Mathematical models of viral transmission and control are important tools for assessing the threat posed by deliberate release of the smallpox virus and the best means of containing an outbreak. Models must balance biological realism against limitations of knowledge, and uncertainties need to be accurately communicated to policy-makers. Smallpox poses the particular challenge that key biological, social and spatial factors affecting disease spread in contemporary populations must be elucidated largely from historical studies undertaken before disease eradication in 1979. We review the use of models in smallpox planning within the broader epidemiological context set by recent outbreaks of both novel and re-emerging pathogens.

Despite eradication, smallpox still presents a risk to public health whilst laboratory stocks of virus remain. One factor crucial to any assessment of this risk is R0, the average number of secondary cases infected by each primary case. However, recently applied estimates have varied too widely (R0 from 1.5 to >20) to be of practical use, and often appear to disregard contingent factors such as socio-economic conditions and herd immunity. Here we use epidemic modelling to show a more consistent derivation of R0. In isolated pre-twentieth century populations with negligible herd immunity, the numbers of cases initially rose exponentially, with an R0 between 3.5 and 6. Before outbreak controls were applied, smallpox also demonstrated similar levels of transmission in 30 sporadic outbreaks in twentieth century Europe, taking into account pre-existing vaccination levels (about 50%) and the role of hospitals in doubling early transmission. Should smallpox recur, such estimates of transmission potential (R0 from 3.5 to 6) predict a reasonably rapid epidemic rise before the implementation of public health interventions, because little residual herd immunity exists now that vaccination has ceased.

The recent spread of highly pathogenic strains of avian influenza has highlighted the threat posed by pandemic influenza. In the early phases of a pandemic, the only treatment available would be neuraminidase inhibitors, which many countries are considering stockpiling for pandemic use. We estimate the effect on hospitalization rates of using different antiviral stockpile sizes to treat infection. We estimate that stockpiles that cover 20%-25% of the population would be sufficient to treat most of the clinical cases and could lead to 50% to 77% reductions in hospitalizations. Substantial reductions in hospitalization could be achieved with smaller antiviral stockpiles if drugs are reserved for persons at high risk.

Pneumonic plague poses a potentially increasing risk to humans in plague nonendemic regions either as a consequence of an aerosolized release or through importation of the disease. Pneumonic plague is person-to-person transmissible. We provide a quantitative assessment of transmissibility based on past outbreaks that shows that the average number of secondary cases per primary case (R0) was 1.3 (variance = 3.1), assuming a geometric probability distribution, prior to outbreak control measures. We also show that the latent and infectious periods can be approximated by using lognormal distributions with means (SD) of 4.3 (1.8) and 2.5 (1.2) days. Based on this parameter estimation, we construct a Markov-chain epidemic model to demonstrate the potential impact of delays in implementing outbreak control measures and increasing numbers of index cases on the incidence of cases in simulated outbreaks.

The ongoing worldwide spread of the H5N1 influenza virus in birds has increased concerns of a new human influenza pandemic and a number of surveillance initiatives are planned, or are in place, to monitor the impact of a pandemic in near real-time. Using epidemiological data collected during the early stages of an outbreak, we show how the timing of the maximum prevalence of the pandemic wave, along with its amplitude and duration, might be predicted by fitting a mass-action epidemic model to the surveillance data by standard regression analysis. This method is validated by applying the model to routine data collected in the United Kingdom during the different waves of the previous three pandemics. The success of the method in forecasting historical prevalence suggests that such outbreaks conform reasonably well to the theoretical model, a factor which may be exploited in a future pandemic to update ongoing planning and response.