Discrete Epidemic Models with Arbitrary Stage Distributions and Applications to Disease Control

Abstract

W. O. Kermack and A. G. McKendrick introduced in their fundamental paper, A Contribution to the Mathematical Theory of Epidemics, published in 1927, a simple deterministic model that captured the qualitative dynamic behavior of single infectious disease outbreaks. A Kermack-McKendrick discrete-time general framework, motivated by the emergence of a multitude of models used to forecast the dynamics of SARS and influenza outbreaks, is introduced in this manuscript. Results that allow us to measure quantitatively the role of classical and general distributions on disease dynamics are presented. The case of the geometric distribution is used to evaluate the impact of waiting-time distributions on epidemiological processes or public health interventions. In short, the geometric distribution is used to set up the baseline or null epidemiological model used to test the relevance of realistic stage-period distribution on the dynamics of single epidemic outbreaks. A final size relationship involving the control reproduction number, a function of transmission parameters and the means of distributions used to model disease or intervention control measures, is computed. Model results and simulations highlight the inconsistencies in forecasting that emerge from the use of specific parametric distributions. Examples, using the geometric, Poisson and binomial distributions, are used to highlight the impact of the choices made in quantifying the risk posed by single outbreaks and the relative importance of various control measures.

Mathematical models of Ebola—Consequences of underlying assumptions

“…However, when isolation of infectious individuals is considered as a control strategy, models with exponentially distributed infectious stages can lead to misleading or even incorrect evaluations of effectiveness (see, e.g., Feng et al [4] ). Similar results hold for discrete-time models, in which the analogue of the exponential distribution is geometric [5,6] . These findings demonstrate some of the drawbacks of ODEs models with exponentially-distributed disease stages.…”

SIS and SIR Epidemic Models Under Virtual Dispersal

“…The study of the role of residence time matrices on the dynamics of a single outbreak within a Susceptible-Infected-Recovered (with immunity) or SIR model without births and deaths is relevant to the development of public disease management measures [14, 26, 36]. Under the parameters and definitions introduced earlier, and making use of the same notation, we arrive at the following system of nonlinear differential equations: $$true\{\begin{array}{cc}left{\stackrel{.}{S}}_i& left=-\left(\frac{{\beta }_i{p}_{ii}^2}{N_i{p}_{ii}+N_j{p}_{ji}}+\frac{{\beta }_j{p}_{1ij}^2}{N_i{p}_{ij}+N_j{p}_{jj}}\right)S_i{I}_i-\left(\frac{{\beta }_i{p}_{ii}{p}_{ji}}{N_i{p}_{ii}+N_j{p}_{ji}}+\frac{{\beta }_j{p}_{ij}{p}_{jj}}{N_i{p}_{ij}+N_j{p}_{jj}}\right)S_i{I}_j,\\ left{\stackrel{.}{I}}_i& left=\left(\frac{{\beta }_i{p}_{ii}^2}{N_i{p}_{ii}+N_j{p}_{ji}}+\frac{{\beta }_j{p}_{1ij}^2}{N_i{p}_{ij}+N_j{p}_{jj}}\right)S_i{I}_i+(\frac{{\beta }_i{p}_{ii}{p}_{ji}}{N_i{p}_{ii}+N_j{p}_{ji}}+\end{array}$$…”

Estimation of discrete survival function for error‐prone diagnostic tests

“…One particular insight, at least within our application, is to model the course of the viral load via a stochastic process. For infectious disease, the gamma distribution is a common building block of stochastic epidemiologic models that are used to study the biology of latent disease stages and infectious disease stages . Such future work could be of benefit in the field of oncology, where disagreements between the investigator and independent central review assessments can occur in the determination of progression‐free survival .…”