Effects of direction-specific training interventions on physical performance and inter-limb asymmetries

This study analyzed the effects of two different training programs on functional performance and inter-limb asymmetries in basketball players. Twenty-four elite youth basketball players were randomly assigned to a training program including variable unilateral horizontal movements (VUH, n = 12) or unilateral lateral movements (VUL, n = 12). Eccentric-overload training (EOT) was performed twice a week for a six-week period. Functional performance assessment included a countermovement jump test, unilateral multidirectional jumping tests (i.e., lateral, horizontal, and vertical), a rebound jump test, a limb symmetry index, a 25 m linear sprint test, and several change of direction (COD) tests. Within-group analysis showed substantial improvements in almost all functional tests in both groups (ES = 0.35−0.89). Furthermore, almost all jumping asymmetries were improved in both groups (ES = 0.38−0.69) except for vertical jumping asymmetry in VUL (ES = −0.04). Between-group analyses showed a substantial and possibly better performance in vertical jumping asymmetry and 5 m in VUH compared to that of VUL, respectively. In contrast, lateral jumping with left (ES = 1.22) and right leg (ES = 0.49) were substantially greater in VUL than in VUH. Specific force-vector training programs induced substantial improvements in both functional performance tests and inter-limb asymmetries, although greater improvements of lateral and horizontal variables may depend on the specific force vector targeted

Modelling 1-month euribor interest rate by using differential equations with uncertainty

[EN] This paper deals with modelling interest rate using continuous models with uncertainty based on Itô-type stochastic differential equations. It is provided an analysis of theoretical aspects that involves the so-called Vasicek s model as well as their practical application. The latter includes model parameter fitting and measurement of goodness-of-fit of the model. The theoretical results are applied to modelling 1-month Euribor interest rate.This work has been partially supported by the Ministerio de Economía y Competitividad grant MTM2013-41765-P.Cortés, J.; Romero, J.; Sánchez Sánchez, A.; Villanueva Micó, RJ. (2015). Modelling 1-month euribor interest rate by using differential equations with uncertainty. Applied Mathematical and Computational Sciences. 7(3):37-50. http://hdl.handle.net/10251/70015S37507

Random Network Models to Predict the Long-Term Impact of HPV Vaccination on Genital Warts

[EN] The Human papillomaviruses (HPV) vaccine induces a herd immunity effect in genital warts when a large number of the population is vaccinated. This aspect should be taken into account when devising new vaccine strategies, like vaccination at older ages or male vaccination. Therefore, it is important to develop mathematical models with good predictive capacities. We devised a sexual contact network that was calibrated to simulate the Spanish epidemiology of different HPV genotypes. Through this model, we simulated the scenario that occurred in Australia in 2007, where 12¿13 year-old girls were vaccinated with a three-dose schedule of a vaccine containing genotypes 6 and 11, which protect against genital warts, and also a catch-up program in women up to 26 years of age. Vaccine coverage were 73% in girls with three doses and with coverage rates decreasing with age until 52% for 20¿26 year-olds. A fast 59% reduction in the genital warts diagnoses occurred in the model in the first years after the start of the program, similar to what was described in the literature.We are grateful for the support from Sanofi Pasteur. The authors would also like to thank M. Diaz-Sanchis from the Institut Catala d'Oncologia (ICO) for her useful comments and the data provided on HPV prevalence. We would also like to thank the ICO for the HPV information centre at http://hpvcentre.net.Diez-Domingo, J.; Sánchez-Alonso, V.; Villanueva Micó, RJ.; Acedo Rodríguez, L.; Moraño Fernández, JA.; Villanueva-Oller, J. (2017). Random Network Models to Predict the Long-Term Impact of HPV Vaccination on Genital Warts. Viruses. 9(10). doi:10.3390/v9100300S91

Uncertainty and sensitivity of the sexual behavior changes to the current human papillomavirus vaccination campaign in Spain

