A mathematical ranking model in learning analytics

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(1)Proceedings of the 16th International Conference on Computational and Mathematical Methods in Science and Engineering Costa Ballena (Rota), Cádiz, Spain July 4th-8th, 2016. Editors J. Vigo-Aguiar Associate Editors P. Schwerdtfeger (New Zealand), W. Sprößig (Germany), N. Stollenwerk (Portugal), Pino Caballero (Spain), J. Cioslowski (Poland), J. Medina (Spain), I. P. Hamilton (Canada), J.A. Alvarez-Bermejo (Spain).

(2) Proceedings of the 16th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE-2016 Costa Ballena (Rota), Cádiz, Spain July 4-8, 2016. Editors J. Vigo-Aguiar Associate Editors P. Schwerdtfeger, W. Sprößig, N. Stollenwerk, Pino Caballero, J. Cioslowski, J. Medina, I. P. Hamilton, J.A. Alvarez-Bermejo.

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(4) Contents: Volume I: An Eco epidemic predator-prey model with prey vaccination Abbona, Francesca; Venturino, Ezio ........................................................ 1 Hub-directed multigraphs and arrowhead matrices Abderramán Marrero, J.; Núñez, Juan; Villar, María Trinidad.......... 13 Fast algorithms for solving general k-tridiagonal matrix linear equations Abderramán Marrero, Jesús .................................................................... 25 A class of tests for the two-sample problem for count data based on the empirical probability generating function Alba-Fernández, M. Virtudes; Batsidis, Apostolos; Jiménez-Gamero, M. Dolores; Jodrá, Pedro ................................................................................ 29 Critical Sections and Software Transactional Memory Comparison in the Context of a TLS Runtime Library Aldea, Sergio; Llanos, Diego R.; González-Escribano, Arturo ............. 35 Fixed Point Theorems for Graph Dynamical Systems Aledo, JA.; Díaz, Luis G.; Sanua, S M.; Valverde, J C. /////////////////48 A fractional Malthusian growth model with variable order using an optimization approach Almeida, Ricardo; Bastos, Nuno R. O.; Monteiro, Teresa M. T. .......... 51 Backward error analysis of almost strictly sign regular matrices Alonso, P.; Peña, J.M.; Serrano, M.L. //////////////////////////////////////////////////77. Real-Time Audio-to-Score Alignment using Multi-Core Architectures Alonso, P.; Vera-Candelas, P.; Cortina, R.; Rodríguez-Serrano, F. J.; Alonso-González, M.; Ranilla, J. .............................................................. 64 A technique to avoid order reduction in the integration of linear initial boundary value problems with Lie-Trotter method ;

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(7) A secure and efficient ECC based method to avoid impersonation for the SIP protocol. Álvarez-Bermejo, J. A.; Lopez-Ramos, J. A. ........................................... 76 Safe Control of Luggage with Homomorphic Cryptography Álvarez-Díaz, Néstor; Caballero-Gil, Pino ............................................. 86 Natural grid numerical methods revisited in cell population balance models with asymmetric division Angulo, O.; López-Marcos, J. C.; López-Marcos, M. A......................... 97 Distribution function estimates from dual frame context Arcos, Antonio; Martínez, Sergio; Rueda, María del Mar; Martínez, Helena........................................................................................................ 102 Convergence of Newton's method under Vertgeim conditions: new extensions using restricted convergence domains Argyros, I. K.; Ezquerro, J. A.; Hernández-Verón, M. A.; Magreñán, Á. A. ................................................................................................................ 114 Improving the domain of parameters for Newton's method Argyros, Ioannis K.; Magreñán, Á. Alberto; Sicilia, Juan Antonio .... 118 FDM for Stochastic Partial Differential Equations Ashyralyev, Allaberen ............................................................................. 124 Bounded Solutions of Nonlinear Parabolic Equations with Time Delay Ashyralyev, Allaberen; Agirseven, Deniz; Ceylan, Burcu ................... 128 Role of Cell Competition in Acquired Chemotherapy Resistance

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(10) ! ................................ 132 Inverse estimation of terminal connections in the cardiac conduction system Barber, F.; Lozano, M.; García-Fernández, I.; Sebastián, R. ............. 142 Parameter Extraction in Electron Devices by means of Polynomial Pattern Analysis Barrera, D.; Ibáñez, M. J.; Roldán, A. M.; Roldán, J. B.; Yáñez, R... 154 Numerical solution of Love's integral equation by quasi-interpolation Barrera, D.; Elmokhtari, F.; Ibáñez, M. J.; Sbibih, D. ........................ 159 Basic reproduction number in a spatially structured model for gut microbiota Barril, Carles; Calsina, Àngel; Ripoll, Jordi ......................................... 164.

(11) A family of second derivative free forth order continuation method for solving nonlinear equations Behl, R.; Maroju, P.; Motsa, S. S............................................................ 170 Performance Evaluation of the Iteratively Reweighted Least Squares Algorithm (IRLS) on a Multi-Core Platform Belloch, Jose A.; Ramiro, Carla; Quintana-Ortí, Enrique S.; Vidal, Antonio M. ................................................................................................ 178 Conversational recommendation to avoid the start-up problem Benito-Picazo, F.; Enciso, M.; Rossi, C.; Guevara, A. .......................... 184 Numerical solution of two-dimensional nonlinear Volterra integral equations Berenguer, María Isabel; Gámez, Domingo .......................................... 191 An efficient method for solving two-dimensional Fredholm integral equations Berenguer, María Isabel; Gámez, Domingo .......................................... 194 Advances in time-dependent current-density functional theory Berger, Arjan ............................................................................................ 198 Parallelization of the 3D Fast Wavelet Transform on a Cluster of Raspberry Pi 2 Boards Bernabé, Gregorio; Hernández, Raúl; Acacio, Manuel E.................... 199 Rational blends of two cones from square-root parameterized medial axis transforms Bizzarri, Michal; Lávicka, Miroslav ...................................................... 211 Approximating Support Function at Inflection Points for CNC Manufacturing Blazková, Eva; Sír, Zbynek..................................................................... 219 The influence of distributed delays on Hes1 gene expression model Bodnar, Marek ......................................................................................... 229 A trade-off between explicit and implicit schemes to solve differential equations on GPUs Bondarenco, Marcelo; Gamazo, Pablo; Ezzatti, Pablo ........................ 239 Auto-Tuning TRSM with an Asynchronous Task Assignment Model on Multicore, GPU and Coprocessor Systems Boratto, Murilo; Alonso, Pedro; San Juan, Pau; Giménez, Domingo 244 Linear and Cyclic Codes over direct product of Finite Chain Rings Borges, J.; Fernández-Córdoba, C.; Ten-Valls, R. ............................... 250 Competition between algae and fungi in a lake: a mathematical model Bulai, Iulia Martina; Venturino, Ezio .................................................... 261.

(12) Mathematical Aspects on Traffic of Incompressible Worms on Simple Circular Structures Buslaev, Alexander P.; Yashina, Marina V. .......................................... 273 QL-fuzzy Implications by Means of Overlap and Grouping Functions Bustince, H.; Elkano, M.; Dimuro, G.; Bedregal, B.; Sesma-Sara, M.; Lucca, G. ................................................................................................... 280 The natural embedding of fuzzy preposet and its residual mapping Cabrera, Inma P.; Cordero, Pablo; Ojeda-Aciego, Manuel ................ 289 Coating of C60 by para-H2 and ortho-D2: revisiting the solvation shell Calvo, Florent; Yurtsever, Ersin ............................................................ 296 Exact solutions and conservation laws of a Generalized Fornberg-Whitham Equation Camacho, J. C.; Bruzón, M. S................................................................. 303 The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term Cao, Xuenian; Cao, Xianxian.................................................................. 306 The correlation attack to LFSRs as a syndrome decoding problem Cardell, Sara D.; Climent, Joan-Josep; Roca, Alicia ............................ 314 On a simple construction of primitive polynomials Cardell, Sara D.; Climent, Joan-Josep ................................................... 322 The modified self-shrinking generator via the generalized self-shrinking generator Cardell, Sara D.; Fúster-Sabater, Amparo............................................ 326. Volume II: Minimal Faithful Upper-Triangular Matrix Representations for Solvable Lie Algebras Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F. .............................. 329 Two-Stage Intra Prediction Algorithm for HEVC Cebrián-Márquez, Gabriel; Martínez, José Luis; Cuenca, Pedro ...... 334 Exploiting Multi-Level Parallelism on a Many-core System for the Application of Hyperheuristics to a Docking Problem Cecilia, J. M.; Cutillas-Lozano, J.M.; Giménez, D.; Imbernón, B. ..... 346.

