Epidemiological modelling and its use to manage COVID-19

#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+I_p. Individuals E are infected, but are not yet excreting and have no symptoms. As for individuals I_p, they begin to excrete before any symptoms appear. We maintain the distinction between a- or pauci-symptomatic individuals (I_a) and symptomatic individuals (I_s), the latter being the only ones at risk of dying from the infection.

This epidemiological model considers 7 states: susceptible (S), latent infected non-shedders (E), incubating infected shedders (without symptoms, I_p), a- or pauci-symptomatic  infected shedders (I_a), symptomatic infected shedders (I_s), cured (R), and dead (M). The strength of infection (λ) takes into account the different contributions of shedder individuals (I_p, I_a, I_s) 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 (I_p and I_a) have the same excretion levels (thus β_p = β_a) 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 (I_p), with a model integrating E and I_p in a single phase I_p2 which then represents the entire incubation period (red curve). Let's update the parameters relating to I_p2 so that the two models are comparable. The duration in I_p2 becomes: d_p2 = d_p1 + d_E, with d_p1 = 1/γ_p and d_E = 1/ε. The transmission rate integrates that E individuals do not shed during d_E: β_p2 = (β_p1d_p1) / (d_E + d_p1).

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.