## Results: 6

1.

Example of the three typical epidemic profiles generated by the SEICWH model, depending on the value of

*R*_{0}. A:*R*_{0}= 1.4, B:*R*_{0}= 4 and C:*R*_{0}= 10. These values reflect the tendency of the contact rate among individuals to increase as the community becomes smaller. Upper panels: contribution of infection and HR to the time series of the prevalence. The dashed line represents the average daily inflow of unprotected individuals (). Lower panels: time course of the effective reproduction number*R*_{e}(*t*). The solid line represents the threshold*R*_{e}= 1, above which the epidemic can grow. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)2.

Detailed analysis of the 10

^{5}realizations of the stochastic SEICWH model for the 1971 TdC epidemic using Gillespie's algorithm (Gillespie, 1977). Upper panel: original incidence data (black dots) and expected incidence (red line) conditioned on non-extinction under the best fit model together with 50 and 95 percentile intervals (red envelopes) due to demographic stochasticity. This figure demonstrates that the best fit of the SEICWH model captures the shape and the dynamics of the data. Lower panel: time course of the extinction probability*p*(*t*) defined as the probability that the epidemic has faded out by time*t*and estimated by the proportion of fade-out realizations at time*t*. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)3.

Dynamics of the immune response, under the SEICWH model, inferred from the 1971 TdC epidemic. At the population level, our framework allows us to reconstruct the proportion of individuals that are infectious (dotted black line), short-term protected thanks to the innate and cellular immunity (dot-dashed black line), protected on the long-term thanks to antibodies (dashed black line) and unprotected (solid black line) by interval since symptom onset. These proportions correspond, respectively, to the probabilities

*P*_{1–4}(*τ*) described in section “Quantities of epidemiological interest”. The dashed red line corresponds to the proportion of individuals seroconverted by interval since symptom onset obtained by Baguelin et al. (2011). This study involved 115 individuals infected during the 2009 A/H1N1 pandemic and the seroconversion interval of each individual was defined as the time taken to reach an HI titre of ≥32. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)4.

Change in the fractions of infected (

*F*_{I.}, upper panel), reinfected (*F*_{II.}, middle panel) and unprotected (*F*_{IS}, lower panel) individuals at the end of the epidemic as a function of*R*_{0}. Each colour refers to an epidemic profile of Fig. 4 (bell: green, tail end: orange, two-peaks: violet). The impact of the reinfection dynamics on*F*_{I.}can be obtained by subtracting the same fraction expected under the SEIR model (solid line, upper panel). The expected fraction of individuals reinfected following a lack of humoral response corresponds to*αF*_{I.}−*F*_{IS}(dashed line, middle panel). The HR threshold is plotted as a dotted line and estimates of*R*_{0}for the 2009 A/H1N1 pandemic (≈1.4, black triangle) and for the 1971 TdC epidemic (≈12, black dot) are also mapped. Note the log-scale on the*x*-axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)5.

Mechanistic modelling of the primary immune response to influenza. (A) Schematized dynamics of the viral load as well as the innate and adaptive immune responses, as described in section “The primary immune response to influenza infection in humans”. (B) The SEICWH model. The six immunological stages are

*S*: susceptible,*E*: exposed,*I*: clinically ill and infectious,*C*: temporarily protected by the cellular response,*W*: temporarily susceptible,*H*: protected on the long-term by the humoral response. The number of sub-compartments in each immunological stages corresponds to the shape of the Erlang distribution for the residence time in this stage (see section “Mechanistic modelling”). The infection force is*λ*=*β*(*I*_{1}+*I*_{2})/*Ω*. A description of the parameters can be found in Table 2. The transition rates used to stochastically simulate the model are provided in Table 1. The set of ordinary differential equations used for deterministic simulations can be found in Text S3.6.

Implication of immunodynamics for the first post-pandemic season. Upper panels: expected fraction of individuals infected at least once by a mutant strain during the post-pandemic season (

*F*_{I.}post-pdm, colour coded), as a function of the relative increase in transmissibility (Δ*R*_{0}/*R*_{0}∈ [0, 1]) and the level of immune escape (*σ*∈ [0, 0.5]) to the pandemic strain (the choice of these parameter ranges is justified in Text S5 based on published studies). Results are given for five reasonable values of*R*_{0}for pandemic scenarios (between 1.2 and 2). For each value of*R*_{0}, the expected fraction of protected () and unprotected () individuals at the end of the pandemic are also given. The fraction is therefore partially protected against the mutant strain by a factor of reduction of susceptibility 1 −*σ*(see also Text S5). Finally, the fraction of infected individuals during the pandemic season (*F*_{I.}pdm) is also mapped as a black dotted isocline for comparison with*F*_{I.}post-pdm (colour-coded). Lower panels: effect of assuming that all infected individuals develop an efficient humoral response during the pandemic and thus a partial-protection against the mutant strain. We compared the SEICWH model with initially partially protected individuals with a SEIR model with as initial condition and plotted the difference (Δ*F*_{I.}post-pdm, colour-coded). Since the SEIR model overestimates the population herd immunity, it predicts a greater invasion threshold for the post-pandemic mutant (isocline*R*_{e}= 1, black solid line) than the SEICWH model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)