(13) A non-singular and positive Bhattacharya method for the numerical modelling of the dewetting process of thin films Chávez-Guzmán, Axel; Medina-Ramírez, I. E.; Macías-Díaz, J. E. ... 354 Rovibrational States of Wigner Molecules in Spherically Symmetric Confining Potentials Cioslowski, Jerzy ...................................................................................... 358 An efficient numerical method to solve 2D parabolic convection-diffusion singularly perturbed problems with turning points Clavero, C.; Vigo-Aguiar, J..................................................................... 363 Multicriteria design of energy-conscious fuzzy rule-based classifiers for embedded devices Cocaña-Fernández, Alberto; Ranilla, José; Sánchez, Luciano; Gil-Pita, Roberto ...................................................................................................... 372 On a sixth-order family for solving nonlinear models combining derivatives and divided differences Cordero, Alicia; Torregrosa, Juan R.; Vassileva, María P. ................. 376 On the dynamics of a class of iterative methods with memory for solving nonlinear equations Cordero, Alicia; Torregrosa, Juan R. ................................................... 385 Approximating the matrix sign function by means of Chebyshev-Halley type method Cordero, Alicia; Soleymani, Fazlollah; Torregrosa, Juan R.; Zaka Ullah, M. ............................................................................................................... 396. Dynamical study of Ostrowski' and Chun's methods for solving nonlinear systems Cordero, Alicia; Maimó, Javier G.; Torregrosa, Juan R.; Vassileva, María P...................................................................................................... 406 Computing the validity of attribute implications in multi-adjoint concept lattices Cornejo, M. E.; Medina, J.; Ramírez-Poussa, E. .................................. 414 On applying a parallel Teaching-Learning-Based optimization procedure for automatic heliostat aiming Cruz, N. C.; Redondo, J. L.; Álvarez, J. D.; Berenguel, M.; Ortigosa, P. M. ............................................................................................................... 424 Empirical Modelling: An Auto-tuning Method for Linear Algebra Routines on CPU+multiGPUs Platforms Cuenca, Javier; García, Luis P.; Giménez, Domingo; Herrera, Francisco J.................................................................................................................. 433.

(14) Fast Intra Mode Decision for an H.264/AVC to HEVC Video Transcoder Diaz-Honrubia, A. J.; Martinez, J. L.; Cuenca, P................................. 443 Phase fitted splitting methods for oscillatory genetic regulatory systems Ehigie, J. O.; Zhang, R.; Hou, X.; You, X. ............................................ 453 A model for Leshmaniasias disease transmission considering asymptomatics and reservoirs Esteva, L.; Vargas, C.; Vargas de León, Cruz ...................................... 457 A faithful functor among algebras and graphs Falcón, Óscar J.; Falcón, Raúl M.; Núñez, Juan; Pacheco, Ana M.; Villar, María Trinidad............................................................................. 466 Trigonometrically fitted explicit symmetric six-step methods Fang, Yonglei ............................................................................................ 478 Effective infection rate in SIR-type models from models with symptomatic and asymptomatic infection Filipe, Raquel; Stollenwerk, Nico; Mateus, Luís; Ghaffari, Peyman; Halstead, Scott; Aguiar, Maíra ............................................................... 483 Lyapunov spectra for torus bifurcations and ways to deterministic chaos in population biology Fuentes Sommer, Pablo; Stollenwerk, Nico; Kooi, Bob; Mateus, Luís; Ghaffari, Peyman; Aguiar, Maíra .......................................................... 491 Black-List Genetic Algorithm Scheduling for Energy Saving in Heterogenous Environments Gabaldon, Eloi; Guirado, Fernando; Lerida, Josep Lluis; Planes, Jordi........................................................................................................... 499 Parallel processing in GPUs for intra-picture prediction in HEVC Galiano, Vicente; Migallón, Héctor; Herranz, Victoria; Piñol, Pablo; López-Granado, Otoniel; Malumbres, Manuel P. ................................ 510 A modified exponential method to approximate positive and bounded solutions of the Burgers-Fisher equation Gallegos, A.; Macías-Díaz, J. E.; Vargas-Rodríguez, H. ...................... 521 Invariant set for third-order switched systems García-Gutiérrez, J. B.; Bénitez-Trujillo, F.; Pérez, C. ....................... 528 The influence of size and temperature dependence on the shape preference of nanoclusters and the implications for heterogeneous catalysis Garden, Anna L.; Pedersen, Andreas; Jónsson, Hannes ..................... 537 Lie symmetries and equivalence transformations for the Barenblatt-Gilman model Garrido, T. M.; Kasatkin, A. A.; Bruzón, M. S.; Gazizov, R. K. ........ 539.

(15) From Clusters to the Liquid State: explaining the anomalous melting temperatures of gallium clusters Gaston, Nicola; Steenbergen, Krista G. ................................................ 543 Solving second order non-linear elliptic partial differential equations using generalized finite difference method Gavete, Luis; Ureña, Francisco; Benito, Juan J.; García, Ángel; Ureña, Miguel; Salete, Eduardo .......................................................................... 546 Using Optimal Control Theory with Mosquito Repellents and Vaccination Applied to Dengue Disease Prevention and Reduction Management, a First Toy Study with Analytically Treatable Models Ghaffari, P.; Wijaya, K. P.;Aguiar, M.; Mateus, L.; Götz, T.;Stollenwerk, N. .................................................................................... 561 Paramagnetic H-related defects in silica: a first-principles investigation. Giacomazzi, Luigi; Martin-Samos, L.; Richard, N. .............................. 570 Two-body interactions and the physics of natural occupation numbers and amplitudes Giesbertz, Klaas J. H.; van Leeuwen, Robert ...................................... 574 Exact solutions of the Schrodinger equation for two electrons on a sphere Gill, Peter M. W.; Loos, Pierre-Francois ............................................... 579 A Hybrid GPU Technique for Real-Time Terrain Visualization González, Cesar; Pérez, Mariano; Orduña, Juan M. ........................... 584 Controlling Oscillations of a Hanging String with a Tip Mass González-Santos, G.; Vargas-Jarillo, C. ............................................... 594 Thwarting randomness reveals in group key agreement González-Vasco, M. Isabel; Pérez del Pozo, Ángel L.; Suárez Corona, Adriana ..................................................................................................... 606 Natural Convection MHD Stokes Flow in a Square Cavity Gürbüz, M.; Tezer-Sezgin, M. ................................................................ 615 Solid State Materials with Transition-Metal Clusters and Fullerenes as Building Blocks Hammerschmidt, Lukas; Schacht, Julia; Gaston, Nicola .................... 624 Cloud implementation of the K-means algorithm for hyperspectral image analysis Haut, Juan Mario; Paoletti, Mercedes; Plaza, Javier; Plaza, Antonio 630 On the Geometric-Arithmetic Index by decompositions Hernández, Juan Carlos; Rodríguez García, José Manuel; Sigarreta, José M........................................................................................................ 642.

(16) Solving algebraic Riccati equations with an efficient iterative process with fourth order of convergence Hernández-Verón, M. A.; Romero, N. ................................................... 652. Volume III: Serial concatenation of a block code and a 2D convolutional code Herranz, V.; Perea, C. ............................................................................. 657 On some properties of colour morphology operators Huidobro, Pedro; Alonso, Pedro; Montes, Susana; Díaz, Irene .......... 665 Tensor Rank Decomposition of the Coulomb Integrals Hummel, Felix; Grüneis, Andreas .......................................................... 672. Insights into the Bonding Situation of Interstitial Gold Clusters and Ligand Stabilized Au(0) Complexes Jerabek, Paul; Frenking, Gernot ............................................................ 673 Efficient Implementation of Morphological Index for Building/Shadow Extraction from Remotely Sensed Images Jiménez, Luis Ignacio; Plaza, Javier; Plaza, Antonio ........................... 676 Multi-Core Implementation of Spatial-Spectral Pre-processing for Hyperspectral Unmixing Jiménez, Luis Ignacio; Bernabé, Sergio; García, Carlos; Plaza, Javier; Martín, Gabriel; Sánchez, Sergio; Plaza, Antonio................................ 688 Widely Linear Quaternion Signal Filter from One-Step Delayed Observations Jiménez-López, J. D.; Fernández-Alcalá, R. M.; Ruiz-Molina, J. C.; Navarro-Moreno, J. ................................................................................. 698 An extension of the Muth distribution Jodrá, P.; Gómez, H. W.; Jiménez-Gamero, M. D.; Alba-Fernández, M. V. ................................................................................................................ 702 Dynamics of a Disk on a Rotating Plane with Friction Karapetyan, A. V. .................................................................................... 708 The Dynamics of a Heavy Ridid Ellipsoid on a Horizontal Plane with Friction Karapetyan, A. V.; Munitsyna, M. A. .................................................... 713 Persistence analysis of the age-structured population model on several patches Kozlov, Vladimir ...................................................................................... 717 M-adic Residue Codes over Fq[v] / (v2-v) and DNA Codes.

