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Solvability of implicit final size equations for SIR epidemic models.
Subekshya Bidari1, Xinying Chen1, Daniel Peters1
1Budapest Semesters in Mathematics, Budapest, Hungary.
Mathematical Biosciences
|December 17, 2016
Summary
This study proves that the implicit equations for final epidemic size in SIR models have unique solutions. Iterative methods are shown to converge to these solutions, aiding epidemic modeling accuracy.
Area of Science:
- Epidemiology
- Mathematical Biology
- Network Science
Background:
- Final epidemic size relations are crucial in mathematical epidemiology but often involve unsolvable implicit equations.
- The solvability of these implicit equations across various complex epidemic models remains less explored.
Purpose of the Study:
- To investigate the solvability of implicit equations for final epidemic size in SIR (Susceptible-Infectious-Recovered) models.
- To develop and analyze iterative methods for solving these equations and assess mean-field approximations.
Main Methods:
- Analysis of homogeneous mean-field, pairwise, and heterogeneous mean-field SIR models.
- Application of a generation-based approach for Markovian SIR epidemic models on finite networks.
- Derivation of explicit analytic formulas and iterative formulas for final size distribution.
Main Results:
- Proved the existence and uniqueness of solutions for the implicit final epidemic size equation in studied SIR models.
- Demonstrated convergence of iterative fixed-point equations to the unique solution.
- Derived explicit formulas for final size distribution on line and star graphs, enabling accuracy assessment of mean-field approximations.
Conclusions:
- The implicit equations for final epidemic size in SIR models are uniquely solvable.
- Iterative methods provide a reliable approach to find these solutions.
- Analytic and iterative formulas enhance understanding of epidemic dynamics on networks and the accuracy of approximations.
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