The SIS and SIR stochastic epidemic models: A maximum entropy approach

Theoretical Population Biology

Volumn 80, Issue 4, 2011, Pages 256-264

  Artalejo, J R ^a   Lopez Herrero, M J ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Epidemiology; Final size; Markov chains; Maximum entropy; Time to extinction

Indexed keywords

DISEASE SPREAD; ENZYME; EPIDEMIC; EPIDEMIOLOGY; HOSPITAL SECTOR; INFECTIOUS DISEASE; MARKOV CHAIN; MAXIMUM ENTROPY ANALYSIS; PLANNING THEORY; SIR; STOCHASTICITY; THEORETICAL STUDY;

ARTICLE; BIOLOGICAL MODEL; DISEASE PREDISPOSITION; ENTROPY; INFECTION; PROBABILITY;

DISEASE SUSCEPTIBILITY; ENTROPY; INFECTION; MARKOV CHAINS; MODELS, BIOLOGICAL;

EID: 81155134674     PISSN: 00405809     EISSN: 10960325     Source Type: Journal    
DOI: 10.1016/j.tpb.2011.09.005     Document Type: Article

References (56)

1. Agmon N., Alhassid Y., Levine R.D. An algorithm for finding the distribution of maximal entropy. J. Comput. Phys. 1979, 30:250-258.
2. Allen L.J.S. An Introduction to Stochastic Processes with Applications to Biology 2003, Prentice-Hall, New Jersey.
3. Andersson H., Britton T. Stochastic Epidemic Models and their Statistical Analysis. Lecture Notes in Statistics 2000, vol. 151. Springer, New York.
4. Artalejo J.R., Economou A., Lopez-Herrero M.J. On the number of recovered individuals in the SIS and SIR stochastic epidemic models. Math. Biosci. 2010, 228:45-55.
5. Artalejo, J.R., Economou, A., Lopez-Herrero, M.J., 2011. Stochastic epidemic models revisited: analysis of some continuous performance measures. J. Biol. Dyn., in press (). doi:10.1080/17513758.2011.552737.
6. Artalejo J.R., Lopez-Herrero M.J. The M/G/1 retrial queue: an information theoretic approach. SORT 2005, 29:119-137.
7. Artalejo J.R., Lopez-Herrero M.J. Quasi-stationarity and ratio of expectations: a comparative study. J. Theoret. Biol. 2010, 266:264-274.
8. Awan I., Ahmad B., Ahmad S. Performance analysis of networks of queues under active queue management scheme. Simul. Modell. Pract. Theory 2007, 15:416-425.
9. Bailey N.T.J. The Mathematical Theory of Infectious Diseases and its Applications 1975, Charles Griffin & Company Ltd., London.
10. Ball F. A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Probab. 1986, 18:289-310.
11. Ball F., Sirl D., Trapman P. Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Probab. 2009, 41:765-796.
12. Becker N.G. Analysis of Infectious Disease Data 1989, Chapman & Hall, London.
13. Becker N.G., Britton T. Statistical studies of infectious disease incidence. J. R. Stat. Soc. Ser. B 1999, 61:287-307.
14. Bolch G., Greiner S., de Meer H., Trivedi K.S. Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications 1998, John Wiley & Sons, New York.
15. Britton T. Stochastic epidemic models: a survey. Math. Biosci. 2010, 225:24-35.
16. Cao C.X., Xu M., Chang C.Y., Xue Y., Zhong S.B., Fang L.Q., Cao W.C., Zhang H., Gao M.X., He Q.S., Zhao J., Chen W., Zheng S., Li X.W. Risk analysis for the highly pathogenic avian influenza in Mainland China using meta-modeling. Chinese Sci. Bull. 2010, 55:4168-4178.
17. Chauchemez S., Ferguson N.M. Likelihood-based estimation of continuous-time epidemic models from time-series data: an application to measles transmission in London. J. R. Soc. Interface 2008, 5:885-897.
18. Chernick M.R. Bootstrap Methods: A Guide for Practitioners and Researchers 2008, John Wiley & Sons, Hoboken.
19. Christakos G., Olea R.A., Yu H.-L. Recent results on the spatiotemporal modelling and comparative analysis of Black Death and bubonic plague epidemics. Public Health 2007, 121:700-720.
20. Cintrón-Arias A., Castillo-Chávez C., Bettencourt L.M.A., Lloyd A.L., Banks H.T. The estimation of the effective reproductive number from disease outbreak data. Math. Biosci. Eng. 2009, 6:261-282.
21. Clancy D., Mendy S.T. The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection. J. Math. Biol. 2010, 61:527-544.
22. Clancy D., O'Neill P.D. Bayesian estimation of the basic reproduction number in stochastic epidemic models. Bayesian Anal. 2008, 3:737-758.
23. Corani G., Gatto M. Structural risk minimization: a robust method for density-dependence detection and model selection. Ecography 2007, 30:400-416.
24. Daley D.J., Gani J. Epidemic Modelling: An Introduction. Cambridge Studies in Mathematical Biology 1999, vol. 15. Cambridge University Press, Cambridge.
25. Demiris N., O'Neill P.D. Computation of final outcome probabilities for the generalised stochastic epidemic. Stat. Comput. 2006, 16:309-317.
26. Gesink Law D.C., Bernstein K.T., Serre M.L., Schumacher C.M., Leone P.A., Zenilman J.M., Miller W.C., Rompalo A.M. Modeling a syphilis outbreak through space and time using the Bayesian maximum entropy approach. Ann. Epidemiol. 2006, 16:797-804.
