key: cord-0011874-vqwstvek authors: Lunelli, Antonella; Pugliese, Andrea title: Evaluating the effectiveness of antiviral treatment in models for influenza pandemic date: 2008-09-17 journal: Math Med Biol DOI: 10.1093/imammb/dqn018 sha: 4816ff7b6b07da5f741e3cbd3f03021c05429962 doc_id: 11874 cord_uid: vqwstvek We study the effectiveness of antiviral treatment in simple susceptible–exposed–infectious–removed models that are at the base of models used for influenza pandemic. The strategy is assessed in terms of the value of the reproductive ratio R(0). We consider a general framework and analyse six different specific cases. The same antiviral strategy is simulated in all models, but they slightly differ in the compartmental structure. These differences correspond to different underlying assumptions concerning the timing of the intervention and the selection of individuals who receive treatment. It is shown that these details can have a strong influence on the predicted effectiveness of the strategy: for instance, with R(0) = 1.8 in absence of treatment, different models predict that with treatment R(0) can become as low as 0.4 or as high as 1.3; still, in all models 70% of infected individuals are treated and the infectiousness of treated individuals is reduced by 80%. A particular assumption that can be included when modelling influenza is time-varying infectivity. We consider a specific model to verify if the predicted effectiveness of antiviral treatment is influenced by the inclusion of this assumption. We compare the results obtained with constant and variable infectivity, in relation also to the time of intervention. It is likely that existing differences in the predictions of the effect of control measures depend on such modelling details. This finding stresses the need for carefully defining the structure of models in order to obtain results useful for policymakers in pandemic planning. The recent emergence of a highly pathogenic avian influenza virus and its subsequent transmission from infected poultry to humans have raised concern about a future pandemic risk. An intensive preparedness planning is occurring in many countries and possible control measures are evaluated, often with the help of mathematical models. Since a pandemic vaccine is unlikely to be promptly available, other control measures have been considered to contain the pandemic in its earliest phases, while waiting for vaccine production and distribution. In this context, antiviral drugs are expected to play a major role both in prevention and in treatment (Balicer et al., 2004; Monto, 2003) . They can be 70-90% effective as prophylaxis and shorten the duration of the infectious period by 1-1.5 days when used in treatment (Cooper et al., 2003; Monto, 2003; Longini et al., 2004; Hayden, 2001; Regoes & Bonhoeffer, 2006; World Health Organization, 2004) . Antiviral use has been widely investigated in mathematical modelling (Arino et al., 2006; Colizza et al., 2007; Cooper et al., 2006; Ferguson et al., 2005 Ferguson et al., , 2006 Flahault et al., 2006; Gani et al., 2005; Germann et al., 2006; Longini et al., 2004 Longini et al., , 2005 Wu et al., 2006) . Some attention has been given also 360 A. LUNELLI AND A. PUGLIESE to the possible emergence of an antiviral-resistant influenza strain (Ferguson et al., 2003 , Lipsitch et al. 2007 , Regoes & Bonhoeffer, 2006 , a real threat to the effectiveness of antiviral-based policies. But results of different studies are often in disagreement: if some authors draw positive conclusions about the possibility of slowing the spread of the infection and reducing the attack rate (Barnes et al., 2007; Colizza et al., 2007; Gani et al., 2005; Germann et al., 2006; Longini et al., 2004 Longini et al., , 2005 Roberts et al., 2007) , even in circumstances in which a resistant strain spreads widely (Lipsitch et al., 2007) , others are more reluctant and suggest that a containment policy based on antivirals alone is unlikely to be successful (Ferguson et al., 2005 (Ferguson et al., , 2006 Flahault et al., 2006) . These differences depend basically on the model considered and the assumptions used in the model regarding the intervention. Since many countries plan to rely on antivirals to face the pandemic during the early months and antivirals will probably be the only pharmaceutic intervention available in the initial phase, there is an evident need to clarify if and how the evaluation of antiviral efficacy is influenced by the model assumptions. Instead of evaluating different strategies, we focus on one strategy and investigate its effectiveness in relation to the structure of the model and its underlying assumptions. We consider the simplest scheme that can be considered to model influenza spread, i.e. a deterministic homogeneous susceptibleexposed-infectious-removed (SEIR) model. Alexander et al. (2008) have recently studied the optimal scheduling of antiviral treatment through the analysis of a homogeneous SEIR model with continuous age of infection. We extend the model in the direction of considering several different options about which infected individuals are treated and when; on the other hand, we subdivide the infectious