Taking into account the public health importance of the human papillomavirus (HPV) control in the future, it is mandatory to assess the effect of the vaccination campaigns on the control of HPV spread and the associated diseases using reliable mathematical models. We propose a computational random network model with the aim of studying the transmission dynamics of HPV infections. This model reflects the herd immunity effect in the heterosexual network more accurately than the classical models. We perform a sensitivity analysis of the sexual behavior changes consisting of increasing the number of men who have sex with men (MSM), increasing the frequency of the intercourses and increasing the number of sexual partners. We find that large changes in the sexual behavior, in some extent, only have minor effects on the decline of the HPV infections in women and men in the current vaccination campaign in Spain (vaccination of young girls with a coverage of 70%). Therefore, the current vaccination program in Spain is robust for the heterosexuals. However, we cannot say the same for MSM, where they do not benefit by the herd immunity effect of the vaccination of girls, and consequently, the circulation of the virus among them remains unchanged. A consequence of the present study is that the effect of other external factors that may affect the transmission dynamics of the HPV, for instance, the tourism or the immigration, does not influence the protection provided by the current Spanish vaccination program.This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. This paper has been supported by the European Union through the Operational Program of the [European Regional Development Fund (ERDF) / European Social Fund (ESF)] of the Valencian Community 2014-2020. Files: GJIDI/2018/A/010 and GJIDI/2018/A/009

Modelling the dynamics of the students academic performance in the German region of North Rhine- Westphalia: an epidemiological approach with uncertainty

This is an author's accepted manuscript of an article published in "International Journal of Computer Mathematics"; Volume 91, Issue 2, 2014; copyright Taylor & Francis; available online at: http://dx.doi.org/10.1080/00207160.2013.813937Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last high school courses do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students academic performance in the German region of North Rhine-Westphalia. This approach is supported by the idea that both, good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years.This work has been partially supported by the Spanish Ministry of Economy and Competitiveness grant MTM2009-08587 and Universitat Politecnica de Valencia grant PAID06-11-2070.Cortés, J.; Ehrhardt, M.; Sánchez Sánchez, A.; Santonja, F.; Villanueva Micó, RJ. (2014). Modelling the dynamics of the students academic performance in the German region of North Rhine- Westphalia: an epidemiological approach with uncertainty. International Journal of Computer Mathematics. 91(2):241-251. https://doi.org/10.1080/00207160.2013.813937S241251912Akaike, H. (1969). Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics, 21(1), 243-247. doi:10.1007/bf02532251Brockwell, P. J., & Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer Texts in Statistics. doi:10.1007/978-1-4757-2526-1Dogan, G. (2007). Bootstrapping for confidence interval estimation and hypothesis testing for parameters of system dynamics models. System Dynamics Review, 23(4), 415-436. doi:10.1002/sdr.362Efron, B. (1979). Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics, 7(1), 1-26. doi:10.1214/aos/1176344552LJUNG, G. M., & BOX, G. E. P. (1979). The likelihood function of stationary autoregressive-moving average models. Biometrika, 66(2), 265-270. doi:10.1093/biomet/66.2.265Martcheva, M., & Castillo-Chavez, C. (2003). Diseases with chronic stage in a population with varying size. Mathematical Biosciences, 182(1), 1-25. doi:10.1016/s0025-5564(02)00184-0J.D. Murray,Mathematical Biology, Springer, New York, 2002.Nelder, J. A., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7(4), 308-313. doi:10.1093/comjnl/7.4.308Yazici, B., & Yolacan, S. (2007). A comparison of various tests of normality. Journal of Statistical Computation and Simulation, 77(2), 175-183. doi:10.1080/10629360600678310M.Á.M. Zabal, P.F. Berrocal, C. Coll, and M. de los Ángeles Melero Zabal,La Interacción Social en Contextos Educativos[Social interaction in educational contexts], Psicología/Siglo XXI de España Editores Series, Siglo XXI de España, 1995

