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"Our model suggests three specific risk factors that favour the emergence and establishment of a vaccine-resistant strain that are intuitively obvious: high probability of initial emergence of the resistant strain, high number of infected individuals54 and low rate of vaccination55. By contrast, a counterintuitive result of our analysis is that the highest risk of resistant strain establishment occurs when a large fraction of the population has already been vaccinated but the transmission is not controlled. Similar conclusions have been reached in a SIR model of the ongoing pandemic56 and a model of pathogen escape from host immunity57. Furthermore, empirical data consistent with this result has been reported for influenza58. Indeed, it seems likely that when a large fraction of the population is vaccinated, especially the high-risk fraction of the population (aged individuals and those with specific underlying conditions) policy makers and individuals will be driven to return to pre-pandemic guidelines59 and behaviours conducive to a high rate of virus transmission60,61. However, the establishment of a resistant strain at that time may lead to serial rounds of resistant strain evolution with vaccine development playing catch up in the evolutionary arms race against novel strains."
Each day and for every individual infected with the wildtype strain, Iwt, there is a small probability p, that a vaccine-resistant strain emerges in that individual. Then this individual switches from state Iwt to state Ir. Conversely, any individual infected with the resistant strain, Ir can revert back to the wildtype strain, Iwt, with the same probability p.
Our extension of the SIR Model features 8 distinct states. Susceptible, S, and recovered, R, individuals are vaccinated over time to become vaccinated, V, or recovered vaccinated, RV, respectively. Susceptible individuals can become infected with the wildtype, Iwt, or the resistant virus strain, Ir. While the vaccinated population is immune to the wildtype, it can be infected by the vaccine-resistant strain, in which case the state is represented by IrV. After a while any infected individual recovers or dies, D. Finally, we assume that the recovered population retains natural immunity towards both strains, but becomes susceptible again with some small rate, μ. In our model, immunity against the wildtype strain gained through vaccination is not lost during the entire model period of 3 years, consistent with current estimates71,72.
We also assume that the immune response provided by the vaccine is more permanent and that immunity provided by infection, is lost at rate μ, on average after 0.5 years2,71,72,74 after recovery. Both of these assumptions influence the model when the number of infected individuals becomes large, which is unlikely for realistic average rates of transmission across the simulated time.