period in discrete stages. We then analyse quantitatively specific subcases of the general model; some of the cases correspond to models used in previous studies. Each case derives from the general model making precise assumptions on the time and way of intervention (in particular on the duration of infectivity before being testable and before the diagnosis), but always assuming the same fraction of infected being treated and the same efficacy of antivirals. All the models considered are easy to analyse mathematically, so that it is possible to quantify the effect of different modelling choices. These models have all constant infectivity. Through the comparison to a model that includes timevarying infectivity, we then investigate the effect of variable infectivity and how it influences the evaluation of antiviral efficacy. Many authors have investigated the effect of different antiviral-based interventions using rather complex models, including social and spatial structures, stochastic fluctuations and other factors. Even if a homogeneous model may be inappropriate to simulate a realistic influenza pandemic, it constitutes the basis of most models considered in the literature. Its transparency allows to evaluate how the structure of the model influences the conclusions about the effectiveness of antiviral treatment. Complex models are definitely more realistic and suitable to simulate a pandemic, but they may obscure the role of underlying assumptions. Our results may be useful when structuring more complex models, such as microsimulation models, and highlight the attention that should be paid to details of the model. Since the SEIR framework is always the skeleton of more complex models, the comparison between the results found with these models may help to understand the role of model assumptions in the evaluation of the efficacy of antiviral-based policies in pandemic containment. In compartmental models with an SEIR structure, the population is divided in four classes according to the disease state: susceptibles (S), contains all the individuals who can be infected; exposed (E), contains the individuals who have been infected but are not infectious yet and do not show symptoms; infectious (I ), contains infected people who can transmit the infection, and immune or removed (R), contains all the individuals who have recovered or, in the worst cases, died. For influenza, the mean latent and infectious period have been estimated to approximately 1 (Ferguson et al., 2005) and 4 (Cauchemez et al., 2004; Hyman & LaForce, 2003; Longini et al., 2004; Mills et al., 2004) days, respectively. When modelling influenza, many authors (Alexander et al., 2008; Arino et al., 2006; Chowell et al., 2007; Colizza et al., 2007; Ferguson et al., 2003; Nuno et al., 2007; Wu et al., 2006) divide the infectious period in phases to allow for asymptomatic stages, differences in infectivity or in symptoms severity. This allows also to structure treatment as administered at certain phases of infection. An alternative, which introduces an element of complexity into the model, would be to explicitly use the time-sinceinfection as a variable (Alexander et al., 2008; Grais et al., 2003; Roberts et al., 2007) . We propose a model with a general structure in which the infectious period is divided in three phases. If no treatment is modelled, individuals progress through three infectious subclasses (I 1 , I 2 and I 3 ) and finally recover. We assume that infected individuals, during the second infectious phase, may be classified (with probability p) as individuals who can receive treatment and therefore enter class Y (individuals potentially selected for treatment) at the end of the period; individuals not classified for treatment enter the third infectious stage (class I 3 ). From class I 3 individuals recover spontaneously. Individuals in class Y (suitable for treatment) have the possibility to be treated, or may recover spontaneously, before actually receiving treatment. The compartmental representation of the model is shown in Fig. 1 . Our model includes, we believe, a great variety of cases considered in the literature. Simpler models with fewer infectious stages can be obtained by formally setting equal to ∞ the exit rate from the missing stages. When simulating antiviral treatment of infected individuals, we ignore preventive antiviral prophylaxis of their contacts, which is generally part of the recommended intervention strategies. Indeed in compartmental models, as the ones we are considering (Colizza et al., 2007; Gani et al., 2005; Germann et al., 2006; Longini et al., 2004 Longini et al., , 2005 , individual contacts are not defined so that such an intervention cannot be modelled exactly, although it can be approximated by an appropriate reduction of withinhousehold transmission rates (Rizzo et al., 2008) . The classification between individuals who can receive treatment and those who cannot could depend on the severity of symptoms or behavioural or social features (geographical isolation, limited access to medical resources, tolerance of disease symptoms). We suppose in what follows that there is no difference in infectiousness between the two