Dynamic Pricing Strategy to Optimally Allocate Vaccines

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1 Proceedings of the 2011 Industrial Engineering Research Conference T. Doolen and E. Van Aken, eds. Dynamic Pricing Strategy to Optimally Allocate Vaccines Mauricio Pommier and Dr. Rubén Proaño Department of Industrial and Systems Engineering Rochester Institute of Technology Rochester, NY 14623, US Abstract In the United States, the Centers for Disease Control and Prevention (CDC) makes recommendations as to which cohorts ought to have higher priority access to vaccines, when supply is insufficient to immunize all susceptible individuals in the country. Typically such cohorts are determined based on the susceptibility to contracting seasonal influenza and the resulting consequences of infection for different age groups. For seasonal influenza, high-risk cohorts commonly include children, teenagers, pregnant women, and people with different chronic diseases. This study proposes the application of revenue management theory to better allocate seasonal influenza vaccines among different risk-based population cohorts. Our model maximizes the number of immunized individuals, by dynamically adjusting the price per dose in each cohort to temporarily discourage vaccination in low-risk cohorts and temporally increase the supply available for high-risk cohorts. The model shows that 12% of infections and deaths may be avoided by implementing this price discrimination policy. Keywords Operations research, revenue management, dynamic programming, health care, vaccines 1. Introduction In the United States, over 40,000 lives are claimed due to the influenza virus [1]. In case of influenza pandemics, the number of cases increases dramatically [2]. For example, the 1918 Spanish Flu pandemic killed 40 million people [3] and infected nearly a quarter of the entire global population at the time. Several epidemiological models have been developed to describe the progression and spread of infectious diseases; see Hethcote as well as Sattenspiel and Lloyd [4, 5]. Such models are useful to plan and establish effective containment interventions including vaccination [6, 7], behavioral interventions such as school closures [8-10], or their combination [11, 12]. Most epidemiological models are compartmentalized systems that describe the disease progression resulting from having different compartmental transition rates [13, 14], in which the progression depends on a deterministic basic reproduction number (R 0 ), representing the number of average people that any one contagious individual infects [15]. Larson criticized such models and proposes an alternative approach that considers that R 0 is dynamic and not necessarily homogeneous [16]. Every year about 80 million seasonal influenza vaccines are available for distribution to less than one-third of the American population [17]. The limitation of the supply of seasonal influenza vaccines results from the availability of eggs used to grow the virus in vaccine manufacturing. Therefore, health care authorities must determine how to distribute the limited supply of seasonal vaccines to provide the most effective protection across the population. The World Health Organization (WHO) refers to the definition of herd immunity in support of a more effective vaccination approach that strategically distributes vaccines to minimize the chances of the disease to spread [18]. In the United States, the Advisory Committee on Immunization Practices (ACIP) makes recommendations on how to divide the population into two distinct cohorts based on the vulnerability toward the disease [18]. ACIP also suggests which cohort should have priority access to available vaccination. This study assumes that only a limited supply of seasonal influenza vaccine is available to the population, and that health care decision-makers need to persuade disease-averse individuals from the low-risk cohort to opt for vaccination later in the flu season. We consider that without an intervention from authorities, such disease-averse individuals will use vaccines that could have been used by people in high-risk cohorts, resulting in a larger group of

2 unvaccinated high-risk individuals. Our study proposes the use of vaccine price as a control measure that can maximize vaccine availability for the high-risk cohorts. Another assumption made in this work is that policy makers aim to minimize the number of infections and deaths over the flu season. 2. Methodology This study proposes an optimization model, which results from modifying the influenza progression model proposed by Larson [16]. The disease is modeled as an expansion of the traditional SIR model with six possible states: Susceptible (S), Exposed (E), Infectious Asymptomatic (A), Infectious Symptomatic (I), Dead (D), and Immune (H), where the immune state may be achieved via recovery from the disease or through vaccination. We calculate the R 0 during each time period that minimizes the difference with the R 0 of the previous period by determining the optimal percentiles of the population for whom the vaccine price per dose is affordable. The data presented by Lee [18] has been adapted to fit the modified data structure from Larson [16]. For example, children s behavior is divided into their interactions at school, in their household, and within their community, and interpreted as interactions with relevant cohorts (i.e. children the same age for school contacts) with the corresponding infectious probability for each particular setting. At first, the entire population is assumed to be susceptible to becoming infected with influenza and may contract the disease when being in contact with an infected individual that is shedding (i.e., a contagious person). Afterwards, there is an incubation period, when the infected individual is asymptomatic and may not infect other individuals. Normally this period is followed by an infectious asymptomatic state and then by an infectious symptomatic state. These last two states are the only ones that present shedding. The number of people who become infected by each new infected individual is a function of the number of contacts the person had during stages (A) and (I), and the probability of infection. It is assumed that an infectious symptomatic person undergoes a voluntary isolation and recovery period, which results in a reduction of its contact rate with individuals in the population. While isolation is not a state of the disease progression, it is modeled as part of the symptomatic state. The transition into recovery from the infectious symptomatic state depends on the probability of recovery and development of natural immunity against the virus. Alternatively, immunity is developed through vaccination, which plays a critical role in this model. The number of people who get vaccinated at a given point in time depends on the number of vaccines available and the number of people willing and able to purchase a vaccine dose at the given price. If an infected patient does not develop immunity, his or her symptoms worsen, which leads to death. Following the nomenclature proposed by Larson [16], we define. N i,k (t) = population from cohort i in state k during time period t. λ i,j,r = rate of contact of a person in cohort i with a person in cohort j, in r different settings. p i,j,r = probability that a susceptible person from cohort i, given contact with an infected, contagious person from cohort j, becomes infected in setting r. p D i = probability that an infected person from cohort i dies due to the disease. Therefore, the probability that in period t a person in cohort i is in contact with an infectious individual is given by the fraction of contacts from the entire population that belongs to contagious individuals, and it is given by β i,j,r (t) = N k A I i k t {, }, ( )* i, j, r N ( t)* k/ D i, k i, j, r (1) The probability of infection of a susceptible individual of cohort i in period t depends on the daily behavior of the individual and his/her susceptibility to the virus. Assuming Poisson-distributed interactions with means λ i,j,r and a probability of contact β i,j,r (t), the probability of infection of an individual from cohort i in period t is given by P I i(t) = j r, * * ( t) p i r i r j, r i, (1 e j,, j, ) (2)

