5 - Heterogeneity of individuals (1/2)

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.

#5 – Why represent the heterogeneity of infected individuals in the model? (1/2)

We have seen that it is often difficult to determine the values of model parameters. However, many of the models mobilized during the COVID19 crisis (as well as for other epidemiological systems) rather finely represent the heterogeneity of infected individuals, thus increasing the complexity of the model.

Let us take the following example: instead of the two states A and I considered in the previous articles (article #1), let's now add a latency phase E and an incubation phase E+Ip. Individuals E are infected, but are not yet excreting and have no symptoms. As for individuals Ip, they begin to excrete before any symptoms appear. We maintain the distinction between a- or pauci-symptomatic individuals (Ia) and symptomatic individuals (Is), the latter being the only ones at risk of dying from the infection.

Schéma de modélisation incluant une phase de latence

This epidemiological model considers 7 states: susceptible (S), latent infected non-shedders (E), incubating infected shedders (without symptoms, Ip), a- or pauci-symptomatic  infected shedders (Ia), symptomatic infected shedders (Is), cured (R), and dead (M). The strength of infection (λ) takes into account the different contributions of shedder individuals (Ip, Ia, Is) to new infections.

This model necessarily has more parameters (9 instead of 6 parameters previously) and more states (7 instead of 5). It is therefore more complicated (it is also said to be less parcimonious). Often, additional assumptions can reduce the complexity to make it more relevant to available knowledge. For example, it can be assumed that shedder individuals without symptoms (Ip and Ia) have the same excretion levels (thus βp = βa) in the absence of further information.

But why complicate the model? Two elements explain it from a methodological point of view:

  • The contribution of infected individuals to new cases may differ according to the stage of infection, which is difficult to take into account without distinguishing between these stages. For example, at the beginning of the infection (stage E), individuals are no longer susceptible (i.e. cannot become infected) but do not shed the pathogen yet (i.e. do not contribute to new cases), which will lead to a delay in the onset of the epidemic.
  • Markov-type compartment models (without memory) assume an exponential distribution of durations in the compartments, which is not very realistic. (article #1). To remedy this rigorously, the underlying mathematical formalism (delay models) would have to be changed. An alternative to reduce the impact of this hypothesis is to consider several successive sub-states, since a sum of durations following exponential distributions converges quickly enough to a quasi-constant duration.

What is the impact of the change in model structure on predictions? 

Let us compare for example the model described in the figure above (black curve below), explicitly considering a latent phase without shedding (E) followed by a pre-symptomatic shedding phase (Ip), with a model integrating E and Ip in a single phase Ip2 which then represents the entire incubation period (red curve). Let's update the parameters relating to Ip2 so that the two models are comparable. The duration in Ip2 becomes: dp2 = dp1 + dE, with dp1 = 1/γp and dE = 1/ε. The transmission rate integrates that E individuals do not shed during dE: βp2 = (βp1dp1) / (dE + dp1).

Nombre de décès cumulés par jour en considérant (en noir) ou non (en rouge) une phase de latence

Prediction of models explicitly including the latency phase E (in black) or not (in red): the epidemic predicted by the red model starts faster than the one predicted by the black model, while the parameters are comparable.

We can clearly see the delaying effect of the latency phase on the epidemic dynamics. It is therefore essential to have as much precise biological knowledge as possible about the course of the infection in the host in order to know whether such a latency phase is relevant or not and how long it lasts.

Article #6 will continue this discussion, taking into account factors that increase the complexity of models in relation to their realism and their use in health management.

Modification date : 10 November 2023 | Publication date : 10 November 2023 | Redactor : DYNAMO