Epidemic Random Network Simulations in a Distributed Computing Environment

We discuss a computational system following the paradigm of distributed computing, which will allow us to simulate the epidemic propagation in random networks with large number of nodes up to one million. This paradigm consists of a server that delivers tasks to be carried out by client computers. When the task is finished, the client sends the obtained results to the server to be stored until all tasks are finished and then ready to be analysed. Finally, we show that this technique allows us to disclose the emergence of seasonal patterns in the respiratory syncytial virus transmission dynamics which do not appear neither in smaller systems nor in continuous systems.This paper has been supported by the Grant from the Universitat Politecnica de Valencia PAID-06-11 ref: 2087 and the Grant FIS PI-10/01433. The authors would like to thank the staff of the Facultad de Administracion de Empresas of the Universidad Politecnica de Valencia, in particular Mara Angeles Herrera, Teresa Solaz, and Jose Luis Real, and the staff of the CES Felipe II of Aranjuez for their help and for letting them use free computer rooms to carry out the Sisifo computations described in this paper. They would also like to acknowledge the BOINC community for its support and the many anonymous volunteers who joined thier project and helped them obtain the results so fast.Villanueva-Oller, J.; Acedo Rodríguez, L.; Moraño Fernández, JA.; Sánchez Sánchez, A. (2013). Epidemic Random Network Simulations in a Distributed Computing Environment. Abstract and Applied Analysis. 2013:1-10. https://doi.org/10.1155/2013/462801S1102013PROULX, S., PROMISLOW, D., & PHILLIPS, P. (2005). Network thinking in ecology and evolution. Trends in Ecology & Evolution, 20(6), 345-353. doi:10.1016/j.tree.2005.04.004Traud, A. L., Mucha, P. J., & Porter, M. A. (2012). Social structure of Facebook networks. Physica A: Statistical Mechanics and its Applications, 391(16), 4165-4180. doi:10.1016/j.physa.2011.12.021Christakis, N. A., & Fowler, J. H. (2008). The Collective Dynamics of Smoking in a Large Social Network. New England Journal of Medicine, 358(21), 2249-2258. doi:10.1056/nejmsa0706154Christakis, N. A., & Fowler, J. H. (2007). The Spread of Obesity in a Large Social Network over 32 Years. New England Journal of Medicine, 357(4), 370-379. doi:10.1056/nejmsa066082Halloran, M. E. (2002). Containing Bioterrorist Smallpox. Science, 298(5597), 1428-1432. doi:10.1126/science.1074674Ahmed, E., & Agiza, H. N. (1998). On modeling epidemics Including latency, incubation and variable susceptibility. Physica A: Statistical Mechanics and its Applications, 253(1-4), 347-352. doi:10.1016/s0378-4371(97)00665-1Martins, M. L., Ceotto, G., Alves, S. G., Bufon, C. C. B., Silva, J. M., & Laranjeira, F. F. (2000). Cellular automata model for citrus variegated chlorosis. Physical Review E, 62(5), 7024-7030. doi:10.1103/physreve.62.7024Hershberg, U., Louzoun, Y., Atlan, H., & Solomon, S. (2001). HIV time hierarchy: winning the war while, loosing all the battles. Physica A: Statistical Mechanics and its Applications, 289(1-2), 178-190. doi:10.1016/s0378-4371(00)00466-0Witten, G., & Poulter, G. (2007). Simulations of infectious diseases on networks. Computers in Biology and Medicine, 37(2), 195-205. doi:10.1016/j.compbiomed.2005.12.002Acedo, L., Moraño, J.-A., & Díez-Domingo, J. (2010). Cost analysis of a vaccination strategy for respiratory syncytial virus (RSV) in a network model. Mathematical and Computer Modelling, 52(7-8), 1016-1022. doi:10.1016/j.mcm.2010.02.041Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599-653. doi:10.1137/s0036144500371907Barabási, A.