An optimal control model to design strategies for reducing the spread of the Ebola virus disease.

In this work, we formulate an epidemiological model for studying the spread of Ebola virus disease in a considered territory. This model includes the effect of various control measures, such as: vaccination, education campaigns, early detection campaigns, increase of sanitary measures in hospital, quarantine of infected individuals and restriction of movement between geographical areas. Using optimal control theory, we determine an optimal control strategy which aims to reduce the number of infected individuals, according to some operative restrictions (e.g., economical, logistic, etc.). Furthermore, we study the existence and uniqueness of the optimal control. Finally, we illustrate the interest of the obtained results by considering numerical experiments based on real data.

State and parameter estimation for a class of schistosomiasis models.

We develop a general framework to estimate the proportion of infected snails and snail-human transmission parameter of a class of models that describes the evolution of schistosomiasis. To do so, we consider simultaneously the dynamics of schistosomiasis, captured by the homogeneous version of the classical MacDonald's model, and the measurable output: the number of female schistosomes per single host. The proposed method consists of designing an auxiliary dynamical system, called observer, whose solutions converge exponentially to those of the system capturing the schistosomiasis model. Moreover, we derive an estimation of the snail-human transmission rate, an unknown but key parameter in the dynamics of schistosomiasis. These estimations are central in two of the strategies of controlling schistosomiasis, namely the use of molluscicides and mass drug administration. To further investigate control strategies on a larger scale, we consider a heterogeneous model which consists of an arbitrary number of human groups or patches and an arbitrary number of fresh-water sources, natural habitats of snails. Provided that the data of infected humans' worm burden in each patch or group is available, we provide a method of estimating the proportion of infected snails in each snail natural habitat, thereby providing a map on where to implement control strategy to mitigate or eliminate Schistosomiasis.

Be-CoDiS: A Mathematical Model to Predict the Risk of Human Diseases Spread Between Countries--Validation and Application to the 2014-2015 Ebola Virus Disease Epidemic.

Ebola virus disease is a lethal human and primate disease that currently requires a particular attention from the international health authorities due to important outbreaks in some Western African countries and isolated cases in the UK, the USA and Spain. Regarding the emergency of this situation, there is a need for the development of decision tools, such as mathematical models, to assist the authorities to focus their efforts in important factors to eradicate Ebola. In this work, we propose a novel deterministic spatial-temporal model, called Between-Countries Disease Spread (Be-CoDiS), to study the evolution of human diseases within and between countries. The main interesting characteristics of Be-CoDiS are the consideration of the movement of people between countries, the control measure effects and the use of time-dependent coefficients adapted to each country. First, we focus on the mathematical formulation of each component of the model and explain how its parameters and inputs are obtained. Then, in order to validate our approach, we consider two numerical experiments regarding the 2014-2015 Ebola epidemic. The first one studies the ability of the model in predicting the EVD evolution between countries starting from the index cases in Guinea in December 2013. The second one consists of forecasting the evolution of the epidemic by using some recent data. The results obtained with Be-CoDiS are compared to real data and other model outputs found in the literature. Finally, a brief parameter sensitivity analysis is done. A free MATLAB version of Be-CoDiS is available at: http://www.mat.ucm.es/momat/software.htm.

On the stock estimation for a harvested fish population.

We consider a stage-structured model of a harvested fish population and we are interested in the problem of estimating the unknown stock state for each class. The model used in this work to describe the dynamical evolution of the population is a discrete time system including a nonlinear recruitment relationship. To estimate the stock state, we build an observer for the considered fish model. This observer is an auxiliary dynamical system that uses the catch data over each time interval and gives a dynamical estimate of the stock state for each stage class. The observer works well even if the recruitment function in the considered model is not well known. The same problem for an age-structured model has been addressed in a previous work.

An observer for a nonlinear age-structured model of a harvested fish population.

We consider an age-structured model of a harvested population. This model is a discrete-time system that includes a nonlinear stock-recruitment relationship. Our purpose is to estimate the stock state. To achieve this goal, we built an observer, which is an auxiliary system that uses the total number of fish caught over each season and gives a dynamical estimation of the number of fish by age class. We analyse the convergence of the observer and we show that the error estimation tends to zero with exponential speed if a condition on the fishing effort is satisfied. Moreover the constructed observer (dynamical estimator) does not depend on the poorly understood stock-recruitment relationship. This study shows how some tools from nonlinear control theory can help to deal with the state estimation problem in the field of renewable resource management.