(17) Kuruz, Ferhat; Segaz Oztas, Elif; Siap, Irfan ....................................... 728 Dynamical of a bouncing ball Lampart, "#$ %&!, Jaroslav ..................................................... 736 Modelling parts of branched skins using rational envelope surfaces Lávicka, Miroslav; Bizzari, Michal ........................................................ 742 An epidemic model for cholera with treatment through quarantine Lemos-Paião, Ana P.; Silva, Cristiana J.; Torres, De!fim F. M............ 752 An overview of canonical Euler splitting methods for nonlinear composite stiff evolution equations Li, Shoufu.................................................................................................. 758 Accuracy analysis of a 2D adaptive mesh refinement method using lid-driven cavity ow and two refinements Li, Zhenquan; Wood, Robert .................................................................. 773 Nodal surfaces in quasi-exactly solvable models Loos, Pierre-Francois............................................................................... 785 How group size influences the efficiency of FMM López-Fernández, J. A.; López-Portugués, M.; Ranilla, José; GonzálezAyestarán, R.; Las-Heras, F. ................................................................... 789 A consistent first order theory about the equilibrium figures in close binary systems López-Ortí, José Antonio; Forner Gumbau, Manuel; Barreda Rochera, Miguel ........................................................................................................ 796 Distributed Group Key Exchanges reusing randomness López-Ramos, J. A.; Rosenthal, J.; Schipani, D.; Schnyder, R. .......... 800 Parallel landing sites detection using LiDAR data on manycore systems Lorenzo, Oscar G.; Martínez, Jorge; Vilariño, David L.; Pena, Tomás F.; Cabaleiro, José C.; Rivera, Francisco F................................................. 805 Stop&Restart vs Resilient MPI applications Losada, Nuria; Martín, María J.; González, Patricia .......................... 817 A deterministic model for the distribution of the stopping time in a stochastic model and its numerical solution Macías-Diaz, Jorge Eduardo; Villa-Morales, José .............................. 825 Relationship of the conflation in the Belnap's logic with the (crisp) stable model semantics Madrid, N.................................................................................................. 829.

(18) Improved convergence analysis for Newton-like methods Magreñán, Á. Alberto; Argyros, Ioannis K.; Sicilia, Juan Antonio .... 837 Some novel and optimal families of King's method with eighth and sixteenthorder of convergence Maroju, P.; Behl, R.; Motsa, S. S............................................................ 841 Dynamical properties of traffic speed '*

(19) + %/3 ................................................................................... 855 Benchmarking of third-order reduced density matrices approximations using the harmonium atom. Matito, Eduard; Rodríguez-Mayorga, Mauricio; Ramos-Córdoba, Eloy; Feixas, Ferran ........................................................................................... 861 Consensus formation in a system of difference equations modelling controversial opinion dynamics with pairwise interactions Medina-Guevara, M. G.; Macías-Díaz, J. E.; Gallegos, A.; VargasRodríguez, H. ............................................................................................ 866 A superconvergent partial differential equation approach to price variance swaps Mehzabeen Jumanah, Dilloo; Yannick, Tangman Désiré .................... 870 Sequences of sums of squares and CATALAN numbers Miana, Pedro J.; Romero, Natalia .......................................................... 889 Evaluation of an Evolutionary Multi-Objective Optimization algorithm on a ARM+GPU system Moreno, Juan José; Ortega, Gloria; Filatovas, Ernestas; Martínez, José Antonio; Garzón, Ester M. ...................................................................... 893 Time series representation using fuzzy logic ARM+GPU system Moreno-Garcia, Antonio; Moreno-Garcia, Juan; Jiménez, Luis; Rodriguez-Benítez, Luis .......................................................................... 898 Application of GFDM: Modelling of Geophysical Methods Muelas, A.; Salete, E.; Benito, J. J.; Ureña, F.; Gavete, L.; Ureña, M. ............................................................................................................... 908 Windows for escaping particles in quartic galactic potentials Navarro, Juan F. ...................................................................................... 920 Accelerating Microrheology models on HPC architectures Ortega, Gloria; Puertas, Antonio M.; Garzón, Ester M. ..................... 932 Argon under High Pressure Pahl, Elke; Wiebke, Jonas; Senn, Florian; Schwerdtfeger, Peter ....... 938.

(20) Isolated in-homogeneities of arbitrary shape with polynomial fields prescribed at infinity. The potential problem in two dimensions. Parnell, William J.; Calvo-Jurado, Carmen.......................................... 940 Global Optimization-Density Functional Theory Study of Tin Oxide Clusters: Structures, Energies, and Trends Paul, Wesley; Fournier, René ................................................................. 944 Stabilization of switched linear systems by using projections Pérez, C.; Benítez-Trujillo, F.; García-Gutiérrez, J. B. ....................... 953 An energy evaluation of data-parallel applications in heterogeneous systems Pérez, Borja; Stafford, Esteban; Bosque, Jose Luis; Beivide, Ramón 965 The Generalized Hyers-Ulam Stability of Additive $\rho$-Functional Inequalities in Random Normed Spaces Phiangsungnoen, Supak; Kumam, Wiyada .......................................... 977. Volume IV: Tile partition analysis for a parallel HEVC encoder Piñol, Pablo; López-Granado, Otoniel; Migallón, Héctor; Galiano, Vicente; Malumbres, Manuel P. ............................................................. 989 Tumour--immune system interaction model with distributed delays Piotrowska, Monika Joanna; Bodnar, Marek ....................................... 999 Explicit and efficient exponential splitting time integrator for the KleinGordon equation with Absorbing Boundary Conditions Portillo, A. M.; Alonso-Mallo, I. ........................................................... 1010 Time Series on Functional Service Life of Buildings using Fuzzy Delphi Method Prieto, A. J.; Chávez, María-José; Garrido-Vizuete, María A.; MacíasBernal, J. M.; Cagigas-Muñiz, Daniel .................................................. 1016 Extension of Newton's method for solving systems of equations when the classical Newton method fails Ramos, Higinio; Monteiro, M. T. T. ..................................................... 1027 Monte Carlo Approach for the Pricing of European Multi-Asset Options Rasulov, A.; Raimova, G. ...................................................................... 1038 On the quasi-positive systems Ricarte, Beatriz; Romero-Vivó, Sergio ................................................ 1047.