-L., & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286(5439), 509-512. doi:10.1126/science.286.5439.509Villanueva-Oller, J., Villanueva, R. J., & Díez, S. (2007). CASANDRA: A prototype implementation of a system of network progressive transmission of medical digital images. Computer Methods and Programs in Biomedicine, 85(2), 152-164. doi:10.1016/j.cmpb.2006.10.002Korpela, E., Werthimer, D., Anderson, D., Cobb, J., & Leboisky, M. (2001). SETI@home-massively distributed computing for SETI. Computing in Science & Engineering, 3(1), 78-83. doi:10.1109/5992.895191Hall, C. B., Powell, K. R., MacDonald, N. E., Gala, C. L., Menegus, M. E., Suffin, S. C., & Cohen, H. J. (1986). Respiratory Syncytial Viral Infection in Children with Compromised Immune Function. New England Journal of Medicine, 315(2), 77-81. doi:10.1056/nejm198607103150201Falsey, A. R., & Walsh, E. E. (2000). Respiratory Syncytial Virus Infection in Adults. Clinical Microbiology Reviews, 13(3), 371-384. doi:10.1128/cmr.13.3.371-384.2000Díez Domingo, J., Ridao López, M., Úbeda Sansano, I., & Ballester Sanz, A. (2006). Incidencia y costes de la hospitalización por bronquiolitis y de las infecciones por virus respiratorio sincitial en la Comunidad Valenciana. Años 2001 y 2002. Anales de Pediatría, 65(4), 325-330. doi:10.1157/13093515ACEDO, L., DÍEZ-DOMINGO, J., MORAÑO, J.-A., & VILLANUEVA, R.-J. (2009). Mathematical modelling of respiratory syncytial virus (RSV): vaccination strategies and budget applications. Epidemiology and Infection, 138(6), 853-860. doi:10.1017/s0950268809991373Weber, A., Weber, M., & Milligan, P. (2001). Modeling epidemics caused by respiratory syncytial virus (RSV). Mathematical Biosciences, 172(2), 95-113. doi:10.1016/s0025-5564(01)00066-9White, L. J., Mandl, J. N., Gomes, M. G. M., Bodley-Tickell, A. T., Cane, P. A., Perez-Brena, P., … Medley, G. F. (2007). Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models. Mathematical Biosciences, 209(1), 222-239. doi:10.1016/j.mbs.2006.08.018Acedo, L., Moraño, J.-A., Villanueva, R.-J., Villanueva-Oller, J., & Díez-Domingo, J. (2011). Using random networks to study the dynamics of respiratory syncytial virus (RSV) in the Spanish region of Valencia. Mathematical and Computer Modelling, 54(7-8), 1650-1654. doi:10.1016/j.mcm.2010.11.068SCHNEEBERGER, A., MERCER, C. H., GREGSON, S. A. J., FERGUSON, N. M., NYAMUKAPA, C. A., ANDERSON, R. M., … GARNETT, G. P. (2004). Scale-Free Networks and Sexually Transmitted Diseases. Sexually Transmitted Diseases, 31(6), 380-387. doi:10.1097/00007435-200406000-00012Lou, J., & Ruggeri, T. (2010). The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network. Journal of Mathematical Analysis and Applications, 365(1), 210-219. doi:10.1016/j.jmaa.2009.10.044Fleming, D. M. (2005). Mortality in children from influenza and respiratory syncytial virus. Journal of Epidemiology & Community Health, 59(7), 586-590. doi:10.1136/jech.2004.026450Meerhoff, T. J., Paget, J. W., Kimpen, J. L., & Schellevis, F. (2009). Variation of Respiratory Syncytial Virus and the Relation With Meteorological Factors in Different Winter Seasons. The Pediatric Infectious Disease Journal, 28(10), 860-866. doi:10.1097/inf.0b013e3181a3e949Welliver, R. C. (2007). Temperature, Humidity, and Ultraviolet B Radiation Predict Community Respiratory Syncytial Virus Activity. The Pediatric Infectious Disease Journal, 26(Supplement), S29-S35. doi:10.1097/inf.0b013e318157da59Dushoff, J., Plotkin, J. B., Levin, S. A., & Earn, D. J. D. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proceedings of the National Academy of Sciences, 101(48), 16915-16916. doi:10.1073/pnas.0407293101Arino, J., Davis, J. R., Hartley, D., Jordan, R., Miller, J. M., & van den Driessche, P. (2005). A multi-species epidemic model with spatial dynamics. Mathematical Medicine and Biology: A Journal of the IMA, 22(2), 129-142. doi:10.1093/imammb/dqi00

