We present a stochastic model for the spread of smallpox after a small number of index cases are introduced into a susceptible population. The model describes a branching process for the spread of the infection and the effects of intervention measures. We discuss scenarios in which ring vaccination of direct contacts of infected persons is sufficient to contain an epidemic. Ring vaccination can be successful if infectious cases are rapidly diagnosed. However, because of the inherent stochastic nature of epidemic outbreaks, both the size and duration of contained outbreaks are highly variable. Intervention requirements depend on the basic reproduction number _{0}

Recently, concerns about a bioterror attack with the smallpox virus or other infectious disease agents have risen (

We investigated which conditions are the best for effective use of ring vaccination, a strategy in which direct contacts of diagnosed cases are identified and vaccinated. We also investigated whether monitoring contacts contributes to the success of ring vaccination. We used a stochastic model that distinguished between close and casual contacts to explore the variability in the number of infected persons during an outbreak, and the time until the outbreak is over. We derived expressions for the basic reproduction number (_{0}_{υ}

The model describes the number of infected persons after one or more index cases are introduced. It simulates a stochastic process in which every infected person generates a number of new infections according to a given probability distribution. This process implies that contacts of different infected persons are independent of each other and that no saturation of the incidence occurs at higher prevalence. The model is applicable for the first few generations of infection, if the outbreak goes unchecked, and for the complete outbreak if it is contained. We summarize the main features of the model; the formal model definition is given in the

The noninfectious state (incubation period plus prodromal phase) lasts 12–15 days (_{I}_{τ} , where τ denotes the day of the infectious period, is high at the beginning and low at the end of the infectious period (

A, the transmission probability per contact by day of the infectious period; B, the probability distribution of the number of contacts with susceptible persons per day; C, the probability of remaining undiagnosed but infectious case by day of the infectious period; and D, the mean (solid line) and the 2.5% and 97.5% percentiles (dotted lines) of the number of infected persons for 500 simulation runs for an epidemic without any intervention after the introduction of one index case at the beginning of this incubation period at t=0.

Transmission takes place in two rings of contacts: 1) household and other close contacts, and 2) more casual face-to-face contacts. We assumed that in the close contact ring the probability of transmission is five times higher than in the casual contact ring (g = 0.2). The number of contacts on day τ of the infectious period in the close contact ring follows a Poisson distribution with mean μ_{τ}^{(1)}, and in the casual contact ring this number follows a a negative binomial distribution with mean μ_{τ}^{(2)}. The values (μ_{τ}^{(1)} = 2 and μ_{τ}^{(2)} = 14.9) were chosen such that the total number of contacts per day was comparable to numbers observed in empirical studies (

Most people can be infected again, and smallpox can develop 10–20 years after vaccination (

The basic reproduction number _{0}

For the baseline parameter values given in the Table, _{0}_{1}_{0}_{1}_{0}

Model parameter | Notation | Baseline value |
---|---|---|

Course of the infection | ||

Maximum duration latent period | _{E} | 15 d |

Probability of transition to infectious state on day τ of the latent period | _{τ} , τ=1,...,D_{E} | 0.0 for τ = 1,...,12 0.3 for τ = 13 0.6 for τ = 14 1.0 for τ = 15 |

Duration infectious period | _{I} | 14 days |

Case-fatality rate | 0.3 | |

Transmission probability per contact | _{τ} |
with _{1} = 0.27, a_{2} = 0.5 |

Ratio of infectiousness of casual contact and close contact ring | 0.2 | |

Contacts | ||

Mean number of contacts in close contact ring (Poisson distribution) | _{τ}^{(1)} | 2 |

Mean number of contacts in the casual contact ring (negative binomial distribution) | _{τ}^{(2)} | 14.9 (sd 8.4) NegBin(4, 0.212) |

Intervention | ||

Probability of diagnosis | _{τ} |
with _{i}_{2}_{2} |

Time needed to trace contact | ^{(i)} | 1 day for i = 1, 3 days for i = 2 (3 days for both rings for first index case) |

Time window during which vaccination is effective | 4 days | |

Vaccination coverage | ^{(i)} | 0.95 for i = 1, 0.5 for i = 2 (0.5 for both rings for first index case) |

Ring vaccination in the model includes complete isolation of diagnosed symptomatic patients with cases of smallpox and vaccination of (a fraction of) all contacts of the diagnosed patient. In our baseline scenario, we assumed that vaccinated contacts are not isolated after vaccination and may therefore transmit the infection to others if they become infectious. In addition, we enhance the baseline intervention by including monitoring of identified contacts. The effectiveness of the intervention therefore is determined by the probability of diagnosis per day of the infectious period, the time needed to identify contacts of the close contact and casual contact ring, the vaccination coverage in the close contact and the casual contact ring, and whether monitoring of contacts is performed. Some of those parameters (speed of diagnosis and time to identifying contacts) differ between the first index case in the population and cases occurring later in the epidemic. In

The time course of events in the process of transmission and intervention. The success of intervention is essentially determined by the time between start of the infectious period and diagnosis of the index case, and the time between the start of contact tracing and the vaccination of the contact.

