Does the Effectiveness of Control Measures Depend on the Influenza Pandemic Profile?

Background Although strategies to contain influenza pandemics are well studied, the characterization and the implications of different geographical and temporal diffusion patterns of the pandemic have been given less attention. Methodology/Main Findings Using a well-documented metapopulation model incorporating air travel between 52 major world cities, we identified potential influenza pandemic diffusion profiles and examined how the impact of interventions might be affected by this heterogeneity. Clustering methods applied to a set of pandemic simulations, characterized by seven parameters related to the conditions of emergence that were varied following Latin hypercube sampling, were used to identify six pandemic profiles exhibiting different characteristics notably in terms of global burden (from 415 to >160 million of cases) and duration (from 26 to 360 days). A multivariate sensitivity analysis showed that the transmission rate and proportion of susceptibles have a strong impact on the pandemic diffusion. The correlation between interventions and pandemic outcomes were analyzed for two specific profiles: a fast, massive pandemic and a slow building, long-lasting one. In both cases, the date of introduction for five control measures (masks, isolation, prophylactic or therapeutic use of antivirals, vaccination) correlated strongly with pandemic outcomes. Conversely, the coverage and efficacy of these interventions only moderately correlated with pandemic outcomes in the case of a massive pandemic. Pre-pandemic vaccination influenced pandemic outcomes in both profiles, while travel restriction was the only measure without any measurable effect in either. Conclusions Our study highlights: (i) the great heterogeneity in possible profiles of a future influenza pandemic; (ii) the value of being well prepared in every country since a pandemic may have heavy consequences wherever and whenever it starts; (iii) the need to quickly implement control measures and even to anticipate pandemic emergence through pre-pandemic vaccination; and (iv) the value of combining all available control measures except perhaps travel restrictions.

To follow the evolution of the disease in each city, two additional state variables are introduced: o W ik (t): number of individuals who become infectious on day t (daily incidence) in class k; o B ik (t): number of new infectious individuals in class k reported to the health authorities on day t (computed as a fraction of W ik (t)); -for each city i and for each sub-group k, the population size (n ik ) is assumed constant over time:  (1) where τ 1 and τ 2 are the maximum length of the latent and infectious periods respectively. Since the simulation horizon was relatively short, no natural demographic dynamic was included.
-the infection distributions (probability of being in a given state or transition probability from one state to another) are defined, by the following discrete probability distributions similarly to [2] o f(τ): the probability for an individual to be in the latent state (τ = 0,1,…, τ 1 , f(0)=1); o g(τ): the probability for an individual to be in the infectious state (τ = 0,1,…, τ 2 , g(0)=0); o h(τ): the probability for an individual to be in the removed state (τ = 0,1,…, o γ(τ): the probability that a latent individual becomes infectious on day τ+1 given that he was latent on day τ (τ = 0,1,…, τ 1 ) : the probability that an infectious individual recovers on day τ+1 given that he was latent on day τ (τ = 0,1,…, τ 2 ) These distributions were fixed at values close to those of Rvachev and Longini [2], calculated to reproduce the 1968 pandemic.
-the infection process is described in each city by a separate but identical set of equations. Some parameters of these equations are equal for all cities; others are specific for each city. -the global spread of influenza is modelled by a symmetric matrix connecting all the cities, its elements being defined as the daily passenger flow from a city to another. Only susceptible and latent individuals travel (infectious individual do not). -the model takes into account the seasonal pattern followed by influenza (high winter and low summer incidence in Northern hemisphere) and the delay of approximately 6 months in influenza activity between the Northern and Southern hemispheres. Thus, cities are divided into three zones according to the geographical position: Northern hemisphere, Southern hemisphere and equatorial zone. Following standard formulation for including seasonality in the transmission rate in influenza models [5], we found the function including harmonic terms that fits original estimates of the transmission parameter values cited in Grais et al [3]. The formulation of this function is as follows: We assume that transmission rates are constant over one month. To distinguish the Northern and the Southern hemispheres we took a shift of phase in cosine arguments but equal values for β 1 . Cities from equatorial zone are not affected by the seasonal trends (β 1 =0). β 0 represents the basic rate of transmission in the absence of any seasonality character of transmission and β 1 is the amplitude of seasonal effect.

Modeling of interventions
Six prevention and control measures are considered in the model. These measures are applied, specifically for each city and for each sub-group inside the city, from a given date or if the number of total reported infectious cases since the beginning of the pandemic is above a predefined threshold.
1. Antiviral prophylaxis reduces the transmission rate and the probability corresponding to a change in state (latent -> infectious) accounting for (i) the reduced susceptibility of treated individuals (ii) the reduction of the probability of an infection to be symptomatic (and hence infectious) for a treated individual and (iii) the reduction of the infectiousness of infected individuals previously prophylactically treated. Susceptible individuals treated are given a single course of antivirals. Once the duration of the antiviral prophylaxis is finished, the individuals are assumed to re-enter the untreated susceptible compartment. We assume that all the prophylactically treated individuals continue to receive antivirals (as therapy) if they become infectious.

