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Epidemiological modelling and its use to manage COVID-19

Insights into mechanistic models, by the DYNAMO team

Over the next few weeks, we will present some key elements of epidemiological modelling through short educational articles. These articles will help you to better understand and decipher the assumptions underlying the epidemiological models that are currently widely used, and how these assumptions can impact predictions regarding the spread of pathogens, particularly SARS-CoV-2. The objective is to discover the advantages and limitations of mechanistic modelling, an approach that is at the core of the DYNAMO team's work. The examples of models will be inspired by models used in crisis, but sometimes simplified to make them accessible.

Before talking about modelling and especially if you are not yet very comfortable with the concepts of epidemiology, we advise you to read this very well done online course : course "Sars cov-2: the epidemic".

#1 – What does an epidemiological model diagram tell ?

Schéma conceptuel d'un modèle SAIRM

Conceptual diagram of an epidemiological model separating individuals into 4 health states
(S: susceptible, A: asymptomatically infected, I: symptomatically infected, R: recovered, shedding no more pathogen) 
and a "dead" state (M). The force of infection (λ) takes into account the different contribution
from individuals A and I to new infections.

What you see directly on the diagram:

  • The population is subdivided into health states (model compartments) : S, A, I, R.
  • Population dynamics (births, deaths not due to disease) are neglected.
  • The living population decreases during the course of the epidemic (due to induced mortality, M).
  • Infected individuals A and I contribute differently to new infections (have their own parameter, βA and βI, in the formula for the strength of infection λ).
  • The rate at which new infections appear in the population (force of infection) is proportional to the size of the population. This means that for the same proportion of infected individuals, there will be a greater proportion of susceptible individuals who will become infected in a large city than in a small village. This notion is called density dependence.
  • After infection, susceptible individuals (S) have a probability p of becoming asymptomatic (A), and a probability (1-p) of becoming symptomatic (I).
  • After infection, there is recovery (R) in all cases for A individuals, and there is either recovery or death for I individuals.
  • There is no return to a susceptible state, recovered individuals are considered immune to this pathogen and cannot reinfect themselves during the continuing of the epidemic. This hypothesis is often unrealistic over a long period of time (immunity induced by the infection is lost over time) but may be appropriate on shorter time scales (a few weeks or months).

Which is not explicit but may be important:

  • Individuals are not explicitly represented, hence cannot have their own characteristics. As a result, all individuals in the same health state are assumed to be identical (on average). For example, S's are all equally sensitive to each other, I's all shed the same amount of virus, and so on.
  • The occurrence of events that make an individual leave a state does not depend on the time already spent by that individual in the state (Markovian model). Thus, the mortality rate is the same for all individuals I, whether they are newly infected or have been infected for a long time.
  • On average, how long does one stay in each health state ?
    Generally, these models assume that the duration in a state follows an exponential distribution, defined by a single parameter: its mean. For example, the mean duration in A is 1/γA units of time (often the day), while the mean duration in I is 1/(γI) units of time. With a γrecovery rate of 0.2, asymptomatic individuals are asymptomatic on average 1/0.2 = 5 days before recovering. However, this type of distribution does not mean that the majority of A individuals stay A for 5 days, but rather that a high proportion of A individuals spends little time in the state, while a low proportion stays there for a very long time. On average the duration in A is indeed the desired duration, but the distribution is unrealistic. It is however a very common assumption because it allows to have only one parameter per state, which is very important when there is little knowledge or observations to estimate the parameters of the model.
  • All individuals can get into contact (unstructured population, and homogeneous mixing hypothesis). This hypothesis is reasonable up to a certain scale, such as the city, but should be reconsidered when considering a model on a national or even global scale.
  • Is the model deterministic (its predictions depend on its structure, parameter values and initial conditions) or stochastic (its predictions also depend on the randomness of certain events)? Similar conceptual diagrams are often used in both cases.

In article #2, we'll discuss what this model can predict...