(21) Strategies for colour mathematical morphology Riesgo, A.; Alonso, P.; Díaz, I.; Montes, S. .......................................... 1051 Assistance Management Application based on IBSC for Emergency Situations Rivero-García, Alexandra; Hernández-Goya, Candelaria; SantosGonzález, Iván; Caballero-Gil, Pino .................................................... 1060 Stability and Optimal Control of a Delayed HIV Model Rocha, Diana; Silva, Cristiana J.; Torres, Delfim F. M. .................... 1071 Analysing criminal networks using Formal Concept Analysis with negative attributes Rodriguez-Jiménez, J. M.; Cordero, P.; Enciso, M.; Mora, A. ......... 1076 A Rendezvous Framework for the Automatic Deployment of Services in Cluster Computing Rodríguez-Quintana, Cristina; Díaz, Antonio F.; Ortega, Julio; Palacios, Raúl H.; Ortiz, Andrés .......................................................................... 1087 Modelling the effects of a differentiated mortality by phenotypic traits on the genotypic distribution Rojas-Castro, Héctor; Córdova-Lepe, Fernando................................ 1093 Energy Consumption of Stencil-Based MPDATA Algorithm Rojek, Krzysztof; Barreda, Maria; Quintana-Ortí, Enrique S.; Wyrzykowski, Roman ............................................................................ 1104 A fuzzy regression approach using Bernstein polynomials for the spreads and an application to a real Economic context Roldán-López de Hierro, A.F.; Martínez-Moreno, J.; Aguilar-Peña, C.; Roldán, C. ............................................................................................... 1108 Symmetry reductions and Conservation laws for a type of Fisher equations Rosa, M.; Gandarias, M. L.................................................................... 1114 An improved class of estimators of a linear parameter using auxiliary information in randomized response surveys Rueda, María del Mar; Cobo, Beatriz ................................................. 1118 A first approach to column updating of Nonnegative Matrix Factorization San Juan Sebastián, P.; Vidal, A. M.; García-Mollá, V. M. .............. 1125 Accelerating Schur Complement Domain Decomposition Method for Wind Field Calculation Sanjuan, Gemma; Margalef, Tomàs; Cortés, Ana ............................. 1132.

(22) Certificate Graph Based Authentication for Communications in Emergency Situations Santos-González, Iván; Caballero-Gil, Pino; Molina-Gil, Jezabel; Rivero-García, Alexandra ..................................................................... 1144 One-fermion picture for Moshinsky-type atoms and significance of generalized Pauli constraints Schilling, Christian................................................................................. 1156 Ab initio calculation of electronically excited states for large molecular systems Schütz, Martin ........................................................................................ 1165 The Extended Lennard-Jones Potential for Cubic Solids Schwerdtfeger, P.; Pahl, E..................................................................... 1167 DRBEM Solution of Biomagnetic Fluid Flow under a Point Source Magnetic Field Senel, P.; Tezer-Sezgin, M. .................................................................... 1172 Numerical simulation of cable truss systems using meshfree RBF method Simonenko, Stanislav; Loya, Jose Antonio; Rodriguez Millan, Marcos; Angot, Philippe ....................................................................................... 1184 Melting mercury with a quantum model - clusters and bulk Steenbergen, K. G.; Pahl, E.; Calvo, F.; Schwerdtfeger, P. ............... 1190 Geometrical Interpretation of Complex Signals as a Tool to Study Fluctuations at Nanoscale Tadic, Bosiljka ........................................................................................ 1193 Taking out even more features from the input subset based on feature ranking Tallón-Ballesteros, Antonio J.; Correia, Luís...................................... 1197 On Cyclic Codes over Zq+uZq Temiz, Fatih; Siap, Irfan ....................................................................... 1203 A DRBEM approach for the Stokes eigenvalue problem Tezer-Sezgin, M.; Türk, Önder ............................................................ 1210 Bias-induced effects in single molecule charge transport Thijssen, Jos; Celis Gil, Jose; de Boer, Josko ...................................... 1220 Factorization and inversion of finite and infinite bordered tridiagonal matrices Tomeo, Venancio .................................................................................... 1222 Quantum entanglement in two-electron systems with emphasis on -Spherium Toranzo, Irene V. ................................................................................... 1234.

(23) Transition-metal oxide clusters: structural and magnetic properties, infrared spectra and perspectives with applications in catalysis Torres, M. B.; Aguado, A.; Aguilera-Granja, F.; Vega, A.; Balbas, L. C. .............................................................................................................. 1238 Golden Dual Fullerenes and their Topological Relationship to Fullerenes Trombach, L.; Rampino, S.; Wang, Lai-S.; Schwerdtferger, P. ....... 1242 Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions Turbiner, A. V.; Escobar-Ruiz, M. A. .................................................. 1244 Application of generalized finite difference method to reflection and transmission problems in seismic SH waves propagation Ureña, M.; Benito, J. J.; Ureña, F.; Salete, E.; Gavete, L.; García, A. .............................................................................................................. 1250 Optimizing a pivot-based algorithm for similarity search on a GPU-based platform Uribe-Paredes, Roberto; Arias, Enrique; Cazorla, Diego; Sánchez, José L. .............................................................................................................. 1262 Contour curves and isophotes on ruled surfaces Vršek, Jan ............................................................................................... 1274 Fast numerical valuation of European options under Merton’s jump-diffusion model Wang, W.; Chen, Y. ............................................................................... 1282 Exact and discretized dissipativity of the nonlinear functional-integrodifferential equations Wen, L.; Liao, Q. .................................................................................... 1289 Quantum Wigner molecules in semiconductor quantum dots and cold-atom optical traps and their mathematical symmetries Yannouleas, Constantine; Landman, Uzi ............................................ 1298 The method of cloud services using for testing in mathematical education Yashina, M.V.; Dotkulova, A.S.; Nakonechniy, I.I. ............................ 1309.

(24) Volume V: Dynamical Study of Gursey Instantons with Bichromatic Force Aydogmus, Fatma; Tosyali, Eren ......................................................... 1328 Surface-induced L10 ordering processes in nanostructured intermetallics with magnetic anisotropy: Monte Carlo simulation Brodacka, Sylwia; Kozlowski, Miroslaw; Kozubski, Rafal; Goyhenex, Christine; Murch, Graeme E. ............................................................... 1337 Advancing Algorithms to Increase Performance of Correlated and Dynamical Electronic Structure Simulations de Jong, Wibe A.; Jacquelin, Mathias; Bylaska, Eric J.; Vogiatzis, Konstantinos; Gagliardi, Laura ........................................................... 1342 Piecewise Modelling and Simulation of a Rotating Extensible Manipulator Link for Base Placement and Path Smoothness Dupac, Mihai .......................................................................................... 1347 A numerical approximation to the solution of the first Painlevé equation of fractional order Erturk, Vedat Suat................................................................................. 1357 Extension of confidence bands based on the exact distribution of the order statistics for Normal S-P Plots Estudillo-Martínez, María Dolores; Castillo-Gutiérrez, Sonia; LozanoAguilera, Emilio ..................................................................................... 1363 Network model for simulating 1-D soil consolidation processes under loadunload conditions García, G.; Alhama, I.; Sánchez, J. F................................................... 1366 On the capabilities of the open-source HEVC Codecs Garcia-Lucas, David; Cebrián-Márquez, Gabriel; Cuenca, Pedro .. 1375 Chirality at the Nanoscale Garzón, I.L.............................................................................................. 1387 Hedonic and spatial analyses applied to the massive assessment of real estate appraisals performed by appraisal companies in the Province of Valencia (Spain) Guadalajara, N.; López, M.A................................................................ 1390 Optical properties of graphene quantum dots: Hawrylak, P. ........................................................................................... 1401.

(25) Study of influence of surface mass coefficient of free chloride in reinforced concrete using Spice code Hidalgo, P.; Sánchez-Pérez, J. F.; Alhama, I....................................... 1403 Smoothed particle hydrodynamics method with partially defined fluid particles Kanetsuki, Yasutomo; Wells, John C.; Nakata, Susumu ................... 1407 Density matrix simulations of quantum electron dynamics in perturbed atoms and electron gas Kitamura, Hikaru .................................................................................. 1419 Solving the Schrödinger Equation of Harmonium Systems with the FreeComplement Local-Schrödinger Equation Method Kurokawa, Yusaku I.; Nakatsuji, Hiroshi ........................................... 1427 Analytical study of labour markets based on graph theory Lloret-Climent, M.; Nescolarde-Selva, J.; Mora, H.; Signes-Pont, M. T. .............................................................................................................. 1433 Detached Simulation of Lateral Jet Interaction flow Ma, J.; Liu, Y.; Liu, Y. .......................................................................... 1437 Stress-strains contours in heterogeneous soils under arbitrary loads at the surface Mena-Requena, M. R.; Morales, J. L.; Alhama, L. ............................ 1442 Vibrational correlation formalism applied to internal conversion rate constants in metal clusters Mineva, Tzonka; Chiodo, Sandro G. ................................................... 1452 Combined use of Geogebra and 3D impression for Geometry learning Orcos, L.; Arís, Nuria ............................................................................ 1455 Design and dissemination of the MENTOR Tutorial Attention Plan in the School of Industrial Engineering of the Universidad de Valladolid Portillo, A. M.; Fernando, M.; Alarcia, E.; Cuello, L.; Díez, P.; Fernández, S.; Fernández, N.