3 In order to effectively postpone the demand of vaccine from low-risk individuals and, thus, increase the available supply of vaccines for high-risk individuals, we propose establishing a price per dose for each cohort. We assume that willingness to pay (WTP) curves for each cohort are known. Such curves describe the percentile of the population, PRICE i (t), that is willing to buy a vaccine for a given price per dose. The percentile is then used to establish the probability that an individual of cohort i in period t will be vaccinated, given by P V i(t) = PRI CE ( t)* VS ( t) i {,, } N, ( )* ( ) k S E A i i k t PRI CE i t (3) where VS(t) = number of available vaccines during time period t The complete description of the state transition and their equations are shown on Figure 1. Figure 1: State transition diagram To illustrate the effect of changes in vaccine price, assume that if the model recommends a price percentile of 0.1 for Population 1 and 0.2 for Population 2, the model interprets this data as 10% of Population 1 wanting the vaccine and being able to afford it. Similarly, 20% of Population 2 want to purchase the vaccine. However, if the stockpile of vaccines is lower than the number of people who want the vaccine, the resulting vaccinations will be distributed according to the population proportion. The example on Figure 2 assumes two cohorts, each with 20 people. With a price percentile of 0.1 and 0.2 during this time period, the number of people that are willing and able to purchase the vaccine is 2 and 4 from each cohort, respectively. They are shown in the green circle. If only 3 vaccines are available during this time period, then only half of the people that want to get vaccinated will get vaccinated. Thus, half from each cohort will get vaccinated, marked by the gray circle the graph.

4 Population 1 Population 2 Figure 2: Diagram depicting an instance of how pricing works The model takes a goal function as an input to determine the optimal price for the current time period. We assume that the goal of policy makers will be to attenuate the spread of the disease, which is given by the basic reproductive number. Therefore, when the value of R 0 is ramping up at the beginning of the disease, we want R 0 s growth rate to be the least steep possible. Similarly, after the disease peaks the goal is to make the steps downward to be as large as possible. Therefore, we minimize the following function at every period, while determining the best price percentiles. F = R 0 (t) R 0 (t-1) (4) A number of vaccines are assumed to be available daily starting on day 10 of the simulation. We follow reports that 91 million people received the flu vaccine during the 2009 pandemic [19], which represents about 30% of the initial susceptible United States population. 3. Experimentation The model was developed using Mathematica 7.0 at Rochester Institute of Technology s Research Computing center. To measure the effectiveness of our approach we only modify the availability of vaccines and the distribution process of the vaccines. None of the parameters established by Larson or Lee are modified in order to maintain consistency across the different settings explored. The number of periods per state used for the experiment is 1 period per non-infectious state being modeled. We assume that the asymptomatic state lasts only half a period and the symptomatic one lasts 3 periods. During the symptomatic state, we imply that the infected individual will reduce its daily contacts to about 10%. Number of Doses in Thousands 1, Vaccine Supply Days Vaccine Supply Figure 3: Assumed daily vaccination of individuals We experiment with three models. The baseline model assumes that no vaccines are available for the population. Therefore, no optimization occurs daily and the only way to get immunity to the virus is through recovery from the

