We exhibit a scaling law for the critical SIS stochastic epidemic: If at time 0 the population consists of square root N infected and N - square root N susceptible individuals, then when time and number currently infected are both scaled by square root N, the resulting process converges, for large N, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first-passage time for the standard Ornstein- Uhlenbeck process. These results are the analogues for the SIS epidemic of results of Martin-Lof for the simple SIR epidemic.
Heterogeneous Markovian Susceptible-Infected-Susceptible (SIS) epidemics with a general infection rate matrix $\widetilde{A}$ are considered. Using a non-negative matrix factorization to approximate $\widetilde{A}$, we are able to identify when a metastable state can be expected, and that the metastable distribution, under certain conditions, will feature a normal distribution with known expectation and covariance. Furthermore, we model a heterogeneous Markovian SIS epidemic, that starts with a fraction of initially infected nodes different from that in the metastable state, by approximating i…
Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size $N$. We study the behaviour of the process as the population size $N$ tends to infinity. Our results cover the entire subcritical regime, including the "barely subcr…
In this paper, we are concerned with the stochastic susceptible-infectious-susceptible (SIS) epidemic model on the complete graph with $n$ vertices. This model has two parameters, which are the infection rate and the recovery rate. By utilizing the theory of density-dependent Markov chains, we give consistent estimations of the above two parameters as $n$ grows to infinity according to the sample path of the model in a finite time interval. Furthermore, we establish the central limit theorem (CLT) and the moderate deviation principle (MDP) of our estimations. As an application of our CLT, reje…
We study continuous-time birth-death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q]=1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n)=λn being a branching process, and α(n)=λn(N-n)/N which corresponds to an SIS epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, A_n, C and S denote the (random) time to extinction, the tota…
Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can…
We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. The spread of each strain in the absence of the other one is described by the stochastic logistic SIS epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (tha…
We study that the breakdown of epidemic depends on some parameters, that is expressed in epidemic reproduction ratio number. It is noted that when $R_0 $ exceeds 1, the stochastic model have two different results. But, eventually the extinction will be reached even though the major epidemic occurs. The question is how long this process will reach extinction. In this paper, we will focus on the Markovian process of SIS model when major epidemic occurs. Using the approximation of quasi--stationary distribution, the expected mean time of extinction only occurs when the process is one step away fr…
We propose a new approximation framework that unifies and generalizes a number of existing mean-field approximation methods for the SIS epidemic model on complex networks. We derive the framework, which we call the Universal Mean-Field Framework (UMFF), as a set of approximations of the exact Markovian SIS equations. Our main novelty is that we describe the mean-field approximations from the perspective of the isoperimetric problem, an insight which results in bounds on the UMFF approximation error. These new bounds provide insight in the accuracy of existing mean-field methods, such as the wi…
We are interested in describing the infected size of the SIS Epidemic model using Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size $M$; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter $α$; and the disease spreads within the population is modelled by a Poisson process with a rate $λ_{I}$. $λ_{I}=βI(1-\frac{I}{M}) $ is similar to the instantaneous rate in the logistic populatio…
Stochastic epidemic models on networks are inherently high-dimensional and the resulting exact models are intractable numerically even for modest network sizes. Mean-field models provide an alternative but can only capture average quantities, thus offering little or no information about variability in the outcome of the exact process. In this paper we conjecture and numerically prove that it is possible to construct PDE-limits of the exact stochastic SIS epidemics on regular and Erdős-Rényi networks. To do this we first approximate the exact stochastic process at population level by a Birth-an…
The decay rate of SIS epidemics on the complete graph $K_{N}$ is computed analytically, based on a new, algebraic method to compute the second largest eigenvalue of a stochastic three-diagonal matrix up to arbitrary precision. The latter problem has been addressed around 1950, mainly via the theory of orthogonal polynomials and probability theory. The accurate determination of the second largest eigenvalue, also called the \emph{decay parameter}, has been an outstanding problem appearing in general birth-death processes and random walks. Application of our general framework to SIS epidemics sh…
We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K $\in$ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K $\rightarrow$ $\infty$, the process composed of the proportions of infectives of each type X^K and the state of the environment $Ξ$^K…
In this paper, we prove functional central limit theorems (FCLTs) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in Forien, Pang and Pardoux (2020).The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epide…
Epidemic spreading has been intensively studied in SIS epidemic model. Although the mean-field theory of SIS model has been widely used in the research, there is a lack of comparative results between different theoretical calculations, and the differences between them should be systematically explained. In this paper, we have compared different theoretical solutions for mean-field theory and explained the underlying reason. We first describe the differences between different equations for mean-field theory in different networks. The results show that the difference between mean-field reaction …
Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. Furthermore, we show the existence and the global stability of a unique endemic equilibrium provided that the migration rates of susceptible and infectious individuals are equal. Finally, we compare the equilibra with those of the homogeneous model, and with those of isolated patches.
Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD…
Cator and Van Mieghem [Cator E, Van Mieghem P., Phys. Rev. E 89, 052802 (2014)] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian SIS and SIR epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the FKG inequality. In this note we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is …