(26) .......................................................... 1461 Calculated Forecast for Technical Obsolescence in Computerised Tomography Equipment Reyes-Santías, F.; Cadarso-Suárez, C.; Espasandin, J....................... 1467 QM/MM simulations of Au nanoclusters and glutathione ligands in water solvent Rojas-Cervellera, Victor; Rovira, Carme; Akola, Jaakko ................. 1474 The nanofluidics of small droplets on hydrophobic surfaces Smith, Alex; Mahelona, Keoni; Hendy, Shaun ................................... 1485.

(27) Ab initio modelling of semiconductor epitaxy processes – gas phase, surface and interfaces Stegmüller, Andreas; Rosenow, Phil; Tonner, Ralf ............................ 1489 Essential Collective Dynamics of Biological Polymers Stepanova, Maria ................................................................................... 1493 Modelling of Nanoparticle-Enzyme Complex Stueker, Oliver; Stepanova, Maria ....................................................... 1496 Comparison of data mining tools Tallón-Ballesteros, A. J.; Benavides-Vallejo, J. E. ............................. 1499 On two optimisation problems related to unsatisfied demand on a time interval Todinov, M. T. ........................................................................................ 1505 Numerical Solutions of GPE under Gaussian Trap Tosyali, Eren; Aydogmus, Fatma ......................................................... 1516 A mathematical ranking model in learning analytics Van der Merwe, A.; Kruger, H. A.; du Toit, J. V. .............................. 1525 Analysis and Prediction of Crossing effect on Inherent Deformation during the Line Heating Process – Part 2 – Multiple Crossed heating lines Vega Saenz, Adan ................................................................................... 1536 A New Method for Calculation of Density of Compressible Flow Wang, T. .................................................................................................. 1546. Late Contributions: An almost fully parallel algorithm for computing the component tree of a binary digital image based on HSF Díaz-del-Rio F.; Real, P. and Onchis, D. .............................................. 1550.

(28) Preface It is a great pleasure to welcome you to the 16th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2016), will be held at Rota, Cádiz (Spain), July 4th-8th, 2016. This consolidate international conference offers the opportunity of discuss on new breakthrough ideas in different applied mathematics disciplines and creates synergies among the attendants in order to advance the frontier of knowledge and, as a consequence, to propose new (European) research projects. Horizon 2020 is the new EU Framework Programme for Research and Innovation. New Societal and Technology problems have been raised where Applied Mathematics seems to be a fundamental field in the vast majority of them. The importance of this discipline in other areas, such as Engineering, Computer Science, Physics and Chemistry, must be a value in order to be part of the funded projects. The different advances are included in the extended abstracts and papers accepted to the conference and will be collected in these proceedings of CMMSE 2016. The proceedings we have pleasure to present here has five volumes, the first four correspond to the articles typeset in LaTeX and the fifth to articles typeset in Word. During CMMSE 2016 you will have the opportunity of discuss about current and new challenges, exchanging ideas, comments and suggestions leading to the improvement and deepening of the papers presented to allow further development of the research occurs. The attendants must also take advantage of the plenary speakers and important attendant researchers, in order to get new ideas and, maybe, propose future collaborations. Twenty symposia show the variety of disciplines considered in the conference, which are formed from high quality accepted papers. The first one, high-performance computing, considers new large-scale problems that arise in fields like bioinformatics, computational chemistry, and astrophysics. Mathematically modeling the future Internet and developing future Internet security technology is a self-explanatory session. The third symposium addresses analytical, numerical and computational aspects of partial differential equations in life and materials science. Computational finance is a session focusing on solving problems related to asset pricing, trading and risk analysis of financial assets that have no analytic solutions under realistic assumptions and thus require computational methods to be resolved. A forum for discussion of the growing impact of new technologies on teaching and the development of new tools to increase learning efficiency is provided in the symposium: new educational methodologies supported by new technologies. The symposium on mathematical models and information-intelligent systems on transport researches in the field of flow-modelling of particles with motivated behavior in complex networks, applied to traffic flows, pedestrian flows, ecology, etc. The seventh symposium studies computational methods for linear and nonlinear optimization and numerical methods for solving nonlinear problems is given in another session. Bio-mathematics studies both theoretical and practical applications of population dynamics, ecoepidemiology, epidemiology of infectious diseases and molecular and antigenic evolution. The 10th symposium will put particular emphasis on species involving Coulombically interacting particles either trapped by external potentials or confined to hypersurfaces: Quasi-Solvable Systems in Quantum Chemistry and Physics. Model interesting problems arisen in Computer Science, considering algebraic and computational (fuzzy) techniques, is the main goal of mathematical models for computer science. The aim of the 12th @CMMSE. Preface- Page iii.

(29) symposium is to obtain a consistent description of the transition from clusters to the solid state, which is a major challenge in computational chemistry and physics. The enormous potential of fixed point theory, which is needed in mathematics, engineering, chemistry, biology, economics, computer science, and other sciences, justifies the great interest in fixed point theory in various abstract spaces and related applications. Computational methods in direct and inverse (systems of) PDE’s covers general phenomena formulated as control problems or inverse problems associated with mathematical models described by partial differential equations (PDE). An overview of mathematical and computational research focusing on corporate or government applications and problems arising from different economic sectors is presented in the 16th mini-symposium: Industrial Mathematics. Parallel implementation using hybrid architectures with accelerators, either GPUs or FPGAs, of numerical methods for solving problems within the following topics of interest: industrial mathematics, fluid mechanics, global optimization, finance, geophysical flows, computational chemistry, electromagnetism, magneto hydrodynamics, atomic physics, relativistic flows; is given in the symposium: Numerical simulation on GPUs. Mathematics in the Information Society aims at presenting recent advances in practical applications of Cryptology, Coding Theory and Development of Technologies that address challenges related to Communication Systems. The computational methods for fluid flow are considered in this symposium, which provides a forum for discussion of current problems and recent advances in the area. Finally, the last symposium, Uncertainty and Imprecision Modelling in Decision Making, introduces new results about the concepts and procedures commonly used in decisionmaking processes where an imprecise knowledge of the data is present. We want to conclude this preface giving thanks to the plenary speakers for their outstanding contributions to research and leadership in their respective fields, including physics, computer science and engineering. We would also like to thank the special session organizers and scientific committee members, who have played a very important part in setting the direction of CMMSE 2016. Finally, we would like to thank the participants because, without their interest and enthusiasm, the conference would not have been possible. We cordially welcome all participants. We hope you enjoy the conference. Costa Ballena, Rota, Cádiz (Spain), July 3rd, 2016. I. P. Hamilton, J. Medina & J. Vigo-Aguiar. @CMMSE. Preface- Page iv.

(30) CMMSE 2016 Mini-symposia Session Title High Performance Computing (HPC) P.D.E.'S in Life and Material Sciences. Type of Session/Organizers J. Ranilla Regular Session. Computational Finance New Educational Methodologies Supported by New Technologies Mathematical Models and InformationIntelligent Systems on Transport Computational Methods for Linear and Nonlinear Optimization Numerical Methods for Solving Nonlinear Problems Bio-mathematics. Regular Session Alexander P. Buslaev & M. Yashenina Maria Teresa Torres Monteiro Juan R. Torregrosa & A. Cordero. Ezio Venturino & Nico Stollenwerk & Maíra Aguiar Quantum Jerzy Cioslowski. Quasi-Solvable Systems in Chemistry and Physics Mathematical Models for Computer Science From clusters to the solid State Computational Linear and NonLinear Algebra Fixed Point Theory in various abstract spaces and related applications Computational methods in direct and inverse (systems of) PDE’s Industrial Mathematics Mathematics in the Information Society Computational methods for fluid flow Uncertainty and Imprecision Modelling in Decision Making. @CMMSE. Jesús Medina & Manuel Ojeda-Aciego Ian Hamilton & Peter Schwerdtfeger P. Alonso Regular Session Regular Session Regular Session P. Caballero Gil & F.J. Lobillo Borrero & J.A. López Ramos & E. Martínez Moro Zhenquan Li & Tiejin Wang P. Alonso, H. Bustince, I. Díaz & S. Montes. Preface- Page v.

