Modeling The Spread of An Infectious Disease
A compartmental model simulates the spread of an infection through a host population that is divided into mutually exclusive compartments based on their infection status: e.g., Susceptible, Exposed, Infectious, Recovered, Vaccinated. Different types of compartmental models can be constructed, e.g., SI, SIS, SIR, SEIR, SIRV. Which compartments to include in the model depends upon the characteristics of the host-pathogen system being modeled and the purpose of the model.
The Susceptible-Infectious-Recovered model is also referred to as the SIR model – this is one of the most basic compartmental models of disease spread.
bl.ocks.org ›
Mechanistic Representation of Disease Spread
SIR and other compartmental models include explicit hypotheses of the biological mechanisms that drive infection dynamics and are therefore categorized as mechanistic models.
For example, the SIR model assumes a closed host population (fixed population size: no births, no deaths, no immigration, no emigration). Infection does not cause mortality and recovered individuals have lifelong immunity. Mixing in the population is homogeneous, i.e. all individuals in the population are assumed to have an equal probability of coming in contact with one another. The hosts are categorized based on their infection status: Susceptibles have had no exposure to the infection, Infectious are currently infected and are able to transmit the infection to susceptibles, and Recovereds have experienced the infection in the past and are currently immune.
Transmission of infection occurs when an infectious host comes in contact with a susceptible host. The susceptible host is then ‘infected’ and transfers to the infectious compartment. The infected host is assumed to be instantly infectious (i.e. able to pass on the infection to a susceptible individual) and remains in the infectious compartment for a period of time before progressing to the recovered (and non-infectious) compartment.
A system of ordinary differential equations represents the dynamic disease spread process. These equations are used to determine the number of hosts in each compartment at time t + 1 based on the number of hosts in each compartment at time t.
dS/dt = – βSI/N ……………………….. (1)
dI/dt = βSI/N – γI ……………………….. (2)
dR/dt = γI ……………………….. (3)
Here, S, I, and R represent the proportion of hosts in the susceptible, infectious, and recovered compartments, respectively. The population size, N, is fixed (S + I + R = N). The main focus in a SIR compartmental model is on the transfer between the three compartments. Two parameters determine the transition of hosts between the three compartments: transmission coefficient, β (also called transmission rate or effective contact rate), and recovery rate, γ.
Transmission Coefficient
β: The average rate at which an infected individual can infect a susceptible individual. It is the product of the contact rate (number of contacts with other individuals in the population per unit time) and the probability of transmission given a contact. Thus, β is a function of the transmissibility of the infection as well as the average number of contacts per individual host.
Note: Direct measurement of β is impossible for most infections, so its value is estimated after determining the force of infection(see next unit), which can be deduced directly or indirectly from epidemiological data ○.
Recovery Rate
γ: The average rate at which infected individuals recover from infection. This is the inverse of the infectious period (1/infectious period) or the average duration a host remains infected (and infectious). For example, measles has an infectious period of 8 days, so the recovery rate is calculated as 1/8 or 0.125/day.
The ratio of these two parameters, β/ γ, is known as the basic reproduction number or R0 (R nought).