From the Guest Editors: Special issue dedicated to Carlos Castillo-Chavez on his 60th birthday

Carlos Castilo-Chavez is a Regents Professor, a Joaquin Bustoz Jr. Professor of Mathematical Biology, and a Distinguished Sustainability Scientist at Arizona State University. His research program is at the interface of the mathematical and natural and social sciences with emphasis on (i) the role of dynamic social landscapes on disease dispersal; (ii) the role of environmental and social structures on the dynamics of addiction and disease evolution, and (iii) Dynamics of complex systems at the interphase of ecology, epidemiology and the social sciences. Castillo-Chavez has co-authored over two hundred publications (see goggle scholar citations) that include journal articles and edited research volumes. Specifically, he co-authored a textbook in Mathematical Biology in 2001 (second edition in 2012); a volume (with Harvey Thomas Banks) on the use of mathematical models in homeland security published in SIAM\u27s Frontiers in Applied Mathematics Series (2003); and co-edited volumes in the Series Contemporary Mathematics entitled ``Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges\u27\u27 (American Mathematical Society, 2006) and Mathematical and Statistical Estimation Approaches in Epidemiology (Springer-Verlag, 2009) highlighting his interests in the applications of mathematics in emerging and re-emerging diseases. Castillo-Chavez is a member of the Santa Fe Institute\u27s external faculty, adjunct professor at Cornell University, and contributor, as a member of the Steering Committee of the ``Committee for the Review of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Curriculum Materials,\u27\u27 to a 2004 NRC report. The CBMS workshop ``Mathematical Epidemiology with Applications\u27\u27 lectures delivered by C. Castillo-Chavez and F. Brauer in 2011 have been published by SIAM in 2013

Using the stochastic Galerkin method as a predictive tool during an epidemic

The ability to accurately predict the course of an epidemic is extremely important. This article looks at an influenza outbreak that spread through a small boarding school. Predictions are made on multiple days throughout the epidemic using the stochastic Galerkin method to consider a range of plausible values for the parameters. These predictions are then compared to known data points. Predictions made before the peak of the epidemic had much larger variances compared to predictions made after the peak of the epidemic. References B. M. Chen-Charpentier, J. C. Cortes, J. V. Romero, and M. D. Rosello. Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation, 219(9):4208 – 4218, 2013. doi:10.1016/j.amc.2012.11.007 B. M. Chen-Charpentier and D. Stanescu. Epidemic models with random coefficients. Mathematical and Computer Modelling, 52:1004 – 1010, 2010. doi:10.1016/j.mcm.2010.01.014 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with individualised parameter distributions. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C160–C176, August 2016. doi:10.21914/anziamj.v57i0.10394 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with uncertainty in the parameters. Mathematical Biosciences, 277:25 – 37, 2016. doi:10.1016/j.mbs.2016.03.012 D. B. Harman and P. R. Johnston. Boarding house: find border. 2019. doi:10.6084/m9.figshare.7699844.v1 D. B. Harman and P. R. Johnston. SIR uniform equations. 2 2019. doi:10.6084/m9.figshare.7692392.v1 H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599–653, 2000. doi:10.1137/S0036144500371907 R.I. Hickson and M.G. Roberts. How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. Journal of Theoretical Biology, 350(0):70 – 80, 2014. doi:10.1016/j.jtbi.2014.01.014 W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772):700–721, August 1927. doi:10.1098/rspa.1927.0118 M. G. Roberts. A two-strain epidemic model with uncertainty in the interaction. The ANZIAM Journal, 54:108–115, 10 2012. doi:10.1017/S1446181112000326 M. G. Roberts. Epidemic models with uncertainty in the reproduction number. Journal of Mathematical Biology, 66(7):1463–1474, 2013. doi:10.1007/s00285-012-0540-y F. Santonja and B. Chen-Charpentier. Uncertainty quantification in simulations of epidemics using polynomial chaos. Computational and Mathematical Methods in Medicine, 2012:742086, 2012. doi:10.1155/2012/742086 Communicable Disease Surveillance Centre (Public Health Laboratory Service) and Communicable Diseases (Scotland) Unit. Influenza in a boarding school. BMJ, 1(6112):587, 1978. doi:10.1136/bmj.1.6112.586 G. Strang. Linear Algebra and Its Applications. Thomson, Brooks/Cole, 2006. D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010

Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters

[EN] In spite of its simple formulation via a nonlinear differential equation, the Gompertz model has been widely applied to describe the dynamics of biological and biophysical parts of complex systems (growth of living organisms, number of bacteria, volume of infected cells, etc.). Its parameters or coefficients and the initial condition represent biological quantities (usually, rates and number of individual/particles, respectively) whose nature is random rather than deterministic. In this paper, we present a complete uncertainty quantification analysis of the randomized Gomperz model via the computation of an explicit expression to the first probability density function of its solution stochastic process taking advantage of the Liouville-Gibbs theorem for dynamical systems. The stochastic analysis is completed by computing other important probabilistic information of the model like the distribution of the time until the solution reaches an arbitrary value of specific interest and the stationary distribution of the solution. Finally, we apply all our theoretical findings to two examples, the first of numerical nature and the second to model the dynamics of weight of a species using real data.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Bevia, V.; Burgos, C.; Cortés, J.; Navarro-Quiles, A.; Villanueva Micó, RJ. (2020). Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters. Chaos, Solitons and Fractals. 138:1-12. https://doi.org/10.1016/j.chaos.2020.109908S112138Golec, J., & Sathananthan, S. (2003). Stability analysis of a stochastic logistic model. Mathematical and Computer Modelling, 38(5-6), 585-593. doi:10.1016/s0895-7177(03)90029-xCortés, J. C., Jódar, L., & Villafuerte, L. (2009). Random linear-quadratic mathematical models: Computing explicit solutions and applications. Mathematics and Computers in Simulation, 79(7), 2076-2090. doi:10.1016/j.matcom.2008.11.008Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Dorini, F. A., Bobko, N., & Dorini, L. B. (2016). A note on the logistic equation subject to uncertainties in parameters. Computational and Applied Mathematics, 37(2), 1496-1506. doi:10.1007/s40314-016-0409-6Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2019). Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 72, 121-138. doi:10.1016/j.cnsns.2018.12.013Calatayud, J., Cortés, J. C., & Jornet, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations. Mathematical Methods in the Applied Sciences, 42(18), 7259-7267. doi:10.1002/mma.5834Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics, 291, 20-35. doi:10.1016/j.cam.2014.11.028Hesam, S., Nazemi, A. R., & Haghbin, A. (2012). Analytical solution for the Fokker–Planck equation by differential transform method. Scientia Iranica, 19(4), 1140-1145. doi:10.1016/j.scient.2012.06.018Lakestani, M., & Dehghan, M. (2009). Numerical solution of Fokker-Planck equation using the cubic B-spline scaling functions. Numerical Methods for Partial Differential Equations, 25(2), 418-429. doi:10.1002/num.20352Mao, X., Yuan, C., & Yin, G. (2005). Numerical method for stationary distribution of stochastic differential equations with Markovian switching. Journal of Computational and Applied Mathematics, 174(1), 1-27. doi:10.1016/j.cam.2004.03.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.015Cortés, J. C., Navarro‐Quiles, A., Romero, J., & Roselló, M. (2019). (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications. Mathematical Methods in the Applied Sciences, 42(17), 5708-5717. doi:10.1002/mma.5440Dorini, F. A., & Cunha, M. C. C. (2011). On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4), 679-690. doi:10.1016/j.matcom.2011.10.008Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Hussein, A., & Selim, M. M. (2013). A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion. Journal of Quantitative Spectroscopy and Radiative Transfer, 125, 84-92. doi:10.1016/j.jqsrt.2013.03.018Hussein, A., & Selim, M. M. (2019). A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique. Journal of Quantitative Spectroscopy and Radiative Transfer, 232, 54-65. doi:10.1016/j.jqsrt.2019.04.034Cortés, J.-C., Jódar, L., Camacho, F., & Villafuerte, L. (2010). Random Airy type differential equations: Mean square exact and numerical solutions. Computers & Mathematics with Applications, 60(5), 1237-1244. doi:10.1016/j.camwa.2010.05.046Bekiryazici, Z., Merdan, M., & Kesemen, T. (2020). Modification of the random differential transformation method and its applications to compartmental models. Communications in Statistics - Theory and Methods, 50(18), 4271-4292. doi:10.1080/03610926.2020.1713372Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Laird, A. K. (1965). Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell. British Journal of Cancer, 19(2), 278-291. doi:10.1038/bjc.1965.32Nahashon, S. N., Aggrey, S. E., Adefope, N. A., Amenyenu, A., & Wright, D. (2006). Growth Characteristics of Pearl Gray Guinea Fowl as Predicted by the Richards, Gompertz, and Logistic Models. Poultry Science, 85(2), 359-363. doi:10.1093/ps/85.2.35

