Reaction diffusion equations and the chronification of liver infections

Last modified: 2014-03-27

#### Abstract

A predator-prey system is used to model the time-dependent virus and lymphocyte population during a liver

infection. We consider a hierarchically family of mathematical models, starting with the two populations

virus as prey and lymphocytes as predator and expanding the model by biochemical signals, by chemotaxis effects,

by the Allee effect or by transport in the blood vessels.

We show mathematically that the resulting reaction-diffusion equations have non-trivial stationary

solutions whenever the underlying domain is sufficiently large or fissured. The charactistic quantity for

the extension of the domain is the smallest eigenvalue of a suitable differential operator with Neumann

boundary condition. The extension of the domain corresponds to the migration speed of the virus and the lymphocytes.

The non-trivial stationary solutions of the model equations are interpreted as chronic liver infections. Thus,

quantitative differences become dispensable for the distinction between acute and chronic hepatitis infections, and

we get a modeling framework for the investigation of the chronification of liver infections. Numerical simulations

for the chronification and for an acute course are presented.

The hierarchical model family is used to discuss possible model selections. Therefore, qualitative observations are

mapped to model components. The set of reproduced observations defines a semi-order or a hierarchy in the model family,

which is used to select robust models.

infection. We consider a hierarchically family of mathematical models, starting with the two populations

virus as prey and lymphocytes as predator and expanding the model by biochemical signals, by chemotaxis effects,

by the Allee effect or by transport in the blood vessels.

We show mathematically that the resulting reaction-diffusion equations have non-trivial stationary

solutions whenever the underlying domain is sufficiently large or fissured. The charactistic quantity for

the extension of the domain is the smallest eigenvalue of a suitable differential operator with Neumann

boundary condition. The extension of the domain corresponds to the migration speed of the virus and the lymphocytes.

The non-trivial stationary solutions of the model equations are interpreted as chronic liver infections. Thus,

quantitative differences become dispensable for the distinction between acute and chronic hepatitis infections, and

we get a modeling framework for the investigation of the chronification of liver infections. Numerical simulations

for the chronification and for an acute course are presented.

The hierarchical model family is used to discuss possible model selections. Therefore, qualitative observations are

mapped to model components. The set of reproduced observations defines a semi-order or a hierarchy in the model family,

which is used to select robust models.

#### Keywords

reaction diffusion equation; model family; liver inflammation; chronification