Disease Propagation Analysis and Mitigation Strategies for Effective Mass Dispensing

Eva K Lee; Chien H Chen; Ferninand Pietz; Bernard Benecke

. 2010 Nov 13;2010:427–431.

Disease Propagation Analysis and Mitigation Strategies for Effective Mass Dispensing^*

Eva K Lee ^1,^2,^3,^#, Chien H Chen ^1,^2,³, Ferninand Pietz ⁴, Bernard Benecke ⁵

PMCID: PMC3041344 PMID: 21347014

Abstract

Mass dispensing of medical countermeasures has been proven to be an effective and crucial means to contain the outbreak of highly infectious disease. The large influx of individuals to the point-of dispensing (POD) centers to receive vaccinations or prophylactic treatment, however, raises the potential risk of serious intra-facility cross-infections. To mitigate the effect, a thorough understanding of how disease propagates during the dispensing under different transmission parameters versus POD design and operational factors is necessary. In this study, we employ a large-scale simulation/optimization decision support system, RealOpt, to analyze the propagation of highly infectious disease within dispensing sites. The simulation results are validated and benchmarked by a mathematical model based on ordinary differential equations. Pros and cons of using analytical versus simulation tools are discussed. We further perform sensitivity analysis on the dynamics of intra-POD disease propagation, and explore feasible mitigation strategies for effective mass dispensing.

1. Introduction

During outbreaks of public health emergencies such as biological attacks or naturally occurring pandemics, it is crucial to launch immediate and aggressive responses of medical countermeasures dispensing¹^,² to reduce potential mass casualties and speedup epidemic eradication.

For highly infectious diseases, research has shown that avoiding possible person-to-person contact is crucial to reduce the infection³^,⁴^,⁵^,⁶ Large-scale dispensing of medical countermeasures (e.g., flu vaccination) creates a large influx of individuals to dispensing centers (point of dispensing facilities, or PODs), which raises the risk of high degree of intra-facility infections. Hence, proper design of dispensing centers, and understanding of operational factors that can mitigate infection are critical.

While there exist many models that study regional disease spread³^,⁴^,⁶^,⁷^,⁸, intra-facility disease propagation and mitigation strategies within clinics remain a critical challenge. In this study, we focus on a local outbreak of a highly infectious disease (e.g., smallpox, pneumonic plague, pandemic flu), analyze intra-facility disease spread, and identify effective mitigation strategies through POD layout design and operational strategies.

We analyze the interaction between the propagation of disease and the operations of medical countermeasure dispensing as an open queueing network. We interlace large-scale optimization and simulation technologies to optimize POD operations. Critical aspects of operations include maximizing throughput, optimal resource allocation, cycle time, queue length, and wait time. Combining these aspects into an integrated simulation/optimization framework opens up the opportunity to incorporate realistic dynamics into intra-facility disease propagation analysis. In addition, a rapid solution time allows for scenario-based analysis which empowers public health and emergency POD designers and epidemiologists to mount dynamic response in real-time to the changing conditions during the dispensing process.

2. Mathematical Model

In this section we present an ordinary differential equations-based (ODE) disease propagation model⁹^,¹⁰. Unlike the traditional SEIR model, we use the novel SEPAIR 6-stage model⁵^,¹¹ to capture more details in the disease development. As shown in Figure 1, two additional disease states, asymptomatic and symptomatic, are considered. People in infectious, asymptomatic, or symptomatic stages are capable of spreading disease. The 6-stage propagation model provides more opportunities to examine the interaction between POD layout design and disease propagation. For example, one can gain a better understanding of triage accuracy on the degree of disease spread.

6-stage disease progression model. S: susceptible; E: exposed; P: infectious; A: asymptomatic; I: symptomatic; R: recovered; V: vaccinated. Individuals in infectious, asymptomatic, and symptomatic stages are capable of spreading diseases. Only susceptible individuals are potentially exposed to disease. People in recovered stage or vaccinated stage are immune. D denotes those who die.

Two forces of infection are studied in the literature: standard incidence and simple mass action incidence¹²^,¹³^,¹⁴. In standard incidence, the infection rate (contact rate) β, defines the average number of transmissions per (infectious, asymptomatic, or symptomatic) person per unit time. In simple mass action incidence, the transmission coefficient (mass action coefficient), δ, defines average number of transmissions per contact per unit time. In this study, we further incorporate outer-POD contact number, γ, to model the average number of people an infectious/asymptomatic/symptomatic individual can contact outside POD. Note that a contact is defined as effective if it can contribute to disease transmission, and hence contacts are not necessarily person-to-person under this definition. In other words, our model can be applied to diseases transmitted via droplet contact transmission or airborne transmission.

