A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold (original) (raw)

Large-Scale Spatial-Transmission Models of Infectious Disease

Science, 2007

During transmission of seasonal endemic diseases such as measles and influenza, spatial waves of infection have been observed between large distant populations. Also, during the initial stages of an outbreak of a new or reemerging pathogen, disease incidence tends to occur in spatial clusters, which makes containment possible if you can predict the subsequent spread of disease. Spatial models are being used with increasing frequency to help characterize these large-scale patterns and to evaluate the impact of interventions. Here, I review several recent studies on four diseases that show the benefits of different methodologies: measles (patch models), foot-and-mouth disease (distance-transmission models), pandemic influenza (multigroup models), and smallpox (network models). This review highlights the importance of the household in spatial studies of human diseases, such as smallpox and influenza. It also demonstrates the need to develop a simple model of household demographics, so ...

Key factors in disease spreading: Spatial heterogeneity, time dependence, and human behavior

Physics of Life Reviews, 2016

The history of mathematical modelling in epidemiology is long and rich. Their origin can be tracked back to the eighteenth century and Bernoulli's mathematical description for smallpox vaccination [1], a model which was recently revisited by Dietz and Heesterbeek [2]. However, and as Sun et al. [3] pointed out in their introduction, it was not until 1927, when Kermack and McKendrick [4] formally presented the SIR compartmental model in the form of coupled non-linear ordinary differential equations, that theoretical epidemiology started to gain momentum. Since then, mathematical modelling in epidemiology developed conceptually and technically with the inclusion of variants like SIS, SIRS, SEIRS, and others, and the consideration of space, vaccination, vectors, age structure, and time dependence. And more recently, with the inclusion of the

A comprehensive spatial-temporal infection model

Chemical Engineering Science, 2021

Motivated by analogies between the spreading of human-to-human infections and of chemical processes, we develop a comprehensive model that accounts both for infection (reaction) and for transport (mobility, advection and diffusion). In this analogy, the three different populations (susceptible, infected and recovered) of infection models correspond to three "chemical species". Areal densities (people/area), rather than populations, emerge as the key variables, thus capturing the effect of spatial density, widely considered important, but ignored or under-represented in existing models. We derive expressions for the kinetics of the infection rates and for the important parameter R 0 , that include areal density and its spatial distribution. Coupled with mobility (through diffusion) the model allows the study of various effects. We first present results for a "batch reactor", the chemical process equivalent of the SIR model. Because density makes R 0 a decreasing function of the process extent, the infection curves are different and smaller than for the standard SIR model, the difference increasing with R 0. We show that the effect of the initial conditions (density of infected individuals) is limited to the onset of the epidemic, everything else being equal. The same invariance is obtained for infection imported into initially non-infected regions. We derive effective infection curves for a number of cases, including a back-and-forth "commute" between regions of low (e.g. "home") and high (e.g. "work") R 0 environments. We then consider spatially distributed systems. We show that diffusion leads to traveling waves, which in 1-D geometries (rectilinear or radial) propagate at a constant speed and with a constant shape, both of which are sole functions of R 0. The infection curves are slightly different than for the batch problem, as diffusion mitigates the infection intensity, thus leading to an effective lower R 0. The dimensional wave speed is found to be proportional to the product of the square root of the diffusivity and of an increasing function of R 0 , confirming the importance of restricting mobility in arresting the propagation of infection. We examine the interaction of infection waves under various conditions and scenarios, and extend the wave propagation analysis to 2-D heterogeneous systems.

The Importance of Being Hybrid for Spatial Epidemic Models:A Multi-Scale Approach

Systems, 2015

This work addresses the spread of a disease within an urban system, defined as a network of interconnected cities. The first step consists of comparing two different approaches: a macroscopic one, based on a system of coupled Ordinary Differential Equations (ODE) Susceptible-Infected-Recovered (SIR) systems exploiting populations on nodes and flows on edges (so-called metapopulational model), and a hybrid one, coupling ODE SIR systems on nodes and agents traveling on edges. Under homogeneous conditions (mean field approximation), this comparison leads to similar results on the outputs on which we focus (the maximum intensity of the epidemic, its duration and the time of the epidemic peak). However, when it comes to setting up epidemic control strategies, results rapidly diverge between the two approaches, and it appears that the full macroscopic model is not completely adapted to these questions. In this paper, we focus on some control strategies, which are quarantine, avoidance and risk culture, to explore the differences, advantages and disadvantages of the two models and discuss the importance of being hybrid when modeling and simulating epidemic spread at the level of a whole urban system.

