Skip to main content


  • Eva María Ortega (a1) and Laureano F. Escudero (a1)

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.

Hide All
1.Abbey H. (1952). An examination of the Reed–Frost theory of epidemics. Human Biology 24: 201.
2.Bailey N.T.J. (1975). The mathematical theory of infectious diseases and its applications. London: Griffin.
3.Ball F. & Clancy D. (1995). The final outcome and temporal solution of carrier-borne epidemic model. Journal of Applied Probability 32: 304315.
4.Ball F. & Clancy D. (1995). The final outcome of an epidemic model with several different types of infective in a large population. Journal of Applied Probability 32: 579590.
5.Becker N.G. & Dietz K. (1995). The effect of the household distribution on transmission and control of highly infectious diseases. Mathematical Biosciences 127: 207219.
6.Becker N.G. & Utev S. (2002). Protective vaccine efficacy when vaccine response is random. Biometrical Journal 44: 2942.
7.Chang C.-S., Shanthikumar J.G. & Yao D.D. (1994). Stochastic convexity and stochastic Majorization. In Yao D.D. (ed.), Stochastic modeling and analysis of manufacturing systems. New York: Springer-Verlag.
8.Denuit M., Lefèvre C. & Utev S. (1999). Generalized stochastic convexity and stochastic orderings of mixtures. Probability in the Engineering and Informational Sciences 13: 275291.
9.Diekmann O. & Heesterbeek J.A.P. (2000). Mathematical epidemiology of infectious diseases. Chichester, UK: Wiley.
10.Donnelly P. (1993). The correlation structure of epidemic models. Mathematical Biosciences 117: 4975.
11.Escudero L.F. & Ortega E.M. (2008). Actuarial comparisons of aggregate claims with randomly right truncated claims. Insurance: Mathematics and Economics 43: 255262.
12.Escudero L.F., Ortega E.M. & Alonso J. (2009). Variability comparisons for some mixture models with stochastic environments in biosciences and engineering. Stochastic Environmental Research and Risk Assessment, onlinefirst, doi:10.1007/s00477-009-0310-6.
13.Fernández-Ponce J.M., Ortega E.M. & Pellerey F. (2008). Convex comparisons for random sums in random environments and applications. Probability in the Engineering and Informational Sciences 22: 389413.
14.Greenwood M. (1931). On the statistical measure of infectiousness, Journal of Hygene Cambridge 31: 336.
15.Haas C.N. (2002). Conditional dose-response relationships for microorganisms: development and application. Risk Analysis 22: 455463.
16.Helander M.E. & Batta R. (1994). A discrete transmission model for HIV. In Kaplan E.H. & Brandeu M.I. (eds.), Modeling the AIDS epidemic: Planning, policy and prediction. New York: Raven Press, pp. 585611.
17.Isham V. (2005). Stochastic models for epidemics: current issues and developments. In: Celebrating Statistics: Papers in honor of Sir David Cox on his 80th Birthday. Oxford: Oxford University Press.
18.Joe H. (1997). Multivariate models and dependence concepts. London: Chapman and Hall.
19.Kegan B. & West R.W. (2005). Modeling the simple epidemics with deterministic differential equations and random initial conditions. Mathematical Biosciences 195: 179193.
20.Lefèvre C. (2005). SIR epidemic models. In: Armitage P. & Colton T. (eds.), Encyclopedia of biostatistics. vol. 7, New York: Wiley, pp. 49604966.
21.Lefèvre C. & Malice M.P. (1988). Comparisons for carrier-borne epidemics in heterogeneous and homogeneous populations. Journal of Applied Probability 25: 663674.
22.Lefèvre C. & Picard P. (1990). A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Advances Applied Probability 22: 2548.
23.Lefèvre C. & Picard P. (1996). Collective epidemic models. Mathematical Biosciences 134: 5170.
24.Lefèvre C. & Picard P. (2005). Nonstationarity and randomization in the Reed–Frost epidemic model. Journal of Applied Probability 42: 950963.
25.Lefèvre C. & Utev S. (1996). Comparing sums of exchangeable Bernoulli random variables. Journal of Applied Probability 33: 285310.
26.Lefèvre C. & Utev S. (1998). On order-preserving properties of probability metrics. Journal of Theoretical Probability 11: 907920.
27.Lloyd-Smith J.O., Schreiber S.J., Kopp P.E. & Getz W.M. (2005). Superspreading and the effect of individual variation disease emergence. Nature 438: 355359.
28.Malice M.P. & Lefèvre C. (1988). On some effects of variability in the Weiss epidemic model. Communications in Statistics- Theory and Methods 17: 37233731.
29.Marinacci M. & Montrucchio L. (2005). Ultramodular functions. Mathematics of Operations Research 30: 311332.
30.Marshall A.W. & Olkin I. (1979). Inequalities: Theory of Majorization and Its Applications. New York: Academic Press.
31.Meester L.E. & Shanthikumar J.G. (1993). Regularity of stochastic processes. A theory based on directional convexity. Probability in the Engineering and Informational Sciences 7: 343360.
32.Meester L.E. & Shanthikumar J.G. (1999). Stochastic convexity on general space. Mathematics of Operations Research 24: 472494.
33.Menezes R.X., Ortega N.R.S. & Massad E. (2004). A Reed–Frost model taking into account uncertainties in the diagnosis of the infection. Bulletin of Mathematical Biology 66: 689706.
34.Müller A. & Scarsini M. (2000). Some remarks on the supermodular order. Journal of Multivariate Analysis 73: 107119.
35.Müller A. & Scarsini M. (2001). Stochastic comparisons of random vectors with a common copula. Mathematics of Operations Research 26: 723740.
36.Nelsen R.B. (1999). An Introduction to Copulas. Springer: New York.
37.O'Neill P.D. & Becker N.G. (2001). Inference for an epidemic when susceptibility varies. Biostatistics 2: 99108.
38.Ortega N.R.S., Santos F.S., Zanetta D.M.T. & Massad E. (2008). A fuzzy Reed–Frost Model for epidemic spreading. Bulletin of Mathematical Biology 70: 19251936.
39.Picard P. & Lefévre C. (1991). The dimension of the Reed–Frost epidemic models with randomized susceptibility levels. Mathematical Biosciences 107: 225233.
40.Ross S.M. (1996). Stochastic processes, 2nd ed.New York: Wiley.
41.Rüschendorf L. (2004). Comparison of multivariate risks and positive dependence. Advances in Applied Probability 41: 391406.
42.Shaked M. & Shanthikumar J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509531.
43.Shaked M. & Shanthikumar G.J. (2007). Stochastic orders New York: Springer.
44.Stoyan D. (1983). Comparisons methods for queues and other stochastic models. New York: Wiley.
45.Tong Y.L. (1997). Some majorization orderings of heterogeneity in a class of epidemics. Journal of Applied Probability 34: 8493.
46.Tuckwell H.C. & Williams R.J. (2007). Some properties of a simple stochastic epidemic model of SIR type. Mathematical Biosciences 208: 7697.
47.Von Bahr B. & Martin-Löf A. (1980). Threshold limit theorems for some epidemic processes. Advances in Applied Probability 12: 319349.
48.Weiss G.H. (1965). On the spread of epidemics by carriers. Biometrics 21: 481491.
49.Wright E.M. (1954). An inequality for convex functions. American Mathematical Monthly 61: 620622.
50.Yi Z. & Weng C. (2006). On the correlation order. Statistics and Probability Letters 76: 14101416.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 72 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.