This paper deals with a stochastic SIR (Susceptible{\textendash}Infected{\textendash}Recovered) model with Erlang(k,μ) distributed infectious period commonly referred as SIkR model. We show that using the total number of remaining Erlang stages as the state variable, we do not need to keep track of the stages of individual infections, and can employ a first step analysis to efficiently obtain quantities of interest. We study the distribution of the total number of recovered individuals and the distribution of the maximum number of individuals who are simultaneously infected until the end of the disease. In the literature, final outbreak size is calculated only for a small population size exactly and derivations of approximate analytic solutions from asymptotic results are suggested for larger population sizes. We numerically demonstrate that our methods are implementable on large size problem instances.

}, keywords = {Erlang distributed infectious period, Final size distribution, Markov models, Maximum size distribution, Stochastic SIR}, issn = {0025-5564}, doi = {https://doi.org/10.1016/j.mbs.2020.108363}, url = {https://www.sciencedirect.com/science/article/pii/S0025556420300547}, author = {Zeynep G{\"o}k{\c c}e {\.I}{\c s}lier and Refik G{\"u}ll{\"u} and Wolfgang H{\"o}rmann} } @article {1161, title = {Efficient simulations for a Bernoulli mixture model of portfolio credit risk}, journal = {Annals of Operations Research}, volume = {260}, year = {2018}, pages = {113{\textendash}128}, author = {Ba{\c s}o{\u g}lu, {\.I}smail and Wolfgang H{\"o}rmann and Sak, Halis} } @article {1162, title = {Mispricing in Option Pricing Models Versus Market Payoffs: An Efficiency-Based Performance Metric}, journal = {Wilmott}, volume = {2017}, year = {2017}, pages = {44{\textendash}57}, author = {Orbay, Berk and Refik G{\"u}ll{\"u} and Wolfgang H{\"o}rmann} } @conference {feichtinger2014distributional, title = {A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups}, year = {2014}, pages = {423{\textendash}446}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, author = {Feichtinger, HG and Wolfgang H{\"o}rmann} } @article {dingecc2013control, title = {Control variates and conditional Monte Carlo for basket and Asian options}, journal = {Insurance: Mathematics and Economics}, volume = {52}, number = {3}, year = {2013}, pages = {421{\textendash}434}, publisher = {North-Holland}, author = {Dinge{\c c}, Kemal Din{\c c}er and Wolfgang H{\"o}rmann} } @article {hormann2013generating, title = {Generating generalized inverse Gaussian random variates}, journal = {Statistics and Computing}, year = {2013}, pages = {1{\textendash}11}, publisher = {Springer US}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @article {bacsouglu2013optimally, title = {Optimally stratified importance sampling for portfolio risk with multiple loss thresholds}, journal = {Optimization}, volume = {62}, number = {11}, year = {2013}, pages = {1451{\textendash}1471}, publisher = {Taylor \& Francis}, author = {Ba{\c s}o{\u g}lu, {\.I}smail and Wolfgang H{\"o}rmann and Sak, Halis} } @article {botts2013transformed, title = {Transformed density rejection with inflection points}, journal = {Statistics and Computing}, volume = {23}, number = {2}, year = {2013}, pages = {251{\textendash}260}, publisher = {Springer US}, author = {Botts, Carsten and Wolfgang H{\"o}rmann and Leydold, Josef} } @article {sak2012fast, title = {Fast simulations in credit risk}, journal = {Quantitative Finance}, volume = {12}, number = {10}, year = {2012}, pages = {1557{\textendash}1569}, publisher = {Routledge}, author = {Sak, Halis and Wolfgang H{\"o}rmann} } @article {dingecc2012general, title = {A general control variate method for option pricing under L{\'e}vy processes}, journal = {European Journal of Operational Research}, volume = {221}, number = {2}, year = {2012}, pages = {368{\textendash}377}, publisher = {North-Holland}, author = {Dinge{\c c}, Kemal Din{\c c}er and