## Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease

A discussion of each choice of parameter values follows this table.

Description |
Symbol | Default value | Plausible range | Special case(s) | ||||
---|---|---|---|---|---|---|---|---|

Basic reproduction number | R_{0} |
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Low |
1.5 |
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Medium |
2.5 |
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High |
3.5 |
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Influenza attack rate for calibrating a | AR |
1968 1957 |
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Adults |
0.34 |
[0.2, 0.5] |
0.37 0.23 | |||||

School children |
0.45 |
[0.35, 0.55] |
0.42 0.46 | |||||

Household escape parameter | θ |
|||||||

R _{0 } = 1.5R_{0} = 2.5R_{0} = 3.5 |
0.82 0.74 0.69 |
0.84 0.84
0.76 0.76 0.71 0.71 |
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Fraction of between-household mixing at work/school | ƒ_{S } |
0.6 |
[0.45, 0.75] |
|||||

Increased mixing between children at school | α |
1.25 |
1.15 1.15 | |||||

Between-household transmission rate | µ |
|||||||

R_{0} = 1.5R_{0} = 2.5R_{0} = 3.5 |
1.05 1.62 2.04 |
0.91 0.91 1.46 1.46 1.84 1.84 |
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Prior immunity in adults | η_{A } |
0 |
0 0.35 | |||||

Prior immunity in children | η_{C } |
0 |
0 0 | |||||

Incubation period | D_{A } |
2 |
[1.8, 2.2] |
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Infectiousness function | ||||||||

Number of travelers from an infected region into Australia on day t |
κ_{t } |
200 |
[10, 400] |
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Probability of detecting a symptomatic traveler at the border on departure | s_{D } |
0.5 |
[0, 1] |
|||||

Probability of detecting a symptomatic traveler at the border on arrival | s_{A } |
0.5 |
[0,1] |
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Relative infectivity while taking Oseltamivir e_{i} = 1: fully infectious, e _{i} = 0: not infectious |
e_{i } |
0.275 |
[0.15, 0.4] |
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Relative infectivity while taking Oseltamivir as treatment e _{t} = 1: fully infectious,e _{t} = 0: not infectious, |
e_{t } |
0.7 |
[0.25,1.0] |
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Relative susceptibility while taking Oseltamivire_{s} = 1: fully susceptible, e_{s} = 0: fully protected |
e_{s } |
0.25 |
[0.1, 0.4] |
|||||

Relative susceptibility due to PPEe_{p} = 1: fully susceptible, e _{p} = 0: fully protected |
e_{p } |
0.23 |
[0.1, 0.8] |
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Mean number of close contacts made by an infective individual | κ |
18.3 |
[15,25] |
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Probability that a general case infects their GP at presentation. | ρ |
0.005 |
[0.001, 0.01] |
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Fraction of their infectivity that an individual spends in the community before being isolated. (For a flat infectiousness function it is the fraction of the infectious period spent in the community before being isolated. In general it is the fraction of the area under the infectiousness function up to the time of isolation.) |
ƒ |
0.8 |
[0.6, 1] |
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Mean number of general public individuals infected by a case when there is no control. | m |
see R _{0 } |
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Fraction of population that is school-aged | π _{C } |
0.18 |
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Fraction of the population that are GPs | π _{D } |
0.001 |
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Fraction of the population acting as influenza-dedicated health-workers | π _{H } |
0.01 |
||||||

Remaining individuals | π _{A } |
0.809 |

## Discussion of data sources

*R*_{0 } – basic reproduction number

[Longini *et al.* (2005): range for *R*_{0 } from 1.1 to 2.4]

[Gani *et al.* (2005): range for *R*_{0 } from 1.28 to 2.0]

[Ferguson *et al.* (2005: *R*_{0 } = 1.8, range from 1.0 to 2.0]

If we calibrate *R*_{0} to influenza attack rates in western countries from past pandemics, we get values between 1.1 and 1.7, even assuming up to 20% prior immunity. However, these attack rates arise under conditions in which many controls were already in place. We consider three scenarios for *R*_{0} – low (1.5), medium (2.5) and high (3.5).

### Attack rate in adults and schoolchildren

Data from the 1957 and 1968-69 pandemics show different age-specific influenza attack rates [Davis *et al.* (1970)]. In 1957, there was a high attack rate in 10-20 year olds, while the attack rate in adults was much lower. In this pandemic, 85% of household outbreaks were initiated by a 5-19 year old. In 1968-69, the attack rate was fairly uniform across age groups, and between 50% and 55% of household outbreaks were initiated by a 5-19 year old.