(31) Acknowledgements We would like to express our gratitude to the University of Cádiz, especially to the group of Prof Medina of the dept. of Mathematics and professor José Antonio Álvarez Bermejo - Universidad de Almería,. We also would like to thank all of the local organizers for their efforts devoted to the success of this conference: o o o o o o o o o o. Pedro Alonso, Universidad Oviedo Jose Ranilla, Universidad Oviedo Raquel Cortina, Universidad Oviedo Carmelo Clavero, Universidad Zaragoza Higinio Ramos, Universidad Salamanca José Antonio Álvarez Bermejo - Universidad de Almería, María Eugenia Cornejo Piñero - Universidad de Cádiz, Juan Carlos Díaz Moreno - Universidad de Cádiz, Eloisa Ramírez Poussa - Universidad de Cádiz, M.T. Torres Monteiro Univerisdade Braga Protugal. CMMSE 2016 Plenary Speakers . Wolfgang Sprößig - TU Bergakademie Freiberg, Germany Nico Stollenwerk - Lisbon University, Portugal Humberto Bustince – University Public of Navarra, Spain Motoko Kotani - Tohoku University, Japan. @CMMSE. Preface- Page vi.

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(33) Proceedings of the 16th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2016 4–8 July, 2016.. An ecoepidemic predator-prey model with prey vaccination. Francesca Abbona1 and Ezio Venturino1 1. Dipartimento di Matematica “Giuseppe Peano”, Universit` a di Torino, via Carlo Alberto 10, 10123 Torino, Italy emails: abbona.francesca@gmail.com, ezio.venturino@unito.it. Abstract In this paper an eco-epidemiological predator-prey model is proposed, in which the presence of a disease in the prey is assumed. Even if the disease does not affect the predators, it can lead to extinction of both populations. Assuming the latter to be a useful resource, the prey vaccination as control measure is introduced in order to protect biodiversity. Boundedness of solutions and equilibria feasibility are obtained. Stability around the different equilibrium points is analyzed through eigenvalues and the Routh-Hurwitz criterion. Simulations are carried out to support the theoretical results. Key words: vaccination, predator-prey model, population models, epidemics MSC 2000: AMS codes: 92D25, 92D10, 92D40. 1. Introduction. Eco-epidemiology is a branch in mathematical biology which considers simultaneously both the ecological and the epidemiological issues. Ecological populations suffer from various infectious diseases, which have a significant role in regulating population sizes. Mathematical studies of such eco-epidemiological models have explored several unknown aspects of ecological population interactions, see e.g. [6]. In ecosystems, the interaction between the predator and their prey is a nonlinear and rather complex process. The effects of disease-induced mortality or disease-reduced reproduction in regulating natural populations, decreasing their population sizes, reducing their natural fluctuations, or causing destabilization of equilibria into oscillations of the population states have been considered in [3, 4]. The findings of [1] indicate that from both the ecological and the mathematical points of view it is important to study ecological systems. c CMMSE. 1. ISBN: 978-84-608-6082-2.

(34) An ecoepidemic predator-prey model with prey vaccination.. subject to epidemiological factors. The aim of this study is to investigate the impact of a vaccine provided on the infected prey population in an eco-epidemiological mathematical predator-prey model. Costs coming from vaccination strategy can be high. However, it is evident that the impact on the system may be beneficial and thereby regulate and preserve the populations in the system. The paper is organized as follows. After a brief introduction, In Section 2 basic assumptions are considered and the model is then formulated. Section 3 investigates first of all the model boundedness and consequently the feasible equilibra, the stability of which is hence examined. Finally Section 4 includes numerical simulations, in order to support and illustrate the analytical result achieved.. 2. Model Formulation. A mathematical model is proposed, involving three populations: the prey population density N , partitioned among susceptible, infected and vaccinated, N (t) = S(t) + I(t) + V (t), and the predator population P (t). The model assumptions are: In the absence of the disease, the prey population N (t) grows logistically with intrinsic growth rate r and carrying capacity k. In the disease presence, prey are partitioned among susceptible S(t) and infected I(t). Susceptible prey become infected by contact with infected prey, at rate λ; they are also removed by predation at rate h. Healed infected prey are not immune and return to the susceptible class at rate γ. Susceptible prey can be vaccinated at rate ν. Newborns from vaccinated prey are susceptible, whereas infected do not reproduce. Susceptible, infected and vaccinated experience intraspecific competition and predation. Predators die at rate n. Thus the model, in which all parameters are nonnegative, reads ⎧ dS S+I +V ⎪ ⎪ + rV − λIS + γI − νS − hSP = rS 1 − ⎪ ⎪ dt k ⎪ ⎪ ⎪ ⎪ dI S+I +V ⎪ ⎨ = λIS − μI − γI − gIP − rI dt k . (1) S+I +V dV ⎪ ⎪ ⎪ = νS − lV P − rV ⎪ ⎪ dt k ⎪ ⎪ ⎪ dP ⎪ ⎩ = −nP + e hS + gI + lV P dt The first equation for susceptibles accounts for their reproduction and intraspecific competition, for vaccinated reproduction, the infection process at rate λ, the vaccination at rate ν, disease recovery at rate γ, and predation, at rate h. The second equation for infected includes new recruitments, through successful contact with a susceptible, natural. c CMMSE. 2. ISBN: 978-84-608-6082-2.

(35) Francesca Abbona, Ezio Venturino. plus disease-related mortality μ, disease recovery, predation and intraspecific competition. The third equation for the vaccinated class accounts for the new entries, the predation at rate l, and intraspecific competition. Finally the fourth equation for the predators considers their mortality at rate n and the benefit of hunting on the three prey classes, scaled via the conversion coefficient e < 1. The Jacobian of (1) is ⎤ ⎡ − kr S − λS + γ − rS J11 k + r −hS ⎢ λI − rI J22 − rI −gI ⎥ k k ⎥ (2) J =⎢ rV ⎣ ν − rV −k J33 −lV ⎦ k ehP egP elP J44 with Sr rI rS rI rV rV − − , J33 = −lP − − −2 , k k k k k k rI rV Sr −2 − , J44 = −n + e (hS + gI + lV ) = λS − μ − γ − gP − k k k. J11 = r − λI − ν − hP − 2 J22. 3. Model analysis. 3.1. Boundedness. To show that the model proposed is well posed, we prove its boundedness. Let U = S +I +V and W = U + P . Then dU r r dW + ηW ≤(r + η)U − U 2 + (η − n)P, + ηU ≤ (r + η)U − U 2 . dt k dt k The right hand side of the second inequality represents a concave parabola Ψ, with maximum k 2 value at U = k(r+η) 2r . Hence, Ψ(U ) = 4r (r + η) and we have U + ηU ≤ Ψ(U ),. W + ηW ≤ Ψ(U ) + (η − n)P.. Since η is arbitrary, we choose it so that 0 < η < n for which it follows W ≤ Ψ(U ) − ηW, giving W ≤. Ψ(U ) (1 − e−ηt ) + W (0)e−ηt . η. As t → ∞, we have W ≤ max{Ψ(U )η −1 , W (0)} = Z, implying that the solutions U , i.e. S, I, V , and P are bounded. All the solutions for the model for any ε > 0 are confined in the region: Γ = (S, I, V, P ) ∈ R4+ : W ≤ Z + ε .. c CMMSE. 3. ISBN: 978-84-608-6082-2.

(36) An ecoepidemic predator-prey model with prey vaccination.. 3.2. Equilibrium points. In addition to the origin E0 , and coexistence, the remaining equilibria Ek = (Pk , Sk , Ik , Rk ) are the following: the predator-and-disease-free point E1 = (S1 , 0, V1 , 0), the disease-free point E2 = (S2 , 0, V2 , P2 ) and the predator-free point E3 = (S3 , I3 , V3 , 0). Specifically, we analyse each one of them below. Equilibrium E1 For the predator-and-disease-free point respectively the third and first equations in system (1) give the two isoclines: S1 =. rV 2 , νk − rV. V1 =. rS + k(ν − r) S. r(k − S). (3). S1 has a positive branch raising up from the origin to infinity at the vertical asymptote V = νkr−1 . Instead V1 has a similar behavior if r < ν, raising up to S = k, or from S0 = k(1 − νr−1 ) < k = S if r > ν. The two curves in the S − V phase subplane meet always in the first quadrant, even when they both go through the origin, since V1 is convex and S1−1 is concave. The intersection occurs in the box which is the cartesian product of < S < k, where we have set m = max{0, S0 }. This the intervals 0 < V < νkr−1 and m ensures that both S1 and V1 are positive. Therefore the feasible existence of E1 is always guaranteed. Equilibrium E2 For the disease-free point from the third equilibrium equation we have S2 =. n l − V, eh h. (4). which entails the feasibility condition: n 0 < V2 < V = . el. (5). The remaining equilibrium populations are the intersection of the following isoclines: 2 1 rl l−h nl n n rn + ek − h V − eh V − h lν − r l − 2 hke r − ν − keh Π(V ) kh h , = P = ϕ1 (V ) = n lV − e l(V − V ) (6) r(l − h)eV 2 − (νlek + rn)V + νnk Γ(V ) P = ϕ2 (V ) = = . (7) kehlV kehlV. c CMMSE. 4. ISBN: 978-84-608-6082-2.

