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A general approach to the integral functionals of epidemic processes

Part of: Markov processes General field theory

Published online by Cambridge University Press:  26 July 2018

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université Lyon 1
*
* Postal address: Université Libre de Bruxelles, Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium. Email address: [email protected]
** Postal address: Université Lyon 1, ISFA, LSAF EA2429, 50 Avenue Tony Garnier, F-69007 Lyon, France. Email address: [email protected]
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Abstract

In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.

Keywords

General epidemicMarkovian epidemic processfinal susceptible sizetotal area under the trajectory of the infectives and the susceptiblesmartingale methodspecial class of polynomials

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D30: Epidemiology
Secondary: 12E10: Special polynomials
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 593 - 609
Copyright
Copyright © Applied Probability Trust 2018 

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References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York. Google Scholar
Ball, F. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310. Google Scholar
Ball, F. and Stefanov, V. T. (2001). Further approaches to computing fundamental characteristics of birth-death processes. J. Appl. Prob. 38, 995–1005. Google Scholar
Ball, F., O'Neill, P. D. and Pike, J. (2007). Stochastic epidemic models in structured populations featuring dynamic vaccination and isolation. J. Appl. Prob. 44, 571–585. Google Scholar
Billard, L. and Zhao, Z. (1993). A review and synthesis of the HIV/AIDS epidemic as a multi-stage process. Math. Biosci. 117, 19–33. Google Scholar
Daley, D. J. and Gani, J. (1999). Epidemic Modelling: An Introduction. Cambridge University Press. Google Scholar
Feng, R. and Garrido, J. (2011). Actuarial applications of epidemiological models. N. Amer. Actuarial J. 15, 112–136. Google Scholar
Flajolet, P. and Guillemin, F. (2000). The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv. Appl. Prob. 32, 750–778. Google Scholar
Gani, J. and Jerwood, D. (1972). The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269. Google Scholar
Gani, J. and Swift, R. J. (2008). A simple approach to the integrals under three stochastic processes. J. Statist. Theory Pract. 2, 559–568. Google Scholar
Greenwood, P. E. and Gordillo, L. F. (2009). Stochastic epidemic modeling. In Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, Dordrecht, pp. 31–52. Google Scholar
Khaluf, Y. and Dorigo, M. (2016). Modeling robot swarms using integrals of birth-death processes. ACM Trans. Autonomous Adaptive Syst. 11, 8. Google Scholar
Kryscio, R. J. and Saunders, R. (1976). A note on the cost of carrier-borne, right-shift, epidemic models. J. Appl. Prob. 13, 652–661. Google Scholar
Lefèvre, C. and Picard, P. (1990). A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 25–48. Google Scholar
Lefèvre, C. and Picard, P. (2015). Risk models in insurance and epidemics: a bridge through randomized polynomials. Prob. Eng. Inf. Sci. 29, 399–420. Google Scholar
Lefèvre, C., Picard, P. and Simon, M. (2017). Epidemic risk and insurance coverage. J. Appl. Prob. 54, 286–303. Google Scholar
O'Neill, P. (1997). An epidemic model with removal-dependent infection rate. Ann. Appl. Prob. 7, 90–109. Google Scholar
Picard, P. (1980). Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599. Google Scholar
Picard, P. and Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 269–294. Google Scholar
Pollett, P. K. (2003). Integrals for continuous-time Markov chains. Math. Biosci. 182, 213–225. Google Scholar
Puri, P. S. (1972). A method for studying the integral functionals of stochastic processes with applications. III. In Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, University of California Press, Berkeley, pp. 481–500. Google Scholar
Severo, N. C. (1969). Right-shift processes. Proc. Nat. Acad. Sci. 64, 1162–1164. Google Scholar
Stefanov, V. T. (1991). Noncurved exponential families associated with observations over finite-state Markov chains. Scand. J. Statist. 18, 353–356. Google Scholar
Stefanov, V. T. and Pakes, A. G. (1997). Explicit distributional results in pattern formation. Ann. Appl. Prob. 7, 666–678. Google Scholar