### Methods

#### Population

We used Sydney's population in 2015,^{[12]} which was estimated using data from the state of New South Wales.^{[13]} The New York population of the same year was derived from the relevant statistical collection.^{[14]} We divided both populations into 5-year age groups up through ages 80–84 years and combined the eldest (persons >84 years of age) into a single group (Figure 1, panel A). Each age group was divided into vaccinated and unvaccinated compartments, which were then further subdivided into 3 categories of immunity: immunocompetent, mildly immunosuppressed, and moderate-to-severely immunosuppressed. We assumed that immunosuppressed persons had no residual immunity from vaccination.

Figure 1.

Characteristics of population used to model smallpox transmission, by age group, New York, NY, USA, and Sydney, New South Wales, Australia. Characteristics (e.g., size, age, immunosuppression rates) of populations from 2015 were used. A) Total population; B) immunosuppressed population.

#### Immunosuppressed Population

We considered common types of immunosuppression estimated in an influenza study.^{[15]} We classified persons into 2 categories of immunosuppression: moderate to severe (called severe in our model) and mild. Severe immunosuppression was defined as a condition in which quantifiable data existed to demonstrate a risk for infection more than twice that of an immunocompetent person. This classification was left as a single category in the absence of reliable methodology to subdivide it. Mild immunosuppression was defined as a condition in which immunosuppression was documented but susceptibility to infection was estimated to be less than twice that of an immunocompetent host. For the analysis, persons with severe immunosuppression were assumed to have 2× and mild immunosuppression 1.5× the susceptibility to infection of a healthy person.^{[16]}

We sourced data for each city, and when only countrywide data were available, we attributed rates from the countrywide data set to the respective fraction of the population in the city. When age-specific immunosuppression prevalence data were not available, we used yearly age-specific incidence data instead to calculate prevalence age distribution.^{[17,18]}

We estimated the populations living with cancer,^{[17,19]} HIV,^{[20,21]} organ transplants,^{[22,23]} respiratory syndromes such as asthma^{[24,25]} and chronic obstructive pulmonary disease,^{[26,27]} dialysis,^{[28,29]} and autoimmune diseases^{[30,31]} and divided these populations into the 2 immunosuppression categories for each city (Technical Appendix Table 1). We acknowledge that many other diseases are associated with immunosuppression. Our method underestimates the amount of immunosuppression in the population.

#### Vaccine-induced Residual Immunity

In the United States, including New York, widespread smallpox vaccination occurred until 1970.^{[32]} In contrast, in the geographically isolated island continent of Australia, quarantine was used to protect the population because smallpox was never endemic.^{[32]} Widespread vaccination never occurred in Australia; only the armed forces and healthcare workers were vaccinated, which occurred until 1979, although reactive vaccination campaigns had been conducted during a smallpox outbreak in Sydney in 1917.^{[33]}

For New York, we assumed 80% of the healthy population 40–69 years of age (born before 1975) were previously vaccinated. For Sydney, we estimated the proportion of persons vaccinated by estimating those born before 1980 in the following groups: healthcare workers in Sydney in 2015,^{[34]} members of the defense forces, and migrants (>30% in the Sydney population),^{[35]} who might have been vaccinated in their country of origin (≈80,000 persons). We estimated that, in Sydney, at most 30% of the total population born before 1980 (persons 35–69 years of age) had been vaccinated. On the basis of a mathematical model^{[36]} that estimated waning immunity against severe smallpox as 1.41% per year after vaccination, we calculated the age-specific residual protection for vaccinated persons 40–69 years of age by multiplying that percentage by the number of years from vaccination and subtracting it from 100% starting protection.

#### Contact Mixing

In our model, we used the heterogeneous age-specific contact rates from the European mixing patterns study.^{[37]} We assumed that persons would greatly reduce their social contacts after becoming symptomatic with smallpox.^{[38]} To account for this change in social contact, we modified the normal contact matrix, multiplying the matrix by a factor (0 < *x* < 1) to reduce the number of contacts per day.^{[39]} Because of the lack of studies quantifying this reduction, we assumed *x* to be 0.5, as in a previous study.^{[39]} Considering severe smallpox types are more substantially prostrating, we applied the reduced contact matrix to hemorrhagic and flat smallpox infections from the first day of illness. For ordinary smallpox, we assumed the behavior change started on the second day and for vaccine-modified smallpox, on the third day.

