PROGRAMA

Horário 13/07 14/07 15/07 16/07 17/07

08h00 – 08h40

Curso

Momo

Curso

Mostafa

Roxana Eduardo Diomar

08h40 – 09h00

Ponencias Ponencias Ponencias 09h00 – 10h00

Inauguración

10h00 – 10h20

Jorge Café + Carteles Café

(10h00 – 10h30) Café + Carteles

Café

10h20 – 11h00

Gustavo Curso

Brenda Curso Joni 11h00 –

12h00 Ponencias Ponencias BedrEddine Roberto

12h00 – 14h00

Almoço Clausura

14h00 – 14h40

Curso

Momo

Curso

Mostafa

Lourdes Mostafa Curso

Brenda

Curso

Joni

14h40 – 15h00

Ponencias

Asamblea SOLABIMA

15h00 – 16h00

Recepção

16h00 – 16h20

Café + Carteles

Café

16h20 – 17h00

Café Rosana

17h00 – 18h00

Sibona Maria Josefina Condat

1

Resumenes

• Comentarios sobre la evolucion de la virulencia deriva-dos de las dinamicas acopladas de infecciones intra- einterhospedero

Jorge X. Velasco HernandezInstituto de Matematicas UNAM, Juriquilla, Mexico

Resumen

Los modelos matematicos han demostrado ser utiles en la explicacion de laevolucion de la virulencia en patogenos. El acoplamiento de las dinamicas intra-e interhospedero ante una enfermedad infecciosa ha generado nuevas perspectivasen las interacciones que ocurren durante el proceso de seleccion natural en las es-calas intra- e interhospedero. En esta charla revisaremos algunos de los resultadosexistentes en este respecto.

• Influence of an external agent in the opinion formationof a community

Gustavo SibonaUniversidad Nacional Cordoba, Argentina

Abstract

We propose a general non linear analytical framework to study the effect of massmedia in an opinion formation model of a population of moving individuals. Weextend the framework presented by Terranova allowing changes in opinion’s indi-viduals while they are in free movement due to a reflection induced by mass media.We obtain a simple self-propelled agent- model in a closed 2D system, where eachagent adopts one of the n possible opinions or states. The scheme allows us to an-alyze the influence of agent dynamics and society density in the opinion formation.In general the equations of the model could be solved numerically, and, for somespecial cases, it is possible to find analytical solutions of the steady states. In orderto check our scheme for different social conventions we implement computationalsimulations, analyzing also two possible real cases, and finding a good agreementbetween theory and simulation results. We found that the effect of mass mediain our model dominates the opinion state in slow dynamics communities. Addi-tionally, for higher active speeds the rate of interactions increase, and the opinionstate is determined by a competition among the reflection/propaganda rate andthe persuasion rate.

• A model to evaluate vaccination strategies in vector-borne diseases

Lourdes EstevaUniversidad Nacional Autonoma de Mexico, Mexico

Abstract

Vector-borne diseases are one of the major public health problems in the worldwith the fastest spreading rate. Control measures have been focused on vectorcontrol, with poor results in most cases. Vaccines should help to reduce the dis-eases incidence, but vaccination strategies should also be defined. In this workwe propose a vector-transmitted SIR disease model with age-structured populationsubject to a vaccination program. We find an expression for the age dependentbasic reproductive number R0. We show that the disease-free equilibrium is locallystable for R0 < 1, and a unique endemic equilibrium exists for R0 < 1. We applythe theoretical results to public data in order to evaluate vaccination strategies,immunization levels, and optimal age of vaccination for dengue disease.