*θ* – probability that an individual avoids being infected by a specific household member

O’Neill *et al.* (2000) estimate *θ* = 0.84 with 99% credible bounds (read from figure) of [ 0.78, 0.89 ]. The data used for these estimates come from a Tecumseh study over influenza epidemic seasons. Infection was serologically confirmed, so should include asymptomatic cases. The overall influenza attack rate in this data is fairly low, so different values of *θ* should be used for the higher values of R_{0 }.

Influenza attack rates in 1957 in Cleveland (47% of individuals and 90% of households infected) also suggest that *θ* should be relatively high.

Clearly *θ* should vary according to the assumed reproduction number, so three values are given.

### ƒ_{s } – proportion of between-household mixing that occurs during work/school hours

α – increased rate of mixing between children at school

µ – baseline mean number of cases an infective infects in the community

*Adults:* Assuming adults are awake for 16 hours/day (=112 hours/week), and have a working week of 40 hours. Assume they spend 3-7 hrs per day at home during the week, and 4-13 hrs per day at weekends, giving a total of 23-61 hrs/week. This would put ƒ_{s } in the range 0.45-0.78.

*School children:* Assuming children are awake for 14 hours/day (=98/week), and have a school week of 32.5 hours. Assume they spend 4-6 hrs per day at home during the week, and 5-11hrs per day at weekends, giving a total of 30-52 hrs/week. This would put ƒ_{s } in the range 0.48-0.71.

The adult/school children values are sufficiently similar that it seems reasonable that one value should be used for both. We adopt a default value of 0.6 with a plausible interval of [0.45,0.75].

Once ƒ_{s } has been fixed, *θ*, *µ* and *α* are calculated to be consistent with *R*_{0} and the attack rates.

*Note that this is calibrated to the SEIR*H* model, which takes account of within-household mixing.. In the SEIR model, ƒ _{s } and α should be adjusted as follows: ƒ_{s } =0.35, α = 1.8, µ = 1.4, 2.33, 3.26 (R0 = 1.5,2.5,3.5)*

*η*_{A},* η*_{C} – prior immunity in adults and children

In some situations, individuals may have some immunity to a pandemic strain arising from recent exposure to influenza. Our default scenario assumes that there is no prior immunity, but we also consider the case where relatively high immunity in adults leads to an age-specific influenza attack rate such as that seen in the 1957 pandemic [Davis *et al.* (1970)]. This is similar to assumptions used by Elveback *et al.* (1976).

*D*A – days from infection until symptom onset (incubation period)

[Gani *et al.* (2005): *D*_{A} = 4.5 days (2 days latent plus 2.5 days infectious and asymptomatic)]

[Longini *et al.* (2004): *D*_{A} = 1.9 days]

[Ferguson *et al.* (2005): *D*_{A} = 1.48 days, σ =0.47]

We assume the default value for *D*_{A} to be 48 hours.

### Infectivity

[Gani *et al.* (2005): latent period of 2 days, infectious period of 4 days]

[Longini *et al.* (2004): latent period of 1.9 days, infectious period of 4.1 days]

[Ferguson *et al.* (2005): assumes infectivity function]

The flat infectivity function assumes a latent period of 1 day, and an infectious period of 5 days. The peaked infectivity function is taken from [Ferguson *et al.* (2005)]*.*

*e*_{i} – relative infectivity when taking Oseltamivir

*e*_{t} – relative infectivity when taking Oseltamivir as treatment

[*e*_{i} = 1: no effect of antivirals on infectivity, *e*_{i} = 0.5: infectivity while on antivirals reduced by half, e_{i} = 0: not infectious at all when on antivirals]

[Hayden *et al.* (1999): *e*_{i} = 0.29, based on area under curve of viral titre]

[Treanor *et al.* (2000): *e*_{i} = 1.0 - ie no effect on infectivity]

[Hayden *et al.* (1996): *e*_{i} = 0.13 if 26 or 32 hrs after innoculation

*e*_{i} = 0.25 if 50 hrs after innoculation]

[Longini *et al.* (2005): *e*_{i} = 0.38, Longini *et al.* (2004): *e*_{i} = 0.2]

We adopt the value *e*_{i} = 0.275.

In some models we also allow for antivirals to be less effective at reducing infectivity if they are taken after onset of symptoms. In these models, we adopt *e*_{t} = 0.7.

