Two positivity preserving flux limited, second‐order numerical methods for a haptotaxis model

In this paper the qualitative properties of certain spatial disease propagation models are investigated. The paper can be considered as a generalization of the papers (Faragóand Horváth, 2016; 2017). The models of these papers assume that the members of the population do not move and that the infection is localized in the sense that only the members in a certain neighbourhood of the infective member can be infected. The considered qualitative properties were: the nonnegativity and the monotonicity of the density functions, and the preservation of the amount of the members. Sufficient conditions for these properties were obtained for the mesh size and the time step in certain finite difference solutions of the model. In these works, the one-dimensional case was investigated only. The present paper extends the above result in two directions: with results for higher dimensional problems and for another disease propagation model given in Capasso (2008). Here the members are allowed to move according to some diffusion law. Similarly to the previous model, sufficient conditions are given that guarantee the validity of the qualitative properties. We focus only on the properties of the discrete models. The results are verified on test problems.

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“…Property P1 is satisfied without any condition. This can be seen easily when we sum up the three equations in (11).…”

Section: Theorem 2 Let Us Suppose That At the Initial State Smentioning

confidence: 87%

“…However, only a little attention is devoted to obtaining qualitatively adequate solutions of different continuous and discrete models (e.g. [10,11]).…”

Section: Tablementioning

confidence: 99%

Qualitative properties of some discrete models of disease propagation

Faragó

Horváth

2018

Journal of Computational and Applied Mathematics

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“…However, only a little attention is devoted to obtaining qualitatively adequate solutions of different continuous and discrete models (e.g. [5,11]). In this paper, we will investigate some consistent discretizations of the system (1.3) from the qualitative point of view.…”

Section: Introductionmentioning

confidence: 99%

Qualitatively adequate numerical modelling of spatial SIRS-type disease propagation

Faragó¹,

Horváth²

2016

Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations (July 1–4, 2015, Szeged, Hungary) Edite

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Abstract. The aim of this paper is the investigation of some discrete iterative models that can be used for modeling spatial disease propagation. In our model, we take into account the spatial inhomogenity of the densities of the susceptible, infected and recovered subpopulations and we also suppose vital dynamics. We formulate some characteristic qualitative properties of the model such as nonnegativity and monotonicity and give sufficient conditions that guarantee these properties a priori. Our discrete model can be considered as some discrete approximation of continuous models of the disease propagation given in the form of systems of partial or integro-differential equations. In this way we will be able to give conditions for the mesh size and the time step of the discretisation method in order to guarantee the qualitative properties. Some of the results are demonstrated on numerical tests.

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