• Mutantes con beneficio invaden residentes con estruc-tura espacial

Maria Josefina HernandezUniversidad Central de Venezuela, Venezuela

Resumen

Los individuos de una poblacion explotan a los individuos de otra cuando elbalance de costos y beneficios involucrados en la interaccion es positivo para losprimeros a expensas de ser negativo para los segundos. La interaccion es mutualistacuando para unos y otros el balance es positivo. Ambas situaciones existen en lanaturaleza, sin embargo, no ha sido facil explicar teoricamente los mecanismos quehan permitido la evolucion y establecimiento de las interacciones de cooperacionentre especies; no sin entrar en argumentos teoricos polemicos. Dos caracterısticaspoblacionales permiten proyectar una posible ruta para la evolucion del mutual-ismo: la estructura espacial de las poblaciones, y la posibilidad de que los costos ybeneficios per capita en la interaccion sean densodependientes. Si en una poblacionresidente de explotadores -con estas caracterısticas- ocurre una mutacion que pro-duce individuos con menores niveles de explotacion hacia la otra especie, o masaun, individuos cooperadores con los de la otra especie, es posible definir las condi-ciones que permitirıan la invasion del mutante en terminos del grado de relacionentre este y el residente. Una ruta posible para la evolucion y establecimiento deuna asociacion mutualista.

• A delayed Leslie-Gower type predator-prey model withstrong Allee effect in prey growth

Roxana Lopez-CruzUniversidad Nacional Mayor de San Marcos, PeruEduardo Gonzales-OlivaresPontificia Universidad Catolica de Valparaıso, Chile

Abstract

This work deals with a modified Leslie-Gower type predator-prey model consid-ering that the prey population is affect by a weak Allee effect and a time delay inprey population growth

(1) Xµ :

dxdt =

(r(

1− x(t−τ)K

)(x−m) − qy

)x

dydt = s

(1 − y

nx

)y

,

where x = x(t) and y = y(t) represent the prey and predators population sizesfor t ≥ 0, respectively, (measured as number of individuals, biomass or density byarea or volume unity), and µ = (r,K, q, s, n,m) ∈ R5

+ × ]−K,K[. The parametershaving different biological meanings, [4], τ indicates the time lag. Using the delay(τ) as a bifurcation parameter, we analize the stability of coexisting equilibriumpoint and show that Hopf-bifurcation can occur when the discrete delay crossessome critical magnitude.

References

[1] Pallav. Jyoti Pal, Tapan. Saha and M. Sen Malay Banerjee, A delayed predator prey modelwith strong Allee effect in prey population growth, Non Linear Dynamics Vol 68, Issue 1-2,pp 23-42, 2012

[2] F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse dependence and the Allee effect,Trends in Ecology and Evolution 14 (1999) 405-410.

[3] F. Courchamp, L. Berec and J. Gascoigne, Allee effects in Ecology and Conservation, OxfordUniversity Press 2008.

[4] E. Gonzalez-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores, Dynamical complexitiesin the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, AppliedMathematical Modelling 35 (2011) 366-381.

[5] C-P. Ho, C-H. Lin and H-N. Huang, On the Lyapunov functional of the Leslie-Gower predator-prey models with time-delay and Hollings’s functional response, Electronic Journal of Quali-tative Theory of Differential Equations 35 (2012) 1-27.

[6] M. Liermann and R. Hilborn, Depensation: evidence, models and implications, Fish andFisheries 2 (2001) 33-58.

[7] A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model withmodified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Analysis: Re-alWorld Applications 7 (2006) 1104-1118.

• Mathematical modeling on resistant Mycobacterium tu-berculosis dynamics to antibiotics

Eduardo Ibarguen-MondragonaUniversidad de Narino, ColombiaLourdes EstevaUniversidad Nacional Autonoma de Mexico, Mexico

Abstract

We propose a system of non linear ordinary differential equations that describesthe population dynamics of Mycobacterium tuberculosis (Mtb) within the host,in order to study the role of macrophages, T cells and antibiotics in the controlof sensitive and resistant Mtb, assuming that acquisition of resistance is throughspontaneous and natural mutations. Conditions for the persistence of sensitiveand resistant bacteria are given in terms of the secondary infections produced bybacteria and macrophages, the immune response, and the antibiotic treatment.Model analysis predicts backward bifurcations for certain values of the parameters.In this case, the dynamics is characterized by the coexistence of two infection stateswith low and high bacteria load, respectively.

