Non-Markovian modelling highlights the importance of age structure on Covid-19 epidemiological dynamics

Abstract

The Covid-19 outbreak was followed by a huge amount of modelling studies in order to rapidly gain insights to implement the best public health policies. Most of these compartmental models involved ordinary differential equations (ODEs) systems. Such a formalism implicitly assumes that the time spent in each compartment does not depend on the time already spent in it, which is unrealistic. To overcome this “memoryless” issue, a widely used solution is to chain the number of compartments of a unique reality (e.g. have infected individual move between several compartments). This allows for greater heterogeneity, but also tends to make the whole model more difficult to apprehend and parameterize. We develop a non-Markovian alternative formalism based on partial differential equations (PDEs) instead of ODEs, which, by construction, provides a memory structure for each compartment. We apply our model to the French 2021 SARS-CoV-2 epidemic and we determine the major components that contributed to the Covid-19 hospital admissions. A global sensitivity analysis highlights a huge uncertainty attributable to the age-structured contact matrix. Our study shows the flexibility and robustness of PDE formalism to capture national COVID-19 dynamics and opens perspectives to study medium or long-term scenarios involving immune waning or virus evolution.

Modelling long-term COVID-19 hospital admission dynamics using empirical immune protection waning data

“…To better assess the impact of vaccination campaigns, we also investigated two additional scenarios in which we implemented a decreased contact rate of -20% compared to the May 2022 value used in the baseline scenario. This value was chosen because it is comparable to others observed in 2021 [Reyné et alii, 2022].…”

“…The sensitivity analyses (Figures S7-S10) highlight the main sources of uncertainty which correspond to factors that are difficult to predict. As discussed by Reyné et alii, 2022, the time-varying contact matrix is impossible to predict — as it depends on government policies, age-specific spontaneous behavioural changes or calendar events such as school holidays— and yet yield an huge uncertainty in the model’s outputs. The seasonality also impacts strongly the results, and also unpredictable.…”

“…The graphs shows the origin of the variance captured by each model parameter using Sobol indices (see Reyné et alii, 2022 for methodological details). A large part of the variations originates from the unknowns in the contact matrix between ages, as in Reyné et alii, 2022. The magnitude of seasonality also matters for the long-term trends, as well as the transmission rate (i.e.…”

“…We show in Table S1 the baseline for the different parameters and the range used for parameters included in the sensitivity analysis. Several parameters were not included in the sensitivity analysis, as they already were included in that in Reyné et alii [2022]. We expect these earlier results to hold and these parameters have limited relevance in the contact of the Omicron epidemic.…”

“…The waning of immune protection is challenging to capture with classical compartmental models because it implies an exponential increase in the number of model compartments, which makes precise parameterization challenging. We extend a non-Markovian approach [Reyné et alii, 2022] that readily accounts for the time spent in each compartment. By using published data, we explore long-term epidemic dynamics in a qualitative way.…”