The force of infection should be weakly dependent on or essentially independent of population size when modeling the disease spreading across an open geographical region⁹ where the population density is roughly identical. However, the simple mass action incidence, which reflects the force of infection being fully dependent on the population size, will be validated to be necessary for modeling intra-POD disease propagation.

For clarification, the population size for intra-POD disease propagation refers to the number of people in a given POD facility. This includes those who are being served and those who are waiting in the queue. The workers, who have received prophylactic medication prior, are not included.

Due to the heterogeneous system behavior, we separate the ODE model into two parts: intra-POD disease propagation and outer-POD disease propagation.

Outer-POD disease propagation

Let N₀ denote the number of living individuals. Specifically, we define the disease state space Φ = {S, E, P, A, I, R, D, V}, and outer-POD population size $N_{0} = \sum_{ϑ \in Φ \ {D}} ϑ_{0}$ , where ϑ₀ is the number of outer-POD individuals at disease stage ϑ. The constant p_S denotes the probability of symptoms development, and p_D represents the mortality probability. μ_E is the rate from exposed stage to infectious stage, μ_P is the rate from infectious stage to (a)symptomatic stage, μ_I (μ_A) is the rate from (a)symptomatic stage to recovered stage, and r_a represents the constant patient arrival rate. The following equations represent the rates of change of populations outside the POD:

\begin{array}{l} \frac{d}{dt} S_{0} = - r_{a} \frac{S_{0}}{N_{0}} - min {N_{0}, γ} \cdot δ \cdot (P_{0} + A_{0} + I_{0}) \frac{S_{0}}{N_{0}} \\ \frac{d}{dt} E_{0} = - r_{a} \frac{E_{0}}{N_{0}} - μ_{E} E_{0} + min {N_{0}, γ} \cdot δ \cdot (P_{0} + A_{0} + I_{0}) \frac{S_{0}}{N_{0}} \\ \frac{d}{dt} P_{0} = - r_{a} \frac{P_{0}}{N_{0}} - μ_{P} P_{0} + μ_{E} E_{0} \\ \frac{d}{dt} A_{0} = - r_{a} \frac{A_{0}}{N_{0}} - μ_{A} A_{0} + (1 - p_{S}) μ_{P} P_{0} \\ \frac{d}{dt} I_{0} = - r_{a} \frac{I_{0}}{N_{0}} - μ_{I} I_{0} + p_{S} μ_{P} P_{0} \\ \frac{d}{dt} R_{0} = - r_{a} \frac{R_{0}}{N_{0}} + (1 - p_{D}) (μ_{A} A_{0} + μ_{I} I_{0}) \\ \frac{d}{dt} D_{0} = p_{D} (μ_{A} A_{0} + μ_{I} I_{0}) \end{array}

Age, social groups and various parameters can be incorporated in this model³^,⁴^,⁵^,⁶^,⁷^,⁸^,⁹^,¹⁰.

Intra-POD disease propagation

We introduce the intra-POD model for this study. For intra-POD disease propagation, assume there are K stations in the POD. Let subscript “i” denote the different stations, each station is manned by n_i workers and each worker can provide same service at nominal service rate λ_i. The population size at each station $N_{i} = \sum_{ϑ \in Φ \ {D}} ϑ_{i}$ . In addition, we define the function $h_{i} (ϑ) = λ_{i} min {n_{i}, N_{i}} \frac{ϑ_{i}}{N_{i}}$ , where ϑ ∈ Φ \ {D}, i={1, 2, ….., K}. Physically, h_i (ϑ) can be interpreted as the real service rate for individuals at disease stage ϑ at station i. The following equations represent the rates of change of populations inside the POD:

\begin{array}{l} \frac{d}{dt} S_{i} = ψ_{i} r_{a} \frac{S_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} (1 - φ_{ji}) q_{ji} h_{j} (S) - h_{i} (S) - δ \cdot (P_{i} + A_{i} + I_{i}) \cdot S_{i} \\ \frac{d}{dt} E_{i} = ψ_{i} r_{a} \frac{E_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} q_{ji} h_{j} (E) - h_{i} (E) - μ_{E} E_{i} + δ \cdot (P_{i} + A_{i} + I_{i}) \cdot S_{i} \\ \frac{d}{dt} P_{i} = ψ_{i} r_{a} \frac{P_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} q_{ji} h_{j} (P) - h_{i} (P) - μ_{P} P_{i} + μ_{E} E_{i} \\ \frac{d}{dt} A_{i} = ψ_{i} r_{a} \frac{A_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} q_{ji} h_{j} (A) - h_{i} (A) - μ_{A} A_{i} + (1 - p_{S}) μ_{P} P_{i} \\ \frac{d}{dt} I_{i} = ψ_{i} r_{a} \frac{I_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} (1 - η_{ji}) q_{ji} h_{j} (I) - h_{i} (I) - μ_{I} I_{i} + p_{S} μ_{P} P_{i} \\ \frac{d}{dt} R_{i} = ψ_{i} r_{a} \frac{R_{0}}{N_{0}} + \sum_{j \in {1, 2, \dots K}} q_{ji} h_{j} (R) - h_{i} (R) + (1 - P_{D}) (μ_{A} A_{i} + μ_{I} I_{i}) \\ \frac{d}{dt} D_{i} = p_{D} (μ_{A} A_{i} + μ_{I} I_{i}) \\ \frac{d}{dt} V_{i} = \sum_{j \in {1, 2, \dots K}} φ_{ji} q_{ji} h_{j} (S) + \sum_{j \in {1, 2, \dots K}} q_{ji} h_{j} (V) - h_{i} (V) \end{array}

Three binary constants are used to represent the operational flow inside the POD: ψ_i =1 means station i is the first station and 0 otherwise. φ_ji =1 means station j is a dispensing/vaccination station and station i is its direct downstream station and 0 otherwise. q_ji is used to model the transition percentage from station j to station i. Lastly, η_ji denotes the triage accuracy from station j to station i, given station j is a triage station. Here, η_ji =1 means perfect inspection rate.

3. Large-scale Simulation-Optimization System

Working with researchers at Centers for Disease Control and Prevention, we have designed a real-time large-scale simulator and decision support system for mass dispensing, RealOpt¹⁵^,¹⁶^,¹⁷^,¹⁸^,¹⁹. The system couples powerful backend simulation and optimization engines with a simple yet elegant front-end graph-drawing tool and graphical user interface. It allows for rapid simulation of dispensing processes, and optimization of staffing. Critical objectives such as maximize throughput, minimize resources, minimize maximum wait-time, minimize queue length (crowd control), equalize utilization (maintain workers’ morale), minimize cycle time (minimize anxiety, and potential social disturbance), and worker fatigue are incorporated. Emergency directors can analyze POD design, perform large-scale virtual drills, perform what-if-scenario analysis, and assess local resource capability and determine gaps. RealOpt has been validated to be a powerful tool in planning large-scale emergency dispensing operations in response to biological threats or infectious-disease outbreaks¹⁵^,¹⁶^,¹⁷^,¹⁸^,¹⁹. For a regional population with hundreds of thousands or millions, it can simulate and optimize POD operations and staff allocation in seconds. Thus, it can be used for strategic and operational planning as well as on-the-fly dynamic changes. Figure 2 shows the modeling and computational schema of the RealOpt system.

Figure 2. — RealOpt modeling and computation schema

Incorporating the disease propagation module into the RealOpt system allows us to relax some of the assumptions in the classical ODE model and provides further insights into the intra-POD disease transmission. Furthermore, the simulation-based disease propagation module enables sensitivity analysis and scenario-based analysis that can assist in testing alternative POD designs and analyzing the tradeoffs between infection risks and operational efficiencies. As shown in Figure 2, the disease propagation module is embedded inside the simulation module to incorporate stochastic system behaviors into consideration.

The disease propagation module in RealOpt incorporates the simple mass action incidence infection force to provide accurate estimates under general POD designs. This assumption reflects clinical observation of interdependency of intra-facility infection and batch sizes at certain process stations. (Batch size refers to the number of people who are processed at a station simultaneously. For example, at orientation people may all view a video.) The design of RealOpt allows for tracking of each single contact during the simulation process, thus avoiding the difficulty in estimating intra-POD contact number, which usually varies between different stations and is highly affected by the service type, e.g., individual service versus group service.

4. Results and Discussions

Users can select POD layouts, operations parameters, infectivity parameters, and simulation duration time for specific scenario-based analysis. In this paper, we report the empirical results of disease propagation in a flu-vaccination clinic. We ran simulations covering three 12-hour shifts, and analyzed the results over the entire 36-hour period. We then made modifications to POD operation parameters or disease propagation parameters, and re-ran the 36-hour simulations. This process was repeated multiple times to gain an understanding of the interplay between intra-POD infectivity and POD layout and operational design.

