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Hierarchy of the presented models. Solid arrows show inheritance, and dashed arrows show unknown inheritance
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Based on an established model for liver infections, we open the discussion on the used reaction terms in the reaction‐diffusion system. The mechanisms behind the chronification of liver infections are widely unknown, therefore we discuss a variety of reaction functions. By using theorems about existence, uniqueness, and nonnegativity, we identify p...
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Citations
... The used parameters and the shape and size of the domain Ω control towards which stationary state the solution is tending. See [8,11,13,14] for further details on the analytical results. ...
... Theorem 16 gives in Eq. (15) an estimate for the L 2 (Ω)-norm of v. By using ξ(t) as solution of Eq. (13) and directly Eq. (14) we get the approximation ...
This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. Non-local effects in the dynamics between the virus and the cells of the immune system lead to an integro-partial differential equation with homogeneous Neumann boundary conditions. Depending on the chosen model parameters, the system shows two types of solutions which are interpreted as different infection courses. Apart from solutions decaying to zero, there are solutions with a tendency towards a stationary and spatially inhomogeneous state. By proving the boundedness of the solution in the $L^1(\Omega)$- and the $L^2(\Omega)$-norms, it is possible to show the global boundedness of the solution. The proof uses the opposite mechanisms in the reaction terms. The gained rough estimates for showing the boundedness in the $L^1(\Omega)$- and the $L^2(\Omega)$-norms are compared numerically with the norms of the solutions.
This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time.
We show the boundedness of the solution in the L1(Ω)- and the L2(Ω)-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the L∞(Ω)-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases.
The gained rough estimates are compared with numerical tests.
