Final Size for Epidemic Models with Asymptomatic Transmission.

The final infection size is defined as the total number of individuals that become infected throughout an epidemic. Despite its importance for predicting the fraction of the population that will end infected, it does not capture which part of the infected population will present symptoms. Knowing this information is relevant because it is related to the severity of the epidemics. The objective of this work is to give a formula for the total number of symptomatic cases throughout an epidemic. Specifically, we focus on different types of structured SIR epidemic models (in which infected individuals can possibly become symptomatic before recovering), and we compute the accumulated number of symptomatic cases when time goes to infinity using a probabilistic approach. The methodology behind the strategy we follow is relatively independent of the details of the model.

Robust control strategy by the Sterile Insect Technique for reducing epidemiological risk in presence of vector migration.

The Sterile Insect Technique (SIT) is a promising technique to control mosquitoes, vectors of diseases, like dengue, chikungunya or Zika. However, its application in the field is not easy, and its success hinges upon several constraints, one of them being that the treated area must be sufficiently isolated to limit migration or re-invasion by mosquitoes from the outside. In this manuscript we study the impact of males and (fertile) females migration on SIT. We show that a critical release rate for sterile males exists for every migration level, in the context of continuous or periodic releases. In particular, when (fertile) females migration is sufficiently low, then SIT can be conducted successfully using either open-loop control or closed-loop control (or a combination of both methods) when regular measurements of the wild population are completed. Numerical simulations to illustrate our theoretical results are presented and discussed. Finally, we derive a threshold value for the females migration rate, when viruses are circulating, under which it is possible to lower the epidemiological risk in the treated area, according to the size of the human population.

Modeling and control of epidemics through testing policies.

Testing is a crucial control mechanism in the beginning phase of an epidemic when the vaccines are not yet available. It enables the public health authority to detect and isolate the infected cases from the population, thereby limiting the disease transmission to susceptible people. However, despite the significance of testing in epidemic control, the recent literature on the subject lacks a control-theoretic perspective. In this paper, an epidemic model is proposed that incorporates the testing rate as a control input and differentiates the undetected infected from the detected infected cases, who are assumed to be removed from the disease spreading process in the population. After estimating the model on the data corresponding to the beginning phase of COVID-19 in France, two testing policies are proposed: the so-called best-effort strategy for testing (BEST) and constant optimal strategy for testing (COST). The BEST policy is a suppression strategy that provides a minimum testing rate that stops the growth of the epidemic when implemented. The COST policy, on the other hand, is a mitigation strategy that provides an optimal value of testing rate minimizing the peak value of the infected population when the total stockpile of tests is limited. Both testing policies are evaluated by their impact on the number of active intensive care unit (ICU) cases and the cumulative number of deaths for the COVID-19 case of France.

Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model.

The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this infinite-horizon problem, and a full characterization of the solution is provided. The best policy consists in applying the maximal allowed social distancing effort until the end of the interval, starting at a date that is not always the closest date and may be found by a simple algorithm. Both theoretical results and numerical simulations demonstrate that it leads to a significant decrease in the epidemic final size. We show that in any case the optimal intervention has to begin before the number of susceptible cases has crossed the herd immunity level, and that its limit is always smaller than this threshold. This problem is also shown to be equivalent to the minimum containment time necessary to stop at a given distance after this threshold value.

How best can finite-time social distancing reduce epidemic final size?

Given maximal social distancing duration and intensity, how can one minimize the epidemic final size, or equivalently the total number of individuals infected during the outbreak? A complete answer to this question is provided and demonstrated here for the SIR epidemic model. In this simplified setting, the optimal solution consists in enforcing the highest confinement level during the longest allowed period, beginning at a time instant that is the unique solution to certain 1D optimization problem. Based on this result, we present numerical essays showing the best possible performance for a large set of basic reproduction numbers and lockdown durations and intensities.

A class of fast-slow models for adaptive resistance evolution.

Resistance to insecticide is considered nowadays one of the major threats to insect control, as its occurrence reduces drastically the efficiency of chemical control campaigns, and may also perturb the application of other control methods, like biological and genetic control. In order to account for the emergence and spread of such phenomenon as an effect of exposition to larvicide and/or adulticide, we develop in this paper a general time-continuous population model with two life phases, subsequently simplified through slow manifold theory. The derived models present density-dependent recruitment and mortality rates in a non-conventional way. We show that in absence of selection, they evolve in compliance with Hardy-Weinberg law; while in presence of selection and in the dominant or codominant cases, convergence to the fittest genotype occurs. The proposed mathematical models should allow for the study of several issues of importance related to the use of insecticides and other adaptive phenomena.

Implementation of control strategies for sterile insect techniques.

In this paper, we propose a sex-structured entomological model that serves as a basis for design of control strategies relying on releases of sterile male mosquitoes (Aedes spp) and aiming at elimination of the wild vector population in some target locality. We consider different types of releases (constant and periodic impulsive), providing sufficient conditions to reach elimination. However, the main part of the paper is focused on the study of the periodic impulsive control in different situations. When the size of wild mosquito population cannot be assessed in real time, we propose the so-called open-loop control strategy that relies on periodic impulsive releases of sterile males with constant release size. Under this control mode, global convergence towards the mosquito-free equilibrium is proved on the grounds of sufficient condition that relates the size and frequency of releases. If periodic assessments (either synchronized with the releases or more sparse) of the wild population size are available in real time, we propose the so-called closed-loop control strategy, under which the release size is adjusted in accordance with the wild population size estimate. Finally, we propose a mixed control strategy that combines open-loop and closed-loop strategies. This control mode renders the best result, in terms of overall time needed to reach elimination and the number of releases to be effectively carried out during the whole release campaign, while requiring for a reasonable amount of released sterile insects.

Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control.

The control of the spread of dengue fever by introduction of the intracellular parasitic bacterium Wolbachia in populations of the vector Aedes aegypti, is presently one of the most promising tools for eliminating dengue, in the absence of an efficient vaccine. The success of this operation requires locally careful planning to determine the adequate number of individuals carrying the Wolbachia parasite that need to be introduced into the natural population. The introduced mosquitoes are expected to eventually replace the Wolbachia-free population and guarantee permanent protection against the transmission of dengue to human. In this study, we propose and analyze a model describing the fundamental aspects of the competition between mosquitoes carrying Wolbachia and mosquitoes free of the parasite. We then use feedback control techniques to devise an introduction protocol that is proved to guarantee that the population converges to a stable equilibrium where the totality of mosquitoes carry Wolbachia.

Behavioral Modulation of Infestation by Varroa destructor in Bee Colonies. Implications for Colony Stability.

Colony Collapse Disorder (CCD) has become a global problem for beekeepers and for the crops that depend on bee pollination. While many factors are known to increase the risk of colony collapse, the ectoparasitic mite Varroa destructor is considered to be the most serious one. Although this mite is unlikely to cause the collapse of hives itself, it is the vector for many viral diseases which are among the likely causes for Colony Collapse Disorder. The effects of V. destructor infestation differ from one part of the world to another, with greater morbidity and higher colony losses in European honey bees (EHB) in Europe, Asia and North America. Although this mite has been present in Brazil for many years, there have been no reports of colony losses amongst Africanized Honey Bees (AHB). Studies carried out in Mexico have highlighted different behavioral responses by the AHB to the presence of the mite, notably as far as grooming and hygienic behavior are concerned. Could these explain why the AHB are less susceptible to Colony Collapse Disorder? In order to answer this question, we have developed a mathematical model of the infestation dynamics to analyze the role of resistance behavior by bees in the overall health of the colony, and as a consequence, its ability to face epidemiological challenges.