Most epidemic models share a common approach on modelling the spread of a disease. The susceptible-infectious-removed (SIR) model is a simple deterministic compartmental model to predict disease spread. An objective population is divided in three groups: the susceptible (S), the infected (I) and the recovered or removed (R). These quantities enter the model as fractions of the total population P.
As a pandemics infects and kills much more quickly than human natural rates of birth and death, the population size is assumed constant except for the individuals that recover or die. Hence, S + I + R = P/P = 1. The pandemics dynamics is modelled as a system of ordinary differential equations which governs the rate of change at which the percentage of susceptible, infected and recovered/removed individuals in a population evolve.
The number of possible transmissions is proportional to the number of interactions between the susceptible and infected compartments, S × I:
Where α / [time] − 1 is the transmission rate of the process which quantifies how many of the interactions between susceptible and infected populations yield to new infections per day.
The population of infected individuals will increase with new infections and decrease with recovered or removed people.
Where β is the percentage of the infected population that is removed from the transmission process per day.
The infectious period, TI / [time] , is defined as the reciprocal of the removal rate:
In early stages of the infection, the number of infected people is much lower than the susceptible population. Hence, S ≈ 1 making dI/dt linear and the system has the analytical solution I(t) = I0exp (α − β)t.
opensir.models.SIR
The SIR-X model extends the SIR model adding two parameters: the quarantine rate κ and the containement rate κ0. This extension allows the model to capture the "decrease" of susceptible population owing containment and quarantine measures.
opensir.models.SIRX