POD Layout and Operations

Upon arrival, individuals are examined in the Triage station, and those with symptoms detected are sent away for medical treatment. Individuals who are eligible for prophylactic treatment are sent to Waiting/Orientation station in batches of 40 where they learn of the disease and treatment procedure. After orientation, half of the individuals are assumed to require form filling assistance, while the rest fill out forms by themselves. In addition, 50% of individuals are assumed to require further medical screening (while the rest go to the Quality Assurance station directly). Among those requiring further screening, 10% are assumed to have alternate medication needs and are sent to other medical facilities, the rest proceed to Quality Assurance Station. After getting vaccinated in Vaccination station, individuals receive final tracking before they exit the system. Users can change all these input parameters for analysis.

Sensitivity Analysis

Time-motion studies that we have performed at various mass vaccination sites have provided data for proper estimation of POD operation parameters. Unfortunately, parameters for disease propagation are usually more difficult to estimate accurately. For example, the accurate estimate of contact number or transmission coefficient is difficult to obtain when the flu strain is new. In addition, to contain the outbreak in time, the initial infectious percentage can only be roughly estimated if the medication dispensing needs to be launched urgently before complete information is gathered. In the sensitivity analysis we vary these parameters to obtain comprehensive understandings of dynamics and magnitudes to compensate for the potential lack of accurate estimates. This also allows us to determine alternative mitigation strategies under different possible scenarios.

We vary the triage accuracy, batch size, and required throughput to derive potential mitigation strategies and test their effectiveness under different scenarios. The results are summarized in Figure 4. From Figure 4, we confirm the significance of implementing accurate triage operation in reducing intra-POD transmission. Under the given parameters, the overall reduction is at least 60% when disease infectivity is high (initial infectious % = 10%, symptomatic proportion = 100%, and transmission coefficient = 1e⁻⁵/min). In addition, we observe that the number of intra-POD infections is relatively insensitive to the batch size (from 40–100). This result allows emergency planners the flexibility to choose the patient flow or patient transportation type that can utilize large batch size to increase operational efficiency and throughput. Lastly, the results reveal that decentralization is in general a better strategy for mitigating disease propagation (i.e., having more facilities with relatively smaller throughput rates).

Figure 4. — Intra-POD infectivity (within 36 hours) with respect to triage effectiveness, batch size, throughput rates.

Conclusion

The intra-POD disease propagation and mitigation strategies within clinics remain a critical challenge. As researchers attempt to incorporate more realistic dynamics into their models (such as non-closed form objectives, non-exponential waiting times, sample-path dependent events, demographical and geographical data, etc), flexible tools such as individual-based stochastic simulations are preferred over classical ODE approaches. However, simulation is much less mathematically tractable. Our study advances the computational power by producing a large-scale simulator/optimizer with rapid solution engines that allow for in-depth investigation of disease propagation and perform scenario-based mitigation strategies. The system has been used by thousands of public health emergency planners to develop dispensing facilities and operations logistics for response to infectious disease outbreaks and biological attacks. The fast engine also helps to derive dynamic responses on-the-fly to mitigate disease propagation, and thus has the potential to reduce casualties.

Figure 3. — A flu dispensing center layout.

Acknowledgments

We acknowledge funding from CDC to conduct the time-motion study and post-event operations analysis, and from the National Institutes of Health for translational biomedical informatics advances.

References

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18.Lee EK, Smalley HK, Zhang Y, Pietz F, Benecke B. Facility Location and Multi-modality Mass Dispensing Strategies and Emergency Response for Biodefense and Infectious Disease Outbreaks. International Journal on Risk Assessment and Management -- Biosecurity Assurance in a Threatening World: Challenges, Explorations, and Breakthroughs. 2009a;12(2/3/4):311–351. [Google Scholar]
19.Lee EK, Chen CH, Pietz F, Benecke B. Modeling and Optimizing the Public Health Infrastructure for Emergency Response. Interfaces -- 2008 Wagner Prize Special Issue. 2009b;39:476–490. [Google Scholar]

[b1-amia-2010_sympproc_0427] 1.Kaplan EH, Craft DL, Wein LM. Emergency response to a smallpox attack: The case for mass vaccination. PNAS. 2002;99(16):10935–10940. doi: 10.1073/pnas.162282799. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b2-amia-2010_sympproc_0427] 2.Wein LM, Craft DL, Kaplan EH. Emergency response to an anthrax attack. PNAS. 2003;100(7):4346–4351. doi: 10.1073/pnas.0636861100. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b3-amia-2010_sympproc_0427] 3.Ferguson NM, Cummings DA, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirthaworn S, Burke DS. Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature. 2005;437:209–214. doi: 10.1038/nature04017. [DOI] [PubMed] [Google Scholar]

