Hamed Yarmand

Optimal two-phase vaccine allocation to geographically different regions under uncertainty

In this article, we consider a decision process in which vaccination is performed in two phases to contain the outbreak of an infectious disease in a set of geographic regions. In the first phase, a limited number of vaccine doses are allocated to each region; in the second phase, additional doses may be allocated to regions in which the epidemic has not been contained. We develop a simulation model to capture the epidemic dynamics in each region for different vaccination levels. We formulate the vaccine allocation problem as a two-stage stochastic linear program (2-SLP) and use the special problem structure to reduce it to a linear program with a similar size to that of the first stage problem. We also present a Newsvendor model formulation of the problem which provides a closed form solution for the optimal allocation. We construct test cases motivated by vaccine planning for seasonal influenza in the state of North Carolina. Using the 2-SLP formulation, we estimate the value of the stochastic solution and the expected value of perfect information. We also propose and test an easy to implement heuristic for vaccine allocation. We show that our proposed two-phase vaccination policy potentially results in a lower attack rate and a considerable saving in vaccine production and administration cost.

Cost-effectiveness analysis of vaccination and self-isolation in case of H1N1

In this research, we have conducted a cost-effectiveness analysis to examine the relative importance of vaccination and self-isolation, with respect to the current H1N1 outbreak. We have developed a continuous-time simulation model for the spread of H1N1 which allows for three types of interventions: antiviral prophylaxis and treatment, vaccination, and self-isolation and mandatory quarantine. The optimization model consists of two decision variables: vaccination fraction and self-isolation fraction among infectives. By considering the relative marginal costs associated with each of these decision variables, we have a linear objective function representing the total relative cost for each control policy. We have also considered upper bound constraints for maximum number of individuals under treatment (which is related to surge capacity) and percentage of infected individuals (which determines the attack rate). We have used grid search to obtain insight into the model, find the feasible region, and conduct the cost-effectiveness analysis.

Identifying optimal mitigation strategies for responding to a mild influenza epidemic

Mathematical models have been developed to simulate influenza epidemics to help public health officials evaluate different control policies. In these models, often severe influenza epidemics with a considerable mortality rate are considered. However, as was the case for the 2009 H1N1 pandemic, some of the influenza epidemics are mild with insignificant mortality rates. In the case of a mild epidemic, the cost of different control policies becomes an important decision factor in addition to disease-related outcomes such as the attack rate. We develop a continuous-time simulation model for the spread of a mild influenza epidemic based on the SEIR model (an epidemiological model with four classes: susceptible, exposed, infective, and recovered) which includes different interventions. To determine the epidemic mitigation policy with the minimum cost, we also develop an optimization model with two decision variables, vaccination and self-isolation fractions, and an upper-bound constraint for the attack rate. We use this model to evaluate the cost-effectiveness of different mitigation policies. Furthermore, we integrate the simulation and optimization models to identify the optimal mitigation policy. Finally, we conduct sensitivity analysis on the key input parameters to ensure robust results. The optimal policy depends on the target population and, as our results show, in general is a combination of vaccination and self-isolation. Further, for low (high) levels of intervention, vaccination (self-isolation) is incrementally more cost-effective. Therefore, public health officials should concentrate on vaccination at the beginning of the epidemic. However, if the epidemic continues to spread, they should promote self-isolation as a more effective intervention.

Analytic solution of the susceptible-infective epidemic model with state-dependent contact rates and different intervention policies

We consider the susceptible-infective (SI) epidemiological model, a variant of the Kermack-McKendrick models, and let the contact rate be a function of the number of infectives, an indicator of disease spread during the course of the epidemic. We represent the resultant model as a continuous-time Markov chain. The result is a pure death (or birth) process with state-dependent rates, for which we find the probability distribution of the associated Markov chain by solving the Kolmogorov forward equations. This model is used to find the analytic solution of the SI model as well as the distribution of the epidemic duration. We use the maximum likelihood method to estimate contact rates based on observations of inter-infection time intervals. We compare the stochastic model to the corresponding deterministic models through a numerical experiment within a typical household. We also incorporate different intervention policies for vaccination, antiviral prophylaxis, isolation, and treatment considering both full and partial adherence to interventions among individuals.

Optimal intervention strategies for an epidemic: A household view

In this research, we identify optimal intervention strategies at the household level in case of an epidemic. We consider an affected household (a household with one initial infective member) and model the effect of different intervention policies, which involve vaccination, antiviral prophylaxis, isolation, and treatment, on disease spread using a variation of Kermack–McKendrick (KM) models. Both full and partial adherence to interventions are considered. An implementation cost is assumed for each intervention policy. We refer to a collection of intervention policies as an intervention strategy. A reward is associated with susceptible members who remain uninfected. We define the effect of the implemented intervention strategy as the total reward earned by all members over the time horizon. We then identify the most cost-effective intervention strategies. In addition, we incorporate a budgetary constraint for the household and find the efficient frontier for the total reward over different upper bounds on the household budget.

SU-F-BRD-08: Guaranteed Epsilon-Optimal Treatment Plans with Minimum Number of Beams for SBRT Using RayStation.

PURPOSE To efficiently find quality-guaranteed treatment plans with the minimum number of beams for stereotactic body radiation therapy using RayStation. METHODS For a pre-specified pool of candidate beams we use RayStation (a treatment planning software for clinical use) to identify the deliverable plan which uses all the beams with the minimum dose to organs at risk (OARs) and dose to the tumor and other structures in specified ranges. Then use the dose matrix information for the generated apertures from RayStation to solve a linear program to find the ideal plan with the same objective and constraints allowing use of all beams. Finally we solve a mixed integer programming formulation of the beam angle optimization problem (BAO) with the objective of minimizing the number of beams while remaining in a predetermined epsilon-optimality of the ideal plan with respect to the dose to OARs. Since the treatment plan optimization is a multicriteria optimization problem, the planner can exploit the multicriteria optimization capability of RayStation to navigate the ideal dose distribution Pareto surface and select a plan of desired target coverage versus OARs sparing, and then use the proposed technique to reduce the number of beams while guaranteeing quality. For the numerical experiments two liver cases and one lung case with 33 non-coplanar beams are considered. RESULTS The ideal plan uses an impractically large number of beams. The proposed technique reduces the number of beams to the range of practical application (5 to 9 beams) while remaining in the epsilon-optimal range of 1% to 5% optimality gap. CONCLUSION The proposed method can be integrated into a general algorithm for fast navigation of the ideal dose distribution Pareto surface and finding the treatment plan with the minimum number of beams, which corresponds to the delivery time, in epsilon-optimality range of the desired ideal plan. The project was supported by the Federal Share of program income earned by Massachusetts General Hospital on C06 CA059267, Proton Therapy Research and Treatment Center and partially by RaySearch Laboratories.

Sensitivity analysis of Matching Pennies game

In this paper, we have discussed the results of sensitivity analysis in a payoff matrix of the Matching Pennies game. After representing the game as a LP model, the sensitivity analysis of the elements of the payoff matrix is presented. The game value and the optimal strategies for different values of parameters are determined and compared.