We denote with δ_{τ} the probability of diagnosis on day τ of the infectious period for those persons who have not been diagnosed before. From those probabilities, one can derive the probability that an infectious person is not yet diagnosed on day τ of his or her infectious period (_{τ}^{(i)}, we denote the probability that a contact in ring i (i = 1 or 2), who was infected on day τ of the index patient’s infectious period, will be vaccinated within 4 days of being infected. In the _{τ} depends on the diagnosis probabilities, the time needed for contact tracing, and the vaccination coverage. Throughout, we assume that the vaccine efficacy is 100%. We can now determine an effective reproduction number _{υ}

A special strategy included in this formula is an intervention where only case isolation is performed without vaccination of contacts. This formula also applies to an intervention where only case isolation is performed without vaccination of contacts, if the vaccination coverage ^{(i)}

If vaccination is ineffective, but there is monitoring of contacts, the monitoring will have the same effect on _{0}_{0}^{(i)}

The outbreak can be controlled if _{υ}

An epidemic starting with one index case in a completely susceptible population without intervention grows exponentially, if it survives early extinction (minor epidemics). The large range of possible courses of the epidemic reflects the stochastic variability (_{υ}_{υ}

If the intervention succeeds in reducing the effective reproduction number _{υ}_{υ}

The distribution of A, the total number of infected persons excluding those infected contacts who were vaccinated on time to prevent disease, and B, the time to extinction for 500 simulation runs with the baseline intervention parameter values and a basic reproduction number of 5.23. For a basic reproduction number of 10.46 and an increase of the vaccination coverage in the casual contact ring to 80% in C, the distribution of the total number of infected persons, and in D, the distribution of the time to extinction, is shown for 500 simulation runs.

To contrast the baseline scenario, in _{0} = 10.46, i.e., twice the value of baseline scenario. To contain the epidemic, we now assumed that 80% of all contacts in the casual contact ring were vaccinated in time. The effective reproduction number _{υ}_{υ}

The initial phase of the epidemic (time before discovery of the first case) is determined by the number of index patients that start the epidemic outbreak and by the time it takes to diagnose the first case. We varied those two variables separately while assuming that after diagnosis intervention took place within the parameters defined in the Table, i.e., with an _{υ}

Results for the sensitivity analyses. The total number of infected persons (excluding successfully vaccinated infected contacts) depends on A, the number of index cases starting the epidemic, and B, the day of the infectious period after which the diagnosis of the first case occurs. The time to extinction is shown for C, different numbers of index cases, and D, the day of the infectious period after which the diagnosis of the first case occurs. The quantiles are taken pointwise for 500 simulation runs.

Among others, the value of _{0}_{0}_{υ}_{0}_{0}_{0}

The effective reproduction number _{υ}_{0}_{0}_{0}

In _{0}

Here the critical vaccination coverage in the casual contact ring is shown as a function of the basic reproduction number _{0}_{0}_{0}_{0}_{0}

Finally, we looked at how differences in intervention effectiveness influence the duration of the epidemic and the cumulative number of infected persons. The effective reproduction number _{υ}_{υ}._{υ}

A, the cumulative number of infected persons (excluding successfully vaccinated infected contacts), and B, the time to extinction are shown for various values of the effective reproduction number _{Rυ}

Our simulation results show that a smallpox epidemic starting from a small number of index cases can be contained by ring vaccination provided the intervention measures are very effective. The time to diagnosis has proven to be an essential and sensitive parameter in determining the intervention effectiveness. The speed of diagnosis is less essential if identified contacts are isolated to prevent them from transmitting further if their vaccination fails. The time window limiting the success of vaccination then loses its importance for determining the effectiveness of intervention. The time to diagnosis of cases and the fraction of contacts found by contact tracing are then the key parameters. Contact tracing would be even more essential if substantial transmission would take place during the prodromal period of infection as is assumed by some authors (

Some limitations of our modeling approach should be kept in mind. First, we only consider epidemics that are started by a small number of index cases. The branching process approach does not allow for overlapping rings of contact, but we implicitly include such an effect by varying the effective transmission probability such that the distribution of transmissions over the infectious period agrees with empirical findings (

In the recent literature, other models, both stochastic and deterministic, of smallpox outbreaks have been introduced to analyze the effects of ring and mass vaccination (_{0}

In a study by Halloran et al. (_{0}

The main difference between the modeling approach of Bozette et al. (_{0}_{υ}_{υ}_{υ}_{0}

Finally, Eichner (

With respect to preparing for a smallpox outbreak, alertness and ability to diagnose quickly are important. Physicians and nurses need to be educated and the public needs to be more aware. Also, since we know little about the timing and effectiveness of identifying infectious persons and their contacts in case of a bioterror attack, obtaining more empirical information about contact patterns and contact tracing will be helpful. Recently, some useful data about contact patterns have been collected during severe acute respiratory syndrome outbreaks, but a more systematic investigation of contact tracing is advisable. Considering the uncertainties connected to all parameter values, we conclude that any contingency plan for use of ring vaccination must also identify the criteria under which switching to large-scale mass vaccination is justified.

Formal Model Definition

We thank three anonymous referees for their helpful and constructive comments.

Dr. Kretzschmar is a senior research scientist at the Department of Infectious Diseases Epidemiology of the National Institute of Public Health and the Environment (RIVM) in Bilthoven, the Netherlands. She is developing mathematical models for the spread and prevention of infectious diseases. She has worked on sexual network structure and the spread and prevention of sexually transmitted diseases and HIV/AIDS and on vaccine-preventable diseases.