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Matrix-geometric methods for the general stochastic epidemic.

J Gani¹, P Purdue

¹Department of Statistics, University of Kentucky.

IMA Journal of Mathematics Applied in Medicine and Biology

|January 1, 1984

Summary

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This study presents a matrix-geometric method for analyzing general stochastic epidemics with complex infection patterns. It determines epidemic survival probabilities and establishes a generalized stochastic threshold theorem.

Area of Science:

Epidemiology
Mathematical Biology
Stochastic Processes

Background:

Stochastic epidemic models are crucial for understanding disease dynamics.
Generalized infection mechanisms present analytical challenges.

Purpose of the Study:

To develop a matrix-geometric formulation for general stochastic epidemics.
To derive methods for calculating epidemic survival probabilities.
To state the stochastic threshold theorem for generalized infections.

Main Methods:

Formulation of a matrix-geometric approach.
Derivation of forward Kolmogorov equations.
Recursive computation of Laplace transforms for state probabilities.

Main Results:

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Obtained probabilities of epidemic survivors.
Established a recursive method for state probability analysis.
Presented the stochastic threshold theorem for generalized infection mechanisms.

Conclusions:

The matrix-geometric formulation provides an effective framework for analyzing complex stochastic epidemics.
The derived methods allow for the calculation of critical epidemic parameters.
The generalized stochastic threshold theorem offers insights into epidemic persistence.