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This vignette describes the modelling approach used in helios. For an
introduction to practical use of the package, see the Get
started vignette.
Population
The population consists of a fixed number of individuals \(i = 1, \ldots, N\). Each individual has properties. Some properties remain constant during the simulation, and others change.
Age structure
The age categories included in the model are children, adults, and the elderly. Each individual has a fixed age category, assumed not to change during the simulation. The age category is sampled from a categorical distribution at initialisation such that \[ \mathbb{P}(\text{age}_i = k) = p_k, \\ k \in \{\text{child}, \text{adult}, \text{elderly}\}. \]
Households
Households are comprised of collections of one or more individuals. We consider two approaches.
Parametric
In the parameteric approach, household sizes \(N_j^\text{household}\) are drawn from a Poisson distribution with rate parameter \(\lambda^\text{household}\) \[ N_j^\text{household} \sim \text{Poisson}(\lambda^\text{household}). \] Individuals are then assigned to households. To ensure that no household contains only children, one adult is paired to each child assigned to a household without any existing adults.
Bootstrap
In the bootstrap approach, households and age distributions are
sampled with replacement from a reference data set. The reference data
baseline_household_demographics are derived from the Office
for National Statistics 2011 survey as used by Hinch et al. (2021). If the bootstrap approach
is used, then its age structure will overwrite the categorical age
structure above.
Schools
All children attend schools. Two approaches to the number of children attending each school are considered.
Parametric
In the parametric approach, the sizes of schools are drawn from a log-normal distribution with mean \(\mu^\text{school}\) and variance \({\sigma^\text{school}}^2\) \[ N^\text{school}_j \sim \text{Lognormal}(\mu^\text{school}, {\sigma^\text{school}}^2). \] Schools \(j\) are drawn until \(\sum_j N^\text{school}_j \geq N^\text{child}\). Any samples \(N^\text{school}_j > N^\text{school}_\text{max} \in \mathbb{N}\) are discarded. The final school size is adjusted such that the sum of school sizes corresponds to the number of children. Assignment of children to particular schools is made using random sampling without replacement.
Workplaces
Workplaces are attended by adults who are not school staff. Following Ferguson et al. (2005), the sizes of workplaces \(N^\text{workplace}_j\) are drawn from an offset truncated power distribution. The cumulative distribution function is \[ \mathbb{P}(N^\text{workplace}_j > N) = \frac{\left( \frac{1 + N^\text{workplace}_\text{max} / a}{1 + m / a} \right) ^ c - 1}{(1 + N^\text{workplace}_\text{max} / a)^c - 1}, \quad \text{for } N \leq N^\text{workplace}_\text{max}, \] where \(a > 0, c > 0\) and \(N^\text{workplace}_\text{max} \in \mathbb{N}\). As with the school sizes, workplace sizes are drawn until their sum exceeds the number of adults who are not school staff. The final workplace is then adjusted downward. Assignment of adults to particular workplaces is done using random sampling without replacement.
Leisure
Leisure venues are attended by all individuals. Leisure venue sizes are drawn from a negative binomial distribution \[ N^\text{leisure}_j \sim \text{NB}(\mu^\text{leisure}, \kappa^\text{leisure}), \] with mean \(\mu^\text{leisure}\) and overdispersion parameter \(\kappa^\text{leisure}\). As above, any samples \(N^\text{leisure}_j > N^\text{leisure}_\text{max} \in \mathbb{N}\) are discarded and the final sample is adjusted such that \(\sum_j N^\text{leisure}_j = N\). The number of leisure visits per person per week is drawn from a Poisson distribution, truncated at 7 corresponding to one leisure visit every day. The locations for each individual’s leisure visits are sampled without replacement with probability proportional to leisure venue size. The day of the week that each individual visits each sampled leisure location are also sampled without replacement.
Disease states
We consider a SEIR compartmental model with susceptible, exposed, infectious, and recovered disease states. At a given time, each individual is in exactly one of the states \(\{S, E, I, R\}\). Transitions between states are simulated as follows.
Transition from state \(S \to E\)
The total force of infection \(\lambda\) for each individual is a sum of household, workplace, school, leisure, and community specific force of infections \[ \lambda = \lambda_h + \lambda_w + \lambda_s + \lambda_l. \] The location specific force of infection terms are given as:
- Household \(\lambda_h\): proportional to the number of infectious individuals in the household multiplied by \(\beta_h\).
- Workplace \(\lambda_w\): proportional to the number of infectious individuals in the workplace multiplied by \(\beta_w\).
- School \(\lambda_s\): proportional to the number of infectious individuals in the school multiplied by \(\beta_s\).
- Leisure \(\lambda_l\): proportional to the number of infectious individuals in the leisure setting multiplied by \(\beta_l\).
- Community \(\lambda_c\): proportional to the total number of infectious individuals multiplied by \(\beta_c\).
The probability of infection in a given time step \(\delta > 0\) is \[ \mathbb{P}(S \to E) = 1 - \exp(- \lambda \delta) \]
Transition from state \(E \to I\)
Individuals progress from exposed to infectious with a random delay \(d \geq 1\) such that \[ c \sim \text{Gamma}(2\gamma, 2), \\ d = \text{round}(c + 1). \] where \(\gamma > 0\) is the duration exposed which we set to 2. The delay \(d\) is a number of time steps \(\delta\).
Transition from state \(I \to R\)
Individuals progress from infectious to recovered after a random disease duration \(f \geq 1\) such that \[ e \sim \text{Gamma}(2\upsilon, 2), \\ f = \text{round}(e + 1). \] where \(\upsilon\) is the duration of infectiousness which we set to 4. The disease duration \(f\) is a number of time steps \(\delta\).