As recently pointed out by the Institute of Medicine, the existing pandemic mitigation models lack the dynamic decision support capability. We develop a large-scale simulation-driven optimization model for generating dynamic predictive distribution of vaccines and antivirals over a network of regional pandemic outbreaks. The model incorporates measures of morbidity, mortality, and social distancing, translated into the cost of lost productivity and medical expenses. The performance of the strategy is compared to that of the reactive myopic policy, using a sample outbreak in Fla, USA, with an affected population of over four millions. The comparison is implemented at different levels of vaccine and antiviral availability and administration capacity. Sensitivity analysis is performed to assess the impact of variability of some critical factors on policy performance. The model is intended to support public health policy making for effective distribution of limited mitigation resources.

As of July 2010, WHO has reported 501 confirmed human cases of avian influenza A/(H5N1) which resulted in 287 deaths worldwide [

The nation's ability to mitigate a pandemic influenza depends on the available emergency response resources and infrastructure, and, at present, challenges abound. Predicting the exact virus subtype remains a difficult task, and even when identified, reaching an adequate vaccine supply can currently take up to nine months [

The existing models on pandemic influenza (PI) containment and mitigation aims to address various complex aspects of the pandemic evolution process including: (i) the mechanism of disease progression, from the initial contact and infection transmission to the asymptomatic phase, manifestation of symptoms, and the final health outcome [

Recently, the modeling efforts have focused on combining pharmaceutical and nonpharmaceutical interventions in search for synergistic strategies, aimed at better resource utilization. Most of such approaches attempt implementing a form of social distancing followed by application of pharmaceutical measures. For significant contributions in this area see [

At the same time, The IOM report [

In an attempt to address the IOM recommendations, we present a simulation optimization model for developing predictive resource distribution over a network of regional outbreaks. The underlying simulation model mimics the disease and population dynamics of each of the affected regions (Sections

The objective of our methodology is to generate a progressive allocation of the total resource availability over a network of regional outbreaks. The methodology incorporates (i) a cross-regional simulation model, (ii) a set of single-region simulation models, and (iii) an embedded optimization model.

We consider a network of regions with each of which classified as either unaffected, ongoing outbreak, or contained outbreak (Figure

Schematic of cross-regional pandemic spread and resource distribution.

In Sections

A schematic of the cross-regional simulation model is shown in Figure

Schematic of cross-regional simulation model.

The single-region model subsumes the following components (see Figure

Schematic of single-region simulation model.

Each region is modeled as a set of population centers formed by

It is assumed that at any point of time, an individual belongs to one of the following compartments (see Figure

Schematic of disease natural history.

Infection transmission occurs during contact events between susceptible and infectious cases, which take place in the mixing groups. At the beginning of every

Let

A schematic of the disease natural history is shown in Figure

Schematic of disease natural history model.

Mortality for influenza-like diseases is a complex process affected by many factors and variables, most of which have limited accurate data support available from past pandemics. Furthermore, the time of death can sometimes be weeks following the disease episode (which is often attributable to pneumonia-related complications [

Mitigation options include pharmaceutical and nonpharmaceutical interventions. Mitigation is initiated upon detection of a critical number of confirmed infected cases [

Both vaccination and antiviral application are affected by a number of sociobehavioral factors, including conformance of the target population, degree of risk perception, and compliance of healthcare personnel [

As presented in Figure

We introduce the following

: Set of all network regions

: Set of regions in which pandemic is contained at the

: Set of ongoing regions at the

: Set of unaffected regions at the

: Set of available types of mitigation resources (

: Amount of resource

: Available amount of resource

: Set of age groups.

: Total number of infected cases in age group

: Total number of infected cases in age group

: Total number of deceased cases in age group

: Total number of person-days of cases in age group

To estimate these measures, we use the following regression models obtained using a single-region simulation of each region

The above relationships between the impact measures and the resource distributions ought to be determined

In addition, we use the following regression model to estimate the

Finally, we calculate the total cost of an outbreak in region

The optimization model has the following form.

The solution algorithm for our dynamic predictive simulation optimization (DPO) model is given below.

Estimate regression equations for each region using the single-region simulation model.

