
 Population includes all sexually active females
 Since there are three doses of the vaccine
there are four susceptible states
 Each iterative step is one year
New Assumptions Cont.
 The probability for returning to the different
susceptible states is the probability for
returning to one susceptible state given by
Barnabus et al. but divided four ways where
4/10 was for returning to S0, 3/10 for S1, 2/10
for S2, and 1/10 for S3
 One can not acquire cancer or pre-cancer
without getting HPV first, thus disregarding
cases of cervical cancer not from HPV
 Curing cervical cancer leads directly back to
the susceptible population
Transition Probability Matrix
General
Model
S0
S1
S2
S3
HPV
CIN I
CIN II
CIN III
CC
S0
.4
.5
0
0
.1
0
0
0
0
S1
0
(.1-x)
.9
0
(x)
0
0
0
0
S2
0
0
(.3-y)
.7
(y)
0
0
0
0
S3
0
0
0
.99
.01
0
0
0
0
HPV
.22
.165
.11
.055
.34
.1
.01
0
0
CIN I
.04
.03
.02
.01
.05
.72
.1
.03
0
CIN II
.072
.054
.036
.018
.02
.25
.49
.06
0
CIN III
.004
.003
.002
.001
.005
0
.01
.974
.001
CC
.22
.165
.11
.055
0
0
0
0
.45
Variable Probabilities
x
y
Cases Considered
 No Vaccine
 One Dose Vaccine
 Variable Probabilities
Is the Model Accurate?
 Although steady states are small my
population is on the scale of hundreds of
millions so even these fractions of a
percent are significant
 In the unvaccinated case there was a
steady state for HPV of 0.1144 meaning
11.44 percent of the population would be
in the HPV state in the long run.
One-Dose Vaccine Discussion
 All cases with any vaccine end with a steady
state for cervical cancer of zero. Leaves us
with HPV as the useful result
 For the case with a one-dose vaccine structure
the steady state for HPV is 0.0153,
corresponding to an 87 percent decrease in
HPV
 In the model with the one-dose vaccine
structure there is an approximate average of a
60 percent decrease in pre-cancer
Three-Dose vs. One-Dose Vaccines
 For the case of the three-dose vaccine
with the base indices, the steady state for
HPV is 0.0173; in the one-dose case the
HPV steady state is 0.0153
How HPV Steady States Change While Varying Transition Probabilities
0.0182
0.018
HPV Steady State
0.0178
0.0176
Varying X
0.0174
Varying Y
0.0172
0.017
0.0168
0.0166
0
0.02
0.04
0.06
Value of X or Y
0.08
0.1
0.12
Why Do the Steady States Increase
Faster when Varying Y?
 Mathematically speaking, only one row of
the transition matrix is varied at each
case
 I put forth that the cause of this
phenomenon is possibly that pS0S1 <
pS1S2
Hypothesis Test
 To test this hypothesis I set
pS0S1=pS1S2=0.5, the original value for
pS0S1 → ΠHPV=0.0178.
 To further test my hypothesis I set
pS0S1=pS1S2=0.9 → ΠHPV=0.0169.
What Do the Results of the
Hypothesis Test Mean?
Percentage of the Population
Hypothesis
Test Comparison
Average HPV
Steady State
When Varying X and Y
1.85
1.8
1.78
1.76
1.75
1.74
Percentage of 1.72
1.7
1.7
Population
1.68
1.66
1.65
1.64
1.6
1.55
Probability of Dose 2 of
0.5
Probability of Dose 2
Original
of 0.5Model
Probability of
Probability
of
VaccineofofVaccine
0.9
0.9
Conclusion
 The model is a good representation of real life
 Implementing a vaccine, especially in a one-dose form,
should heavily curtail cases of HPV
 It is advisable to refrain from sexual activity during the
vaccination period because we see that HPV cases in
the long term are variable with unknown transition
probabilities in between doses
 Vaccinating more women in the beginning, and
ensuring that more women, who begin the series,
complete it, results in a reduction of overall HPV cases
and greatly decreases variance in unknown transition
probabilities in between doses
Further Research
 More attention to which S state people
return to after CC
 More elaborate model tracking cervical
cancer
 Make a model for genital warts caused
by HPV
Thank You
 The entire Math Department, faculty and
students…and of course everyone who
came to watch
 Professor Tamas Lengyel
 Professor Mickey McDonald
 Professor Ron Buckmire
 Last, but definitely not least, Professor
Angela Gallegos