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Age groups and contacts

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People of different ages mix differently: children have the most contacts, and most of them with other children. An age-structured model splits each state by age group, and a contact matrix says how many contacts a day a person in each group has with people in each other group. With contacts, β is the rate of infection per contact, and the force of infection on group i is β times the sum over groups j of the contacts of i with j times the fraction of j who are infectious. This page builds an age-structured SIR with a contact matrix written by hand, as in the age-structured models of Keeling and Rohani (2008, chapter 3). Then it uses a real country’s population and contact matrices, one for each setting (home, school, work, and the rest of the community), and closes schools by removing one of them.

The model

Parameter Value Meaning
β 0.05 per contact Rate of infection per contact with an infectious person, per day
Infectious period 10 days Mean time in I (exponential)
Age groups 0–14, 15–64, 65+ With 30%, 50%, and 20% of people
Contacts see the matrix Contacts a day of a person in each group (rows) with each group (columns)
Population 100,000
Infectious at the start 0.1% In each group

The contact matrix, with the rows and columns in the order 0–14, 15–64, 65+:

0–14 15–64 65+
0–14 8 5 2
15–64 3 6 2
65+ 3 5 3 
import starsim as ss

S, I, R = ss.states('S I R')
infection = S >> I @ 0.05*I
recovery  = I >> R @ ss.days(10)
sir = ss.Disease(infection, recovery, init=dict(I=0.001), name='sir')
sir.stratify(age={0: 0.3, 15: 0.5, 65: 0.2})  # The lower edge of each age group, and its fraction of people

contacts = ss.contacts(ages=[0, 15, 65], matrix=[[8, 5, 2], [3, 6, 2], [3, 5, 3]])

The diagram, from diagram(), is the same as an SIR’s: the age groups split each state, and don’t change the transitions.

sir.diagram()

Result

sim = ss.Sim(sir, networks=contacts, total_pop=100_000, stop=ss.days(100))
sim.run()
sim.results.sir.I.by('age').plot()
Sim: sir, 100,000 people, as an ODE, days 0 to 100
  sir: R0 = 6.29, with 100 infectious at the start
  Done in 0.0024 s; sir: 99,592 infections; at most 53,597 infectious, on day 17

R0 is about 6, set by the largest eigenvalue of the matrix of contacts weighted by β and the infectious period, not by any one group’s contacts. Children, who have the most contacts, drive the epidemic, but with an R0 this high, it reaches almost everyone in every age group. The total infected in each age group:

sim.results.sir.infection.by('age').summarize()
0-14     29953.119963
15-64    49742.041450
65+      19896.816580
Name: infection, dtype: float64

Of 30,000 children, 50,000 adults under 65, and 20,000 people aged 65 and over, these are all but a few hundred in each group.

Advanced: a real population, with contacts by setting

This part is an extension of the example above, with a larger model. Closing schools is one of the first measures considered against a new respiratory infection. It removes the contacts children make at school, but not those at home or elsewhere, so whether it helps depends on how much of the spread happens at school, and on when the schools close and reopen. A model of it needs contacts by setting. This part adapts the school closure example of the Epidemics R package: an SEIR model of an influenza-like infection in the population of the United Kingdom, in three age groups, with the contact matrices of Prem et al. (2021) for each setting.

Parameter Value Meaning
β 0.016 per contact Rate of infection per contact with an infectious person
Latent period 3 days Mean time in E (exponential)
Infectious period 7 days Mean time in I (exponential)
Age groups 0–19, 20–39, 40+ With the fractions of the United Kingdom’s population
Contacts Prem et al. (2021) One matrix each for home, school, work, and the rest of the community
Infectious at the start One in a million
Closure Days 200 to 260 School contacts removed; the others go on

Rather than writing the fractions of each age group and the matrix by hand, ss.population() gives the country’s population by age, and ss.contacts() its matrix for each setting, for the age groups of the model:

S, E, I, R = ss.states('S E I R')
flu = ss.Disease(
    infection = S >> E @ 0.016*I,
    onset     = E >> I @ ss.days(3),
    recovery  = I >> R @ ss.days(7),
    init      = dict(I=1e-6),
    name      = 'flu',
)
flu.stratify(age=[0, 20, 40])

home, school, work, community = [ss.contacts('United Kingdom', layer=layer) for layer in ['home', 'school', 'work', 'community']]

R0 is about 1.5.

Closing schools

The closure is a modifier of the school contacts: from day 200 to day 260 they are multiplied by zero, and when it ends, they are back as they were, since a modifier never changes the contacts themselves.

closure = ss.modify(school, multiply=0, start=ss.days(200), stop=ss.days(260))
pars = dict(people=ss.population('United Kingdom'), networks=[home, school, work, community], stop=ss.days(600))
sims = {
    'Schools open': ss.Sim(flu, **pars),
    'Schools closed for 60 days': ss.Sim(flu, interventions=closure, **pars),
}
msim = ss.parallel(sims)
msim.plot('flu_I')
MultiSim: 2 runs of 2 scenarios (Schools open, Schools closed for 60 days)
  Done in 0.95 s

The closure doesn’t prevent the epidemic. It delays the peak by about six weeks and lowers it by about a fifth, which could matter for hospitals, but most of those it spares are infected after schools reopen, since the epidemic resumes among people who are still susceptible: the number infected in all is only about 3% lower.

msim.summarize().loc['flu_infection']
value
scenario
Schools open 33,987,071
Schools closed for 60 days 32,886,651

By age, the number infectious falls in every group while schools are closed, and most among children, who make most of the contacts at school:

msim.sims['Schools closed for 60 days'].results.flu.I.by('age').plot() # The run sim, from the MultiSim

The Epidemics package says a closure another way: as a fraction by which the contacts of each age group fall (e.g. by half for those under 20). That works in Starsim too, as a modifier of each group’s susceptibility and transmissibility, e.g. ss.modify(flu.rel_sus, multiply=0.5, who=flu.age < 20, start=..., stop=...). Removing the school contacts says more directly what a closure does, when the contacts by setting are known.

Going further