The six models with six different reinfection mechanisms can be retrieved by adding to three common skeletons (in black) the transitions corresponding to the indicated colour. All models have five epidemiological states in common: susceptible (*S*), exposed (*E*), clinically ill and infectious (*I*), temporarily removed from the transmission process (*R*) and protected in the long-term against reinfection by the same strain (*L*). To improve biological realism, durations of the states *E*, *I* and *R* are all gamma-distributed (electronic supplementary material, text S1.1). (*a*) The 2Vi (blue) and Mut (red) models implement a widely used history-based formalism [17,18] with (*i*,*j*) *ε* {1,2}^{2} and *i* ≠ *j*. Upper index stands for the infective strain, bottom index for the already-immunized strain, *λ*_{i} = *β*_{i}(*I*^{i} + *I*_{j}^{i})/*N* is the force of infection of strain *i* and both strains are supposed to have the same mean latent, infective and temporary removed periods (electronic supplementary material, text S1.2). Hosts recovered from strain *i* enter the *L*_{i} class and become completely protected against reinfection by strain *i* while remaining susceptible to the other circulating strain *j*. For the Mut model, the two strains are supposed to have the same transmissibility (*β*_{1} = *β*_{2}, see electronic supplementary material, text S1.2) and to interact through a cross-immunity parameter *σ* *ε* [0,1] that acts by reducing the susceptibility to the other strain (electronic supplementary material, text S1.3). The dashed red arrow indicates that at time *T*_{mut} if *I*^{1}> 0, one infectious host with the initial strain (*i* = 1) becomes infectious with the mutated strain (*j* = 2). (*b*,*c*) For the AoN, PPI, InH and Win models, *λ* = *β**I*/*N* is the force of infection of the single strain. In the AoN model (red), we assume that hosts acquire full protection against reinfection with probability *α*, otherwise they re-enter the *S* class. In the PPI model (blue), we assume that all hosts develop long-term immunity that partially reduces the level of susceptibility through a protection factor *σ* *ε* [0,1]. In the InH model (green), we assume that infected hosts are able to clear the viral load with probability *α*, otherwise they suffer from an intra-host reinfection and, after some time, re-enter the *I* state. In the Win model (*c*), we assume host heterogeneity in the waiting time for acquisition of a completely protective immunity [9]: if *some* hosts re-enter the transmission process before protection is effective, they fall into a time window of susceptibility to reinfection (*W*). We simply assume that *all* hosts remain in the *W* state for a duration that is exponentially distributed: this distribution has a positive density in zero, thus enabling *some* hosts to immediately enter the *L* class (electronic supplementary material, text S1.4). Parameter descriptions can be found in . The transition rates to simulate the six stochastic models are provided in electronic supplementary material, text S1.5.

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