groups. Several authors (Colizza et al., 2007; Alexander et al., 2008) have assumed that individuals not selected for treatment are asymptomatic infectives and that they have a lower infectiousness; on the other hand, we stress the relevance of the potential presence of infectives who are as infectious as the others, but cannot be reached by treatment. Asymptomatic FIG. 1. Compartmental representation of the general model considered. Individuals are divided in classes according to the disease state: S (susceptibles), E (exposed), I 1 (infectious during the first stage), I 2 (infectious during the second stage), I 3 (infectious during the third stage), Y (infectious who can receive treatment), T (treated) and R (removed). A. LUNELLI AND A. PUGLIESE infectives with low infectivity add little to the reproduction ratio of the infection, so that ignoring them does not affect strongly the results. The reproductive ratio of the model can be easily computed using the method of Diekmann & Heesterbeek (2000) and van den Driessche & Watmough (2002) and it is given by where S 0 is the fraction of individuals initially susceptible, r represents the reduction in the transmission due to treatment (corresponding to AVE I in Longini et al. (2004) , 80% in the numerical example), β is the transmission rate, γ 1 , γ 2 , γ 3 and γ Y are the exit rates from class I 1 , I 2 , I 3 and Y , respectively, α is the treatment rate of selected individuals and λ is the recovery rate of treated individuals. We also assume that antivirals shorten the infectious period of treated individuals (by 1 day in the numerical example). The model can be viewed as an age of infection epidemic model of the type proposed by Kermack & McKendrick (1927) and analysed using the approach suggested by Brauer (2005) . The analysis shows that R 0 is a threshold value: if R 0 1, starting from any initial state S 0 , only a few new infections will occur, without a major epidemic; if R 0 > 1, starting from a large enough susceptible fraction S 0 , a major epidemic will occur; during the outbreak the number of susceptibles can only decrease and, when the epidemic dies off, will finally settle to an equilibrium value that depends on the value of R 0 . The larger the R 0 the smaller is the number of individuals who escape the infection (Diekmann & Heesterbeek, 2000) . It seems therefore adequate to judge the efficacy of antiviral treatment through the resulting reduction of R 0 , as computed from (1). An antiviral treatment is generally measured by the reduction in infectivity (r ), the reduction of the period of infectivity (that will be parameterized later), and by the fraction P of the infected who are treated. A standard computation shows that this is given by It is, however, clear from (1) that P, r and the length of the period of infectivity of treated individuals are not sufficient to obtain R 0 . In order to understand better which are the factors leading to larger or smaller reductions of R 0 , we have considered several submodels, most of which have been chosen by other authors to investigate the effect of antivirals. The compartmental representation of each model is shown in Fig. 2 . Although the formulae given above apply to the general model, all the cases we will consider in detail belong to one of two model structures: either p is equal to 1, so that all infected individuals enter the class Y and can be selected for treatment; or γ Y is equal to 0, so that all individuals entering class Y (those potentially selected for treatment) are actually treated. It will be seen later that choosing one structure or the other changes substantially the estimate of the efficacy of antiviral treatment. In the first model, all individuals are potentially treatable: when they leave the latent class they go directly to class Y , the only infectious class. From then on, they either enter the class of treated individuals (at rate α) or recover (at rate γ Y ). This is a special case of the general model and may be obtained by letting γ 1 and γ 2 go to infinity and setting p = 1. According to (2), the overall probability of being treated is given by α/(α + γ Y ). A model with this structure has been previously used by Flahault et al. (2006) to simulate antiviral treatment of cases. In the second model, we assume that as the individuals leave the latent class, they are immediately classified either (with probability p) as individuals who will be treated (subgroup Y ) or (with probability 1 − p) not (subgroup I 3 ). Individuals in subgroup Y will enter the group of treated individuals at rate α, FIG. 2. Compartmental representation of the six models considered. Individuals are divided in classes according to the disease state: S (susceptibles), E (exposed), I 1 , I 2 and I 3 (infectious in different stages), Y (selected for treatment), T (treated) and R (removed). The transmission rate β is assumed to be constant. while those in subgroup I 3 will recover at rate γ 3 . This model may be obtained by letting γ 1 and γ 2 go to infinity and setting γ Y = 0 in the general model. It has the same structure as the models proposed by Alexander et al. (2008) and Chowell et al. (2006) , even if in their