Linear models.- Strongly nonlinear models.- Quasimonotone systems. Positive feedback systems. Cooperative systems.- Spatial heterogeneity.- Age structure.- Optimization problems.

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Frobenius Theory of Positive Operators: Comparison Theorems and Applications

Mathematical structures of epidemic systems

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.

Threshold and stability results for an age-structured epidemic model.

Introduction.- Aspects of stochastic control theory.- Optimal stabilization of linear stochastic systems.- Linear mappings on ordered vector spaces.- Newtons method.- Solution of the Riccati equation.

Rational Matrix Equations in Stochastic Control

Although its usefulness and possibility of the well-known definition of the basic reproduction number R0 for structured populations by Diekmann, Heesterbeek and Metz (J Math Biol 28:365-382, 1990) (the DHM definition) have been widely recognized mainly in the context of epidemic models, originally it deals with population dynamics in a constant environment, so it cannot be applied to formulate the threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R0 to the case of a periodic environment. In particular, the definition of R0 in a periodic environment by Bacaer and Guernaoui (J Math Biol 53:421-436, 2006) (the BG definition) is most important, because their definition of periodic R0 can be interpreted as the asymptotic per generation growth rate, which is an essential feature of the DHM definition. In this paper, we introduce a new definition of R0 based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment. Using the generation evolution operator, we show that the DHM definition and the BG definition completely allow the generational interpretation and, in those two cases, the spectral radius of GEO equals the spectral radius of the next generation operator, so it gives the basic reproduction number. Hence the new definition is an extension of the DHM definition and the BG definition. Finally we prove a weak sign relation that if the average Malthusian parameter exists, it is nonnegative when R0>1 and it is nonpositive when R0<1.

https://www.ms.u-tokyo.ac.jp/~inaba/inaba_JMB_2012_R0.pdf

Frobenius Theory of Positive Operators: Comparison Theorems and Applications

Citations

Mathematical structures of epidemic systems

Threshold and stability results for an age-structured epidemic model.

Rational Matrix Equations in Stochastic Control

On a new perspective of the basic reproduction number in heterogeneous environments.

Theory of cones

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