Realistic Distributions of Infectious Periods in Epidemic Models: Changing Patterns of Persistence and Dynamics (original) (raw)
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Ecological Monographs, 2002
Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ϳ300 000-500 000. We develop a model, the TSIR (Time-series Susceptible-Infected-Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era (1944-1966). The 60 cities analyzed span a size range from London (3.3 ϫ 10 6 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (''true mass action''). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R 0 , is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus, efficient in ''regulating'' the susceptible population-even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly ''deterministic'' through most of the growth and decline phase.
The Mathematics of Infectious Diseases
Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number R 0 , the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R 0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R 0 and σ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
2012
Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, will also be discussed.
Applied Mathematics and Computation, 2005
The aim of this paper is to discuss the new class of epidemic models proposed by Satsuma et al., which are characterized by incidence rates which are nonlinearly dependent on the number of susceptibles as follows: infection rate (S, I) = g(S)I. By adding the biologically plausible constraint g 0 (S) > 0, we study the SIR and the SEIR models with vital dynamics with such infection rate, and results are done on the global asymptotic stability of the disease free and of the endemic equilibria, similarly to the ones of the classical models, also in presence of traditional and pulse vaccination strategies. Relaxing the constraint g 0 (S) > 0, we show that the epidemic system may exhibit multiple endemic equilibria.
First Principles Modeling of Nonlinear Incidence Rates in Seasonal Epidemics
PLoS Computational Biology, 2011
In this paper we used a general stochastic processes framework to derive from first principles the incidence rate function that characterizes epidemic models. We investigate a particular case, the Liu-Hethcote-van den Driessche's (LHD) incidence rate function, which results from modeling the number of successful transmission encounters as a pure birth process. This derivation also takes into account heterogeneity in the population with regard to the per individual transmission probability. We adjusted a deterministic SIRS model with both the classical and the LHD incidence rate functions to time series of the number of children infected with syncytial respiratory virus in Banjul, Gambia and Turku, Finland. We also adjusted a deterministic SEIR model with both incidence rate functions to the famous measles data sets from the UK cities of London and Birmingham. Two lines of evidence supported our conclusion that the model with the LHD incidence rate may very well be a better description of the seasonal epidemic processes studied here. First, our model was repeatedly selected as best according to two different information criteria and two different likelihood formulations. The second line of evidence is qualitative in nature: contrary to what the SIRS model with classical incidence rate predicts, the solution of the deterministic SIRS model with LHD incidence rate will reach either the disease free equilibrium or the endemic equilibrium depending on the initial conditions. These findings along with computer intensive simulations of the models' Poincaré map with environmental stochasticity contributed to attain a clear separation of the roles of the environmental forcing and the mechanics of the disease transmission in shaping seasonal epidemics dynamics.
Mathematical Biosciences
We investigate an SIR epidemic model with discrete age groups to understand the transmission dynamics of an infectious disease in a host population with an age structure. We derive the basic reproduction number 0 and show that it is a sharp threshold parameter. If 1, 0 the disease-free equilibrium E 0 is globally stable. If > 1, 0 E 0 is unstable, the model is uniformly persistent, and an endemic equilibrium exists. The global stability of the endemic equilibrium when > 1 0 is established under a sufficient condition. The model is then used to analyze the measles data in India and evaluate the effectiveness of several vaccination strategies for the control of measles epidemics in India.
A Sir Epidemic Model Structured by Immunological Variables
Journal of Biological Systems, 2013
Standard mathematical models for analyzing the spread of a disease are usually either epidemiological or immunological. The former are mostly ordinary differential equation (ODE)-based models that use classes like susceptibles, recovered, infectives, latently infected, and others to describe the evolution of an epidemic in a population. Some of them also use structure variables, such as size or age. The latter describe the evolution of the immune system/pathogen in the infected host — evolution that usually results in death, recovery or chronic infection. There is valuable insight to be gained from combining these two types of models, as that may lead to a better understanding of the severity of an epidemic. In this article, we propose a new type of model that combines the two by using variables of immunological nature as structure variables for epidemiological models. We prove the well-posedness of the proposed model under some restrictions and conclude with a look at a practical a...
Demographic population cycles and ℛ0 in discrete-time epidemic models
Journal of Biological Dynamics
We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, R 0 , to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where k ≥ 1. When R 0 < 1 and the demographic equation (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove the local asymptotic stability of the disease-free period k cycle. That is, the disease goes extinct whenever R 0 < 1. Under the same period k demographic assumption but with R 0 > 1, we prove that the disease-free period k population cycle is unstable and the disease persists. Using the Ricker recruitment function, we apply our results to discrete-time infectious disease models that are formulated for Susceptible-Infectious-Recovered (SIR) infections with and without vaccination, and Infectious Salmon Anemia Virus (ISAv) infections in a salmon population. When R 0 > 1, our simulations show that the disease-free period k cycle dynamics drives the SIR disease dynamics, but not the ISAv disease dynamics.