Wolfgang H{\"o}rmann} } @conference {dingecc2012new, title = {New control variates for L{\'e}vy process models}, booktitle = {Proceedings of the Winter Simulation Conference}, year = {2012}, pages = {15}, publisher = {Winter Simulation Conference}, organization = {Winter Simulation Conference}, author = {Dinge{\c c}, Kemal Din{\c c}er and Wolfgang H{\"o}rmann} } @article {leydold2011generating, title = {Generating generalized inverse Gaussian random variates by fast inversion}, journal = {Computational Statistics \& Data Analysis}, volume = {55}, number = {1}, year = {2011}, pages = {213{\textendash}217}, publisher = {North-Holland}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @article {dingecc2011using, title = {Using the continuous price as control variate for discretely monitored options}, journal = {Mathematics and Computers in Simulation}, volume = {82}, number = {4}, year = {2011}, pages = {691{\textendash}704}, publisher = {North-Holland}, author = {Dinge{\c c}, Kemal Din{\c c}er and Wolfgang H{\"o}rmann} } @article {hormann2010t, title = {t-Copula generation for control variates}, journal = {Mathematics and Computers in Simulation}, volume = {81}, number = {4}, year = {2010}, pages = {782{\textendash}790}, publisher = {North-Holland}, author = {Wolfgang H{\"o}rmann and Sak, Halis} } @conference {Derflinger;Hoermann;Leydold;Sak:2009a, title = {Efficient Numerical Inversion for Financial Simulations}, booktitle = {Monte Carlo and Quasi-Monte Carlo Methods 2008}, year = {2009}, pages = {297{\textendash}304}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Heidelberg}, author = {Derflinger, Gerhard and Wolfgang H{\"o}rmann and Leydold, Josef and Sak, Halis} } @inbook {Hoermann;Leydold:2009a, title = {Sampling from Linear Multivariate Densities}, booktitle = {Advancing the Frontiers of Simulation: A Festschrift in Honor of George Samuel Fishman}, year = {2009}, pages = {143{\textendash}152}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Heidelberg}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @article {Hoermann:2007a, title = {A simple Generator for the t-Distribution}, journal = {Computing}, volume = {81}, year = {2007}, author = {Wolfgang H{\"o}rmann} } @conference {Hoermann;Leydold:2006a, title = {Black-Box Algorithms for Sampling from Continuous Distributions}, booktitle = {Proceedings of the 2006 Winter Simulation Conference}, year = {2006}, pages = {129{\textendash}136}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @article {Derflinger;Hoermann:2005a, title = {Asymptotically Optimal Design Points for Rejection Algorithms}, journal = {Communications in Statistics: Simulation and Computation}, volume = {34}, number = {4}, year = {2005}, pages = {879-893}, author = {Derflinger, Gerhard and Wolfgang H{\"o}rmann} } @conference {Hoermann;Leydold:2005a, title = {Monte Carlo Integration Using Importance Sampling and Gibbs Sampling}, booktitle = {Proceedings of the International Conference on Computational Science and Engineering}, year = {2005}, pages = {92{\textendash}97}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @book {724, title = {Automatic Nonuniform Random Variate Generation}, year = {2004}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Berlin Heidelberg}, author = {Wolfgang H{\"o}rmann and Leydold, Josef and Derflinger, Gerhard} } @article {Tirler;etal:2004a, title = {An Error in the {Kinderman-Ramage} Method and How to Fix It}, journal = {Computational Statistics and Data Analysis}, volume = {47}, number = {3}, year = {2004}, pages = {433{\textendash}440}, author = {Tirler, G{\"u}nter and Dalgaard, Peter and Wolfgang H{\"o}rmann and Leydold, Josef} } @article {Leydold;Hoermann:2004b, title = {Smoothed Transformed Density Rejection}, journal = {Monte Carlo Methods and Applications}, volume = {10}, number = {3{\textendash}4}, year = {2004}, pages = {393{\textendash}402}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @article {Leydold;etal:2003a, title = {An Automatic Code Generator for Nonuniform Random Variate Generation}, journal = {Mathematics and Computers in Simulation}, volume = {62}, number = {3{\textendash}6}, year = {2003}, pages = {405{\textendash}412}, author = {Leydold, Josef and Derflinger, Gerhard and Tirler, G{\"u}nter and Wolfgang H{\"o}rmann} } @article {Hoermann;Leydold:2003a, title = {Continuous Random Variate Generation by Fast Numerical Inversion}, journal = {ACMTOMACS}, volume = {13}, number = {4}, year = {2003}, pages = {347{\textendash}362}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @article {Hoermann;Derflinger:2002a, title = {Fast Generation of Order Statistics}, journal = {ACMTOMACS}, volume = {12}, number = {2}, year = {2002}, pages = {83{\textendash}93}, abstract = {Generating a single order statistic without generating the full sample can be an important task for simulations. If the density and the CDF of the distribution are given, then it is no problem to compute the density of the order statistic. In the main theorem it is shown that the concavity properties of that density depend directly on the distribution itself. Especially for log-concave distributions, all order statistics have log-concave distributions themselves. So recently suggested automatic transformed density rejection algorithms can be used to generate single order statistics. This idea leads to very fast generators. For example for the normal and gamma distributions, the suggested new algorithms are between 10 and 60 times faster than the algorithms suggested in the literature.

}, keywords = {black box, order statistics, T-concave, TDR}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann:2002a, title = {A note on the Performance of the {\textquoteleft}{\textquoteleft}{A}hrens Algorithm{\textquoteright}{\textquoteright}}, journal = {Computing}, volume = {69}, year = {2002}, pages = {83{\textendash}89}, author = {Wolfgang H{\"o}rmann} } @conference {Leydold;Janka;Hoermann:2002a, title = {Variants of Transformed Density Rejection and Correlation Induction}, booktitle = {Monte Carlo and Quasi-Monte Carlo Methods 2000}, year = {2002}, pages = {345{\textendash}356}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Heidelberg}, author = {Leydold, Josef and Janka, Erich and Wolfgang H{\"o}rmann} } @conference {Leydold;Hoermann:2001a, title = {Universal Algorithms as an Alternative for Generating Non-Uniform Continuous Random Variates}, booktitle = {{M}onte {C}arlo {S}imulation}, year = {2001}, note = {Proceedings of the International Conference on {M}onte {C}arlo Simulation 2000}, pages = {177{\textendash}183}, publisher = {A. A. Balkema}, organization = {A. A. Balkema}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @article {Hoermann:2000a, title = {Algorithm 802: An Automatic Generator for Bivariate Log-Concave Distributions}, journal = {ACMTOMS}, volume = {26}, number = {1}, year = {2000}, pages = {201{\textendash}219}, abstract = {Different automatic (also called universal or black-box) methods have been suggested to sample from univariate log-concave distributions. Our new automatic algorithm for bivariate log-concave distributions is based on the method of transformed density rejection. In order to construct a hat function for a rejection algorithm the bivariate density is transformed by the logarithm into a concave function. Then it is possible to construct a dominating function by taking the minimum of several tangent planes, which are by exponentiation transformed back into the original scale. The choice of the points of contact is automated using adaptive rejection sampling. This means that points that are rejected by the rejection algorithm can be used as additional points of contact. The paper describes the details how this main idea can be used to construct Algorithm ALC2D that can generate random pairs from all bivariate log-concave distributions with known domain, computable density and computable partial derivatives.