• Controllability in Biology and Medicine

Bedreddine AinsebaUniversite Bordeaux Segalen, France

Abstract

In this talk we will discuss the use of controllability technics in Biology andMedicine. After a brief presentation of the state of the art we will show how wecan solve classical epidemiological or demographical control problems. The caseof structured models with respect to classical variables present in epidemiology orpopulation dynamics will also be discussed. The Tuberculosis control program andthe classical electrocardiography inverse problem will be presented.

• Mathematical Analysis and Numerical Approximationof a Coupled System Modeling Cardiac Electromechan-ics

Mostafa BendahmaneUniversite Bordeaux Segalen, France

Abstract

This talk is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric po-tential and subsequent deformation of the cardiac tissue. The problem consists ina reaction-di?usion system governing the dynamics of ionic quantities, intra andextra-cellular potentials, and the elasticity equations are adopted to describe themotion of an incompressible material. The coupling between muscle contraction,biochemical reactions and electric activity is introduced with a so-called activestrain decomposition framework, where the material gradient of deformation issplit into an active (electrophysiology-dependent) part and an elastic (passive) one.We prove existence of weak solutions to the underlying coupled reaction-di?usionsystem and uniqueness of regular solutions. A ?nite element formulation is alsointroduced, for which we establish existence of discrete solutions and show con-vergence to a weak solution of the original problem. We close with a numericalexample illustrating the behavior of the method and some features of the model.

• Automatos celulares modelando a dinamica do HIV

Rosana Sueli da Motta JafeliceUniversidade Federal de Uberlandia, Brasil

Resumo

A Sındrome de Imunodeficiencia Adquirida (AIDS) e uma doenca infecciosa econtagiosa causada pelo Vırus de Imunodeficiencia Humana (HIV), que causa pro-gressiva perda da imunidade. Esta sındrome e caracterizada por um conjunto desinais e sintomas apresentados pelos indivıduos infectados devido a taxa reduzida delinfocitos T CD4+, que sao celulas de suma importancia para o perfeito funciona-mento do sistema imunologico humano. Especialistas da ONU afirmaram que, em2013, 35 milhoes de pessoas estavam contaminadas com o HIV em todo o mundo; naAmerica Latina, 1.6 milhoes de pessoas estao infectadas, o que indica que a nossasociedade enfrenta um problema biologico de grande repercussao social e economica[5]. O HIV e um retrovırus esferico composto por RNA. Tres proteınas do vırussao de suma importancia para o processo de replicacao: a Transcriptase Reversa,a Integrase e a Protease. O tratamento antirretroviral funciona inibindo estasproteınas. Estes inibidores tentam impedir que partıculas virais livres infectem oslinfocitos T CD4+ e atrasem a replicacao viral, permitindo que o organismo reajanaturalmente [3]. O sistema de equacoes diferenciais ordinarias apresentado em [4]nao consegue descrever todas as fases da dinamica do HIV, sem tratamento comantiretrovirais. Desta forma, apresentamos o estudo de dois modelos de automatoscelulares. O primeiro modelo diz respeito a dinamica do HIV em indivıduos semtratamento com terapia antirretroviral. O segundo modelo considera os indivıduossob terapia antirretroviral. Os efeitos da terapia antirretroviral no automato celu-lar sao modelados usando um sistema baseado em regras fuzzy. A simulacao erealizada para tres indivıduos HIV hipoteticos; com diferentes adesoes ao trata-mento e potencias variadas do medicamento. Inclusive, obtemos a taxa de infeccaodo linfocito T CD4+ pelo HIV para esses indivıduos. Os resultados de ambos osmodelos de automatos celulares estao de acordo com as informacoes relatadas naliteratura sobre o assunto [1] e [2].