Assessing ecosystem services from multifunctional trees in pastures using Bayesian belief networks

A Bayesian belief network (BBN) was developed to assess preferred combinations of trees in live fences and on pastures in silvopastoral systems. The BBN was created with information from Rivas, Nicaragua, using local farmer knowledge on tree species, trees' costs and benefits, farmers' expressed needs and aspirations, and scientific knowledge regarding tree functional traits and their contribution to ecosystem services and benefits. The model identifies combinations of trees, which provide multiple ecosystem services from pastures, improving their productivity and contribution to farmer livelihoods. We demonstrate how the identification of portfolios of multifunctional trees can satisfy a profile of desired ecosystem services prioritized by the farmer. Diagnostics using Bayesian inference starts with an identification of farmer needs and ‘works backwards’ to identify a silvopastoral system structure. We conclude that Bayesian belief networks are a promising modeling technique for multi-criteria decisions in farm adaptation processes, where interventions must be adapted to specific contexts and farmer preferences

Pontiella desulfatans gen. nov., sp. nov., and Pontiella sulfatireligans sp. nov., two marine anaerobes of the Pontiellaceae fam. nov. producing sulfated glycosaminoglycan-like exopolymers

Recently, we isolated two marine strains, F1T and F21T, which together with Kiritimatiella glycovorans L21-Fru-ABT are the only pure cultures of the class Kiritimatiellae within the phylum Verrucomicrobiota. Here, we present an in-depth genome-guided characterization of both isolates with emphasis on their exopolysaccharide synthesis. The strains only grew fermentatively on simple carbohydrates and sulfated polysaccharides. Strains F1T, F21T and K. glycovorans reduced elemental sulfur, ferric citrate and anthraquinone-2,6-disulfonate during anaerobic growth on sugars. Both strains produced exopolysaccharides during stationary phase, probably with intracellularly stored glycogen as energy and carbon source. Exopolysaccharides included N-sulfated polysaccharides probably containing hexosamines and thus resembling glycosaminoglycans. This implies that the isolates can both degrade and produce sulfated polysaccharides. Both strains encoded an unprecedently high number of glycoside hydrolase genes (422 and 388, respectively), including prevalent alpha-L-fucosidase genes, which may be necessary for degrading complex sulfated polysaccharides such as fucoidan. Strain F21T encoded three putative glycosaminoglycan sulfotransferases and a putative sulfate glycosaminoglycan biosynthesis gene cluster. Based on phylogenetic and chemotaxonomic analyses, we propose the taxa Pontiella desulfatans F1T gen. nov., sp. nov. and Pontiella sulfatireligans F21T sp. nov. as representatives of the Pontiellaceae fam. nov. within the class Kiritimatiellae.ERC -European Research Council(024.002.002)info:eu-repo/semantics/publishedVersio

Non-parametric probabilistic forecasting of academic performance in Spanish high school using an epidemiological modelling approach

Academic underachievement is a concern of paramount importance in Europe, and particularly in Spain, where around of 30% of the students in the last two courses in high school do not achieve the minimum knowledge academic requirement. In order to analyze this problem, we propose a mathematical model via a system of ordinary differential equations to study the dynamics of the academic performance in Spain. Our approach is based on the idea that both, good and bad study habits, are a mixture of personal decisions and influence of classmates. Moreover, in order to consider the uncertainty in the estimation of model parameters, a bootstrapping approach is employed. This technique permits to forecast model trends in the next few years using confidence intervals. Unfortunately, the obtained results do not suggest improvement in academic performance for the coming years.Cortés López, JC.; Sánchez Sánchez, A.; Santonja, FJ.; Villanueva Micó, RJ. (2013). Non-parametric probabilistic forecasting of academic performance in Spanish high school using an epidemiological modelling approach. Applied Mathematics and Computation. 221(15):648-661. doi:10.1016/j.amc.2013.06.070S6486612211

