Bounds for the expected value of one-step processes

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{0,1,\ldots,N\}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.Comment: 14 pages, 4 figures, revise

SIS epidemic propagation on hypergraphs

Mathematical modeling of epidemic propagation on networks is extended to hypergraphs in order to account for both the community structure and the nonlinear dependence of the infection pressure on the number of infected neighbours. The exact master equations of the propagation process are derived for an arbitrary hypergraph given by its incidence matrix. Based on these, moment closure approximation and mean-field models are introduced and compared to individual-based stochastic simulations. The simulation algorithm, developed for networks, is extended to hypergraphs. The effects of hypergraph structure and the model parameters are investigated via individual-based simulation results

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE

Monte Carlo simulation and analytic approximation of epidemic processes on large networks

Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways. © 2012 Versita Warsaw and Springer-Verlag Wien

Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells

The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to �nd the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the Parametric Representation Method, and this enables us to determine numerically the parameter domain where bistability occurs that is important from the chemical point of view

On bounding exact models of epidemic spread on networks

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that closures or approximations at individual or node level lead to mean-field models that bound the exact stochastic process from above. This will be done in the context of modelling epidemic spread on networks and the proof of the result relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively). The results in the paper hold for Markovian epidemics and arbitrary weighted and directed networks. Furthermore, we cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models. While deterministic initial conditions are key to obtain the negative correlation result we show that this condition can be relaxed as long as extra conditions on the edge weights are imposed

Dynamic control of modern, network-based epidemic models

In this paper we make the first steps to bridge the gap between classic control theory and modern, network-based epidemic models. In particular, we apply nonlinear model predictive control (NMPC) to a pairwise ODE model which we use to model a susceptible-infectious-susceptible (SIS) epidemic on nontrivial contact structures. While classic control of epidemics concentrates on aspects such as vaccination, quarantine, and fast diagnosis, our novel setup allows us to deliver control by altering the contact network within the population. Moreover, the ideal outcome of control is to eradicate the disease while keeping the network well connected. The paper gives a thorough and detailed numerical investigation of the impact and interaction of system and control parameters on the controllability of the system. For a certain combination of parameters, we used our method to identify the critical control bounds above which the system is controllable. We foresee that our approach can be extended to even more realistic or simulation-based models with the aim of applying these to real-world situations

Super compact pairwise model for SIS epidemic on heterogeneous networks

In this paper we provide the derivation of a super compact pairwise model with only 4 equations in the context of describing susceptible-infected-susceptible (SIS) epidemic dynamics on heterogenous networks. The super compact model is based on a new closure relation that involves not only the average degree but also the second and third moments of the degree distribution. Its derivation uses an a priori approximation of the degree distribution of susceptible nodes in terms of the degree distribution of the network. The new closure gives excellent agreement with heterogeneous pairwise models that contain significantly more differential equations