Abstract
We consider a random dynamical system, where the deterministic dynamics are driven by a finite-state space Markov chain. We provide a comprehensive introduction to the required mathematical apparatus and then turn to a special focus on the susceptible-infected-recovered epidemiological model with random steering. Through simulations we visualize the behaviour of the system and the effect of the high-frequency limit of the driving Markov chain. We formulate some questions and conjectures of a purely theoretical nature.
Highlights
The spread of an epidemic and its related characteristics in a large population may be efficiently described by deterministic models
The Infected dynamics are controlled by a random Markov process—i.e. from a randomly behaving population, we obtain observed infected individuals
The common features of these models is that the dynamics is randomly interrupted so that the internal state of the systems is changed, according to a law that depends on the state just before this intervention
Summary
The spread of an epidemic and its related characteristics in a large population may be efficiently described by deterministic models. Are not fixed anymore (they do depend on a particular ∈ ), and our control functions (⋅) = ◦( ) may vary frequently We describe this notion in all details by introducing first: Definition 5 Let ( , F, P) be a (complete) probability space. In the case, when the control phase space ∗ is finite, continuous time homogeneous Markov chains (CTHMC) are very simple and intuitive stochastic objects. Their dynamics and random evolution is fully governed by the so-called Q matrices Norris (2007).
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