Stability and Optimal Control of Delayed Epidemiological Models
Delfim F. M. Torres

We consider some mathematical models that are given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the number of infectious and/or latent individuals, as well as the cost of interventions. Delays are introduced in the models, representing, e.g., the time delay on the diagnosis and commencement of treatment of individuals, incubation and/or pharmacological delays. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are studied. Some open questions are formulated.

The talk is based on several works done with Cristiana J. Silva and collaborators: see [1, 2, 3].

Research partially supported by project TOCCATA, reference PTDC/EEI-AUT/2933/2014, funded by Project 3599 - Promover a Produção Científica e Desenvolvimento Tecnológico e a Constituição de Redes Temáticas (3599-PPCDT) and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionalização (POCI), and by national funds through Fundação para a Ciência e a Tecnologia (FCT) and CIDMA, within project UID/MAT/04106/2013.

[1] D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., in press. DOI:10.1002/mma.4207
[2] C. J. Silva, H. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control Delays, Math. Biosci. Eng. 14 (2017), no. 1, 321-337.
[3] C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4639-4663.

The Scaling Of Species Diversity
Luís Borda-de-Água

Species abundance distributions are central to the description of the diversity of a community and have played a major role in the development of theories of biodiversity and biogeography [1]. However, most work on species abundance distributions has focused on one single scale, typically a spatial scale. Instead, here we look at the evolution of species abundance distributions as a function of area and describe its scaling properties. A practical consequence of being able to describe how species abundance distributions evolve as a function of area is to predict how they look at larger scales, which we do by looking at the scaling properties of its moments. The reasoning is the following: if we know how the moments behave as a function of area then we can extrapolate the moments, then, if we know the moments of a distribution, we can reconstruct its probability density function. There are two venues to reconstruct the probability density function. One is if we consider one specific distribution and know how its parameters relate to the moments. The other is non parametric, and it is based on results from probability theory, which tells us that the moments are the coefficients of the Maclaurin expansion of the characteristic function [2]. The latter approach, however, is not practical in real situations and we use here a method based on discrete orthonormal Tchebichef moments [3]. To exemplify this procedure we use data on tree and shrub species from a 50ha plot of tropical rain forest in Barro Colorado Island, Panama [4]. First, we assess the application of the method within the 50 ha plot and, then, we predict the species abundance distribution for larger areas up to 500ha. We predict that this approach will be of major importance in conservation biology studies because it allows extrapolation of the relative species abundance distribution to larger areas and not only of the number of species [5].
This talk is based on a joint work with Henrique M. Pereira, Stephen P. Hubbell and Paulo A. V. Borges.

[1] Hubbell, S.P., The unified neutral theory of biodiversity and biogeography, Princeton University Press, Princeton NJ USA, 2001.
[2] Feller, W. An introduction to probability theory and its applications, Wiley, London, 1971.
[3] Mukundan, R., Ong, S. H. and Lee, P.A., Image analysis by Tchebichef moments, IEEE Trans. Image Proc., Vol. 10 (2001), pp. 1357-1364.
[4] Condit, R., Tropical forest census plots: methods and results from Barro Colorado Island, Panama and a comparison with other plots. Springer and R. G. Landes Company, Georgetown TX USA, 1998.
[5] Borda-de-Água, L., Borges, P.A.V., Hubbell, S.P. and Pereira, H.P., Spatial scaling of species abundance distributions, Ecography, Vol. 35, (2012), pp. 549-556.

Population Responses To Harvesting In A Discrete-Time Seasonal Model
Eduardo Liz

Population dynamics of many species are influenced by seasonality, and seasonal interactions have the potential to modify important factors such as population abundance and population stability [1]. We consider a discrete semelparous population model with an annual cycle divided into a breeding and a non-breeding season, and introduce harvesting into the model following [2]. We report some interesting phenomena such as conditional and non-smooth hydra effects [3], coexistence of two nontrivial attractors, and hysteresis. Our results highlight the importance of several often underestimated issues that are crucial for management, such as census timing and intervention time.

[1] I. I. Ratikainen et al., When density dependence is not instantaneous: theoretical developments and management implications, Ecol. Lett., Vol. 11 (2008), pp. 184-198.
[2] N. Jonzen and P. Lundberg, Temporally structured density dependence and population man- agement, Ann. Zool. Fennici, Vol. 36 (1999), pp. 39-44.
[3] P. A. Abrams, When does greater mortality increase population size? The long story and diverse mechanisms underlying the hydra effect, Ecol. Lett., Vol. 12 (2009), pp. 462-474.

Recent Results on Epidemiological Models and on Prey-Predator Models
Carlota Rebelo

Mathematical analysis is a useful tool to give insights in very different mathematical biology problems.
In this talk we will present two examples of this fact.
First of all we consider a simple epidemiological model with heterogeneity and discuss the relation between variance in the susceptibility of the individuals and prevalence of infection.
Then we consider predator-prey models. Using the notion of basic reproduction number R_0, given by Nicolas Bacaer in the case of periodic models we prove uniform persistence when R_0 > 1. We will give some examples such as models including competition among predators, prey-mesopredator-superpredatormodels and Leslie-Gower systems.
This talk is based in joint works with N. Bacaer, M. Garrione, M.G.M.Gomes and A. Margheri.

[1] A. Margheri, C.Rebelo and M.G.M. Gomes, On the correlation between variance in individual susceptibilities and infection prevalence in populations, Journal of Math. Biol., 71, (2015) 1643-1661.
[2] M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Analysis: Real World Applications, 30, (2016) 73-98.
[3] C. Rebelo, A. Margheri and N. Bacaer, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64, (2012) 933-949.

Modelling Organisms With Dynamic Energy Budgets
Gonçalo Marques

One of the basic requirements of quantitative research is the creation and use of mathematical models, both in the design of experiments and in the analysis of their results. Dynamic energy budget (DEB) theory [1] is a framework where the full life cycle of individual organisms can be modelled and its energetics can be quantified. All the key processes are included, such as feeding, digestion, storage, maintenance, growth, development, reproduction, product formation, respiration and aging. The theory amounts to a set of simple process-based rules for the uptake and use of substrates (food, nutrients, light) by individuals. It has far-reaching implications for population dynamics and metabolic organization.
In this framework the individual can effectively be modelled in terms of a dynamical system and is defined by a set of parameters. One of the crucial first steps when using DEB is to estimate the parameters for the species of interest. We will start by presenting the standard estimation procedure and the resulting Add-my-pet collection with more than 400 species [2]. In parallel we will show the case of a model for a parasitic wasp [3].
Finally we will discuss the challenges and the latest developments implemented to make the estimation/optimization process more user-friendly.
This talk is based in a joint work with A. L. Llandres, J.Casas, D. Lika, S. Augustine, L. Pecquerie, S.A.L.M. Kooijman and T. Domingos.

[1] S.A.L.M. Kooijman, Dynamic Energy Budget Theory for Metabolic Organization, Cambridge University Press, Cambridge, 2010.
[2] Add-my-pet collection, curated by D. Lika, G.M. Marques S. Augustine, L. Pecquerie, S.A.L.M. Kooijman, http://www.bio.vu.nl/thb/deb/deblab/add_my_pet, 2016
[3] A. L. Llandres, G. M. Marques, J. Maino, S.A.L.M. Kooijman, M. R. Kearney, J. Casas, A Dynamical Energy Budget for the whole life-cycle of holometabolous insects. Ecological Monographs 85, (2014), 353–371