#### Disease Types

We categorized smallpox disease into 4 different types defined by infectivity (R_{0}) and CFRs: hemorrhagic, flat, ordinary, and vaccine-modified. Age-specific and other model parameters (Technical Appendix Table 2) as well as further model details are explained in the Technical Appendix.

#### Smallpox Disease Type Distribution

We assumed infected persons had different probabilities of developing each disease type, depending on their age and immunologic status. The incidence of each disease type within each age group for healthy unvaccinated persons was drawn from historical outbreaks^{[9]} (Technical Appendix Table 3). For healthy unvaccinated persons, hemorrhagic smallpox ratios ranged from 7 cases/1,000 persons infected (in the 5–9-year age group) to 200 cases/1,000 persons infected (in the >85-year age group). Flat smallpox age-specific rates were lowest for the 10–14-year age group (30 cases/1,000 persons infected) and reached 180 cases/1,000 persons infected for the oldest age group. For the mildly immunocompromised population, we doubled the age-specific probability of hemorrhagic and flat smallpox. We assumed 100% of severely immunocompromised persons would develop hemorrhagic disease. We assumed the vaccinated subgroup had reduced susceptibility and rates of severe smallpox types. We estimated that 25.3% of vaccinated persons would get vaccine-modified smallpox.^{[9]} We applied a waning immunity function over time at a rate of 1.41% per year from vaccination^{[36]} and assumed the rates of hemorrhagic and flat smallpox would increase with time from vaccination while rates of vaccine-modified smallpox would decrease with time from vaccination (Technical Appendix Table 4).

#### Mathematical Model

We constructed a modified SEIR (susceptible, exposed, infected, recovered) model for smallpox transmission (Technical Appendix Table 2). The population was divided into vaccinated and unvaccinated compartments, which were then further subdivided into 3 categories of immunity: immunocompetent, mildly immunosuppressed, and moderate-to-severely immunosuppressed. The model used ordinary differential equations to move populations into epidemiologic states related to their smallpox infectious status: susceptible, infected, prodromal, infectious, recovered, or dead. Once infected, populations were moved into the next state on the basis of disease duration rates. To obtain the age-specific force of infection (i.e., the rate at which susceptible persons acquire smallpox), we used the Euler approximation to make discrete contact rates, assuming the rates were proportional to the patterns observed in the United Kingdom. Then, to account for the different infectivity rates of different smallpox types, we estimated the transmission parameter β (i.e., the probability of getting infected from a contact) for each smallpox disease type to calculate the R_{0} for hemorrhagic, flat, ordinary, and vaccine-modified smallpox. Finally, we multiplied the force of infection by a parameter (α_{1,} α_{2,} α_{3,} α_{4}; Technical Appendix Table 2) to account for the different susceptibility levels of different populations.

The model ran for 100 simulated days. We assumed an attack in a crowded public space, such as an airport, and started the epidemic with 51 infected in New York and 29 in Sydney to reflect the same attack rate for each population. We assumed a dynamic population updated each day using the birth^{[40]} and age-specific death rates^{[41,42]} from 2014 for both cities.

#### Sensitivity Analysis

We conducted a sensitivity analysis on the assumption of waning immunity, reducing immunity by 0.7% per year (approximately half the value used in the base case scenario [i.e., the first scenario discussed]). We present results for 3 different assumptions about residual vaccine immunity: no residual immunity, base case immunity (1.41% waning immunity per year), and high residual immunity (0.7% waning immunity per year). We also conducted a sensitivity analysis to test the model outputs without considering population immunosuppression, which has been the approach in most past models.^{[43]}

Emerging Infectious Diseases. 2018;24(4):646-653. © 2018 Centers for Disease Control and Prevention (CDC)

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