[b4-amia-2010_sympproc_0427] 4.Ferguson NM, Cummings DA, Fraser C, Cajka JC, Cooley PC, Burke DS. Strategies for mitigating an influenza pandemic. Nature. 2006;442:448–452. doi: 10.1038/nature04795. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b5-amia-2010_sympproc_0427] 5.Wu JT, Riley S, Fraser C, Leung GM. Reducing the impact of the next influenza pandemic using household-based public health interventions. PLoS Medicine. 2006;3(9):1532–1540. doi: 10.1371/journal.pmed.0030361. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b6-amia-2010_sympproc_0427] 6.Fraser C, Riley S, Anderson RM, Ferguson NM. Factors that make an infectious disease outbreak controllable. PNAS. 2004;101(16):6146–6151. doi: 10.1073/pnas.0307506101. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b7-amia-2010_sympproc_0427] 7.Germann TC, Kadau K, Longini IM, Jr, Macken CA. Mitigation strategies for pandemic influenza in the United States. PNAS. 2006;103:5935–5940. doi: 10.1073/pnas.0601266103. [DOI] [PMC free article] [PubMed] [Google Scholar]

[b8-amia-2010_sympproc_0427] 8.Das TK, Savachkin AA, Zhu Y. A large-scale simulation model of pandemic influenza outbreaks for development of dynamic mitigation strategies. IIE Transactions. 2008;40:893–905. [Google Scholar]

[b9-amia-2010_sympproc_0427] 9.Anderson RM, May RM, Anderson B. Infectious diseases of humans: dynamics and control. Oxford University Press; 1992. [Google Scholar]

[b10-amia-2010_sympproc_0427] 10.Diekmann O, Heesterbeek JAP. Mathematical epidemiology of infectious disease: model building, analysis and interpretation. John Wiley and Sons; 2000. [Google Scholar]

[b11-amia-2010_sympproc_0427] 11.Longini IM, Jr, Halloran ME, Nizam A, Yang Y. Containing pandemic influenza with antiviral agents. American Journal of Epidemiology. 2004;159(7):623–633. doi: 10.1093/aje/kwh092. [DOI] [PubMed] [Google Scholar]

[b12-amia-2010_sympproc_0427] 12.Kermack WO, McKendrick AG. Contributions to the mathematical theory of epidemics—I. Proceedings of the Royal Society. 1927;115A:700–721. [Google Scholar]

[b13-amia-2010_sympproc_0427] 13.Anderson RM. Discussion: the Kermack-McKendrick epidemic threshold theorem. Bulletin of mathematical biology. 1991;53(1–2):3–32. doi: 10.1016/s0092-8240(05)80039-4. [DOI] [PubMed] [Google Scholar]

[b14-amia-2010_sympproc_0427] 14.Hethcote HW. The Mathematics of Infectious Diseases. SIAM Review. 2000;42(4):599–653. [Google Scholar]

[b15-amia-2010_sympproc_0427] 15.Lee EK, Maheshwary S, Mason J. Real-Time staff allocation for emergency treatment response of biologic threats and infectious disease outbreak; INFORMS William Pierskalla Best Paper Award on research excellence in HealthCare and Management Science; Nov, 2005. [Google Scholar]

[b16-amia-2010_sympproc_0427] 16.Lee EK, Maheshwary S, Mason J, Glisson W. Large-scale dispensing for emergency response to bioterrorism and infectious-disease outbreak. Interfaces. 2006a;36(6):591–607. [Google Scholar]

[b17-amia-2010_sympproc_0427] 17.Lee EK, Maheshwary S, Mason J, Glisson W. Decision support system for mass dispensing of medications for infectious disease outbreaks and bioterrorist attacks. Annals of Operations Research. 2006b;148(1):25–53. [Google Scholar]

[b18-amia-2010_sympproc_0427] 18.Lee EK, Smalley HK, Zhang Y, Pietz F, Benecke B. Facility Location and Multi-modality Mass Dispensing Strategies and Emergency Response for Biodefense and Infectious Disease Outbreaks. International Journal on Risk Assessment and Management -- Biosecurity Assurance in a Threatening World: Challenges, Explorations, and Breakthroughs. 2009a;12(2/3/4):311–351. [Google Scholar]

[b19-amia-2010_sympproc_0427] 19.Lee EK, Chen CH, Pietz F, Benecke B. Modeling and Optimizing the Public Health Infrastructure for Emergency Response. Interfaces -- 2008 Wagner Prize Special Issue. 2009b;39:476–490. [Google Scholar]

PERMALINK

Disease Propagation Analysis and Mitigation Strategies for Effective Mass Dispensing^*

Eva K Lee, PhD

Chien H Chen, MS

Ferninand Pietz, MS

Bernard Benecke, MPH

Abstract

1. Introduction

2. Mathematical Model