Predicting cocaine consumption in Spain: A mathematical modelling approach

This is an author's accepted manuscript of an article published in “Drugs: Education, Prevention, and Policy "; Volume 18, Issue 2, 2011; copyright Taylor & Francis; available online at: http://dx.doi.org/10.3109/09687630903443299In this article, we analyse the evolution of cocaine consumption in Spain and we predict consumption trends over the next few years. Additionally, we simulate some scenarios which aim to reduce cocaine consumption in the future (sensitivity analysis). Assuming cocaine dependency is a socially transmitted epidemic disease, this leads us to propose an epidemiological-type mathematical model to study consumption evolution. Model sensitivity analysis allows us to design strategies and analyse their effects on cocaine consumption. The model predicts that 3.5% of the Spanish population will be habitual cocaine consumers by 2015. The simulations carried out suggest that cocaine consumption prevention strategies are the best policy to reduce the habitual consumer population. In this article, we show that epidemiological-type mathematical models can be a useful tool in the analysis of the repercussion of health policy proposals in the short-time future. © 2011 Informa UK Ltd.Sánchez, E.; Villanueva Micó, RJ.; Santonja, FJ.; Rubio, M. (2011). Predicting cocaine consumption in Spain: A mathematical modelling approach. Drugs: Education, Prevention, and Policy. 18(2):108-115. doi:10.3109/09687630903443299S108115182Blower, S. M., & Dowlatabadi, H. (1994). Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: An HIV Model, as an Example. International Statistical Review / Revue Internationale de Statistique, 62(2), 229. doi:10.2307/1403510Dutra, L., Stathopoulou, G., Basden, S. L., Leyro, T. M., Powers, M. B., & Otto, M. W. (2008). A Meta-Analytic Review of Psychosocial Interventions for Substance Use Disorders. American Journal of Psychiatry, 165(2), 179-187. doi:10.1176/appi.ajp.2007.06111851Gorman, D. M., Mezic, J., Mezic, I., & Gruenewald, P. J. (2006). Agent-Based Modeling of Drinking Behavior: A Preliminary Model and Potential Applications to Theory and Practice. American Journal of Public Health, 96(11), 2055-2060. doi:10.2105/ajph.2005.063289Jódar, L., Santonja, F. J., & González-Parra, G. (2008). Modeling dynamics of infant obesity in the region of Valencia, Spain. Computers & Mathematics with Applications, 56(3), 679-689. doi:10.1016/j.camwa.2008.01.011JOHNSON, B., ROACHE, J., AITDAOUD, N., JAVORS, M., HARRISON, J., ELKASHEF, A., … BLOCH, D. (2006). A preliminary randomized, double-blind, placebo-controlled study of the safety and efficacy of ondansetron in the treatment of cocaine dependence. Drug and Alcohol Dependence, 84(3), 256-263. doi:10.1016/j.drugalcdep.2006.02.011Levin, F. R., Evans, S. M., Brooks, D. J., & Garawi, F. (2007). Treatment of cocaine dependent treatment seekers with adult ADHD: Double-blind comparison of methylphenidate and placebo. Drug and Alcohol Dependence, 87(1), 20-29. doi:10.1016/j.drugalcdep.2006.07.004Marino, S., Hogue, I. B., Ray, C. J., & Kirschner, D. E. (2008). A methodology for performing global uncertainty and sensitivity analysis in systems biology. Journal of Theoretical Biology, 254(1), 178-196. doi:10.1016/j.jtbi.2008.04.011Martcheva, M., & Castillo-Chavez, C. (2003). Diseases with chronic stage in a population with varying size. Mathematical Biosciences, 182(1), 1-25. doi:10.1016/s0025-5564(02)00184-0Nelder, J. A., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7(4), 308-313. doi:10.1093/comjnl/7.4.308Olsson, A., Sandberg, G., & Dahlblom, O. (2003). On Latin hypercube sampling for structural reliability analysis. Structural Safety, 25(1), 47-68. doi:10.1016/s0167-4730(02)00039-5Santonja, F. J., Tarazona, A. C., & Villanueva, R. J. (2008). A mathematical model of the pressure of an extreme ideology on a society. Computers & Mathematics with Applications, 56(3), 836-846. doi:10.1016/j.camwa.2008.01.001Schmitz, J. M., Stotts, A. L., Rhoades, H. M., & Grabowski, J. (2001). Naltrexone and relapse prevention treatment for cocaine-dependent patients. Addictive Behaviors, 26(2), 167-180. doi:10.1016/s0306-4603(00)00098-8Sharomi, O., & Gumel, A. B. (2008). Curtailing smoking dynamics: A mathematical modeling approach. Applied Mathematics and Computation, 195(2), 475-499. doi:10.1016/j.amc.2007.05.012Stotts, A. L., Mooney, M. E., Sayre, S. L., Novy, M., Schmitz, J. M., & Grabowski, J. (2007). Illusory predictors: Generalizability of findings in cocaine treatment retention research. Addictive Behaviors, 32(12), 2819-2836. doi:10.1016/j.addbeh.2007.04.020White, E., & Comiskey, C. (2007). Heroin epidemics, treatment and ODE modelling. Mathematical Biosciences, 208(1), 312-324. doi:10.1016/j.mbs.2006.10.00