Epidemiological modelling and its use to manage COVID-19

#4 – The importance of the (often unobserved) beginning of an epidemic

This article is part of a series and uses the SAIRM model presented in the previous articles.

In a deterministic framework, i.e. when the randomness of the modelled processes is not taken into account, as is generally the case in large populations, the predictions of a model are based on three components: (1) the structure of the model (i.e. its diagram and associated equations, cf. article#1), (2) the value of its parameters (cf. article#3), and (3) its initial conditions. Here we discuss this 3^rd component and the effect that these initial conditions may have on the model's predictions in the short term (transient regime) or in the long term (asymptotic regime).

First of all, what do we mean by initial conditions? In an epidemiological model, it is the initial situation that will potentially lead to the spread of a pathogen in the host population. In order for the model to be able to predict the population size by health states over time, it must be told where to start from, i.e. how the host population in these states is initially distributed.

The most common assumption is to initially introduce (at t ) a single infected individual into a fully susceptible population. However, the actual conditions may be very different! The exact date of introduction of the pathogen is usually unknown, especially at the beginning of an epidemic. Besides, the pathogen may be introduced into the population several times, and each introduction may be caused by a single individual or by the simultaneous arrival of several infected individuals. Moreover, the infected individuals are not identical: introducing an asymptomatic (A) or a symptomatic (I) individual may lead to different dynamics! Finally, the population may not be totally naive, but may have been previously exposed to the infection and therefore have some level of population immunity (which we will discuss later).

The following figures illustrate how a change in initial conditions impacts model predictions. In the long run, the dynamics are similar, since the equilibrium reached by a deterministic model does not depend on the initial conditions. However, transient regimes (i.e., the dynamics of the system before reaching equilibrium, which can be a long period) differ widely depending on initial conditions.

Epidemic dynamics up to 8 months (1^st figure) or only the beginning until March 16 (2^nd figure) as predicted by the SAIRM model defined in the previous articles for 6 initial conditions: 1 vs. 100 asymptomatic (A) or symptomatic (I) individuals introduced in t , or 10 A vs. I individuals introduced each day for 10 days.

Thus, there is a one-month gap between the scenarios tested before reaching the cumulative number of deaths observed as of March 16. Yet, 1 or 100 individuals compared to 70 million seem only a drop in the ocean! But that drop of water makes the work of the modellers more complex when they try to fit their models to the available data, especially at the beginning of an epidemic...

Article#5 will discuss the advantages and drawbacks of taking greater account of the heterogeneity of infected individuals in the model.