5 Pommier and Proaño disease. The second model assumes that vaccines are available, in accordance with Figure 3. However, this model assumes that vaccines are distributed throughout the population at no cost to the individual, as is the case in the United States during pandemics. In the third model, i.e. the one we propose, a percentile is calculated to offset the demand of vaccines from low-risk cohorts. 4. Results The effectiveness of the model can be shown by the decrease in the total number of infections and deaths. Table 1 presents the results from our model and compares them to the other two scenarios. Table 1: Summary of results Scenario Pandemic Length Total Infections Total Deaths (periods) (% population) (individuals) Model 1: No Vaccination % 144,582 Model 2: Free Vaccination % 128,499 Model 3: Priced Vaccination % 113,472 The scenario with no vaccination has the highest spread of the disease within the population but the disease got under control the fastest. The model where there is no preference in vaccination and the one we proposed had similar lengths of the disease until nobody from the population was infected. However, our proposed approach reduced the number of infections and deaths by 12%, saving 15,027 lives. Number of Individuals in Thousands Number of Infections No Vaccines Free Vaccines Priced Vaccines Days Figure 4: Spread of the disease over time It can be noted that the infection spreads through the population at a faster rate when no vaccination is used. The peak of the disease also happens earlier than when countermeasures are presented, as shown in Figure Conclusions Policy makers are constantly looking for strong evidence for how to effectively mitigate the spread of a disease. During the 2009 flu pandemic there were discrepancies when distributing the vaccines to people of different cohorts. Some vaccination clinics strictly enforced the recommendations from the ACIP regarding who should have preferential access to vaccination while others administered the vaccine at normal cost to the entire population. We propose an innovative approach where the option of vaccination is not stripped away from the population but rather discouraged through pricing. Our results demonstrate that benefits can be gained by temporarily enforcing the vaccination of high-risk individuals by postponing vaccination of those at lesser risk through pricing control.

6 Acknowledgements The authors would like to acknowledge the support of the Research Computing center at Rochester Institute of Technology. We also thank anonymous reviewers for their helpful comments. References 1. Dushoff, J., Plotkin, J., Viboud, C., Earn, D., and Simonsen, L., 2006, "Mortality due to influenza in the United States -an annualized regression approach using multiple-cause mortality data," American Journal of Epidemiology, vol. 163, p Reid, A., and Taubenberger, J., 2003, "The origin of the 1918 pandemic influenza virus: a continuing enigma," Journal of General Virology, vol. 84, p Johnson, N., and Mueller, J., 2000, "Updating the accounts: global mortality of the " Spanish" influenza pandemic," Bulletin of the History of Medicine, vol. 76, pp Hethcote, H., 2000, "The mathematics of infectious diseases," SIAM review, vol. 42, pp Sattenspiel, L., and Lloyd, A., 2009, The geographic spread of infectious diseases: models and applications: Princeton Univ Pr. 6. Cahill, E., Crandall, R., Rude, L., and Sullivan, A., 2005, "Space-time influenza model with demographic, mobility, and vaccine parameters." 7. De Jong, M.C.M., and Bouma, A., 2001, "Herd immunity after vaccination: how to quantify it and how to use it to halt disease," Vaccine, vol. 19, pp Blendon, R.J., Koonin, L.M., Benson, J.M., Cetron, M.S., Pollard, W.E., Mitchell, E.W., Weldon, K.J., and Herrmann, M.J., 2008, "Public Response to Community Mitigation Measures for Pandemic Influenza," Emerging Infectious Diseases, vol. 14, pp Araz, O.M., Fowler, J.W., Lant, T.W., and Jehn, M., 2010, "A Pandemic Influenza Simulation Model for Preparedness Planning." 10. Davey, V.J., and Glass, R.J., 2008, "Rescinding Community Mitigation Strategies in an Influenza Pandemic," Emerging Infectious Diseases, vol. 14, pp Carrat, F., Luong, J., Lao, H., Sallé, A., Lajaunie, C., and Wackernagel, H., 2006, "A'small-world-like' model for comparing interventions aimed at preventing and controlling influenza pandemics," BMC medicine, vol. 4, p Davey, V.J., Glass, R.J., Min, H.J., Beyeler, W.E., and Glass, L.M., 2008, "Effective, Robust Design of Community Mitigation for Pandemic Influenza: A Systematic Examination of Proposed US Guidance," PLoS ONE, vol. 3, p. e Jacquez, J.A., and Simon, C.P., 1993, "Qualitative Theory of Compartmental Systems," SIAM Review, vol. 35, pp Nokes, D., and Anderson, R., 1988, "The use of mathematical models in the epidemiological study of infectious diseases and in the design of mass immunization programmes," Epidemiology and infection, vol. 101, pp Van den Driessche, P., and Watmough, J., 2002, "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission," Mathematical Biosciences, vol. 180, pp Larson, R.C., 2007, "Simple Models of Influenza Progression Within a Heterogeneous Population," Operations Research, vol. 55, p Longini, I., and Halloran, M., 2005, "Strategy for distribution of influenza vaccine to high-risk groups and children," American Journal of Epidemiology, vol. 161, p Lee, B.Y., Brown, S.T., Cooley, P.C., Zimmerman, R.K., Wheaton, W.D., Zimmer, S.M., Grefenstette, J.J., Assi, T., Furphy, T.J., Wagener, D.K., and Burke, D.S., 2010, "A computer simulation of employee vaccination to mitigate an influenza epidemic," American journal of preventive medicine, vol. 38, pp Drummond, K., 2010, "Once Hard to Get, H1N1 Vaccines Now Getting Dumped," ed.

Dynamic pricing strategy to optimally allocate vaccines

Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-10-2012 Dynamic pricing strategy to optimally allocate vaccines Mauricio Pommier Gasser Follow this and additional