models only a fraction of individuals selected for treatment are actually treated (i.e. γ Y = 0) and individuals in class I 3 are considered as asymptomatic with reduced infectivity. Influenza is characterized by a short incubation period, a high attack rate and a lack of diseasespecific symptoms (Balicer et al., 2004) . All these epidemiological characteristics can impose difficulties in identifying cases promptly when they enter the infectious class and may cause a delay in treatment. This aspect has been considered in several studies (Ferguson et al., 2005 (Ferguson et al., , 2006 Germann et al., 2006; Longini et al., 2005) , in which intervention has been postponed to the second or third day after symptoms onset. Therefore, all the following models include a delay in treatment, assuming that some time is needed to identify cases and organize treatment. This period might be considered also as an infectious but asymptomatic stage. In the third model, we assume that at the end of this phase, infectious individuals are either identified and treated or enter the class I 3 and will not be treated. A similar model has been proposed by Gani et al. (2005) , and Ferguson et al. (2003) have introduced an analogous mild asymptomatic infection stage in their model. The model is obtained by letting γ 1 and α go to infinity. Models 4 and 5 integrate the presence of the first phase of unrecognized infection with the treatment scheme used in Models 1 and 2, respectively. In Model 4, after a first infectious phase, all individuals are potentially treatable (class Y ). Then they either enter the treated class (at rate α) or recover (at rate γ Y ). To obtain this model, we have set p = 1 and let γ 2 go to infinity in the general model. In Model 5, after a first infectious phase, individuals are assigned either to subgroup I 3 or to subgroup Y . Individuals in subgroup Y will enter the group of treated individuals at rate α, while those in subgroup I 3 will recover at rate γ 3 . This is obtained by letting γ 2 go to infinity and setting γ Y = 0. A model with this structure has been considered by Wu et al. (2006) (with hospitalized instead of treated individuals) to include an initial asymptomatic phase of the infectious period, while similar infectious stages, but in a more complex model, have been proposed by Nuno et al. (2007) . As Models 4 and 5 correspond, respectively, to Models 1 and 2, Model 6 is comparable to Model 3. After the phase I 1 , individuals enter class I 2 during which they are either identified and treated or enter the third stage of the infectious period and then recover. This model is obtained from the general model by letting α go to infinity. For each model, we have computed the average infectious period T I in absence of intervention, the average infectious period T AV for a treated individual and the average infectious period T noAV for a non-treated individual. We will use their mathematical expressions, given in Table 1 , for parameter calibration. The reproductive ratio R 0 is computed directly from (1), using the assumptions on the parameters made for each model. In order to compare the values for R 0 found in different models that include different parameters, it is necessary to properly calibrate the parameters. We have estimated the values of the parameters to investigate the effect of intervention on the value of R 0 and how this effect vary when we consider different models. First of all, we require that, in absence of antiviral treatment, the mean infectious period has to be the same (4 days) in all models. This implies the condition T I = 4, where T I is the mean infectious period in absence of treatment and is given, for each model, in Table 1 . Secondly, the probability of receiving treatment, computed using (2), is the same (P = 0.7) in all models. In Models 1 and 4, this probability is given by α/(α + γ Y ), and thus determines the value of α, while in Models 2, 3 and 5 it is represented by p and so we are free to set α. In Models 1 and 2, individuals receive treatment, on average, 1/α days after leaving class E; therefore, in Model 2 we have kept α as in Model 1. Analogously, in Model 5 we have taken it as in Model 4 (individuals receive treatment, on average, 1/α days after leaving class I 1 ). In Model 3, we assume that individuals have the possibility to be treated 1 day after becoming infectious. This could be due, e.g. to a first asymptomatic phase. Therefore, we set 1/γ 2 = 1. Then from the relation 1/γ 2 + 1/γ 3 = 4, we can estimate γ 3 . In Models 4-6, we have introduced a 1-day delay in treatment administration, which gives 1/γ 1 = 1. Using the assumption of a 4-day natural infectious period, we can estimate γ Y and γ 3 . As for the effect of antivirals, we assume that the infectiousness is reduced by 80% by antivirals, hence r = 0.2. Finally, we need to establish the value of λ, reflecting the shortening of the infectiousness period of treated individuals. A commonly used assumption (see, e.g. Colizza et al., 2007) is that the infectious period of treated individuals T AV is 1 day shorter than T noAV , the infectious period of untreated individuals. However, Table 1 shows that, for Models 