Box 1 Definitions (see [90, 97,98,99])

Population-based models: although the overall population is made up of individual units, population-based models group all individual units of the same state together, without distinction between individual units belonging to the same subgroup. The number (or proportion) of the epidemiological units in each subgroup (e.g., susceptible, infectious, recovered) is tracked, but not the individual state of each unit (for instance, we know how many individual units are infectious, but not which ones). These models are also called compartmental models.

Individual-based models: contrary to population-based models, individual-based models monitor explicitly the state of each individual unit in the overall population. Therefore, it is possible to track both which individual units are in each state, and also the number of epidemiological units in each state.

Epidemiological unit: the unit of interest and the smallest entity of the model. It could be an individual animal, a group of animals, herds, or populations in regions or countries. The epidemiological unit can be aggregated and modeled as a number or proportion of the overall population in each state, or modeled as individuals whose status is tracked.

Frequency-dependence (sometimes referred to as “pseudo-mass action”): with this assumption, the contact rate (the number of contacts made by each epidemiological unit per unit time, where contacts are of an appropriate type for transmission to be possible) is constant irrespective of the population size \(N\left(t\right)\). In that case, the force of infection is of the form \(\beta \frac{I\left(t\right)}{N\left(t\right)}\), where \(\beta\) is the effective contact rate (in time⁻¹), i.e., the number of effective contacts made by each epidemiological unit per unit time. An effective contact is a contact that would effectively lead to transmission if the contact is between an infectious and a susceptible unit.

Density-dependence (sometimes referred to as “mass action”): with this assumption, the contact rate increases with the population size \(N\left(t\right)\). In that case, the force of infection is of the form \(\beta I\left(t\right)\), where \(\beta\) (in unit⁻¹.time⁻¹) is the per capita rate at which two specific epidemiological units come into effective contact per unit time. Note that the density-dependent \(\beta\) is not equivalent to the frequency-dependent \(\beta\) because of these different assumptions on the contact rate.