Stochastic effects in a seasonally forced epidemic model

The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity, is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effects such as switching between coexisting attractors induced by stochasticity often mentioned in the literature as playing an important role in the dynamics of childhood infectious diseases are also investigated. The main conclusion is that stochastic amplification, rather than these effects, is the key ingredient to understand the observed incidence patterns.

A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion

An epidemic model is formulated by a reactionâeuro"diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.

Statistical Analysis of an SEIR Epidemic Model

This thesis was focussed on statistical analysis methods and proposes the use of Bayesian inference to extract information contained in experimental data by estimating Ebola model parameters. The model is a system of differential equations expressing the behavior and dynamics of Ebola. Two sets of data (onset and death data) were both used to estimate parameters, which has not been done by previous researchers in (Chowell, 2004). To be able to use both data, a new version of the model has been built. Model parameters have been estimated and then used to calculate the basic reproduction number and to study the disease-free equilibrium. Estimates of the parameters were useful to determine how well the model fits the data and how good estimates were, in terms of the information they provided about the possible relationship between variables. The solution showed that Ebola model fits the observed onset data at 98.95% and the observed death data at 93.6%. Since Bayesian inference can not be performed analytically, the Markov chain Monte Carlo approach has been used to generate samples from the posterior distribution over parameters. Samples have been used to check the accuracy of the model and other characteristics of the target posteriors.

A stochastic spatially structured epidemic model with diffusive processes

We developed a stochastic lattice model to describe the vector-borne disease (like yellow fever or dengue). The model is spatially structured and its dynamical rules take into account the diffusion of vectors. We consider a bipartite lattice, forming a sub-lattice of human and another occupied by mosquitoes. At each site of lattice we associate a stochastic variable that describes the occupation and the health state of a single individual (mosquito or human). The process of disease transmission in the human population follows a similar dynamic of the Susceptible-Infected-Recovered model (SIR), while the disease transmission in the mosquito population has an analogous dynamic of the Susceptible-Infected-Susceptible model (SIS) with mosquitos diffusion. The occurrence of an epidemic is directly related to the conditional probability of occurrence of infected mosquitoes (human) in the presence of susceptible human (mosquitoes) on neighborhood. The probability of diffusion of mosquitoes can facilitate the formation of pairs Susceptible-Infected enabling an increase in the size of the epidemic. Using an asynchronous dynamic update, we study the disease transmission in a population initially formed by susceptible individuals due to the introduction of a single mosquito (human) infected. We find that this model exhibits a continuous phase transition related to the existence or non-existence of an epidemic. By means of mean field approximations and Monte Carlo simulations we investigate the epidemic threshold and the phase diagram in terms of the diffusion probability and the infection probability.

The SEIQS stochastic epidemic model with external source of infection

This paper deals with a stochastic epidemic model for computer viruses with latent and quarantine periods, and two sources of infection: internal and external. All sojourn times are considered random variables which are assumed to be independent and exponentially distributed. For this model extinction and hazard times are analyzed, giving results for their Laplace transforms and moments. The transient behavior is considered by studying the number of times that computers are susceptible, exposed, infectious and quarantined during a period of time (0, t] and results for their joint and marginal distributions, moments and cross moments are presented. In order to give light this analysis, some numerical examples are showed.

Nature of the Epidemic Threshold for the Susceptible-Infected-Susceptible Dynamics in Networks

We develop an analytical approach to the susceptible-infected-susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks. We find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold. In particular, small-world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit.

Optimal control strategy for fed-batch enzymatic hydrolysis of lignocellulosic biomass based on epidemic modeling

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants.

In this paper we examine the time T to reach a critical number K0 of infections during an outbreak in an epidemic model with infective and susceptible immigrants. The underlying process X, which was first introduced by Ridler-Rowe (1967), is related to recurrent diseases and it appears to be analytically intractable. We present an approximating model inspired from the use of extreme values, and we derive formulae for the Laplace-Stieltjes transform of T and its moments, which are evaluated by using an iterative procedure. Numerical examples are presented to illustrate the effects of the contact and removal rates on the expected values of T and the threshold K0, when the initial time instant corresponds to an invasion time. We also study the exact reproduction number Rexact,0 and the population transmission number Rp, which are random versions of the basic reproduction number R0.

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible--infected--removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback-Leibler divergence for the two fitted models to these data.

Threshold behaviour and final outcome of an epidemic on a random network with household structure

This paper considers a stochastic SIR (susceptible-infective-removed) epidemic model in which individuals may make infectious contacts in two ways, both within 'households' (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically-motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly-sized finite populations. The extension to unequal sized households is discussed briefly.

Robustness against large-scale failures in communications networks

This thesis studies robustness against large-scale failures in communications networks. If failures are isolated, they usually go unnoticed by users thanks to recovery mechanisms. However, such mechanisms are not effective against large-scale multiple failures. Large-scale failures may cause huge economic loss. A key requirement towards devising mechanisms to lessen their impact is the ability to evaluate network robustness. This thesis focuses on multilayer networks featuring separated control and data planes. The majority of the existing measures of robustness are unable to capture the true service degradation in such a setting, because they rely on purely topological features. One of the major contributions of this thesis is a new measure of functional robustness. The failure dynamics is modeled from the perspective of epidemic spreading, for which a new epidemic model is proposed. Another contribution is a taxonomy of multiple, large-scale failures, adapted to the needs and usage of the field of networking.

Random walks systems with killing on Z

We study random walks systems on Z whose general description follows. At time zero, there is a number N >= 1 of particles at each vertex of N, all being inactive, except for those placed at the vertex one. Each active particle performs a simple random walk on Z and, up to the time it dies, it activates all inactive particles that it meets along its way. An active particle dies at the instant it reaches a certain fixed total of jumps (L >= 1) without activating any particle, so that its lifetime depends strongly on the past of the process. We investigate how the probability of survival of the process depends on L and on the jumping probabilities of the active particles.

RUMOR PROCESSES ON N

We study four discrete-time stochastic systems on N, modeling processes of rumor spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumor. The appetite for spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand-based on the distribution of the random variables-whether the probability of having an infinite set of individuals knowing the rumor is positive or not.

NONHOMOGENEOUS RANDOM WALKS SYSTEMS ON Z

We consider a random walks system on Z in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on Z, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to I but not fast enough.