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 , we assume that

during term time and

during school holidays . 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 , 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,

.

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