(37) Francesca Abbona, Ezio Venturino. Study of the curve ϕ1 ϕ1 is defined for all V ≥0, except for V = V . It intersects the vertical axis at −1 P0 = h r 1 − ν − n(ekh)−1 , whereas the intersections with the horizontal one, existing if its discriminant Δ1 is nonnegative, are the roots V2± of the parabola Π(V ) in the numerator. The latter is convex if and only if l > h. Let P0 = n(eh)−1 r 1 − n(ekh)−1 = Π(0) denote the parabola intersection with the vertical axis. Then P0 = −ne−1 P0 . There are several cases depending on the convexity of Π and on the location of its roots with respect to V . Only in some of them φ1 admits a positive branch, which are discussed below. The ranges for a viable V are specified. We begin with the cases for Δ1 > 0 and l > h: [1] V− < V+ ≤ 0 < V ≤ V ; here φ1 has a branch coming down on the right of the vertical asymptote at V to a minimum positive value and then tending to +∞ as V → +∞. [2] V− ≤ 0 ≤ V ≤ V+ < V ≤ V ; there is an arc joining (0, P0 ) with (V+ , 0) and further also a branch coming down on the right of the vertical asymptote at V to a minimum positive value and then tending to +∞ as V → +∞. [3] 0 ≤ V− ≤ V ≤ V+ < V < V ; there is a positive concave arc joining (V− , 0) and (V+ , 0) and a branch coming down on the right of the vertical asymptote at V to a minimum positive value and then tending to +∞ as V → +∞. [4] V− < 0 ≤ V < V < V+ ≤ V ; there are two branches raising up, the first one to the vertical asymptote from (0, P0 ), the second one going from the point (V+ , 0) to +∞ as V → +∞. [5] 0 < V− ≤ V < V < V+ ≤ V ; two branches raise up, the first one to the vertical asymptote from (V− , 0), the second one going from the point (V+ , 0) to +∞ as V → +∞. [6] 0 < V < V ≤ V− < V+ ≤ V ; again there are two branches, one coming down on the right of the vertical asymptote to (V− , 0), the second one going from the point (V+ , 0) to +∞ as V → +∞. Next, the cases for Δ1 > 0 and l < h: [7] V− < V+ ≤ 0 ≤ V < V ; here φ1 has a branch raising up from (0, P0 ) to the vertical asymptote at V . [8] V− ≤ 0 ≤ V+ ≤ V < V ; there is an arc raising up from (V+ , 0) to the vertical asymptote at V . [9] 0 ≤ V ≤ V− ≤ V+ ≤ V < V ; there are two branches, the first one coming down from (0, P0 ) to (0, V− ), the second one going from the point (V+ , 0) to the vertical asymptote at V . [10] V− < 0 ≤ V+ ≤ V < V ; there is an arc raising up from (V+ , 0) to the vertical asymptote at V . [11] 0 ≤ V ≤ V− < V < V ≤ V+ ; two branches come down, the first one from (0, P0 ) to (V− , 0), the second one coming down from the vertical asymptote to the point (V+ , 0). [12] 0 ≤ V < V ≤ V− ≤ V ≤ V+ ; one branch raises up from (0, P0 ) to the vertical asymptote. c CMMSE. 5. ISBN: 978-84-608-6082-2.

(38) An ecoepidemic predator-prey model with prey vaccination.. at V , and then a concave arc joining (V− , 0) with (V+ , 0). Finally we consider the case for Δ1 < 0 and l < h, as the opposite one cannot arise: [13] 0 < V < V there is a branch coming down on the right of the vertical asymptote at V to a minimum positive value and then tending to +∞ as V → +∞. Study of the curve ϕ2 ϕ2 is defined for all positive values of V , with a vertical asymptote at the origin. The intersections V ± of Γ with the horizontal axis exist depending on its discriminant Δ2 = (rn + νlek)2 − 4rnkν(l − h). [A] For l > h and Δ2 < 0, Γ > 0 everywhere, so that ϕ2 from the vertical asymptote at the origin comes down to a minimum positive value and then raises up to +∞ as V → +∞. [B] For l < h and Δ2 > 0, since Γ(0) > 0, Γ is positive for 0 < V < V + , so that ϕ2 comes down from the vertical asymptote at the origin to the value zero at V = V + . For l > h and Δ2 > 0, Γ is positive for V < V − and V > V + , and since Γ(0) > 0, there are two cases: [Ca] V − < V + < 0, for which ϕ2 is convex and comes down from the vertical asymptote at the origin to a minimum positive value and then tends to +∞ as V → +∞; [Cb] 0 < V − < V + ; in this situation ϕ2 comes down from the vertical asymptote at the origin to the point (V − , 0) and then raises up from (V + , 0) to +∞. Study of the intersections of the curves ϕ1 and ϕ2 We now combine the above situations in order to see whether an intersection between ϕ1 and ϕ2 is guaranteed. There are other cases in which with further conditions other intersections may occur, but we do not explore them. The intersection exists unconditionally in the following cases. We give also the interval in which the abscissa V∗ lies. For Δ1 > 0, Δ2 > 0 and l > h: [4Ca] (V∗ < V ), [5Ca] (V− < V∗ < V ), [6Ca] (V < V < V− ). For Δ1 > 0, Δ2 < 0 and l > h: [1Cb] (max{V + , V } < V ∗ ), [2Cb] (max{V + , V } < V ∗ ), [3Cb] (max{V + , V } < V ∗ ), [4Cb] (V ∗ < min{V − , V }), ; for [5Cb] the intersection exists if V− < V ∗ < min{V − , V }. For Δ1 > 0, Δ2 > 0 and l < h: for [7B] and [12B] the location is V ∗ < min{V + , V }; in the next three cases the intersection exists conditionally, namely if V+ < V ∗ < min{V + , V }: for [8B], [9B] and [10B]. For [11B] the intersection exists if max{V + , V } < V ∗ < V + . The case Δ1 < 0, Δ2 < 0 and l > h instead does not guarantee an intersection without further stricter conditions, which we do not explore. Equilibrium E3 For the predator-free point from the second equation I = η1 (S, V ) =. c CMMSE. λk k − 1 S − V − (μ + γ). r r. 6. (8). ISBN: 978-84-608-6082-2.

