Extinction times for closed epidemics: the effects of host spatial structure (original) (raw)

Predicting extinction rates in stochastic epidemic models

Journal of Statistical Mechanics: Theory and Experiment, 2009

We investigate the stochastic extinction processes in a class of epidemic models. Motivated by the process of natural disease extinction in epidemics, we examine the rate of extinction as a function of disease spread. We show that the effective entropic barrier for extinction in a susceptible-infectedsusceptible epidemic model displays scaling with the distance to the bifurcation point, with an unusual critical exponent. We make a direct comparison between predictions and numerical simulations. We also consider the effect of non-Gaussian vaccine schedules, and show numerically how the extinction process may be enhanced when the vaccine schedules are Poisson distributed.

Slow epidemic extinction in populations with heterogeneous infection rates

Physical Review E, 2013

We explore how heterogeneity in the intensity of interactions between people affects epidemic spreading. For that, we study the susceptible-infected-susceptible model on a complex network, where a link connecting individuals i and j is endowed with an infection rate β ij = λw ij proportional to the intensity of their contact w ij , with a distribution P (w ij ) taken from face-to-face experiments analyzed in Cattuto et al. (PLoS ONE 5, e11596, 2010). We find an extremely slow decay of the fraction of infected individuals, for a wide range of the control parameter λ.

Mathematical Modeling of Infectious Disease Transmission Dynamics in a Metapopulation

IOSR Journals , 2019

Epidemic modeling is an important theoretical approach for investigating the transmission dynamics of infectious diseases. It formulates mathematical models to describe the mechanisms of disease transmissions and dynamics of infectious agents and then informs the health control practitioners the likely impact of the control methods. In this paper we investigate the spread of an infectious disease in a human population structured into n-patches. The population is initially fully susceptible until an infectious individual is introduced in one of the patches. The interaction between patches is dominated by movement of individuals between patches and also the migration of individuals and therefore any infection occurring in one patch will have a force of infection on the susceptible individuals on the other patches. We build a mathematical model for a metapopulation consisting of í µí±› patches. The patches are connected by movement of individuals. For í µí±› = 2, we obtained the basic reproduction number and obtained the condition under which the disease free equilibrium will be asymptotically stable. We further described in terms of the model parameters how control methods could be applied to ensure that the epidemic does not occur and validated the results by the use of the numerical simulation. We showed that the global basic reproduction number cannot exceed one unless the local basic reproduction number is greater than one in at least one of the sub-populations. We further showed that the control of the epidemic in this case can be achieved by applying a control method that decreases the transmission parameters in patches where the local basic reproduction number is greater than one.

Spatial evolutionary epidemiology of spreading epidemics

Proceedings of the Royal Society B: Biological Sciences, 2016

Most spatial models of host–parasite interactions either neglect the possibility of pathogen evolution or consider that this process is slow enough for epidemiological dynamics to reach an equilibrium on a fast timescale. Here, we propose a novel approach to jointly model the epidemiological and evolutionary dynamics of spatially structured host and pathogen populations. Starting from a multi-strain epidemiological model, we use a combination of spatial moment equations and quantitative genetics to analyse the dynamics of mean transmission and virulence in the population. A key insight of our approach is that, even in the absence of long-term evolutionary consequences, spatial structure can affect the short-term evolution of pathogens because of the build-up of spatial differentiation in mean virulence. We show that spatial differentiation is driven by a balance between epidemiological and genetic effects, and this quantity is related to the effect of kin competition discussed in pr...

Multiscale, resurgent epidemics in a hierarchical metapopulation model

Proceedings of The National Academy of Sciences, 2005

Although population structure has long been recognized as relevant to the spread of infectious disease, traditional mathematical models have understated the role of nonhomogenous mixing in populations with geographical and social structure. Recently, a wide variety of spatial and network models have been proposed that incorporate various aspects of interaction structure among individuals. However, these more complex models necessarily suffer from limited tractability, rendering general conclusions difficult to draw. In seeking a compromise between parsimony and realism, we introduce a class of metapopulation models in which we assume homogeneous mixing holds within local contexts, and that these contexts are embedded in a nested hierarchy of successively larger domains. We model the movement of individuals between contexts via simple transport parameters and allow diseases to spread stochastically. Our model exhibits some important stylized features of real epidemics, including extreme size variation and temporal heterogeneity, that are difficult to characterize with traditional measures. In particular, our results suggest that when epidemics do occur the basic reproduction number R 0 may bear little relation to their final size. Informed by our model's behavior, we suggest measures for characterizing epidemic thresholds and discuss implications for the control of epidemics. math model ͉ population structure

Modelling the Spread of Infectious Diseases in Complex Metapopulations

Mathematical Modelling of Natural Phenomena, 2010

Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially, and, in the second one, both processes occur simultaneously. Here we follow the second approach and give a necessary and sufficient condition for the instability of the disease-free equilibrium in generic networks under the assumption of limited (or frequency-dependent) transmission. Moreover, for uncorrelated networks and under the assumption of non-limited (or density-dependent) transmission, we give a bounding interval for the dominant eigenvalue of the Jacobian matrix of the model equations around the disease-free equilibrium. Finally, for this latter case, we study numerically the prevalence of the infection across the metapopulation as a function of the patch connectivity.

The role of initial inoculum on epidemic dynamics

Journal of Theoretical Biology, 2006

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.

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.

Evolution in a spatially structured population subject to rare epidemics

Physical Review E, 2001

We study a model that gives rise to spatially inhomogeneous population densities in a system of host individuals subject to rare, randomly distributed disease events. For stationary hosts that disperse offspring over short distances, evolutionary dynamics can lead to persistent populations with a variety of spatial structures. A mean-field analysis is shown to account for the behavior observed in simulations of a one-dimensional system, where the evolutionarily stable state corresponds to the solution of a straightforward optimization problem. In two dimensions, evolution drives the system to a stable critical state that is less well understood.

The vector–host epidemic model with multiple strains in a patchy environment

Journal of Mathematical Analysis and Applications, 2013

Spatial heterogeneity plays an important role in the distribution and persistence of infectious diseases. In this article, a vector-host epidemic model is proposed to explore the effect of spatial heterogeneity on the evolution of vector-borne diseases. The model is a Ross-McDonald type model with multiple competing strains on a number of patches connected by host migration. The multi-patch basic reproduction numbers R j 0 , j = 1, 2, • • • , l are respectively derived for the model with l strains on n discrete patches. Analytical results show that if R j 0 < 1, then strain j cannot invade the patchy environment and dies out. The invasion reproduction numbers R j i , i, j = 1, 2, i = j are also derived for the model with two strains on n discrete patches. It is shown that the invasion reproduction numbers R j i , i, j = 1, 2, i = j provide threshold conditions that determine the competitive outcomes for the two strains. Under the condition that both invasion reproduction numbers are lager than one, the coexistence of two competing strains is rigorously proved. However, the two competing strains cannot coexist for the corresponding model with no host migration. This implies that host migration can lead to the coexistence of two competing strains and enhancement of pathogen genetic diversity. Global dynamics is determined for the model with two competing strains on two patches. The results are based on the theory of type-K monotone dynamical systems.