Begin the cross-regional simulation model.

Initialize the sets of regions:

Select randomly the initial outbreak region

Update sets of regions:

Solve the resource distribution model for region

If

For each ongoing region, implement a next day run of its single-region simulation.

Check the containment status of each ongoing region. Update sets

For each unaffected region, calculate its outbreak probability.

Based on the outbreak probability values, determine if there is a new outbreak region(s)

For each new outbreak region

Increment

Update sets

Re-estimate regression equations for each region

Solve the resource distribution model for region

Update the total resource availabilities.

Calculate the total cost for each contained region and update the overall pandemic cost.

To illustrate the use of our methodology, we present a sample H5N1 outbreak scenario including four counties in Fla, USA: Hillsborough, Miami Dade, Duval, and Leon, with populations of 1.0, 2.2, 0.8, and 0.25 million people, respectively. A basic unit of time for population and disease dynamics models was taken to be

Demographic and social dynamics data for each region [

Each infected person was assigned a daily travel probability of 0.24% [

Interregional travel probabilities.

Origin | Interregional Travel Probability | |||

Hillsborough | Miami D. | Duval | Leon | |

Hillsborough | 0.00 | 0.60 | 0.27 | 0.13 |

Miami D. | 0.74 | 0.00 | 0.16 | 0.10 |

Duval | 0.61 | 0.29 | 0.00 | 0.10 |

Leon | 0.52 | 0.31 | 0.17 | 0.00 |

The instantaneous force of infection applied to contact

The values of age-dependent base instantaneous infection probabilities were adopted from [

Instantaneous infection probabilities.

Age group | 0–5 | 6–19 | 20–29 | 31–65 | 66–99 |
---|---|---|---|---|---|

0.156 | 0.106 | 0.205 | 0.195 | 0.344 |

Base mortality probabilities (

Mortality probabilities for different age groups.

Age group | % HRC | % Mortality in HRC | |
---|---|---|---|

0–19 | 6.4 | 9.0 | 0.007 |

20–64 | 14.4 | 40.9 | 0.069 |

65+ | 40.0 | 34.4 | 0.162 |

Single-region simulation models were calibrated using two common measures of pandemic severity [

Mitigation resources included stockpiles of vaccines and antiviral and administration capacities (Section

The vaccination risk group included healthcare providers [

A version of the CDC guidance for quarantine and isolation for Category 5 was implemented (Section

An outbreak was considered contained, if the daily infection rate did not exceed five cases, for seven consecutive days. Once contained, a region was simulated for an additional 10 days for accurate estimation of the pandemic statistics. A 2^{5} statistical design of experiment [

The simulation code was developed using C++. The running time for a cross-regional simulation replicate involving over four million inhabitants was between 17 and 26 minutes (depending on the initial outbreak region, with a total of 150 replicates) on a Pentium 3.40 GHz with 4.0 GB of RAM.

The performance of the DPO and myopic policies is compared at different levels of resource availability.

Table

Total and regional resource requirements.

Region (population) | Resource requirements by region | ||||

Hillsb. | Miami D. | Duval | Leon | Total | |

(1,007,916) | (2,209,702) | (852,168) | (248,761) | (4,318,547) | |

Resource | |||||

Vaccine stockpile | 305,036 | 679,181 | 241,522 | 76,007 | 1,301,745 |

Antiviral stockpile | 415,294 | 749,058 | 460,393 | 105,307 | 1,730,052 |

No. antiv. nurses | 650 | 1,104 | 786 | 166 | 2,706 |

No. vacc. nurses | 1,059 | 2,358 | 839 | 264 | 4,520 |

Values of pandemic impact measures (societal and economic costs).

Pandemic impact measure (age group, years) | Value US$ |
---|---|

Average cost of lost lifetime productivity of a deceased case (0–19) | $1,336,347.86 |

Average cost of lost lifetime productivity of a deceased case (20–64) | $1,370,987.28 |

Average cost of lost lifetime productivity of a deceased case (65–99) | $98,959.24 |

Average cost of lost productivity and medical expenses of a recovered/deceased case (0–19) | $5,078.48 |

Average cost of lost productivity and medical expenses of a recovered/deceased case (20–64) | $10,466.68 |

Average cost of lost productivity and medical expenses of a recovered/deceased case (65–99) | $11,566.09 |

Average daily cost of lost productivity of a non-infected quarantined case (20–99) | $432.54 |

Comparison of the two strategies is done at the levels of 20%, 50%, and 80% of the total resource requirement shown in Table

Comparison of DPO and myopic policies (average number infected 6(a) and deaths 6(b)).

