RESUMO
The Covid-19 pandemic 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 at odds with the clinical data. To overcome this "memoryless" issue, a widely used solution is to increase and chain the number of compartments of a unique reality (e.g. have infected individual move between several compartments). This allows for greater heterogeneity and thus be closer to the observed situation, 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 thereby allowing us to limit the number of compartments. We apply our model to the French 2021 SARS-CoV-2 epidemic and, while accounting for vaccine-induced and natural immunity, we analyse and determine the major components that contributed to the Covid-19 hospital admissions. The results indicate that the observed vaccination rate alone is not enough to control the epidemic, and 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.
RESUMO
In an epidemic, individuals can widely differ in the way they spread the infection, for instance depending on their age or on the number of days they have been infected for. The latter allows to take into account the variation of infectiousness as a function of time since infection. In the absence of pharmaceutical interventions such as a vaccine or treatment, non-pharmaceutical interventions (e.g. social distancing) are of great importance to mitigate the pandemic. We propose a model with a double continuous structure by host age and time since infection. By applying optimal control theory to our age-structured model, we identify a solution minimizing deaths and costs associated with the implementation of the control strategy itself. This strategy depends on the age heterogeneity between individuals and consists in a relatively high isolation intensity over the older populations during a hundred days, followed by a steady decrease in a way that depends on the cost associated to a such control. The isolation of the younger population is weaker and occurs only if the cost associated with the control is relatively low. We show that the optimal control strategy strongly outperforms other strategies such as uniform constant control over the whole populations or over its younger fraction. These results bring new facts the debate about age-based control interventions and open promising avenues of research, for instance of age-based contact tracing.
RESUMO
SARS-Cov-2 virus has spread over the world creating one of the fastest pandemics ever. The absence of immunity, asymptomatic transmission, and the relatively high level of virulence of the COVID-19 infection it causes led to a massive flow of patients in intensive care units (ICU). This unprecedented situation calls for rapid and accurate mathematical models to best inform public health policies. We develop an original parsimonious model that accounts for the effect of the age of infection on the natural history of the disease. Analysing the ongoing COVID-19 in France, we estimate the value of the key epidemiological parameters, such as the basic reproduction number [Formula], and the efficiency of the national control strategy. We then use our deterministic model to explore several scenarios posterior to lock-down lifting and compare the efficiency of non pharmaceutical interventions (NPI) described in the literature.
RESUMO
Since Dec 2019, the COVID-19 epidemic has spread over the globe creating one of the greatest pandemics ever witnessed. This epidemic wave will only begin to roll back once a critical proportion of the population is immunised, either by mounting natural immunity following infection, or by vaccination. The latter option can minimise the cost in terms of human lives but it requires to wait until a safe and efficient vaccine is developed, a period estimated to last at least 18 months. In this work, we use optimal control theory to explore the best strategy to implement while waiting for the vaccine. We seek a solution minimizing deaths and costs due to the implementation of the control strategy itself. We find that such a solution leads to an increasing level of control with a maximum reached near the 16th month of the epidemics and a steady decrease until vaccine deployment. The average containment level is approximately 50% during the 25-months period for vaccine deployment. This strategy strongly out-performs others with constant or cycling allocations of the same amount of resources to control the outbreak. This work opens new perspectives to mitigate the effects of the ongoing COVID-19 pandemics, and be used as a proof-of-concept in using mathematical modelling techniques to enlighten decision making and public health management in the early times of an outbreak.
