The parameter denotes the background birth rate, the background death rate, is the transmission rate from infected individuals I_j to susceptible individuals S_i, and is the average length of infectiousness, where i, j = 1 represents primary infections and i, j = 2 represents repeat infections. To mimic the opening and closing of schools, which affects transmission between children [40], we assume that during term time and during school holidays [41]. In addition, because we would like to focus on the relative infectiousness of repeat infections to primary infections (), and the relative magnitude of the contact rates, we rewrite the transmission rates as , and , where is the average transmission rate from individuals with a primary infection to naive individuals. Following the work of Nguyen & Rohani [42], we assume that the infectious period is gamma-distributed with shape parameter n = 4. The parameter represents the probability that susceptible (but previously infected or vaccinated) individuals, upon exposure, boost their immunity instead of becoming infectious. In the basic model, and in the immune-boosting model, .

Panel A illustrates results for the pre-vaccine era, panel B for the vaccine era assuming that , and panel C for the vaccine era fixing the average duration of vaccine-induced immunity at 10 years (). The diameter of the marker reflects the proportion of the 50 largest cities (data) or 1000 simulations (model: 50 realizations are generated for 20 different population sizes) for which spectral analysis of weekly case reports reveals that period to be the dominant signal. Any dominant period not significant at the 95% level is denoted as having a period of 0 years. The length of time series analyzed is 13 years in the pre-vaccine era, and 15 years in the vaccine era. The percentages displayed to the right of each panel are the overlap between the data and the model output. Parameter values for the model are given in Table 1, with . The population size, N, is varied from 75,000 to 1.5 million. To allow for the reintroduction of infection following extinction in a single population, we include a background force of infection of 50/million/yr (results are similar if we assume 10/million/yr). The axis representing the average duration of immunity is not to scale between 100 years () and permanent immunity ().

Panels A and B illustrate analyses of weekly fade-outs in the stochastic model in the pre-vaccine and vaccine era as the average duration of immunity () is varied. Panel C shows fade-out analyses for the England and Wales data in the pre-vaccine (blue) and vaccine (red) eras: open circles denote data points and solid lines the best-fit exponential curve. Panel D demonstrates the results of fitting model output to the fade-out curves shown in C, as assessed by the square of the residuals: the blue lines represent the pre-vaccine era; the red lines represent the vaccine era assuming that vaccine-induced immunity is lost at the rate . Solid lines denote averages and dashed lines indicate the 90% confidence envelope. For the stochastic model, weekly fade-outs are calculated as the average number of weeks per year with zero case reports, assuming a 15% primary reporting rate and 1% secondary reporting rate, averaged over 50 realizations for each population size. Parameter values for the model are given in Table 1, with . In panels A, B and D, the axis representing the average duration of immunity is not to scale between 100 years () and permanent immunity ().

All other parameters are defined in Table 1. The mathematical formula for R₀ is given in Appendix S1.

Panel A illustrates results for the pre-vaccine era, panel B for the vaccine era assuming that , and panel C for the vaccine era fixing the average duration of vaccine-induced immunity at 10 years (). The diameter of the marker reflects the proportion of the 50 largest cities (data) or 1000 simulations (model: 50 realizations are generated for 20 different population sizes) for which spectral analysis of weekly case reports reveals that period to be the dominant signal (note: as we show in Figure S11, increasing the number of stochastic realizations does not qualitatively affect our findings). Any dominant period not significant at the 95% level is denoted as having a period of 0 years. The average normalized power for each dominant signal is illustrated in Figure S1. The length of time series analyzed is 13 years in the pre-vaccine era, and 15 years in the vaccine era. The percentages displayed to the right of each panel are the overlap between the data and the model output. Parameter values for the model are given in Table 1, with . The population size, N, is varied from 75,000 to 1.5 million. To allow for the reintroduction of infection following extinction in a single population, we include a background force of infection of 50/million/yr (results are similar if we assume 10/million/yr). The axis representing the average duration of immunity is not to scale between 100 years () and permanent immunity ().

Panels A and B illustrate analyses of weekly fade-outs in the stochastic model in the pre-vaccine and vaccine era as the average duration of immunity () is varied. Panel C shows fade-out analyses for the England and Wales data in the pre-vaccine (blue) and vaccine (red) eras: open circles denote data points and solid lines the best-fit exponential curve. Panel D demonstrates the results of fitting model output to the fade-out curves shown in C, as assessed by the square of the residuals: the blue lines represent the pre-vaccine era; the red lines represent the vaccine era assuming that vaccine-induced immunity is lost at the rate . Solid lines denote averages and dashed lines indicate the 90% confidence envelope. For the stochastic model, weekly fade-outs are calculated as the average number of weeks per year with zero case reports, assuming a 15% primary reporting rate and 1% secondary reporting rate, averaged over 50 realizations for each population size. Parameter values for the model are given in Table 1, with . In panels A, B and D, the axis representing the average duration of immunity is not to scale between 100 years () and permanent immunity ().

The gamma-distributed periods are both with shape parameter k = 2, equivalent to assuming two sub-classes.