References

[1] Jafelice, R.M., Bechara, B.Z., Barros, L.C., Bassanezi, R.C., Gomide, F. Cellular Automatawith Fuzzy Parameters in Microscopic Study of Positive HIV Individuals Mathe-matical and Computer Modelling, 50 32-44 (2009).

[2] Jafelice, R.M., Silva, P.N., Studies on Population Dynamics Using Cellular Au-tomata, Intech, chapter 6, 105-130, In: Cellular Automata-Simplicity Behind Complexity,Alejandro Salcido, editor (2011).

[3] Ministerio da Saude. http://www.aids.gov.br (accesso: Outubro 2013).[4] Nowak, M.A., Bangham,C.R.M. Population Dynamics of Immune Responses to Per-

sistene Viruses, Science, 272 (1996).[5] UNAIDS. http://www.unaids.org (acesso: Outubro 2014).

• Stabilization times for self-propelled microorganism dis-tributions confined to shallow chambers

Carlos Alberto CondatUniversidad Nacional Cordoba, Argentina

Abstract

We calculate the characteristic time for the stabilization of a low-density dis-tribution of self-propelled microorganisms injected in a shallow chamber. Thesetimes are found to depend on the nature of the surrounding wall, i.e. whether itis smooth or petal-shaped, and on the shape of the microbial trajectories. In thecase of sperm cells, which follow trajectories where quasi-linear and quasi circularstretches alternate, we obtain an equation that allows us to estimate the stabiliza-tion time in terms of the rotational diffusion coefficient and the duration of thequasi-linear stretches.

• Efeitos de ruıdos sobre a dinamica espaco-temporal deuma populacao de presas e seus predadores1

Diomar Cristina MistroUniversidade Federal de Santa Maria, Brasil

Resumo

O risco de extincao de uma populacao e uma questao de grande importanciaem Ecologia. Variacoes estocasticas (chamadas ruıdos) podem provocar a extincaomesmo quando a taxa de crescimento media e positiva. Muitos autores tem explo-rado os efeitos de variacoes estocasticas sobre a persistencia e risco de extincao depopulacoes e ha, atualmente, uma vasta literatura que explora os tipos de ruıdos ea maneira como afetam o crescimento e extincao das populacoes.

Nesta palestra, apresentamos um sistema presa-predador discreto espacialmenteestruturado descrito via Redes de Mapas Acoplados no qual a populacao de presasapresenta Efeito Allee e os predadores sao especialistas. Analisamos os efeitos dediferentes tipos de ruıdos sobre o risco de extincao local e global das populacoes esobre a formacao e estabilidade de padroes espaciais.

1Trabalho realizado em colaboracao com Luiz Alberto Dıaz Rodrigues (UFSM), Horst Malchow (Uni-versity of Osnabruck) e Joice Chaves Marques (PPGMat/UFSM) e apoiado pela FAPERGS (Processo0414-2551/14-0).

• A stochastic microscopic model for the dynamics of anti-genic variation

Gustavo GuerberoffUniversidad de la Republica, Uruguay

Abstract

I present a novel model that describes the within-host evolutionary dynamics ofparasites undergoing antigenic variation. The approach uses a multi-type branchingprocess with two types of entities defined according to their relationship with theimmune system: clans of resistant parasitic cells (i.e. groups of cells sharing thesame antigen not yet recognized by the immune system) that may become sensitive,and individual sensitive cells that can acquire a new resistance thus giving riseto the emergence of a new clan. The simplicity of the model allows analyticaltreatment to determine the subcritical and supercritical regimes in the space ofparameters. By incorporating a density-dependent mechanism the model is ableto capture additional relevant features observed in experimental data, such as thecharacteristic parasitemia waves. The model can be easily adapted to cope withbroader and more complex situations.

