Limits of a multi-patch SIS epidemic model

 We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site (`local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to ∞. Two limit procedures are considered, according to the order in which M and N go to ∞; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x ^* , the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts. Received: 18 July 2001 / Revised version: 16 March 2002 / Published online: 26 September 2002 Mathematics Subject Classification (2000): 92D30, 60F99 Key words or phrases: SIS epidemic – Metapopulation – Markov population processes – Weak convergence of measures     

Extinction times in the subcritical stochastic SIS logistic epidemic

Journal of Mathematical Biology - Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such...     

A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

Outbreak analysis of an SIS epidemic model with rewiring

This paper is devoted to the analysis of the early dynamics of an SIS epidemic model defined on networks. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), is based on the pair-approximation formalism and assumes that, at a given rewiring rate, susceptible nodes replace an infected neighbour by a new susceptible neighbour randomly selected among the pool of susceptible nodes in the population. The analysis uses a triple closure that improves the widely assumed in epidemic models defined on regular and homogeneous networks, and applies it to better understand the early epidemic spread on Poisson, exponential, and scale-free networks. Two extinction scenarios, one dominated by transmission and the other one by rewiring, are characterized by considering the limit system of the model equations close to the beginning of the epidemic. Moreover, an analytical condition for the occurrence of a bistability region is obtained.     

Evolution of unconditional dispersal in periodic environments

Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against 'unconditional' dispersal in spatially heterogeneous environments. 'Unconditional' means individuals disperse at a rate independent of their location. We prove that if within-patch fitness varies spatially and between two values temporally, then there is selection for unconditional dispersal: any evolutionarily stable strategy (ESS) or evolutionarily stable coalition (ESC) includes a dispersive phenotype. Moreover, at this ESS or ESC, there is at least one sink patch (i.e. geometric mean of fitness less than one) and no sources patches (i.e. geometric mean of fitness greater than one). These results coupled with simulations suggest that spatial-temporal heterogeneity is due to abiotic forcing result in either an ESS with a dispersive phenotype or an ESC with sedentary and dispersive phenotypes. In contrast, the spatial-temporal heterogeneity due to biotic interactions can select for higher dispersal rates that ultimately spatially synchronize population dynamics.     

Multiple attractors, saddles, and population dynamics in periodic habitats

Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.     

Community robustness and limiting similarity in periodic environments

Temporal environmental variation has long been considered as one of the potential factors that could promote species coexistence. A question of particular interest is how the ecology of fluctuating environments relates to that of equilibrium systems. Equilibrium theory says that the more similar two species are in their modes of regulation, the less robust their coexistence will be; that is, the volume of external parameters for which all populations persist shrinks with increasing similarity. In this study, we will attempt to generalize these results to temporally varying situations and establish the precise mathematical relationship between the two. Our treatment considers unstructured populations in continuous time with periodic attractors of fixed period length, where the periodic behavior is due to external forcing. Within these conditions, our treatment is general. We provide a coherent theoretical framework for defining measures of species similarity and niche. Our main conclusion is that all factors that function to regulate population growth may be considered as separate regulating factors for each moment of time. In particular, a single resource becomes a resource continuum, along which species may segregate in the same manner as along classical resource continua. Therefore, we provide a mathematical underpinning for considering fluctuation-mediated coexistence as temporal niche segregation.     

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

Journal of Mathematical Biology - Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many...     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

Resonance of the epidemic threshold in a periodic environment

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold. Resonance occurs when the Euler-Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.     

Asymptotic results for a multi-type contact birth-death process and related SIS epidemic

Exact results concerning the asymptotic speed of propagation of infection have recently been obtained for the multi-type SIS epidemic in continuous space when the contact distributions are assumed to be symmetric with the Laplace transforms finite for all entries. There is a link between the equations for this epidemic and the equations for a multi-type contact birth-death process. This enables methods developed for the epidemic to be used to obtain the asymptotic speed of translation for the contact birth-death process. Symmetry of the contact distributions is required but no existence constraint is placed on their Laplace transforms. The method for removing this constraint may also be used for the SIS epidemic. Results are given for both processes when the basic reproduction ratio is at most one.     

Multiple stable subharmonics for a periodic epidemic model

The S → I → R epidemic model of K. Dietz with annual oscillation in the contact rate is shown to have multiple stable subharmonic solutions of different integral year periods.     

A stochastic SIS epidemic with demography: initial stages and time to extinction

We study an open population stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The model describes an SIS (susceptible–infective–susceptible) epidemic where all individuals, including infectious ones, reproduce at a given rate. An approximate expression for the outbreak probability is derived using a coupling argument. Further, we analyse the behaviour of the model close to quasi-stationarity, and the time to disease extinction, with the aid of a diffusion approximation. In this situation the number of susceptibles and infectives behaves as an Ornstein–Uhlenbeck process, centred around the stationary point, for an exponentially distributed time before going extinct.     

Bifurcation analysis of periodic SEIR and SIR epidemic models

The bifurcations of the periodic solutions of SEIR and SIR epidemic models with sinusoidally varying contact rate are investigated. The analysis is carried out with respect to two parameters: the mean value and the degree of seasonality of the contact rate. The corresponding portraits in the two-parameter space are obtained by means of a numerical continuation method. Codimension two bifurcations (degenerate flips and cusps) are detected, and multiple stable modes of behavior are identified in various regions of the parameter space. Finally, it is shown how the parametric portrait of the SEIR model tends to that of the SIR model when the latent period tends to zero.     

Reproductive numbers for nonautonomous spatially distributed periodic SIS models acting on two time scales

In this work we deal with a general class of spatially distributed periodic SIS epidemic models with two time scales. We let susceptible and infected individuals migrate between patches with periodic time dependent migration rates. The existence of two time scales in the system allows to describe certain features of the asymptotic behavior of its solutions with the help of a less dimensional, aggregated, system. We derive global reproduction numbers governing the general spatially distributed nonautonomous system through the aggregated system. We apply this result when the mass action law and the frequency dependent transmission law are considered. Comparing these global reproductive numbers to their non spatially distributed counterparts yields the following: adequate periodic migration rates allow global persistence or eradication of epidemics where locally, in absence of migrations, the contrary is expected.     

A periodic SEIRS epidemic model with a time-dependent latent period

Journal of Mathematical Biology - Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes...