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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