• Human population size and the risk of epidemics of vector-borne diseases

Roberto Andre KraenkelUnesp, Brazil

Abstract

We formulate a model for vector-borne diseases taking into account the vectorpopulation dynamics and modelling the interaction of vectors with human hosts.We allow the vector bite-rate to depend on the total population numbers of humansand vectors. The model departs from the strict frequency dependent models byassuming functional forms coming from host-parasite ecological theory. We builda framework which allows us to compute R0 under several assumptions.

We focus on the impact of host population size on disease invasibility and wediscuss situations where departures from frequency-dependent models should beexpected, as for instance, in forest fringe malaria.

Short Courses• Recent progress in inverse problems in electrocardiology

Mostafa BendahmaneUniversite Bordeaux Segalen, France

Abstract

In this course, we will study an homogeneous and non-homogeneous inverse prob-lem in electrocardiology representing the heart, lungs surfaces, and torso model.Our goal is to reconstruct the electrical potentials on the surface of the heart fromthe information obtained non invasively on the torso surface. The existence anduniqueness of solution for the heart-torso problem and the related inverse problemis assessed, and the primal and dual problems are discretized using a finite elementmethod. We present some preliminary numerical experiments using an efficient im-plementation of the proposed scheme in homogeneous and nonhomogeneous cases.

• Procesos no lineales en biologıa: problemas, estrategias,modelos, ejemplos

Fernando Roberto MomoUniversidad Nacional de General Sarmiento, Argentina

Contenido

– Respuestas no lineales, bifurcaciones reales, analisis de fluctuaciones. ¿Quetipo de respuestas no lineales son comunes en fenomenos biologicos? ¿Quesignifica una bifurcacion en terminos biologicos? Ejemplos. ¿Como sabemossi estamos ante un fenomeno no lineal? Exponente de Hurst, exponentes deLyapunov, signos de criticalidad, signos de complejidad.

– Transiciones crıticas y cambios bruscos ¿Se pueden modelar? Modelos deriva-dos de teorıa de catastrofes. Las catastrofes elementales y sus aplicacionesbiologicas. Otros tipos de transiciones crıticas. Bifurcaciones y caos en bi-ologıa. Ejemplos. Herramientas de analisis.

– Transiciones crıticas y cambios bruscos ¿Se pueden predecir? ¿Que le pasa a lossistemas cuando estan cerca del cambio brusco? Propiedades de escalamiento.Propiedades dinamicas. Propiedades ecologicas y homeostaticas. Cuando loscambios bruscos no significan cambio de regimen. Ejemplos.

• Dos tecnicas de estabilidad global en modelos de quimiostatos

Brenda Tapia SantosUniversidad Veracruzana, Mexico

Contenido

– Bases matematicas∗ Estabilidad de sistemas de ecuaciones diferenciales autonomos.∗ Criterio de Routh Hurwitz.∗ Teorema Pi-Buckinham.∗ Funcion de Liapunov.

– Modelos en quimiostato∗ Modelo consumidor–recurso.∗ Modelo de competencia en un quimiostato.∗ Modelo de competencia con inhibidores internos.∗ Modelo de competencia con inhibicion interna y resistencia.

• Modelos biomatematicos en ecuaciones diferenciales par-ciales

Joao Frederico da Costa Azevedo MeyerUnicamp, BrasilPaulo Cesar Carmona TabaresUniversidad del Quindıo, ColombiaIrene Duarte GandicaUniversidad del Quindıo, Colombia

Contenido

– Introduccion a los modelos en ecuaciones diferenciales parciales: ecuacion deconservacion, flujos en 2 y 3 dimensiones, formas clasicas del flujo (adveccion-conveccion, atraccion-repulsion, difusion-dispersion).

– Solucion numerica de ecuaciones diferenciales parciales: metodos de diferenciasfinitas, metodo de Crank-Nicolson.

– Ejemplos de modelos en ecuaciones diferenciales parciales: clasicos (pestebubonica, rabia), dispersion de contaminantes, propagacion del toxoplasma.