Nine challenges for deterministic epidemic models

Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where superinfection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.

Epidemiological model for the inhomogeneous spatial spreading of COVID-19 and other diseases

2020

We suggest a mathematical model for the spread of an infectious disease in human population, with particular attention to the COVID-19. Common epidemiological models, e.g., the well-known susceptible-exposed-infectious-recovered (SEIR) model, implicitly assume fast mixing of the population relative to the local infection rate, similar to the regime applicable to many chemical reactions. However, in human populations, especially under different levels of quarantine conditions, this assumption is likely to fail. We develop a continuous spatial model that includes five different populations, in which the infectious population is split into latent (or pre-symptomatic) and symptomatic. Based on nearest-neighbor infection kinetics, we arrive into a “reaction-diffusion” model. Our model accounts for front propagation of the infectious population domains under partial quarantine conditions, which is present on top of the common local infection process. Importantly, we also account for the v...

Stochastic modeling of animal epidemics using data collected over three different spatial scales

Epidemics, 2011

A stochastic, spatial, discrete-time, SEIR model of avian influenza epidemics among poultry farms in Pennsylvania is formulated. Using three different spatial scales wherein all the birds within a single farm, ZIP code, or county are clustered into a single point, we obtain three different views of the epidemics. For each spatial scale, two parameters within the viral-transmission kernel of the model are estimated using simulated epidemic data. We show that simulated epidemics modeled using data collected on the farm and ZIP-code levels behave similar to the actual underlying epidemics, but this is not true using data collected on the county level. Such analyses of data collected on different spatial scales are useful in formulating intervention strategies to control an ongoing epidemic (e.g., vaccination schedules and culling policies).

Spatial Heterogeneity in Epidemic Models

Journal of Theoretical Biology, 1996

Spatial heterogeneity is believed to play an important role in the persistence and dynamics of epidemics of childhood diseases because asynchrony between populations within different regions allows global persistence, even if the disease dies out locally. A simple multi-patch (metapopulation) model for spatial heterogeneity in epidemics is analysed and we examine conditions under which patches become synchronized. We show that the patches in non-seasonal deterministic models often oscillate in phase for all but the weakest between patch coupling. Synchronization is also seen for stochastic models, although slightly stronger coupling is needed to overcome the random effects. We demonstrate that the inclusion of seasonal forcing in deterministic models can lead to the maintenance of phase differences between patches. Complex dynamic behaviour is observed in the seasonally forced spatial model, along with the coexistence of many different behaviours. Compared to the non-spatial model, chaotic solutions are observed for weaker seasonal forcing; these solutions have a more realistic minimum number of infectives.

An S--I--R Model for Epidemics with Diffusion to Avoid Infection and Overcrowding

A model for an epidemic of S--I--R type is described in which susceptibles move to avoid the infection and all individuals move away from overcrowded regions. Some qualitative properties of the mathematical model are discussed. and exhibited with results from numerical simulations. 1 Introduction The Kermack--McKendrick model is the first one to provide a mathematical description for the kinetic transmission of an epidemic in an unstructured population [3]. In this model the total population is assumed to be constant and divided into three classes: susceptible, infected, and removed (or recovered). The propagation of an infection governed by this simple model, which does not incorporate structure due to age, sex, degree of infectivity, or spatial position, is well-known. Several extensions of the model have been considered. Webb [7] proposed and analyzed a model structured by spatial position in a bounded one-dimensional environment, [0; L], L ? 0. The spatial distribution is assume...

Analysis of spatial dynamics and time delays in epidemic models

2014

for providing me with the conducive atmosphere where this work was carried out, for their patience, assistance and guidance through out this research, and from whom I acquired a great deal of skills and knowledge. Their contributions towards the completion of this thesis are unmeasurable. I will forever remain indebted to them. To other members of the Sussex faculty including (but not limited to) Drs. Anotida Madzvamuse, Istvan Kiss, Yuliya N. Kyrychko, I thank them for their useful ideas and helps. Also, I am indebted to my postgraduate colleagues whom include Hussaini Sal