Figure

Average number of regional outbreaks for DPO and myopic policies.

Total resource availability | |||

Policy | 20% | 50% | 80% |

DPO | 1.75 | 1.66 | 1.44 |

Myopic | 2.40 | 1.77 | 1.50 |

DPO versus myopic (total cost).

It can be observed that the values of all impact measures exhibit a downward trend, for both DPO and myopic policies, as the total resource availability increases from

An increased total resource availability not only helps alleviating the pandemic impact inside the ongoing regions but also reduces the probability of spread to the unaffected regions. For both policies, as the total resource availability approaches the total resource requirement (starting from approximately 60%), the impact numbers show a converging behavior, whereby the marginal utility of additional resource availability diminishes. This behavior can be explained by noting that the total resource requirements were determined assuming the worst case scenario when

In this section, we assess the marginal impact of variability of some of the critical factors. The impact was measured separately by the change in the total pandemic cost and the number of deaths (averaged over multiple replicates), resulting from a unit change in a decision factor value, one factor at a time. Factors under consideration included: (i) antiviral efficacy, (ii) social distancing conformance, and (iii) CDC response delay. We have used all four regions, separately, as initial outbreak regions for each type of sensitivity analysis. The results (patterns) were rather similar. Due to limited space, we have opted to show the results for only one initial region, chosen arbitrarily, for each of the three types of sensitivity studies. While Duval County was selected as the initial outbreak region to show the sensitivity results on antiviral efficacy, Hillsborough and Miami Dade were used as the initial regions to show the results on, respectively, social distancing conformance and CDC response delay.

Figure

Sensitivity analysis for antiviral efficacy.

It can be noted that the performance of both policies is somewhat identical for low antiviral efficacy (between

Reduction of the contact intensity through quarantine and social distancing has proven to be one of the most effective containment measures, especially in the early stages of the pandemic [

Figure

Sensitivity analysis for quarantine conformance.

The CDC response delay corresponds to the interval of time from the moment an outbreak is detected to a complete deployment of mitigation resources. Depending on the disease infectivity, CDC response delay may represent one of the most critical factors in the mitigation process.

Figure

Sensitivity analysis for CDC response delay.

As recently pointed by the IOM, the existing models for PI mitigation fall short of providing

The model supports dynamic predictive resource distribution over a network of regions exposed to the pandemic. The model aims to balance both the ongoing and potential outbreak impact, which is measured in terms of morbidity, mortality, and social distancing, translated into the cost of lost productivity and medical expenses. The model was calibrated using historic pandemic data and compared to the myopic strategy, using a sample outbreak in Fla, USA, with over 4 million inhabitants.

In the testbed scenario, the DPO strategy on average outperformed the myopic policy. As opposed to the DPO strategy, the myopic policy is reactive, rather than predictive, as it allocates resources regardless of the remaining availability and the overall cross-regional pandemic status. In contrast, the DPO model distributes resources trying to balance the impact of actual outbreaks and the expected impact of potential outbreaks. It does so by exploiting region-specific effectiveness of mitigation resources and dynamic reassessment of pandemic spread probabilities, using a set of regression submodels. Hence, we believe that in scenarios involving regions with a more heterogeneous demographics, the DPO policy will likely to perform even better and with less variability than the myopic strategy. We also note that the difference in the model performance was particularly noticeable at lower levels of resource availability, which is in accordance with a higher marginal utility of additional availability at that levels. We thus believe that the DPO model can be particularly useful in scenarios with very limited resources.

We also developed a decision-aid simulator which is made available to the general public through our web site at

The authors would like to acknowledge with thanks the many helpful suggestions made by Professor Yiliang Zhu, Department of Epidemiology and Biostatistics at the University of South Florida, Tampa, Fla, USA.