(39) Francesca Abbona, Ezio Venturino. From (8) I is nonnegative if: V3 <. λk k − 1 S3 − (μ + γ). r r. (9). Substituting (8) into the other equations, two functions are obtained, depending on S: V = η2 (S) = V = η3 (S) =. ν S, λS − (μ + γ). (10). Ω(S) λ2 kS 2 − {r(r − ν + μ) + λk[γ + r(μ + γ)]} S + kγ(μ + γ) . = r (λS + r − γ) rλ(S − S) (11). Study of the curve η2 The function η2 goes through the origin, it has a vertical asymptote at S = (μ + γ)λ−1 It is always decreasing, having an always negative but it is positive only when S > S. derivative μ+γ S = −ν . (12) [λS − (μ + γ)]2 Therefore there is only one possibility: [I] η2 it comes down from the asymptote at S = S to the horizontal asymptote V = νλ−1 . Study of the curve η3 The isocline η3 is feasible for S ≥ 0, it has a vertical asymptote at S = (γ − r)λ−1 . It crosses the vertical axis at the height V0 = kγ(μ + γ)[r(r − γ)]−1 . The parabola Ω is convex, with positive height at the origin, Ω(0) = kγ(μ + γ) > 0. The roots of Ω are complex if its discriminant is negative, Δ3 < 0. Then for η3 there is: (a) a positive branch from (0, V0 ) tending to +∞ as S → +∞ if S < 0, i.e. r > γ; (b) a positive branch coming down from the right of the vertical asymptote at S = S to a minimum and then tending to +∞ as S → +∞ if S > 0, i.e. r < γ. For Δ3 > 0 since Ω(0) > 0, both roots have the same sign. For negative roots we have: (c) S < S− < S+ < 0, (d) S− < S < S+ < 0, (e) S− < S+ < S < 0, all imply that for η3 there is a positive branch from (0, V0 ) tending to +∞ as S → +∞; (f) S− < S+ < 0 < S the positive branch of η3 comes down from the right of the vertical asymptote at S to a positive minimum and then raises up to +∞ as S → +∞. When the roots are positive instead we find: (g) S < 0 < S− < S+ gives to two positive branches for η3 , one coming down from (0, V0 ) to (S− , 0), the other one raising up from (S+ , 0) to +∞ as S grows; (h) 0 < S < S− < S+ gives to two positive branches for η3 , one coming down from the right. c CMMSE. 7. ISBN: 978-84-608-6082-2.

(40) An ecoepidemic predator-prey model with prey vaccination.. of the vertical asymptote at S to (S− , 0), the other one from (S+ , 0) to +∞ as S grows; (i) 0 < S− < S < S+ gives to two positive branches for η3 , one raising up from (S− , 0) to the vertical asymptote, the other one raising up from (S+ , 0) to +∞ as S grows; (j) 0 < S− < S+ < S gives two positive branches for η3 , a concave one joining (S− , 0) with (S+ , 0), the other one coming down from the right of the vertical asymptote to a positive minimum and then raising up to +∞ as S → +∞. Study of the intersections of the curves η2 and η3 Combining these cases, a feasible intersection always occurs at the abscissa S ∗ for: For [Ia], [Ic], [Id], [Ie] the intersection lies in S < S ∗ ; for [Ig], [Ih] it lies in S+ < S ∗ ; [Ib], [If], [Ij], all give an intersection S < S ∗ if S < S; ∗ [Ii] gives an intersection S+ < S if S < S and a second one S < S ∗ < S if S < S. Equilibrium E3 is feasible if η2 and η3 intersect as discussed above and (9) holds. Finally, the coexistence equilibrium will be analyzed numerically.. 3.3. Stability analysis. Equilibrium E0 The Jacobian here shows two negative eigenvalues, Λ1 = −(μ + γ) and Λ2 = −n. The other ones are the roots of the equation Λ2 − tr(J(E0 ))Λ + det(J(E0 )) = 0 coming from a minor J of the Jacobian, where tr(J(E0 )) and det(J(E0 )) provide the stability condition tr(J) = r − ν < 0 but the remaining one cannot be satisfied, det(J) = −rν < 0 so that E0 is always unstable. Equilibrium E1 At E1 we find r r Λ1 = J22 |E1 = λS − S − V − μ − γ, k k. Λ2 = J44 |E1 = −n + ehS + elV. The other two eigenvalues do not affect the result. In fact, the Routh-Hurwitz conditions on the remaining 2 by 2 minor J are satisfied, 2 2 1 )) = r r V + ν S + rV + νS > 0. 1 )) = − r (S + V ) − r V − ν S < 0, det(J(E tr(J(E k S V k S V Thus the stability conditions are: hn λk − r k −S >V > S − (μ + γ) l eh r r. (13). The second condition (13) is exactly the opposite of (9), thereby showing a transcritical bifurcation between E3 and E1 , while comparing the first one with (4), it follows that whenever E2 is feasible, E1 must be unstable. Thus, when equilibrium E1 is stable, equilibria. c CMMSE. 8. ISBN: 978-84-608-6082-2.

(41) Francesca Abbona, Ezio Venturino. E2 and E3 do not exist. On the other hand, if at least one of them is feasible, then E1 is unstable. Equilibrium E2 Here one eigenvalue is explicitly obtained, Λ2 = J22 |E2 = λS − (μ + γ) − gP − rk −1 (S + V ). The characteristic polynomial is the cubic a 0 Λ3 + a 1 Λ2 + a 2 Λ + a 3 = 0 with 2 2 r r S S 2V 2S + (l − h)2 SV, a1 = (S + V ) + ν + r , a3 = hl(νS + rV ) + rl + νh k V V S V k 2 2 V r r2 r S r − (νS + rV ) + eh2 SP. +ν a0 = 1, a2 = 2 2 SV + el2 V P + 2rν + k k S V k Stability occurs when the Routh-Hurwitz conditions hold, but since a1 > 0 and a3 > 0, we need only to require the last one, as well as the negativity of the eigenvalue Λ2 < 0: a 1 a 2 > a3 ,. r λS2 < (μ + γ) + gP2 + (S + V ). k. (14). Equilibrium E3 At E3 = (S3 , I3 , V3 , 0), again one eigenvalue is Λ4 = J44 |E3 = −n+e (hS + gI + ). UslV 3 ing the Routh-Hurwitz criterion for the remaining cubic characteristic equation i=0 bi Λi , with b0 = 1 r V r I S b1 = − I +r − (S + V ) + γ + ν k S k S V r IV rν(S + V )S b2 = (γ + r) + + νr + λγI+ k S kV r(rV 2 + γI 2 ) νγ rγ r2 (S + V )I 2 + +I λ S+ + − k2 kS V k 3 2 2 2 2 r V S rνS rλ S rνγ + + + + νλ2 + b3 = I k2 S kV k V k 2 r γIV rγλV S r3 2 r2 νS 2 rλνS −I + + λνγ + 3 S + 2 + k2 S k V k k V k Stability is achieved when Λ4 < 0, and the Routh-Hurwitz conditions hold, i.e. explicitly. r k. . hS + gI + lV < ne , r2. c CMMSE. 2. V k S. . b1 b2 > b3 , S2. + ν VS + λ2 S 2 + νγ + νλγ V >. r V k I+ r S r r IV k kγ S. 9. < kr (S + V ) + γ SI + ν VS , +. r2 k2. S 2 + kr ν. S2 V. . (15). + γλV + λνS .. ISBN: 978-84-608-6082-2.

(42) An ecoepidemic predator-prey model with prey vaccination.. 4. Simulations. The simulations, implemented with the Matlab solver ode45, are run for selected sets of parameter values, in part obtained from the literature and in part taken as hypothetical. They are summarized in Table1. Parameter r K λ μ n ν h g l γ e. Value 11.2 10000 0.4 0.3 0.1 0.3 0.1 0.25. Explanation intrinsic growth rate carrying capacity susceptible to infected rate of conversion infected mortality due to disease predators mortality vaccination rate predation of susceptible predation of infected predation of vaccinated infected to susceptible conversion rate conversion factor of prey into new predators. Source [5],[6] hypothetical simulated [5],[6] [7] simulated hypothetical hypothetical hypothetical simulated [5],[6]. Table 1: Model parameters. Figure 4, left frame, represents the endemic state: the four populations coexist. In this way the infection is not eradicated. Coexistence is feasible with the parameters choice: λ = 5, ν = 0.3, γ = 11. In this way only a very small portion of the susceptible population is treated with vaccine. Moreover, it is assumed that the disease spreads easily, but also infected recover at high rate. The same situation is reproduced with λ = 9: infection is assumed to have a higher incidence with the same portion of vaccinated as in the previous situation, in the right frame of Figure 4. The equilibrium E2 is stably attained for the following choice of parameters: λ = 1.2, ν = 10, γ = 3. Initial conditions are S = 100, I = 500, M = 10, N = 50. Infected prey become quickly extinct and susceptible prey, vaccinated and predators show decaying oscillations before they reach the stable values, 4 left frame. Even on the assumption of having a greater number of contacts λ = 5 between susceptible and infected prey, with the same rate of vaccination equilibrium E2 is achieved, see right frame of Figure 4.. 5. Conclusions. In this paper, a nonlinear predator-prey ecoepidemic model is proposed and analyzed in order to study the effect of a vaccine on the prey population suffering from a disease. Apart from the extinction point, other four equilibria have been shown to be feasible. c CMMSE. 10. ISBN: 978-84-608-6082-2.

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