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A general approach to the integral functionals of epidemic processes. (English) Zbl 1401.60148

Summary: 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 \(S_T\), the area under the trajectory of the infectives \(A_T\), and the area under the trajectory of the susceptibles \(B_T\). 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.

MSC:

60J28 Applications of continuous-time Markov processes on discrete state spaces
92D30 Epidemiology
12E10 Special polynomials in general fields
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