}, url = {http://www.acm.org/pubs/citations/journals/toms/2000-26-1/p201-hormann/}, author = {Wolfgang H{\"o}rmann} } @conference {Hoermann;Leydold:2000a, title = {Automatic Random Variate Generation for Simulation Input}, booktitle = {Proceedings of the 2000 Winter Simulation Conference}, year = {2000}, pages = {675{\textendash}682}, author = {Wolfgang H{\"o}rmann and Leydold, Josef} } @conference {Leydold;Hoermann:2000a, title = {Black Box Algorithms for Generating Non-Uniform Continuous Random Variates}, booktitle = {COMPSTAT 2000. Short Communications and Posters}, year = {2000}, pages = {53{\textendash}54}, publisher = {Statistics Netherlands}, organization = {Statistics Netherlands}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @conference {Leydold;Leeb;Hoermann:2000a, title = {Higher Dimensional Properties of Non-Uniform Pseudo-Random Variates}, booktitle = {Monte Carlo and Quasi-Monte Carlo Methods 1998}, year = {2000}, pages = {341{\textendash}355}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Berlin, Heidelberg}, author = {Leydold, Josef and Leeb, Hannes and Wolfgang H{\"o}rmann} } @conference {Hoermann;Bayar:2000a, title = {Modelling Probability Distributions from Data and its Influence on Simulation}, booktitle = {Proceedings IMACS Symposium on Mathematical Modeling}, year = {2000}, pages = {429-435}, author = {Wolfgang H{\"o}rmann and Bayar, Onur} } @article {Leydold;Hoermann:1998a, title = {A Sweep-Plane Algorithm for Generating random tuples in simple polytopes}, journal = {Mathematics of Computation}, volume = {67}, number = {224}, year = {1998}, pages = {1617{\textendash}1635}, keywords = {black box, multivariate log-concave distributions, polytope, rejection method, sweep-plane algorithm, T-concave, TDR, uniform distributions, universal method}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @conference {Leydold;Hoermann:1997a, title = {The automatic generation of one- and multi-dimensional distributions with transformed density rejection}, booktitle = {Proceedings of the 15th IMACS World-Congress, Berlin, Vol 2}, year = {1997}, pages = {757{\textendash}760}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @conference {Hoermann;Derflinger:1997a, title = {An automatic generator for a large class of unimodal discrete distributions}, booktitle = {ESM 97}, year = {1997}, pages = {139{\textendash}144}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann;Derflinger:1996a, title = {Rejection-Inversion to Generate Variates from Monotone Discrete Distributions}, journal = {ACMTOMACS}, volume = {6}, number = {3}, year = {1996}, pages = {169{\textendash}184}, keywords = {monotone discrete distributions, Poisson distribution, random number generation, rejection-inversion, T-concave, universal algorithm, Zipf distribution}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann:1995a, title = {A Rejection Technique for Sampling from {T}-Concave Distributions}, journal = {ACMTOMS}, volume = {21}, number = {2}, year = {1995}, pages = {182{\textendash}193}, keywords = {black box, log-concave, nonuniform RNG, rejection algorithm, rejection method, T-concave, t-distribution, transformed density rejection, unimodal distributions}, author = {Wolfgang H{\"o}rmann} } @conference {Hauser;Hoermann;Kunst;Lenneis:1994a, title = {A note on generation, estimation and prediction of stationary processes}, booktitle = {Compstat, Proceedings in Computational Statistics}, year = {1994}, pages = {323{\textendash}328}, publisher = {Physica-Verlag}, organization = {Physica-Verlag}, address = {Heidelberg}, author = {Hauser, M.A. and Wolfgang H{\"o}rmann and Kunst, R.M. and Lenneis, J.} } @article {Hoermann:1994a, title = {A Note on the Quality of Random Variates Generated by the Ratio of Uniforms Method}, volume = {4}, number = {1}, year = {1994}, pages = {96{\textendash}106}, keywords = {linear congruential generators, lower bounds, one-dimensional distribution of pseudorandom numbers, ratio of uniforms method, uniform random numbers}, author = {Wolfgang H{\"o}rmann} } @conference {Hoermann:1994c, title = {The quality of non-uniform random numbers}, booktitle = {Operations Research Proceedings 1993}, year = {1994}, pages = {329{\textendash}335}, publisher = {Springer Verlag}, organization = {Springer Verlag}, address = {Berlin}, author = {Wolfgang H{\"o}rmann} } @article {Hoermann;Derflinger:1994a, title = {The transformed rejection method for generating random variables, an alternative to the ratio of uniforms method}, journal = {Commun. Stat., Simulation Comput.