*e*_{s} – relative susceptibility while taking Oseltamivir

[*e*_{s} = 1: fully susceptible when on antivirals, *e*_{s} = 0.5: susceptibility while on antivirals reduced by half, *e*_{s} = 0: completely protected when on antivirals]

[Hayden *et al.* (1999): *e*_{s} = 0.57, based on prob. infection,

*e*_{s} = 0.0, based on prob. of shedding Note small n.]

[Hayden *et al.* (1996): *e*_{s} = 0.18, based on prob. infection.

*e*_{s} = 0.04, based on prob. shedding. ]

[Longini *et al.* (2005) and Longini *et al.* (2004): es = 0.7 ]

[Hayden *et al.* (1999): *e*_{s} = 0.50, based on lab confirmed infection]

[Welliver *et al.* (2001): *e*_{s} = 0.11, lab. confirmed, post-exposure prophylaxis within 48 hours of symptom onset of a household member (index case not given antivirals)]

[Cochrane review of all Neuraminidase Inhibitors: *e*_{s} = 0.26 (naturally occurring), *e*_{s} = 0.4 (lab. confirmed)]

We adopt the value *e*_{s} = 0.25.

*e*_{p} – reduction in susceptibility due to personal protective equipment

[Loeb *et al.* (2004): *e*_{p}=0.22]

[Yen *et al.* (2006): *e*_{p}=0.23]

Both of these trials have P-values of around 0.02-0.03, which suggests that the 95% confidence intervals are wide. We assume *e*_{p} = 0.23.

### κ - the mean number of close contacts made by an infective individual

[Edmunds *et al.* (1997): κ= 18.3]

*p* – probability that a general case infects a GP at presentation.

Data suggest that health care professionals are more likely than general individuals to be infected during the influenza season [Gamage *et al.* (2005)]. If we assume that every infected individual consults a GP, there will be 300-400 consultations per 1000 individuals over the course of the pandemic. Australia has around 1-1.2 GPs per 1000 individuals, so this suggests that each GP will have an average of 250-400 consultations from individuals infected with pandemic flu.

We calculate that:

if *p* = 0.001 GPs have a 22-33% chance of being infected by a patient

if *p* = 0.01, GPs have a 92-98% chance of being infected by a patient

However, the assumption that every infected individual consults their GP is probably unrealistic. The UK pandemic plan assumes there will be 50 consultations per 1000 individuals over the course of the pandemic (based on data from 1969). Under this assumption, each GP will have 42-50 consultations from infected individuals, and

*p* = 0.001 GPs have a 4-5% chance of being infected by a patient

*p* = 0.01 GPs have a 34-39% chance of being infected by a patient

Note that in each case this is not the attack rate for GPs, since they will also have a force of infection acting on them from their household and the general community – this just gives us the extra infection attributable to their role as a GP.

We adopt the value *p* = 0.05 as the default value of this parameter.

*ƒ* – fraction of the infectivity an individual spends in the community before being isolated

(for the flat infectiousness function *ƒ* is the fraction of the infectious period spent in the community)

During the SARS outbreak, the shortest time from onset of symptoms to isolation that was achieved was around 2 days in Singapore. For influenza, isolation two days after onset of symptoms would correspond to *ƒ* = 0.6. We adopt a default value of *ƒ* = 0.8, and consider the range [0.6,1.0].

*m* – mean number of general cases infected by a general case if no isolation

This parameter is used in the model with GPs and HCWs but without children as a separate class. As *p* is small, we can assume that this is equal to *R*_{0}.

*π*_{C}, *π*_{D}, *π*_{H} and *π*_{A} – fraction of the population in different classes.

*π*_{C} is calculated from census data, *π*_{D} from the AIHW medical labour force survey (which estimates around 20,000 GPs for Australia). Total essential/emergency services personnel constitute around 5% of the population, but any one individual should not be on prophylaxis for more than six weeks. If we assume that 10-20% are on prophylaxis at any one time, and the remainder are not any more likely to be exposed than a general individual, then 100,000-200,000 essential service personnel would be on prophylaxis at any time, using up 1,000-2,000 courses per day.

In particular, we assume that *π*_{H} is some fraction of the HCW population that is on prophylaxis, and that is at high risk of encountering the virus (say because they are treating patients in a hospital). Total health worker numbers (including GPs etc) are around 750,000 – that is between 4% and 5% of the population. We assume that around 20-25% of these are on prophylaxis, giving *π*_{H} = 0.1.

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