Qualitatively adequate numerical modelling of spatial SIRS-type disease propagation

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.

Keywords

• differential equations
• epidemic models
• qualitative properties of systems of pdes
• nonnegativity
• finite difference method