2.
Masks use applied to susceptible and latent people, as prophylactic intervention, reduces the transmission rate, illustrating the decrease in the probability of becoming infected for an individual using a mask given contact with an infectious person. Once the use of masks implemented, independently of the use of other prophylaxis interventions, this measure is assumed to be applied during the entire duration of the pandemic.

3.
Vaccination as prophylactic measure was modeled by a parameter diminishing the number of susceptible individuals. Two policies of administration were considered here. First, vaccination with pre-pandemic influenza vaccines was globally modeled by a coefficient affecting the number of susceptible individuals and representing the global effect of the policy in population. Second, a pandemic vaccination campaign (with vaccine updated for matching pandemic circulating strains) was introduced by taking into account vaccination coverage and vaccine efficacy. During vaccination campaign, the proportion of population to be vaccinated is specified daily.

4.
Limitation of air travel between cities was modeled by the reduction of the entries of the transportation matrix, specifically for each origin-destination city pair.

5.
Antiviral therapy diminishes, for infectious treated individuals, the transmission rate (illustrating the reduction of infectiousness of those individuals) and the length of the infectious period and thus the probability of the transition infectious -> recovered. 6.
Isolation was applied to non-treated and treated infectious individuals but not to latent individuals (who are not symptomatic). Isolated individuals do not spread infection. In the mathematical formulation, the implementation of these interventions results in the introduction of five supplementary state variables for every sub-group within each city: Three additional compartments (V ik , S M ik and S PM ik ) explicitly represented in the Figure  1 of the main text are only implicitly calculated, as part of S ik , S P ik , E ik , E P ik dynamics. As the model includes all six interventions described above, simulations may be performed including all, none or several of the control measures implemented. When a specific measure is not implemented all the corresponding parameters are set to zero.
In the current analysis the parameters related to interventions were equal for all cities.

New infections within each city
The infection process is generated using the standard mass action formulation. The number of newly infected individuals (in latent state) on day t in each class k of every city i is calculated as the product of the number of susceptibles, number of infectious individuals and the transmission rate, β ijk (taken identical for all sub-groups in a city) affected by coefficients modelling the interventions: Another class of latents, E P , those coming from susceptibles treated by prophylaxis, is also incremented each day by new infections. Since an infected individual in latent state -here equivalent to incubating -does not exhibit any symptom, he continues to be considered susceptible and hence to be given propylaxis if that was the case before infection.
where P ik c is the coverage of antiviral prophylaxis.

Travel between cities, transportation operator
The transportation network is quantified by passengers flows between cities: σ ilk represents the average daily passenger from city i to city l belonging to the class k.
Susceptible and latent individuals are assumed to travel proportionally to their fraction in each city. As in [2], a transportation operator is applied to the dynamics of the susceptible and latent persons (equations 6 and 7): where L represents the number of cities, here L = 52. In both equations Tr ik c and Tr lk c represent the proportions of transportation that is stopped in corresponding cities.
The same operator is applied to the susceptibles and latents treated by prophylaxis (S P and E P respectively).
For each pair of cities of the network, the distribution of travellers in sub-groups is assumed to be equal to the mean of the two demographic distributions (those of the origin and de destination cities).

Infection dynamics within and between cities
Modelling the potential natural immunity acquired during previous infections with similar strains, the initial number of susceptible individuals in class k of city i is assumed to be a proportion α ik of the population in this subgroup of the city, n ik . The global potential effect of the pre-pandemic vaccination is integrated by a supplementary parameter v ik affecting the number of susceptible individuals.
The dynamics of susceptibles treated by prophylaxis is obtained in a similar manner: In the computational algorithm, the initial number of latent individuals is strictly positive for the city representing the pandemic source only and is zero for all other cities.
The dynamic of the latent individuals receiving or not antivirals as prophylaxis is given by equations (11) and (12): operator Ω) who become infectious plus the number of infectious persons at the previous time step who remain infectious and minus those who are isolated. If the delay from infection is between the maximum length of the latent period, τ 1 , and the maximum length of the infectious period, τ 2 , the equation only includes the infectious individuals at the previous time that did not recover minus those who are isolated. The initial number of infectious individuals is zero for all cities. e represents the efficacy of the prophylaxis on the probability of illness given infection. In equations 14 and 16, transition probability I->R is different for individuals receiving treatment ( T ik δ ) from that of those untreated, as a consequence of an assumed shorter infectious period (by one day) under therapy.
The incidence on day t+1 (equation 17) is calculated as sum of all new infectious (viewed as the product of the transportation operator applied to the latent individuals on day t and the transition probability from the latent to the infectious state). The daily reported incidence is simply a fraction of the daily incidence, ρ ik denoting the reporting rate (equation 18