Diagnosis and Antiviral Intervention Strategies for Mitigating an Influenza Epidemic

Background Many countries have amassed antiviral stockpiles for pandemic preparedness. Despite extensive trial data and modelling studies, it remains unclear how to make optimal use of antiviral stockpiles within the constraints of healthcare infrastructure. Modelling studies informed recommendations for liberal antiviral distribution in the pandemic phase, primarily to prevent infection, but failed to account for logistical constraints clearly evident during the 2009 H1N1 outbreaks. Here we identify optimal delivery strategies for antiviral interventions accounting for logistical constraints, and so determine how to improve a strategy's impact. Methods and Findings We extend an existing SEIR model to incorporate finite diagnostic and antiviral distribution capacities. We evaluate the impact of using different diagnostic strategies to decide to whom antivirals are delivered. We then determine what additional capacity is required to achieve optimal impact. We identify the importance of sensitive and specific case ascertainment in the early phase of a pandemic response, when the proportion of false-positive presentations may be high. Once a substantial percentage of ILI presentations are caused by the pandemic strain, identification of cases for treatment on syndromic grounds alone results in a greater potential impact than a laboratory-dependent strategy. Our findings reinforce the need for a decentralised system capable of providing timely prophylaxis. Conclusions We address specific real-world issues that must be considered in order to improve pandemic preparedness policy in a practical and methodologically sound way. Provision of antivirals on the scale proposed for an effective response is infeasible using traditional public health outbreak management and contact tracing approaches. The results indicate to change the transmission dynamics of an influenza epidemic with an antiviral intervention, a decentralised system is required for contact identification and prophylaxis delivery, utilising a range of existing services and infrastructure in a “whole of society” response.


S1.1 Changed definitions and new parameters
The model presented here assumes that asymptomatic and symptomatic individuals are equally infectious (this corresponds to χ = 1 in the original model [2]), rendering the distinction between symptomatic and asymptomatic infections redundant.
Accordingly, the I and A states distinguish between presenting and non-presenting cases, unlike [2] where they distinguished between symptomatic and asymptomatic infections.
The parameter α is the proportion of cases that present (to hospitals or to outpatient facilities). The basic reproduction number (R 0 ) of the pandemic influenza strain and the inverse infectious period (γ) are explicit parameters of this model; the number of infections per unit time made by an infectious individual (β) is not.
In recognition that not all contacts of an infectious individual can be identified and provided with post-exposure prophylaxis, we introduce a parameter σ that defines the proportion of contacts that are potentially identifiable. Accordingly, the proportion of all contacts that receive prophylaxis ( ) cannot exceed σ.
Finally, the fraction of presenting cases that receive treatment (ψ) and the fraction of contacts that receive prophylaxis ( ) are functions of time, since they are both affected by the logistical constraints introduced in this model and do not remain constant throughout an epidemic.

S1.2 Presentation, diagnosis and antiviral deployment
The proportion of all infected cases that present (α) is the sum of the severe cases (all of which present) and the proportion (α M ) of the remaining (i. e. mild) cases that present:  Figure 1: The flow between the state variables in the model where ψ and are functions of time, unlike in [2]. The contact classes C np and C p are labels for tracking contact status and are orthogonal to the SEIR states; see [1] for further details.
We have assumed that α M is dependent on the severity of the epidemic (η); the probability distribution for α M is a function of η and is presented in Section S2.2.
All severe cases present at hospitals and receive timely diagnosis and treatment; only mild cases (i. e. outpatient presentations) are subject to constraints on diagnosis and treatment. The rate at which mild cases present is denotedṖ M : Given the rate of mild presentations and an estimate of the proportion of influenzalike illness (ILI) presentations that are infected with the pandemic strain (ILI % ), the rate of ILI presentations (Ṗ ILI ) can be calculated: The rate at which positive diagnoses for pandemic influenza (Ḋ P ) are returned from outpatient ILI presentations is the sum of the true positives (i. e. pandemic cases) and the false positives, subject to a maximal diagnosis rate of MAX D . The parameters s N and s P are the fraction of true positives and true negatives that are identified across all outpatient ILI presentations, respectively.
The rate of true positives is given byḊ T P : Antivirals are deployed as treatment only to positively-diagnosed ILI presentations who did not receive antivirals for prophylaxis, subject to a maximal delivery rate of MAX T : The rate at which effective treatment is delivered to mild cases (Ṫ M ) is the fraction of the antiviral deployment that is delivered to pandemic cases, where the efficacy of antivirals delivered as a result of general practice (GP) presentations (f GP ) is reduced by e GP to account for delays inherent in analysing samples at external labs: The fraction of all presenting cases that receive treatment (ψ) is the sum of the mild cases that receive effective treatment and the severe cases (all of which receive effective treatment): Antivirals are deployed as prophylaxis to a fraction σ of the contacts of all severe cases and of all ILI presentations that return a positive diagnosis (Ḋ P ), subject to a maximal delivery rate of MAX P : The rate at which prophylaxis is delivered to contacts of pandemic cases (Ṗ T P ) is the fraction of the antiviral deployment that is delivered to contacts of pandemic cases. We approximate this fraction to beḊ T Ṗ D P , which discounts the fact that all severe cases are correctly diagnosed; the justification for this approximation is that mild cases represent the bulk of the pandemic infections and are the key to controlling transmission in the community. As per the delivery of effective treatment, the efficacy of antivirals delivered as a result of GP presentations is reduced by e GP to account for delays inherent in analysing samples at external labs: The fraction of all contacts that receive prophylaxis ( ) is the rate at which prophylaxis is delivered to these contacts, divided by the total number of contacts: Finally, the antiviral stockpile is depleted due to the total number of antiviral doses distributed for treatment and for prophylaxis:

S1.3 Vaccine distribution and infection
The original SEIR model [2] introduced Θ p and Θ np , which define the proportion of susceptible contacts in the population: Here we introduce similar variables Θ V p and Θ V np , which define the proportion of susceptible vaccinated contacts, who have a reduced susceptibility e v due to successful seroconversion: The force of infection (λ) arises from the five infectious classes just as in the original SEIR model [2], given the number of infections per unit time made by an infectious individual (β): We introduce a new state S V for vaccinated susceptibles (shown in Figure 1). People move from S to S V in proportion to the rate of seroconversion v SC : The vaccine seroconversion rate v SC is held at zero until week 20 of the epidemic, under the assumption that a vaccine becomes available 18 weeks into the epidemic and that seroconversion occurs two weeks after receiving a single dose of vaccine [3].

S2 Model parameters
The model parameters are now presented in detail. We begin with an estimate of the proportion of ILI presentations infected with pandemic influenza at any point in an epidemic. This is followed by a discussion of the basic reproduction number and how it compares to estimates from the 2009 pandemic. Finally, we provide the probability distributions that were used by the Latin hypercube sampling (LHS) algorithm to select parameter values.

S2.1 Estimating ILI presentations from pandemic presentations
Assuming a finite diagnosis capacity, the number of pandemic cases that are diagnosed depends on the proportion of ILI presentations that are infected with the pandemic strain (ILI % ). Victorian surveillance data from the 2009 epidemic indicates that this proportion is almost 0% early in the epidemic, raising to as high as 65% at the epidemic peak [4,5]. Combining surveillance data with hospitalisation data [6]-shown in Figure 2-permits a linear model relating ILI % to pandemic presentations to be fitted, assuming that the hospitalised cases represent a fixed proportion of pandemic presentations (we assumed 0.5% to fit the linear model presented here). We add the constraint ILI % ∈ [3%, 65%] to avoid stiffness issues with the MATLAB ODE solver, and arrive at the following relationship:

S2.2 Probability distributions for model parameters
The model parameters can be divided into two categories: those that are independent of the chosen diagnosis strategy and those that are specific to the chosen di-  Table 2 ILI presentations diagnosed as true positives s P see Table 2 ILI presentations diagnosed as true negatives Table 2 Maximum number of outpatient diagnoses (per day) e GP see Table 2 Antiviral efficacy for GP-diagnosed cases and contacts Table 1: Probability distributions for the model parameter; each parameter is associated with a beta distribution, a uniform distribution, or a single value.
agnosis strategy. The probability distributions for the strategy-independent parameters are listed in Table 1, while the values of the strategy-specific parameters are listed in Table 2. The probability distribution for α M is a function of η and is shown in Figure 3.

S2.3 Basic reproduction number
Our value of R 0 ≈ 1.4 is smaller than a conservative estimate for the early growth phase of the 2009 Victorian epidemic, where R 0 ≈ 1.6 when correcting for undetected transmission, but the same study found that R 0 < 1 except for youth-to-youth transmissions [7]. Our R 0 value is consistent with whole-wave estimates from the UK [8] and there is consistent serological evidence to suggest that the observed attack rates were low due to significant amounts of preexisting immunity [9]. Here we have assumed that the entire population is initially susceptible, which explains why our model produces more severe epidemics with R 0 ≈    Table 2: Values for parameters specific to a diagnosis strategy; MAX D = N reflects a capacity sufficient to diagnose all outpatient ILI presentations.