}, volume = {23}, number = {3}, year = {1994}, pages = {847-860}, keywords = {normal distribution, random variate generation, t-distribution, transformed rejection method}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann:1994b, title = {A universal generator for discrete log-concave distributions}, journal = {Computing}, volume = {52}, number = {1}, year = {1994}, pages = {89{\textendash}96}, keywords = {log-concave distribution, random number generation, rejection method, simulation}, author = {Wolfgang H{\"o}rmann} } @conference {Hoermann;Derflinger:1994b, title = {Universal generators for correlation induction}, booktitle = {Compstat, Proceedings in Computational Statistics}, year = {1994}, pages = {52{\textendash}57}, publisher = {Physica-Verlag}, organization = {Physica-Verlag}, address = {Heidelberg}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann:1993a, title = {The generation of binomial random variates}, journal = {J. Stat. Comput. Simulation}, volume = {46}, number = {1{\textendash}2}, year = {1993}, pages = {101{\textendash}110}, keywords = {binomial distribution, binomial random variates, fast rejection algorithm, random numbers, transformed rejection method}, author = {Wolfgang H{\"o}rmann} } @inbook {Hoermann1993, title = {New generators of normal and {P}oisson deviates based on transformed rejection}, booktitle = {Operations Research Proceedings 1992}, year = {1993}, note = {Proc. DGOR-Tagung Aachen Sept. 92}, pages = {334{\textendash}341}, publisher = {Springer}, organization = {Springer}, address = {Berlin}, author = {Wolfgang H{\"o}rmann} } @article {Hoermann;Derflinger:1993a, title = {A Portable Random Number Generator Well Suited for the Rejection Method}, volume = {19}, number = {4}, year = {1993}, pages = {489{\textendash}495}, abstract = {Up to now, all known efficient portable implementations of linear congruential random number generators with modulus $2^{31} - 1$ have worked only with multipliers that are small compared with the modulus. We show that for nonuniform distributions, the rejection method may generate random numbers of bad qualify if combined with a linear congruential generator with small multiplier. A method is described that works for any multiplier smaller than $2^{30}$. It uses the decomposition of multiplier and seed in high-order and low-order bits to compute the upper and lower half of the product. The sum of the two halves gives the product of multiplier and seed modulo $2^{21} - 1$. Coded in ANSI-C and FORTRAN77 the method results in a portable implementation of the linear congruential generator that is as fast or faster than other portable methods.

}, keywords = {algorithms, linear congruential generator, portability, quality of nonuniform random numbers, rejection method, uniform random number generator}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann:1993b, title = {The transformed rejection method for generating {P}oisson random variables}, journal = {Insurance: Mathematics and Economics}, volume = {11}, year = {1993}, pages = {1{\textendash}7}, author = {Wolfgang H{\"o}rmann} } @conference {Afflerbach;Hoermann:1992a, title = {Nonuniform random numbers: A sensitivity analysis for transformation methods}, booktitle = {Lecture Notes in Econom. Math. Systems}, series = {Lect. Notes Econ. Math. Syst.}, volume = {374}, year = {1992}, note = {Proc. Int. Workshop Comput. Intensive Methods Simulation Optimization, Laxenburg/Austria 1990}, pages = {135{\textendash}144}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {New York}, abstract = {-?-

}, keywords = {-?-}, author = {Afflerbach, Lothar and Wolfgang H{\"o}rmann} } @article {Hoermann;Derflinger:1990a, title = {The ACR method for generating normal random variables}, journal = {OR Spektrum}, volume = {12}, number = {3}, year = {1990}, pages = {181{\textendash}185}, keywords = {acceptance-complement method, ACR method, decomposition method, normal random number generator, ratio of uniform method}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} }