Controllability of evolution equations with memory

First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 4, 2017, Pages 2437-2459, published by the Society for Industrial and Applied Mathematics (SIAM)This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimatesF. W. Chaves-Silva was partially supported by the ERC project Semi- Classical Analysis of Partial Di erential Equations, ERC-2012-ADG, project number: 320845. X. Zhang was supported by the NSF of China under grant 11231007 and the Chang Jiang Scholars Program from the Chinese Education Ministry. This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (ANR-2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain

Effect of the numerical scheme resolution on quasi-2D simulation of an automotive radial turbine under highly pulsating flow

Automotive turbocharger turbines usually work under pulsating flow because of the sequential nature of engine breathing. However, existing turbine models are typically based on quasi-steady assumptions. In this paper a model where the volute is calculated in a quasi-2D scheme is presented. The objective of this work is to quantify and analyse the effect of the numerical resolution scheme used in the volute model. The conditions imposed upstream are isentropic pressure pulsations with different amplitude and frequency. The volute is computed using a finite volume approach considering the tangential and radial velocity components. The stator and rotor are assumed to be quasi-steady. In this paper, different integration and spatial reconstruction schemes are explored. The spatial reconstruction is based on the MUSCL method with different slope limiters fulfilling the TVD criterion. The model results are assessed against 3D U-RANS calculations. The results show that under low frequency pressure pulses all the schemes lead to similar solutions. But, for high frequency pulsation the results can be very different depending upon the selected scheme. This may have an impact in noise emission predictions.The authors are indebted to the Spanish Ministerio de Economia y Competitividad through Project TRA 2012-36954. The authors also wish to thank Mr. Roberto Navarro for his invaluable work during CFD simulations.Galindo, J.; Climent, H.; Tiseira Izaguirre, AO.; García-Cuevas González, LM. (2016). Effect of the numerical scheme resolution on quasi-2D simulation of an automotive radial turbine under highly pulsating flow. Journal of Computational and Applied Mathematics. 291:112-126. https://doi.org/10.1016/j.cam.2015.02.025S11212629

Multiangle social network recommendation algorithms and similarity network evaluation

Multiangle social network recommendation algorithms (MSN) and a new assessmentmethod, called similarity network evaluation (SNE), are both proposed. From the viewpoint of six dimensions, the MSN are classified into six algorithms, including user-based algorithmfromresource point (UBR), user-based algorithmfromtag point (UBT), resource-based algorithm fromtag point (RBT), resource-based algorithm from user point (RBU), tag-based algorithm from resource point (TBR), and tag-based algorithm from user point (TBU). Compared with the traditional recall/precision (RP) method, the SNE is more simple, effective, and visualized. The simulation results show that TBR and UBR are the best algorithms, RBU and TBU are the worst ones, and UBT and RBT are in the medium levels