27. Gómez-Corral, A., López-García, M., 2011. Extinction times and size of the surviving species in a two-species competition process. J. Math. Biol., in press (). doi:10.1007/s00285-011-0414-8.
28. Hao Z., Singh V.P. Entropy-based parameter estimation for extended Burr XII distribution. Stoch. Environ. Res. Risk Assess. 2009, 23:1113-1122.
29. Jiang L., Walrand J. Scheduling and Congestion Control for Wireless and Processing Networks. Synthesis Lectures on Communication Networks 2010, vol. 6. Morgan & Claypool Publishers.
30. Kapur J.N., Kesavan H.K. Entropy Optimization Principles with Applications 1992, Academic Press, Boston.
31. Keeling M.J., Ross J.V. On methods for studying stochastic disease dynamics. J. R. Soc. Interface 2008, 5:171-181.
32. Kouvatsos D.D. Entropy maximization and queueing network models. Ann. Oper. Res. 1994, 48:63-126.
33. Krishnarajah I., Cook A., Marion G., Gibson G. Novel moment closure approximations in stochastic epidemics. Bull. Math. Biol. 2005, 67:855-873.
34. Kroese D.P., Taimre T., Botev Z.I. Handbook of Monte Carlo Methods. Wiley Series in Probability and Statistics 2011, John Wiley & Sons, New York.
35. Meester R., Koning J., Jong M.C.M., Diekmann O. Modeling and real-time prediction of classical swine fever epidemics. Biometrics 2002, 58:178-184.
36. Menéndez M.L., Pardo J.A., Pardo M.C. Estimators based on sample quantiles using (h, φ)-entropy measures. Appl. Math. Lett. 1998, 11:99-104.
37. Mkwawa I.-H.M., Kouvatsos D.D., Brandstätter W., Horak G., Geier A., Cavadias C. Performance modelling and evaluation of a mobility management mechanism in IMS-based networks. Lecture Notes in Computer Science 2011, vol. 5233:594-611. Springer-Verlag, Berlin.
38. Nåsell I. The threshold concept in stochastic epidemic and endemic models. Epidemic Models: Their Structure and Relation to Data 1995, 71-83. Cambridge University Press, Cambridge. D. Mollison (Ed.).
39. Nåsell I. Extinction and quasi-stationarity in the Verhulst logistic model. J. Theoret. Biol. 2001, 211:11-27.
40. Nåsell I. Stochastic models of some endemic infections. Math. Biosci. 2002, 179:1-19.
41. Noonan J.P., Basu P. On estimation error using maximum entropy density estimates. Kybernetes 2007, 36:52-64.
42. O'Neill P.D. A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods. Math. Biosci. 2002, 180:103-114.
43. Phillips S.J., Anderson R.P., Schapire R.E. Maximum entropy modeling of species geographic distributions. Ecol. Model. 2006, 190:231-259.
44. Press W.H., Teuklosky S.A., Vetterling W.T., Flannery B.P. Numerical Recipes in Fortran: The Art of Parallel Scientific Computing 1996, Cambridge University Press, Cambridge.
45. Ross J.V., House T., Keeling M.J. Calculation of disease dynamics in a population of households. PloS One 2010, 5(3):e9696.
46. Ross J.V., Pagendam D.E., Pollett P.K. On parameter estimation in population models II: multi-dimensional processes and transient dynamics. Theor. Popul. Biol. 2009, 75:123-132.
47. Ross J.V., Taimre T., Pollett P.K. On parameter estimation in population models. Theor. Popul. Biol. 2006, 70:498-510.
48. Sharkey K.J. Deterministic epidemic models on contact networks: correlations and unbiological terms. Theoret. Popul. Biol. 2011, 79:115-129.
49. Shore J.E., Johnson R.W. Properties of cross-entropy minimization. IEEE Trans. Inform. Theory 1981, 27:472-482.
50. Sumi A., Kamo K.-I., Ohtomo N., Mise K., Kobayashi N. Time series analysis of incidence data of influenza in Japan. J. Epidemiol. 2011, 21:21-29.
51. Václavík T., Kanaskie A., Hansen E.M., Ohmann J.L., Meentemeyer R.K. Predicting potential and actual distribution of sudden oak death in Oregon: prioritizing landscape contexts for early detection and eradication of disease outbreaks. For. Ecol. Manag. 2010, 260:1026-1035.
52. van Doorn E.A., Pollett P.K. Survival in a quasi-death process. Linear Algebra Appl. 2008, 429:776-791.
53. van Doorn, E.A., Pollett, P.K., 2011. Quasi-stationary distributions. Memorandum 1945. ISSN: 1874-4850. Available from: http://www.math.utwente.nl/publications.
54. Welch D., Bansal S., Hunter D.R. Statistical inference to advance network models in epidemiology. Epidemics 2011, 3:38-45.
55. Yu H.-L., Yang S.-J., Yen H.-J., Christakos G. A spatio-temporal climate-based model of early dengue fever warning in southern Taiwan. Stoch. Environ. Res. Risk Assess. 2011, 25:485-494.
56. zu Dohna H., Pineda-Krch M. Fitting parameters of stochastic birth-death models to metapopulation data. Theoret. Popul. Biol. 2010, 78:71-76.