Estimating the state of the Covid-19 epidemic in France using a non-Markovian model

Abstract

In this paper, we use a deterministic non-Markovian epidemic model to estimate the state of the Covid-19 epidemic in France. This model allows us to consider realistic distributions for the exposed and infectious periods in a SEIR model, contrary to standard ODE models which only consider exponentially distributed exposed and infectious periods. We present theoretical results linking the (unobserved) parameters of the model to various quantities which are more easily measured during the early stages of an epidemic. We also stress the main quantitative differences between the non-Markovian and the Markovian (ODE) model. We then apply these results to estimate the state of the Covid-19 epidemic in France by analyzing three regions: the Paris region, the northeast regions and the rest of the country, based on current knowledge on the infection fatality ratio and the exposed and infectious periods distributions for Covid-19. Our analysis is based on the hospital data published daily by Santé Publique France (daily hospital admissions, intensive care unit admissions and hospital deaths).

Estimating the state of the COVID-19 epidemic in France using a model with memory

“…Note that an analysis similar to the one conducted in this paper can be carried out for a model where the infectivity of each individual is a random function of the time since infection, those random functions for the various individuals being i.i.d., see [ 20 ]. This would yield a model which could account for the variability, both in time and between individuals, of the infectivity of each infected individual, something which is apparent for COVID-19 for example in [ 17 ].…”

“…It seems that asymptomatic individuals are less infectious than symptomatic ones (see [ 19 ]). We could have included in the model different values of infectivity depending upon the duration of the infectious period of each individual, using the theory developed in [ 20 ], but we lack quantitative information about the various levels of infectivity to make serious predictions, while the available data do not allow us to estimate many parameters. Note also that, as shown in [ 20 ], during the early phase of the epidemic, the product of the number of asymptomatic patients and their infectivity determines the evolution of the epidemic.…”

“…We could have included in the model different values of infectivity depending upon the duration of the infectious period of each individual, using the theory developed in [ 20 ], but we lack quantitative information about the various levels of infectivity to make serious predictions, while the available data do not allow us to estimate many parameters. Note also that, as shown in [ 20 ], during the early phase of the epidemic, the product of the number of asymptomatic patients and their infectivity determines the evolution of the epidemic. Overestimating the infectivity leads to an underestimation of the number of cases, which has no significant impact on the dynamics of the epidemic during the early phase, but may lead to underestimation of the proportion of the population which has acquired immunity at the end of this early phase.…”

A parsimonious approach for spatial transmission and heterogeneity in the COVID-19 propagation

“…Thus, they seem more adapted for our purpose. However, we note that according to recent studies [ 49 ], stochastic SIR models are Markovian only if the infectious periods follow independent exponential distributions, which may not be realistic. Other more general assumptions, with e.g.…”

“…Other more general assumptions, with e.g. a bimodal distribution of the sojourn time in the infectious compartment lead to more complex macroscopic integro-differential models [ 49 ]. Nonetheless, the estimated parameter values (e.g.…”