1 and 4, T AV > T noAV ; hence, it is not possible to require T AV = T noAV − 1. In other words, the time spent in the infectious class by a treated individual is on average longer than the infectious time of an individual who does not receive treatment. Nevertheless, if 1/λ = 1/γ Y , i.e. if treatment has no effect on the duration of the infectious period, on average the individuals will stay in the infectious class 1/γ Y days, as one would expect; however, those being treated stay there longer than the average, while those not being treated less than the average. This apparently bizarre fact comes from the assumption that being treated and recovering are two competing risks; hence, individuals who receive treatment are those who naturally would have a longer infectious period. To overcome this problem, we choose to assume, following Colizza et al. (2007) , T AV = T I − 1, which allows us to find λ. For Models 2, 3, 5 and 6, since T noAV = T I , this makes no difference, and we can see from Table 1 that this relation can be used to obtain λ as long as 1/γ 3 − 1/α > 1 (for Models 2 and 5) or 1/γ 3 > 1 (for Models 3 and 6). Similarly, for Models 1 and 4, provided that 1/γ Y − 1/(α + γ Y ) > 1, the relation T AV = T I − 1 allows us to find λ. In Models 1 and 4, we actually treat individuals with an infectious period longer than the average T I , so the assumption T AV = T I − 1 may be too optimistic. Another possibility would be to consider T * AV , TABLE 1 Results found by analysing Models 1 to 6. T I , T AV and T noAV are, respectively, the average infectious period in absence of intervention and of treated and untreated individuals with intervention. R 0 is the reproductive ratio of the model, found from (1) as β S 0 times the expression reported in the last column; β is the transmission rate of untreated individuals and S 0 is the initial fraction of susceptible individuals; other parameters can be seen in Fig. 2 Model no. defined as the infectious period of treated individuals when treatment has no effect (i.e. λ = γ Y ), and to require T AV = T * AV − 1, which corresponds to 1/λ = 1/γ Y − 1. Results do not differ significantly; the highest variation is found with Model 1: if we assume a reproductive ratio of 1.8 in absence of treatment, we obtain R 0 = 0.65 with the hypothesis T AV = T I − 1 and R 0 = 0.74 with the hypothesis T AV = T * AV − 1. Therefore, we present only the results obtained with the first hypothesis. The parameter values used are given in Table 2 . The general model without intervention is characterized by an average infectious period 1/γ = 1/γ 1 + 1/γ 2 + 1/γ 3 , the sum of the average duration of each infectious phase. Its reproductive ratio is given by R 0 = β S 0 /γ (i.e. (1) with p = 0), where β is the transmission rate and S 0 the initial fraction of susceptible individuals. Using the parameter values given in Table 2 , we have evaluated the effectiveness of the same intervention strategy when implemented in different ways, as described by the models considered. As discussed above, this evaluation has been made in terms of R 0 : we have computed the ratio between the R 0 of the model without intervention and the reproductive ratio of the model with intervention for each of the models investigated. This ratio tells how much the reproductive ratio is reduced under antiviral treatment. Results are given in Table 2 and show how the reduction is very sensitive to the assumptions made when modelling the intervention. In Models 1 and 2, individuals have the same probability of receiving treatment and, on average, they are treated after 1.7 days in both models. But in Model 1 the intervention seems to be much more effective. With an hypothetical R 0 of 1.8, a value commonly used to simulate a future pandemic (Ferguson et al., 2005) , using Model 1 we would conclude that antivirals are able to contain the pandemic, reducing the value of R 0 below 1. The same conclusion is not reached using Model 2. As expected, the introduction of a delay in antivirals administration reduces significantly the effectiveness of the control measure. This can be observed by comparing Models 1, 2 and 3 with Models 4, 5 and 6, which are their respective refinements. Assuming hypothetically R 0 = 1.8, antiviral treatment 367 would reduce it to 0.65, 1.17 and 0.99 if simulated with Model 1, 2 or 3 and to 0.9, 1.3 and 1.24 with Model 4, 5 or 6, where a 1-day delay has been included. We have assumed a treatment delay of 1 day, but some authors (Ferguson et al., 2005) have considered a delay of 2 days. To investigate the effect of a longer delay, we can compare, for example, Model 3 and Model 6. With a 1-day delay (Model 3), we have R 0 = 2.2β S 0 . A 2-day delay (Model 6) gives R 0 = 2.76β S 0 . As expected, a longer delay in antivirals administration reduces significantly the effectiveness of the intervention. Isselbacher et al. (1994) have observed the natural course of influenza and have reported its clinical characteristics in an otherwise healthy 28-year-old male. According to them, the virus shed is maximal 2 days after the onset of illness and then decreases and reaches a minimum on day 5. Taking these results into consideration, it