We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.
Suppose that initially there are only a few infective vertices. We prove that there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak may occur. We prove that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. In contrast to earlier results for this model, our results only require basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex.
We also study the regime just above the threshold: we determine the probability that a large epidemic occurs and the size of a large epidemic.
This is joint work with Svante Janson and Peter Windridge.

Data driven decision-making is particularly challenging during a quickly evolving outbreak such as the 2013-2015 Ebola virus disease (EVD) epidemic in West-Africa. In this talk, I will present several examples of how dynamical models, both from our group and others, were used to analyse the Ebola outbreak and make predictions that influenced decision-makers. In particular, I will discuss several modelling (deterministic, stochastic) and statistical (Kalman filter, particle MCMC) approaches used for:

• Real-time modelling of Ebola transmission and prediction of bed demands in treatment centres.
• Measuring the impact of the control-measures in Sierra-Leone.
• Assessing the feasibility of vaccine trials in a declining epidemic.
• Analysing past-Ebola outbreaks in the Democratic Republic of Congo.

The talk is based on two recent publications:

A. Yu. Veretennikov, On the rate of convergence for infinite server Erlang–Sevastyanov’s problem, Queueing Systems, 2014, 76(2), 181-203

A. Yu. Veretennikov and G. A. Zverkina, Simple Proof of Dynkin’s Formula for Single-Server Systems and Polynomial Convergence Rates, Markov Processes Relat. Fields, 2014, 20, 479–504,

and on one new conference presentation in preparation.

In 1909 in the first paper on queueing, Erlang established a stationary distribution for some basic "telephone type system" with the exponential distribution of service times. Erlang himself and many researches after him attacked the general case trying also to prove any kind of convergence to the stationary regime. This general case as well as the problem of convergence remained unsolved for nearly 50 years until the paper by Sevastyanov in 1957. The question of estimating the rate of convergence for general distributions remained open for another period of several decades. This problem will be discussed in the talk. In reliability theory, the object of the major interest is the "coefficient of readiness", which is just a probability that the system is in the working state. The same problem of estimating convergence rate of the non-stationary version of this coefficient to its stationary value will be discussed for certain models.

The work is partly joint with Galina Zverkina, MIIT, Moscow.