arXiv subjects

Markus Kirkilionis

Publications and source records attributed to Markus Kirkilionis.

3 recordsLinked to original sources

arXiv2024

A Rule-Based Epidemiological Modelling Framework

Motivated by chemical reaction rules, we introduce a rule-based epidemiological framework for the systematic mathematical modelling of future pandemics. Here we stress that we do not have a specific model in mind, but a whole collection of models which can be transformed into each other, or represent different aspects of a pandemic, and these aspects can change during the course of the emergency, as happened during the Covid-19 pandemic. As conditions for outbreaks in the modern world change on different time-scales, some rapidly, epidemiology has few 'laws', besides perhaps the fundamental infection process described by Kermack-McKendrick. Each single of our variety of models, called framework, is based on a mathematical formulation that we call a rule-based system. They have several advantages, for example that they can be both interpreted stochastically and deterministically, without changing the model structure. Rule-based systems should be easier to communicate to non-specialists, when compared to differential equations. Due to their combinatorial nature, the rule-based model framework we propose is ideal for systematic mathematical modelling, systematic links to statistics, data analysis in general and also machine learning leading to artificial intelligence.

q-bio.PE↗

arXiv2022

Generalised Gillespie Algorithms for Simulations in a Rule-Based Epidemiological Model Framework

Rule-based models have been successfully used to represent different aspects of the COVID-19 pandemic, including age, testing, hospitalisation, lockdowns, immunity, infectivity, behaviour, mobility and vaccination of individuals. These rule-based approaches are motivated by chemical reaction rules which are traditionally solved numerically with the standard Gillespie algorithm proposed in the context of molecular dynamics. When applying reaction system type of approaches to epidemiology, generalisations of the Gillespie algorithm are required due to the time-dependency of the problems. In this article, we present different generalisations of the standard Gillespie algorithm which address discrete subtypes (e.g., incorporating the age structure of the population), time-discrete updates (e.g., incorporating daily imposed change of rates for lockdowns) and deterministic delays (e.g., given waiting time until a specific change in types such as release from isolation occurs). These algorithms are complemented by relevant examples in the context of the COVID-19 pandemic and numerical results.

q-bio.PE↗

arXiv2010

An Averaging Principle for Combined Interaction Graphs. Part I: Connectivity and Applications to Genetic Switches

Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given 'interaction diagrams' is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both deterministic and stochastic hybrid dynamics. The conventions in genetics are well-known, and therefore we use this field for illustration purposes. In [25] and [26] the authors showed that with the help of a multi-scale analysis stochastic systems with both continuous variables and finite state spaces can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordinary differential equations (ODEs) and Markov chains. The leading order term in these dynamical systems is called average dynamics and turns out to be an adequate concept to analyse a class of simplified hybrid systems. Once the dynamics is defifined the mutual interaction of both ODEs and Markov chains can be analysed through the (reverse) introduction of the so called Interaction Graph, a concept originally invented for time-continuous dynamical systems, see [5]. Here we transfer this graph concept to the average dynamics, which itself is introduced as an heuristic tool to construct models of reaction or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models in terms of connectivity and (feedback) loop formation.

q-bio.MN↗