is reasonable to assume that the infectivity of an individual varies in time, determining a variability in the transmission rate. To assess the importance of considering different levels of infectivity, we have designed a specific model that allows us to compare the results obtained by assuming constant or varying infectivity. The model follows basically the structure of Model 6, but we assume that treated individuals follow an infection path similar to the untreated ones, with two phases characterized by different infectivity. Although other model structures are certainly possible, this allows us to understand the interaction of treatment timing with variable infectivity. Precisely, we make the following assumptions: without treatment, after the latent period individuals go through three infectious stages, each characterized by a specific infectivity, and then recover. According to the results of Isselbacher et al. (1994) , we assume a first low-infectivity stage lasting 1/γ 1 = 1 day, followed by a second stage with high infectivity lasting 1/γ 2 = 2 days and by a third stage again with low infectivity lasting 1/γ 3 = 1 day. Varying infectivity is translated in non-constant transmission rates. Mimicking the results of Isselbacher et al. (1994) , we have assumed β 1 = β 3 = 3/5β 2 , thus representing lower infectivity during the first and third stage. β 1 , β 2 and β 3 are the transmission rates during the three infectious stages, respectively. Infected individuals can receive treatment at the end of the first stage (with probability p 1 ), thus entering class T 2 , or after the second stage (with probability p 2 ), entering class T 3 . The transmission rate of treated individuals is reduced by a factor r , as in previous models, and therefore it will be equal to rβ 2 in class T 2 and rβ 3 in class T 3 . We further assume that individuals stay in class T 2 1.5 days before advancing to class T 3 , while individuals treated after the end of the second infectious stage recover after 0.5 days. The compartmental representation of the model is given in Fig. 3. FIG. 3. Compartmental representation of the model considered to include varying infectivity. Individuals are divided in classes according to the disease state: S (susceptibles), E (exposed), I 1 , I 2 and I 3 (infectious in different stages), T 2 (treated at the end of the first infectious stage), T 3 (treated at the end of the second infectious stage) and R (removed). The reproductive ratio of the model is given by where λ 1 and λ 2 are the recovery rates of treated individuals. The probability of receiving treatment in the model considered is given by P = p 1 + (1 − p 1 ) p 2 and we have set it equal to 0.7, coherently with the previous numerical examples. Defining Q = p 1 P as the proportion of individuals treated after the first infectious phase, we have investigated the dependence of the effect of antiviral treatment on the timing of intervention. Namely, varying Q between 0 and 1 we change from a scenario where all the treated individuals receive prophylaxis after the second infectious stage (very late) to a scenario where treatment is administered to all selected individuals after the first infectious stage (i.e. 2 days in advance). Further, for a given Q, we can compare results obtained with varying and constant infectivity, obtained by setting β 1 = β 2 = β 3 . Figure 4 shows that introducing variable infectivity can influence the results, although to a limited extent quantitatively, and makes the time of intervention even more crucial in the evaluation of the effectiveness of antiviral treatment. As expected, the higher the proportion of individuals treated after the first phase, the more effective the intervention is, both with variable and with constant infectivity. For example, with varying infectivity, assuming R 0 = 1.8 in absence of treatment, we obtain R 0 = 0.92 if we treat all the selected individuals after the first phase (Q = 1) and R 0 = 1.6 if we treat all the selected individuals 2 days later (Q = 0). From Q = 0 to Q = 1, R 0 decreases linearly. In the case of constant infectivity, the results are analogous, but R 0 varies only from 0.97 to 1.51. FIG. 4 . Relative effectiveness of antiviral treatment, computed as the ratio between the reproductive rate with intervention (model in Fig. 3 ) and the reproductive rate of the plain SEIR model. In the model, individuals may be treated at the end of the first or second infectious phase. Q represents the proportion of treated individuals who receive prophylaxis after the first phase. The graph shows results for varying and constant (i.e. β 1 = β 2 = β 3 = β T = β) infectivity. We have considered different models for an epidemic with antiviral treatment. All models have an SEIR structure and derive from the same general model. We have shown that details in the model assumptions can strongly influence the evaluation of antiviral treatment as a containment measure for pandemic influenza. It must be remarked that although the compartmental structure of some models considered may appear unusual, they are all quite natural and suitable to simulate the intervention; some have indeed been used in previous