Finding optimal vaccination strategies for pandemic influenza using genetic algorithms
Introduction
Influenza is a major public health concern. Influenza spreads rapidly in seasonal epidemics which cost society a considerable amount in terms of health care expenses, lost productivity, and loss of life. Globally, influenza annually costs from $71 and $167 billion and results in 250 000 and 500 000 deaths (World Health Organization, 2004). Annual influenza epidemics occur partially due to strains of influenza genetically drifting from year to year. The influenza vaccine produced before the influenza season is targeted against the strains that are predicted to circulate in the coming season. However, a major antigenic shift can occur with little warning, resulting in pandemic influenza (Kilbourne, 1975). The two most recent pandemics were the “Asian Flu” A (H2N2) of 1957–1958 (Elveback et al., 1976, Jordan, 1961, Longini et al., 1978) and “Hong Kong Flu” A (H3N2) of 1968–1969 (Davis et al., 1970, Elveback et al., 1976, Longini et al., 1978, Sharrar, 1969). The Asian Flu is estimated to have caused 70 000 deaths in the US, while the Hong Kong Flu is estimated to have caused 34 000 deaths in the US. In addition, the Hong Kong pandemic of 1968–1969 is estimated to have cost society 3.8 billion dollars in the US alone (Kavet, 1972). Should a major antigenic shift occur, there may be time to produce only a limited amount of vaccine efficacious against the new strain and ensuing pandemic. Knowledge on how to distribute the limited supply of vaccine optimally among different age groups would help us to minimize the impact of the epidemic. This impact can be measured in many ways, two of which are number of illnesses and loss of life.
Optimization methods have been developed for deterministic simulation models (Anderson and May, 1991, Greenhalgh, 1986, Hethcote and Waltman, 1973, Wickwire and Guest, 1976), and optimization studies have been done to find the vaccine distributions for influenza using a simple deterministic model (Longini et al., 1978). However, they have not been done with more complex stochastic simulation models. Influenza is transmitted in a complex way from person to person. In addition, given an introduction of influenza into a population, the probability of a major epidemic and the possible size of an epidemic are highly variable. Thus, the mathematical models for influenza epidemics should have a detailed contact structure and be stochastic. In addition, the epidemic process is non-linear since the incidence of new infections depends on the current number of both infectives and susceptibles in the population at a particular time. All of these factors make optimization based on traditional gradient methods, such as the Newton–Raphson method, difficult or even prohibitive. Robbins and Munro (1951) developed a stochastic approximation method whose convergence is guaranteed under mild conditions. The method, however, requires knowledge of the analytic gradient of the considered objective function (Weisstein, 2004). Kiefer and Wolfowitz (1952) developed an extension to the Robbins–Munro algorithm. However, in terms of simulation optimization, the drawback to both of these methods remains the unavailability of gradients. In the case of our stochastic simulation multi-dimensional optimization problem, we consider genetic algorithms (GA) and random mutation hill climbing (RMHC) as stochastic optimization methods.
In this paper, we find optimal distributions of a limited supply of vaccine in the event of pandemic influenza generated by a stochastic simulation model using GA (Holland, 1975) as well as RMHC (Forrest and Mitchell, 1993). We configure the model to simulate the baseline illness attack rates consistent with the past patterns of Asian and Hong Kong pandemic influenza. We find optimal vaccine distributions in terms of minimizing influenza illness or death. These optimization methods are generally applicable to other infectious diseases and population structures.
Section snippets
The simulation model
We use a discrete time, stochastic simulation model of influenza spread within a structured population of 10 000 individuals to estimate the effectiveness of various distributions of a given amount of vaccine. This model is a direct extension of an earlier model developed for influenza intervention studies (Halloran et al., 2002a, Longini et al., 2004). The model simulates the stochastic spread of influenza in a population where the age and family structure approximate information from the US
Initialization
- 1.
Randomly generate an initial set of 50 individuals. We generate the initial pool of individuals such that (3) is randomly determined for each individual under the condition that (2) is satisfied at equality.
Iteration
- 2.
Evaluate each of the 50 individuals in the pool by pre-vaccinating the proportion of the population according to the individual being evaluated, then running the stochastic simulator 20 times. The resulting fitness of each individual is the mean ofover the 20 runs of the simulator. A smaller value of this fitness function represents a more fit individual, as this fitness function is essentially a loss function.
- 3.
Sort the 50 individuals according to their fitness.
- 4.
We select the best 25
Convergence
- 6.
We repeat this process from Step 2 to Step 5 until the best individual does not change for four consecutive generations. The best individual in the final generation will yield the optimal vaccine distribution given the quantity of available influenza vaccine, the particular population structure, the illness attack rate pattern, and the objective of the optimization.
Initialization
- 1.
Generate a random individual as defined in the GA approach which satisfies (2) at equality.
- 2.
Evaluate by pre-vaccinating the proportion of the population according to running the stochastic simulator 20 times. The resulting fitness of is the mean ofover the 20 runs of the simulator.
Iteration
- 3.
Randomly mutate by first creating a copy of , , and mutating in the following manner:
(a) Select one age group, i, at random, and set where U is randomly chosen from a Uniform distribution. A Gaussian distribution is commonly used to mutate continuous genes, however we noticed no difference in the quality of resulting individuals and running time of the algorithm using the uniform distribution. The advantage of using a uniform distribution in this case is
Convergence
- 6.
We continue this process until 100 consecutive mutations of do not reveal a more fit individual than .
Optimal vaccine distributions
Table 2, Table 3, Table 4, Table 5 show the results given by the GA. The results from RMHC are similar to those from the GA, and are thus not in the tables. For example, the RMHC result for the Asian-like flu case when minimizing illness attack rate and is . We do not expect exact results from each algorithm as the results are approximations of the optimal distributions. The two algorithms converging to significantly different distributions of the vaccine
Discussion
The optimal vaccine distribution is sensitive to the nature of the spread of the influenza agent, the objective for control, and also the amount of vaccine available. GA, as well as RMHC, in conjunction with a properly calibrated simulation model would allow us to obtain an approximation of the optimal vaccine distribution given that we understand the behavior of the next pandemic agent and that we determine an objective for the control of the agent.
It is of interest to compare the optimal
Acknowledgment
This work was partially supported by grant R01-AI32042 from the National Institute of Allergy and Infectious Diseases.
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