How Effective is the HPV Vaccine?

1. How Effective is the HPV Vaccine? Aaron Steinberg 10 April 2008 Math Comps

2. Overview • HPV, the genital HPV infection, and Gardasil • Previous models • Markov Models • My previous work • New Markov model • Results and discussion of new model

3. 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

4. Previous Models • Models by R.V. Barnabus et al. and E.J. Dasbach et al. explore the affects of Gardasil on HPV

5. 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

6. My Previous Work • A simple Markov model with three states • Concerned with HPV and Cervical Cancer • Population is 101 million women

7. Transition Matrices for Previous Model

8. Graphed Results of Previous Model

9. 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

10. New Markov Model

11. 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

12. 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

13. Transition Probability Matrix

14. Variable Probabilities x y

15. Cases Considered • No Vaccine • One Dose Vaccine • Variable Probabilities

16. 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.

17. 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

18. 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

20. 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

21. 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.

22. What Do the Results of the Hypothesis Test Mean?

23. 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

24. 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

25. 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