studies. As discussed in Section 2, there is an implicit difference between Models 1 and 4, on one side, and Models 2, 3, 5 and 6, on the other: in Model 1 (and 4), the individuals who do not get treatment are those who recover faster than they can be targeted for treatment; this has the consequence, already discussed, that the average infection period of untreated individuals is shorter than the average infectious period in the absence of intervention. In Model 2 (and 3, 5 and 6), it is assumed that infectives can be in principle distinguished between those who will be treated and those who will be not; the average infection period of untreated individuals (as well as their infectivity) is exactly the same as the average infectious period in absence of interventions. From the results shown in Table 2 , it can be seen that there is indeed a corresponding difference in the reduction of R 0 because of antiviral treatment between the two groups of models. This can also be seen in the formula for R 0 : in Models 1 and 4, the probability of receiving treatment is given by P = α α+γ Y and the mathematical expression of R 0 can be rewritten as γ 1 in Model 4 . In Models 2 and 5, P = p and R 0 = β S 0 (1 − P) 1 γ 3 + P( 1 α + r λ ) + β S 0 1 γ 1 in Model 5 . Considering that γ Y = γ 3 in Models 1 and 4, we can see that the difference between them is in the term Pβ S 0 α , the force of infection of treated individuals during the period before treatment starts. In other words, the value of R 0 in Models 1 and 4 looks as if we were ignoring the fact that treated individuals are infectious before receiving treatment. These results show that the question of who is treated is decisive: it is very different if treated individuals are those, for one reason or another, outside the reach of the health system, if they are asymptomatic with low infectivity, or those who recover faster. These assumptions are often implicitly included into the structure of the model that should therefore be chosen carefully. A second factor strongly affecting the effectiveness of intervention is the timing of treatment. This can be seen by comparing Models 1, 2 and 3, on one side, with Models 4, 5 and 6, that are analogous, except that a first infectious period is added, where no treatment is possible. Clearly, the inclusion of a time delay in drug administration reduces significantly its impact on the dynamics of the epidemic. Time-varying infectivity makes timing of intervention even more crucial. In fact, if infectivity is lower in the first and last stage of the infectious period and higher in the middle stage, a late intervention is even less effective than in the case of constant infectivity: at the end of the middle stage, an individual will have already infected almost all the individuals he would eventually infect. On the other hand, missing treatment in the first infectious stage is less crucial, since few individuals would be infected anyway during that stage. This can be seen from Fig. 4 that shows the effectiveness of intervention as a function of the proportion Q of individuals treated after the first stage: there exists a threshold value Q t (in the numerical example Q t ≈ 0.53) such that if Q < Q t , the intervention is more effective if infectivity is constant than if it is variable (most individuals treated after the second stage), while it is less effective if Q > Q t (most individuals treated after the first stage). Our study shows that when studying the effectiveness of antiviral treatment, much attention should be paid to the assumptions (often implicit) about the timing of intervention and the individuals who get treatment: even if the same intervention is apparently being modelled, different models can lead to different conclusions. The detailed structure of the model is very relevant and should be carefully evaluated and specified when assessing the importance of the results. Although the models considered are all SEIR-type models for a homogeneous population, the results immediately translate to more complex SEIR models used to simulate an influenza pandemic. In fact, R 0 for an epidemic in a metapopulation is strongly influenced by the value of R 0 in each population in isolation (Diekmann & Heesterbeek, 2000) and may even be the same under some special choices of the contact matrix (Colizza et al., 2007) . Individual-based models (Ferguson et al., 2005) are more flexible and can incorporate detailed assumptions about the timing of infectiousness and antiviral use, as well as allowing for antiviral prophylaxis of case contacts. Still, the results of this paper stress the need of making consistent and realistic choices when building any kind of model, and especially of making them transparent. Different results on the evaluation of containment strategies may depend on hidden assumptions in the model structure. Hence, the structure of models has to be carefully defined in order to obtain results that can be useful for policymakers in pandemic planning. 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This work has been partially supported by Provincia Autonoma di Trento, Italy, under the project EPIdemics description and COntrol (EPICO).