Table 1 Main model parameters and their default values

Population size

100,000 persons

chosen for computational purposes

Transmission-probability rate

Ν(mean 0.03; SD^∗ 0.001) per contact-day

calibrated to match French influenza-like illnesses data [35]

Reporting probability of symptomatic cases

20%

Initial percentage of immune people each year

23%

Incubation period

Γ(mean 2; var 0.1)

[33, 34]

Symptomatic period

Γ(mean 4; var 1)

[26, 27, 34, 49]

Shedding period

(1 day after asymptomatic state onset; 2 days after symptomatic state onset)

[27, 33, 34]

Asymptomatic period duration

Γ(mean 21; var 25)

[18, 37]

Infection-probability rate

4.2e-5 per day

calibrated to obtain an average annual incidence of 220 PI cases per 100,000 [38]

Symptomatic case-reporting probability

100%

all PI cases are assumed to be reported

Symptomatic period duration

Γ(mean 12; var 16)

[38]

Shedding onset in the asymptomatic phase

Day 0

[17, 39, 40]

Shedding end in the symptomatic phase

Day 2

due to severity, PI cases are assumed to be isolated after 2 days of symptoms

Acquisition interaction (α₁₂, α₂₁)

(1, 1)

to be varied

Transmission interaction (θ₁₂, θ₂₁)

(1, 1)

to be varied

Carriage rate in the population (% of asymptomatic individuals)

20%

[17, 39, 40]

Immunity-period duration (immunity reinitialized at the end of every year)

300 days

no immunity assumed for pneumococcus

Pathogenicity interaction (π₁₂)

1

to be varied

Beginning of the 2nd pathogen epidemic

U(30 – 60) days after influenza epidemic onset

can be varied according to the chosen pathogen

No. of cases when the 2nd pathogen epidemic starts

U(20 – 30)

Initial percentage of immune people

25%

Cross-immunity interaction (μ₁₂, μ₂₁)

(1, 1)

to be varied